Dissipativity in Control Engineering: Applications in Finite- and Infinite-Dimensional Systems 9783110677942, 9783110677935

Table of contents :
Preface
Contents
About the author
List of Figures
Part I: Introduction and motivation
1 Motivation and problem formulation
Part II: Theoretical foundations
2 Stability, dissipativity and some system-theoretic concepts
3 Dissipativity-based observer and feedback control design
Part III: Application examples
Introduction
4 Finite-dimensional systems
5 Infinite-dimensional systems
6 Conclusions and outlook
A Lemmata on quadratic forms
B Kalman decomposition for observer design
C The algebraic Riccati equation, optimality and dissipativity
D Kernel derivations for the backstepping approach
Bibliography
Index

Citation preview

Alexander Schaum Dissipativity in Control Engineering

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Alexander Schaum

Dissipativity in Control Engineering |

Applications in Finite- and Infinite-Dimensional Systems

Author PD Dr. habil. Alexander Schaum Chair of Automatic Control, Kiel University Kaiserstr. 2 24143 Kiel Germany [email protected]

ISBN 978-3-11-067793-5 e-ISBN (PDF) 978-3-11-067794-2 e-ISBN (EPUB) 978-3-11-067809-3 Library of Congress Control Number: 2021933820 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2021 Walter de Gruyter GmbH, Berlin/Boston Cover image: Gettyimages / andsun Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface The purpose of the present study is to present recent results on dissipativity-based design methods for some classes of finite- and infinite-dimensional systems. It does not aim to present a complete summary of dissipativity-based approaches or a general treatment of dissipative systems theory. For such a purpose the reader is referred to e. g. the scholarly book of Brogliato et al. (2007), the path-breaking original papers of Willems (1972a,b) and Moylan (1974), Hill and Moylan (1976, 1980) or the seminal book of van der Schaft (2017) to name some central treatises within a long list of studies on this beautiful subject. Dissipativity has attracted attention due to its inherent connections to stability of solutions of dynamical systems. By establishing quantitative dissipation measures a direct connection to Lyapunov stability theory paves the way to analyze nonlinear and infinite-dimensional systems using straight-forward approaches. In the authors understanding, what the dissipativity framework provides is thus more than a specific design approach. It is like a complete field of possible ways of interpreting the (control or observer) error dynamics and subsequently decide on the basis of the particular structure of the system which design approach to follow in order to achieve a desired closed-loop behavior. To make this clear, the author tried, whenever possible and adequate, to draw connections to well-known design methods for linear and nonlinear systems. The dissipativity framework provides interesting views on such design approaches in terms of energy dissipation and extends the class of systems to which these approaches are applicable to. This is achieved by allowing that they can be applied only on a part of the system, providing design criteria which ensure the functioning of the complete dynamics due to an interpretation in terms of interconnected systems. What is exactly meant by this hopefully becomes clear during the lecture of this text. A main purpose of this study is to bring together recent approaches for observer and control design within a common dissipativity framework, yielding simple and constructive closed-loop stability criteria. Such criteria are derived for a rather general class of systems including nonlinear finite-dimensional systems described by nonlinear ordinary differential equations (odes) and semi-linear infinite dimensional systems described by parabolic partial-differential or hyperbolic partial integrodifferential equations with nonlinearities. The present treatise summarizes the authors research results in the area of dissipativity-based feedback control and observer design over the last years which have been part of the german habilitation process for achieving the facultas docendi and the venia legendi1 in the german university teaching system. Even though the main results presented here are the fruit of independent research on this subject, the 1 Faculty and Permission of reading. https://doi.org/10.1515/9783110677942-201

VI | Preface author is deeply grateful for the accompanying discussion with colleagues beginning with Prof. Jaime Moreno from Universidad Nacional Autónoma de México in Mexico City, who was the one who introduced him to this interesting subject during the dissertation (doctoral thesis), as well as Prof. Jesus Alvarez from Universidad Autónoma Metropolitana – Iztapalapa in Mexico City with whom the author was working intensively during a post-doctoral stay. Finally, special thanks go to Prof. Thomas Meurer from Kiel University, who gave me the opportunity of doing my research at the Chair of Automatic Control and has been accompanying me with fruitful discussions on this subject. Of course I also thank all my colleagues for their support and the nice discussions. I also want to express special thanks to the reviewers who helped to improve the present work with their comments, suggestions and corrections. Additionally I due particular thanks to my parents Rolf and Hannelore Schaum, my wife Romina Elizabeth for always supporting and encouraging me, our two sons, Matthias Federico and Bastian Gabriel and our daughter Anna Sophia. Further thanks to all my family, friends and colleagues. Finally, going to the deep truth that I feel about this, I would like to join the famous quote often used by Johann Sebastian Bach: Soli deo gloria. Kiel, March 1, 2021

Contents Preface | V About the author | XI List of Figures | XIII

Part I: Introduction and motivation 1 1.1 1.2 1.3

Motivation and problem formulation | 3 Considered classes of systems | 5 Problem formulation and state of the art | 8 Main contributions | 13

Part II: Theoretical foundations 2 2.1 2.1.1 2.1.2 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.3 2.3.1 2.3.2 2.4

Stability, dissipativity and some system-theoretic concepts | 17 A brief review of stability theory | 17 Lyapunov stability | 17 The direct method of Lyapunov | 19 Dissipative systems | 22 Definitions and general aspects | 22 Finite-dimensional systems: necessary and sufficient conditions | 26 Linear systems with quadratic storage functions | 28 Exponential stability and dissipativity | 30 Static (memory-less) subsystems and sector nonlinearities | 34 Interconnections of dynamic and static subsystems | 43 System theoretic concepts | 47 Observability and detectability | 48 Stabilizability and relative degree | 51 Summary | 54

3 3.1 3.1.1 3.1.2 3.1.3

Dissipativity-based observer and feedback control design | 55 Dissipative observers | 55 Linear dynamic subsystems with linear correction scheme | 56 Partially observable finite-dimensional systems | 58 Further results for state-dependent gain matrices in finite dimensional systems | 63 State-feedback control design | 66

3.2

VIII | Contents 3.2.1 3.2.2 3.2.3 3.3 3.4 3.4.1 3.4.2 3.5

Dissipation through feedback control in finite dimensions | 66 Linear feedback control for linear subsystems | 68 Nonlinear finite-dimensional systems | 70 Dissipativity-based stabilization of dynamic-static system interconnections | 72 Observer-based feedback control | 74 Linear correction and control structure | 74 Observer-based feedback subsystem-linearization in finite dimensions | 80 Summary | 84

Part III: Application examples 4 4.1 4.2 4.3 4.4 4.5 4.6

Finite-dimensional systems | 89 Two-state linear-nonlinear system interconnection | 89 Ship course control | 95 Continuous stirred-tank reactor with exothermic reaction | 101 Nonlinear Luenberger observer for bioreactor monitoring | 106 Inverted pendulum with applied torque | 110 Nonlinear state-feedback control of a MIMO system | 115

5 5.1 5.2 5.3 5.4 5.4.1 5.4.2

Infinite-dimensional systems | 120 Spectral measurement injection dissipative observer | 120 Point-wise measurement injection observer | 128 In-domain output-feedback control of a semi-linear heat equation | 137 Backstepping-based dissipative control and observer design | 148 Control of a semi-linear heat equation | 149 Control of a semi-linear partial integro-differential equation | 161

6

Conclusions and outlook | 169

A A.1 A.2

Lemmata on quadratic forms | 171 Quadratic forms | 171 Schur complement | 172

B B.1 B.2 B.3 B.4

Kalman decomposition for observer design | 175 The Kalmann observability criterion | 175 Detectability | 176 A permutated observer canonical form | 179 The linear Luenberger observer | 181

Contents | IX

B.5 B.5.1 B.5.2 B.5.3

Nonlinear systems | 183 Local observability | 184 Local Kalman decomposition | 186 Detectability analysis | 190

C

The algebraic Riccati equation, optimality and dissipativity | 191

D D.1 D.1.1 D.1.2 D.2 D.3 D.4 D.5

Kernel derivations for the backstepping approach | 195 Forward transformation from Subsection 5.4.1 | 195 Derivation of the kernel pde | 195 Solution of the kernel equation (D.2) | 197 Backward transformation in Subsection 5.4.1 | 201 Observation error forward transformation from subsection 5.4.1 | 203 Transformation from Subsection 5.4.2 | 205 Inverse transformation from Section 5.4.2 | 213

Bibliography | 217 Index | 227

About the author Alexander Schaum received the diploma in Technical Cybernetics from the University of Stuttgart, Germany in 2006, and the Ph. D. degree from the National University of Mexico (Universidad Nacional Autónoma de México, UNAM) in Mexico City in 2009. From 2009 to 2010, he worked as a Post-Doc at the Institute of Engineering of the UNAM, and from 2010 to 2011 in the Chemical Engineering Department of the Metropolitan University (Universidad Autónoma Metropolitana, UAM) in Iztapalapa in Mexico City. From 2011 to 2014 he was guest professor at the Applied Mathematics and Systems Department of UAM Cuajimalpa, also located in Mexico City. Since 2014 he is working at the Chair of Automatic Control at Kiel University in Germany, where he finished the Habilitation Control engineering and control theory in December 2020. His actual research focuses on monitoring and control design for nonlinear finite- and infinite-dimensional systems, with applications in process control, networks and stochastic systems.

https://doi.org/10.1515/9783110677942-202

List of Figures Figure 1.1 Figure 1.2 Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 3.1 Figure 4.1 Figure 4.2 Figure 4.3

Figure 4.4 Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11

Closed-loop system with plant Σ and feedback control u = u∗ − ϖ(x, y). | 9 Closed-loop system with plant Σ, observer Σ̂ and feedback control u = u∗ − ϖ(x,̂ y). | 12

Sector condition for the static map y = sin(u) in (2.42). | 35 Sector condition (2.47) for the nonlinear map y = sin(u) with [k1 , k2 ] = [−0.22, 1]. | 36 Sector condition for the function φ(u) = u sin(ku) with k = 10, included in the (non-Lipschitz) sector [−1, 1]. | 40 Input-Feedforward configuration of a nonlinear map α(u) and a constant matrix M. | 42 Negative feedback interconnection (2.62) of nonlinear dynamic and static systems. | 43 Interconnected control and observation error dynamics (3.60) in Lur’e configuration. | 76 Graph of the nonlinearity φ(σ) given in (4.3). | 90 Closed-loop response of the nonlinear system (4.1)–(4.3) with the linear dissipative state-feedback control (4.6). | 93 Closed-loop response of the nonlinear system (4.1)–(4.3) with the linear dissipative observer-based output-feedback control (4.12). The observed state is represented by the dashed lines. | 95 Shape of the maneuvering characteristics H(ψ)̇ (4.14) with b0 = 0, b1 = −1, b2 = 0, b3 = 1. | 96 Simulation results for the closed-loop behavior of the ship model (4.13) with the state-feedback control (4.17) for a constant set point ψd = 50∘ : heading angle ψ (top) and ψ̇ (middle) and rudder angle δ (bottom). | 100 Simulation results for the closed-loop behavior of the ship model (4.13) with the observer-based output-feedback (4.26) with (4.20): heading angle ψ (top) and ψ̇ (middle) and rudder angle δ (bottom). The observed state is represented by the discontinuous line. | 101 Simulation results for the closed-loop behavior with the state (upper two plots, continuous line) and estimate (upper two plots, discontinuous line) and rudder angle (bottom plot). | 102 Simulation results for the open-loop (dotted lines) and closed-loop reactor (continuous lines) and the observer states (discontinuous lines) with concentration c (top), temperature T (center) and cooling temperature Tc (bottom). | 106 Simulation results for the bioreactor example with the extended Luenberger observer (dashed lines), the actual reactor trajectory (continuous lines) and the substrate estimate of an asymptotic observer (dotted line) with biomass b (top) and substrate s (bottom). | 110 Simulation results for the closed-loop operation of the pendulum (4.44) for the (dissipative) feedback-linearizing control (4.52) with dissipative observer (4.50) without measurement noise (left) and with measurement noise (right). Upper plots: state (continuous line) and observed state (discontinuous line). Bottom: control inputs. | 115 Simulation results for the open-loop dynamics (4.56), showing the convergence to a three-dimensional relaxation oscillation. | 116

XIV | List of Figures

Figure 4.12 Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 Figure 5.8 Figure 5.9 Figure 5.10 Figure 5.11 Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.15 Figure 5.16

Figure 5.17 Figure 5.18 Figure 5.19 Figure 5.20

Simulation results for the closed-loop dynamics (4.56) with the feedback controller (4.62). | 118 State evolution (first row), state estimate (second row) and observation error (third row) without measurement injection. | 128 State estimate (first row) and observation error (second row) of the proposed observer with measurement injection in the first k = 3 modes. | 129 Comparison of the approximated L2 -norm for the open-loop simulation (dashed line) and the proposed observer (continuous line). | 129 System state profile evolution starting from the initial condition x0 (z). | 135 Simulation without measurement injection starting from the initial condition x0̂ (z). | 135 Point-wise injection observer evolution. The dark lines indicate the sensor location. | 136 Observation error evolution for the pointwise injection observer (5.23). The dark lines indicate the sensor location. | 136 Comparison of the approximated L2 -norms without measurement injection (dashed line) and with pointwise measurement injection (continuous line). | 137 Open-loop profile evolution for the unstable heat equation (5.35) with (5.58). | 147 Closed-loop profile evolution for the unstable heat equation (5.35) with (5.58) and output-feedback control u = −20σ. | 147 Closed-loop control input signal for k = 20. | 148 Open-loop (dashed line) and closed-loop (continuous line) approximated L2 norms. | 148 Open-loop behavior of the solutions of the semi-linear heat equation (5.59) with (5.81). | 156 Closed-loop behavior of the solutions of the semi-linear heat equation (5.59) with (5.81) and the backstepping controller (5.67) with β = 3. | 156 State estimation (left) and observation error (right) of the backstepping observer for the semi-linear heat equation (5.59) with (5.81) in open-loop. | 160 State profile evolution (left) and state estimation (right) of the closed-loop system (5.59) with parameters (5.81) in closed-loop with the observer-based implementation of the feedback control (5.67) with βo = 10 and β = 3. | 161 Illustration of the relation between finite-time convergence in norm and exponential stability according to (5.102). | 164 Open-loop unstable state profile evolution of (5.98) with (5.105), (5.107). | 167 Closed-loop state profile evolution of (5.98) with (5.105), (5.107) with the state-feedback controller 5.106. | 168 Time behavior of the control input for the simulation example. | 168

|

Part I: Introduction and motivation

1 Motivation and problem formulation Systems theory has evolved over the last century as a wide area of research having important impacts in engineering, social and natural sciences. Mathematical systems theory has established a platform for studying, analyzing and describing system behavior as well as designing specific mechanisms for adapting systems in such a way that they behave in a desired manner. Control theory, as a particular field within (applied) systems theory has experienced important breakthroughs from 1930 to 1950 where most of the basis for its developments had been established (Bennet, 1993). After the rapid evolution of computer science and technology, many concepts for controlling and monitoring complex systems became practically feasible and great technological achievements have become possible on the basis of the theoretical platform and its computational implementation. Nonlinear systems and control theory (Isidori, 1995, 1999; Slotine and Li, 1991; Adamy, 2014; Röbenack, 2017) and the theory of distributed parameter systems (Curtain and Zwart, 1995; Tucsnak and Weiss, 2009; Krstic and Smyshlyaev, 2008a; Meurer, 2013a; Christofides, 2001) have also made great advances over the last decades. Even though the most significant basis have already been developed, providing means to solve many or most of the actual engineering problems, the theory itself and possible further applications still have some open questions to be answered and problems to be solved. One very basic of such problems is the closed-loop stability of observer-based feedback control for nonlinear systems. This in particular also applies for nonlinear (or semi-linear1 ) distributed parameter systems, i. e. those systems described by partial differential equations. The main problem resides in the fact that the separation principle does not hold for nonlinear systems and thus, even though both feedback control and observer function well, the combination of both may fail to achieve the desired behavior. Beyond the open questions in the actual control engineering area, it is quite sure that future technological developments will confront engineers and scientists with more and more challenging tasks, for which theory is still to be developed. Anyway, the stability of a desired solution behavior will always be one of the main foci of control engineering, besides concepts like optimality, which on the other hand can be related to the stability question in many cases. A way to assess the stability properties of dynamical systems in general is the theory introduced by Aleksandr Lyapunov and named after him (Lyapunov, 1992; Fuller, 1992; LaSalle and Lefschetz, 1961; Zubov, 1964). Lyapunov theory in combination with the constructive control approach, which exploits passivity properties to design nonlinear control schemes with inherent robustness and stability features has 1 Semi-linear systems are modeled by partial differential equation with nonlinearities for which the highest order state derivative appears linearly. https://doi.org/10.1515/9783110677942-001

4 | 1 Motivation and problem formulation been one of the most significant tools in nonlinear control theory, besides the geometric techniques which have led to exact linearization and flatness-based approaches. Lyapunov-based stability assessments are also employed in the context of dissipativity theory and dissipativity-based design approaches (Brogliato et al., 2007; Willems, 1972a; Moylan, 1974; Hill and Moylan, 1976, 1980; Moreno, 2004, 2005). The present text provides some extensions of these approaches and connections with important design techniques which are widely used in applications as well as in theoretical studies. In particular these include (partial) input-output linearization and the backstepping design which provide elegant means for the design of feedback stabilization mechanisms.

Notation In the following A, B are normed metric spaces. – 𝒞 k (A, B) denotes the space of k-times continuously differentiable functions mapping from A to B. In particular 𝒞 0 (A, B) is denoted by 𝒞 (A, B). – ℬ(A, B) denotes the set of bounded functions from A to B. – L(A, B) denotes the set of linear bounded transformations T : A → B. – L2 (0, 1) denotes the Hilbert space of real-valued functions w : [0, 1] → ℝ which are 1 quadratically Lebesgue integrable, i. e. ∫0 w2 (z)dz < ∞, equipped with the inner product 1

⟨v, w⟩L2 = ∫ vwdz,

v, w ∈ L2 (0, 1)

0 2

–

that induces the standard L norm ‖ ⋅ ‖L2 = √⟨⋅, ⋅⟩L2 . The product space (L2 (0, 1))n is a real-valued Hilbert space with inner product 1

⟨v, w⟩ = ∫ v T wdz 0

– – – – – – –

for v, w ∈ L2 (0, 1) and the induced norm ‖ ⋅ ‖ = √⟨⋅, ⋅⟩. The space H k (0, 1) is the Sobolev space of k times weakly differentiable functions with k-th weak derivative in L2 (0, 1). 2 The space of square summable sequences (an )n∈ℕ0 such that ∑∞ n=0 an < ∞ is denoted by l2 . For a complex number λ ∈ ℂ, λ̄ denotes its complex conjugate. R(λ) denotes the real part of a complex number λ. σ(P) denotes the spectrum of a matrix or an operator P. The operator norm of P : A → A, with A being a normed vector space is denoted . by ‖P‖O = supx∈A,‖x‖=0̸ ‖Px‖ ‖x‖ The interior of a set M is denoted by int(M).

1.1 Considered classes of systems | 5

1.1 Considered classes of systems In this section the system classes are introduced which are considered in the sequel. The first class of systems is given by finite-dimensional input-affine nonlinear control systems that are modeled with ordinary differential equations of the form p

ẋ = Ax + f (x) + ∑ bi (x)ui ,

t > 0,

y = h(x),

t ≥ 0,

i=1

x(0) = x 0

(1.1a) (1.1b)

with the state vector x(t) ∈ ℝn at time t ≥ 0, a constant matrix A : ℝn → ℝn , smooth bounded vector fields f : ℝn → ℝn , bi : ℝn → ℝn , i = 1, . . . , p and bounded inputs ui ∈ ℬ([0, ∞), ℝ). The existence and uniqueness of a solution x : [0, t1 ) → ℝn for some t1 > 0, satisfying x(0) = x 0 is ensured by Carathéodory’s existence theorem (Hale, 1980), because the right-hand side of (1.1) as a function of (t, x) is continuous in the second argument for any t, measurable in the first argument for any x, locally Lipschitz continuous in the second argument given the smoothness of the vector fields f , bi , i = 1, . . . , p, and bounded, given the boundedness of f , bi and ui . In order to highlight the dependency of the value of the solution x at time t ≥ 0 on the initial condition x 0 and the input u sometimes the notation x(t; x 0 , u) is used instead of x(t). A second important class of systems are semi-linear distributed parameter systems. In this work two different types are considered, namely systems described by parabolic partial-differential equations (pdes) or hyperbolic partial integro-differential equations. The first type includes semi-linear diffusion-convection-reaction systems with indomain and/or boundary control and measurements modeled by the equation set 𝜕t x = D𝜕z2 x − v𝜕z x + r(x) + bud ,

t > 0,

z ∈ (0, 1)

(1.2a)

with initial condition x(z, 0) = x0 (z) for z ∈ [0, 1] and boundary conditions b01 𝜕z x(0, t) + b02 x(0, t) = b0c u0 (t), b11 𝜕z x(1, t) + b12 x(1, t) = b1c u1 (t),

t≥0

t≥0

(1.2b) (1.2c)

associated outputs yi (t) = x(ζi , t),

i = 1, . . . , m,

t ≥ 0.

(1.2d)

In (1.2) t ≥ 0 denotes the time, z ∈ [0, 1] the space, x(⋅, t) ∈ L2 (0, 1) the state, D, v > 0 the diffusion and convection coefficients, respectively, and b01 , b02 , b11 , b12 , b0c , b1c ∈ ℝ are constants. The function r : 𝒞 1 (ℝ, ℝ) denotes the reaction rate. External control inputs are considered with in-domain control support b ∈ L2 (0, 1), bounded, smooth inputs ud , u0 , u1 ∈ 𝒞 2 ([0, ∞), ℝ), and pointwise outputs yi (t) ∈ ℝ that correspond to the value

6 | 1 Motivation and problem formulation of x(⋅, t) at the measurement location ζi ∈ [0, 1] for i = 1, . . . , m. Sufficient conditions for the existence and uniqueness of a solution for such systems (including sufficiently smooth time- and space-varying coefficients and coupling of several such equations) with initial condition x0 ∈ H 2 (0, 1) are provided for example in Henry (1981), Fridman and Orlov (2009), Pazy (1983), Schaum et al. (2013b). The second type includes systems that are modeled by semi-linear first-order partial integro-differential equations with boundary control of the form z

𝜕t x = 𝜕z x + ∫ fxdζ + φ(x),

t > 0,

z ∈ (0, 1)

(1.3a)

0

with initial condition x(⋅, 0) = x0 , boundary condition x(1, t) = u(t),

t≥0

(1.3b)

and pointwise outputs yi (t) = x(ζi , t),

ζi ∈ [0, 1],

i = 1, . . . , m,

t ≥ 0.

(1.3c)

In (1.3) t ≥ 0 denotes the time, z ∈ [0, 1] the space, x(⋅, t) ∈ L2 (0, 1) the state at time t ≥ 0, f : [0, 1] × [0, 1] → ℝ and φ ∈ 𝒞 1 (ℝ, ℝ) are smooth functions of their arguments, and u ∈ 𝒞 1 ([0, ∞), ℝ) is a bounded input. The outputs yi (t) ∈ ℝ are defined by the sensor locations ζi ∈ [0, 1], i = 1, . . . , m. Existence and uniqueness of a solution for such systems with initial condition x0 ∈ H 1 (0, 1) has been established e. g. in Beniich et al. (2018) using general theory of operator semigroups on Banach (and Hilbert) spaces (Pazy, 1983; Nagel, 1986) and in Krstic and Smyshlyaev (2008a,b) in the context of backstepping-based control design. Given that (1.2) and (1.3) consider states in function spaces they correspond to infinite-dimensional systems. Using the concept of abstract differential equations in Hilbert spaces (Curtain and Zwart, 1995; Banks, 1983; Tröltzsch, 2009), these infinitedimensional systems can be modeled in a way that formally resembles the one of (1.1). To introduce this concept consider the model ẋ = Ax + Bu + f (x), y = Cx,

t > 0,

t≥0

x(0) = x 0

(1.4a) (1.4b)

with the state x(t) ∈ 𝕏 at time t ≥ 0, the state space 𝕏, A : 𝒟(A) ⊂ 𝕏 → 𝕏 an operator with domain 𝒟(A), smooth function f : 𝕏 → 𝕏, bounded operator B : ℝp → 𝕏, bounded input u : [0, ∞) → ℝp and output y(t) ∈ ℝm with the associated output operator C : 𝕏 → ℝm . The local existence and uniqueness of solutions to the abstract Cauchy problem (1.4) are typically ensured for x 0 ∈ 𝒟(A) (see, e. g., Curtain and Zwart, 1995; Banks, 1983). In the sequel an initial condition x 0 for which a unique solution is ensured is called an admissable initial condition and an input u that is sufficiently

1.1 Considered classes of systems | 7

regular so that a unique solution exists is called an admissable input with 𝒰 denoting the set of admissable inputs. In some places in the subsequent discussions for the case that f = 0 the system (1.4) will be denoted as Σ(A, B, C). Example 1.1.1. Considering (1.2) with u0 , u1 = 0 and p = m = 1 it turns out that (1.2) can be written in this abstract form with the state x(t) = x(⋅, t) ∈ L2 (0, 1) at time t ≥ 0, the state space 𝕏 = L2 (0, 1), the parabolic operator A = D𝜕z2 − v𝜕z with domain 2

𝒟(A) = {x ∈ H (0, 1) | b01 𝜕z x(0) + b02 x(0) = 0, b11 𝜕z x(1) + b12 x(1) = 0}, 1

and the output operator C : L2 (0, 1) → ℝ with Cx = ∫0 δ(z − ζ1 )x(z)dz. The input operator B : ℝp → L2 (0, 1) is then defined by Bv = bv with b ∈ L2 (0, 1) and v ∈ ℝ. The function f : L2 (0, 1) → L2 (0, 1) in (1.4) is then defined by f (ξ )(z) = r(ξ (z)) for all ξ ∈ L2 (0, 1), z ∈ [0, 1]. ⬦ Example 1.1.2. In the case that p in-domain inputs are present, i. e., u(t) ∈ ℝp the input operator B in the preceding example can be defined by Bu(t) = ∑pk=1 bk uk (t) ∈ L2 (0, 1) provided that bk ∈ L2 (0, 1), k = 1, . . . , p. ⬦ Example 1.1.3. For a bounded distributed input u ∈ ℬ([0, ∞), L2 (0, 1)), the associated input operator B : L2 (0, 1) → L2 (0, 1) can be defined by Bu(t) = bu(t), b ∈ L2 (0, 1). ⬦ Example 1.1.4. The model (1.3) with u = 0 can be written in the form (1.4) with the state x(t) ∈ 𝕏 = L2 (0, 1), the hyperbolic integro-differential operator z

Ax := 𝜕z x + ∫ fxdζ

1

𝒟(A) = {x ∈ H (0, 1) | x(1) = 0},

0

the output operator of Example 1.1.1, and the function f : L2 (0, 1) → L2 (0, 1) in (1.4) such that f (ξ )(z) = φ(ξ (z)) for all ξ ∈ L2 (0, 1) and z ∈ [0, 1]. ⬦ Example 1.1.5. Coupled pdes can be represented accordingly in the product state space 𝕏 = (L2 (0, 1))n . ⬦ Note that in the preceding examples it is also possible to consider the control inputs on the boundary. Nevertheless, boundary control systems require a set of additional considerations in the abstract setup (1.4) to ensure an adequate definition of the operator B. For a further discussion of this subject the reader is referred to Curtain and Zwart (1995). Example 1.1.6. The finite-dimensional system (1.1) with state-dependent input gain can also be written in a similar way considering 𝕏 = ℝn and B : 𝕏 → 𝕏 with B(x) = [b1 (x), . . . , bp (x)] ,

u1 .. ] ] . ]. [up ]

[ u=[ [

⬦

8 | 1 Motivation and problem formulation In consequence, the representation in form of an abstract differential equation motivates to address several general discussions about the input-output behavior of finite- and infinite-dimensional systems in the present work using the common notation (1.4) and indicating the particular state space, i. e., for example 𝕏 = ℝn in the case of (1.1) or 𝕏 = L2 (0, 1) in the cases (1.2) or (1.3). Besides the general considerations and depending on the particular needs in the developments of the subsequent chapters either representations of the form (1.2), (1.3) or (1.4) will be used. Whenever no particular indication of the state space is given, it is claimed that the result holds true for systems in the Hilbert space (L2 (0, 1))n and systems in the Euclidean space ℝn , provided the according definition of the operators.

1.2 Problem formulation and state of the art Motivated by the discussion in the preceding section, for the problem formulation and presentation of the related state of the art the following general model is considered p

ẋ = Ax + f (x) + ∑ bk (x)uk ,

t > 0,

y = Cx,

t≥0

k=1

x(0) = x 0

(1.5a) (1.5b)

with state x(t) ∈ 𝕏 at time t ≥ 0, operator A : 𝒟(A) ⊂ 𝕏 → 𝕏, smooth functions f , bk : 𝕏 → 𝕏, k = 1, . . . , p, bounded inputs uk : [0, ∞) → ℝ, and output operator C : 𝕏 → ℝm . Let x ∗ be a steady-state for the system (1.5) when the inputs are set to their nominal values uk = u∗k ∈ ℝ, k = 1, . . . , p, i. e. p

0 = Ax ∗ + f (x ∗ ) + ∑ bk (x ∗ )u∗k .

(1.6)

k=1

The problems addressed in the present text are related to the stabilization of systems of the form (1.5) by designing either an output feedback control scheme uk (t) = u∗k − ϖk (y(t)),

t ≥ 0,

ϖk (Cx ∗ ) = 0,

k = 1, . . . , p

(1.7a)

ϖk (x ∗ ) = 0,

k = 1, . . . , p

(1.7b)

or state feedback control scheme uk (t) = u∗k − ϖk (x(t)),

t ≥ 0,

with nominal values u∗k , so that the state x(t) converges (exponentially or at least asymptotically2 ) to a desired steady-state x ∗ ∈ 𝕏, i. e. 󵄩 󵄩 lim 󵄩󵄩x(t) − x ∗ 󵄩󵄩󵄩 = 0.

t→∞ 󵄩

2 See Chapter 2 for formal definitions of these concepts.

1.2 Problem formulation and state of the art | 9

The norm in the preceding equation depends on the particular state space (ℝn or (L2 (0, 1))n ). The associated stabilization problem involves the analysis of the stability properties of the state x ∗ in the closed-loop system consisting of the nonlinear (or semi-linear) dynamics (1.5) with the feedback laws (1.7). This is schematically illustrated in Figure 1.1 where Σ represents the system modeled by (1.5).

Figure 1.1: Closed-loop system with plant Σ and feedback control u = u∗ − ϖ(x, y).

Introducing the state deviation ̃ = x(t) − x ∗ , x(t)

t≥0

this can be carried out by analyzing the stability of the origin of the dynamics p

ẋ̃ = Ax + f (x) + ∑ bi (x)(u∗k − ϖk (x, y)) k=1 p

p

= Ax + f (x) + ∑ bk (x)(u∗k − ϖk (x, y)) − (Ax ∗ + f (x ∗ ) + ∑ bk (x ∗ )u∗k ) k=1

p

k=1 p

= Ax̃ + f (x ∗ + x)̃ − f (x) + ∑ [bk (x ∗ + x)̃ − bk (x ∗ )]u∗k − ∑ bi (x)ϖk (x, y). k=1

k=1

The stability analysis of this kind of nonlinear finite-dimensional systems and semilinear distributed parameter systems is quite non-trivial and systematic approaches exist only for special cases. For finite-dimensional systems which are completely input-to-state or inputoutput linearizable with asymptotically stable zero dynamics (Isidori, 1995, 1999; Slotine and Li, 1991; Adamy, 2014; Röbenack, 2017), the control can be designed in such a way that the error dynamics become linear and the stability analysis can be carried out using methods for linear systems, e. g. by checking if all eigenvalues (of the inputoutput dynamics) are contained in the left-half of the complex plane, i. e. the spectrum

10 | 1 Motivation and problem formulation satisfies σ ⊂ ℂ− . Alternatively, the constructive control approach (Sepulchre et al., 1997) can be employed using passivity-based methods (Brogliato et al., 2007; van der Schaft, 2017) or integrator backstepping (Krstic et al., 1995; Freeman and Kokotovic, 1996; Sepulchre et al., 1997). Besides these systematic approaches, for some types of systems it can also be proven that classical proportional (P), proportional-integral (PI) or proportional-integral-derivative (PID) control schemes achieve global stabilization (see e. g. Alvarez-Ramirez et al., 2011; Alvarez et al., 2011; Schaum et al., 2013a; Diaz-Salgado et al., 2012; Schaum, 2013). For linear distributed parameter systems a spectral decomposition approach (Curtain and Zwart, 1995; Curtain, 1982), operator semi-group methods (Tucsnak and Weiss, 2009), backstepping control (Krstic and Smyshlyaev, 2008a; Meurer, 2013a; Vazquez and Krstic, 2008a,b; Coron et al., 2013; Deutscher, 2016), Lyapunov-based approaches with linear matrix inequalities (Yang and Dubljevic, 2014; Fridman, 2001), or optimal control theory (Callier and Winkin, 1992; Khurshudyan, 2014; Frerik et al., 2014) can be used to ensure the stability of the error dynamics. Approximation-based methods for distributed parameter systems include also all finite-dimensional approximations, via finite-differences, proper orthogonal decomposition, the finite-element method, approximate inertial manifolds or similar approaches to obtain a model approximation in the form (1.1) and apply some of the above mentioned methods. These are known as early-lumping whereas methods based on the complete pde description are called late-lumping. For more details on early lumping approaches, also for semi-linear distributed parameter systems see e. g. (Christofides, 2001) and references therein. The design and analysis efforts become much larger when semi-linear distributed parameter systems are considered within a late-lumping approach. Particular studies have employed, e. g. matrix inequalities (Hagen and Mezic, 2003; Hagen, 2006; Fridman and Orlov, 2009), absolute stability theory (Curtain et al., 2003, 2004; Logemann and Curtain, 2000), sliding modes (Orlov and Dochain, 2002; Orlov, 2009), and passivity-based constructive control approaches (Franco de los Reyes et al., 2019b,a,c). Backstepping control has been extended to semilinear systems with Volterra nonlinearities (Vazquez and Krstic, 2008a,b), hyperbolic quasilinear pdes (Vazquez et al., 2011; Coron et al., 2013) and semi-linear parabolic and hyperbolic pdes (Hasan, 2015, 2016; Hu et al., 2015). First connections with dissipativity theory have been recently proposed in Schaum and Meurer (2019a) for a class of semilinear partial integro-differential equations as those given in (1.3). Furthermore, dissipativity-based analysis of in-domain actuation for a scalar heat equation with a non-locally interacting output-feedback nonlinearity has been analyzed considering classical proportional feedback control in Schaum and Meurer (2019) and PI control in Schaum et al. (2020) including compensation of actuator uncertainties. Given that the outputs and inputs are the only indicators of what happens in the system and the complete state information is normally not available, state reconstruction methods known as observers have to be employed in order to be able to imple-

1.2 Problem formulation and state of the art | 11

ment the control law (1.7). Besides the purpose of feedback control implementation, observers also provide a powerful means to system monitoring with state reconstruction, data assimilation, fault detection and process prediction capabilities. Given the nonlinear effects, phenomena like steady-state multiplicity or bifurcations have to be considered and the unknown initial system state x 0 can lead to completely erroneous model predictions. Thus, adequate measurement injection mechanisms must be provided in combination with model predictions for the observer design ensuring that the erroneous initial guess is compensated and the state estimate converges to the actual system state. Observers and filters for linear finite-dimensional systems have been thoroughly studied by Luenberger (1964, 1971) and Kalman (1960), Gelb (1978) and extensions to nonlinear systems have been provided using different kind of approaches including the high-gain observer (Gauthier et al., 1992), extended Luenberger observer (Zeitz, 1987; Röbenack, 2005), Lipschitz observer (Rajamani, 1998), sliding mode observers (Spurgeon, 2008; Salgado et al., 2011), asymptotic observers (Dochain et al., 1992; Dochain, 2003), circle-criterion based observers (Arcak and Kokotovic, 2001) and dissipativity-based observers (Moreno, 2004, 2005, 2008; Schaum and Moreno, 2006). For linear and semi-linear infinite-dimensional systems important milestones have been reached during the last decades. Particular approaches for linear systems include modal designs (Curtain, 1982; Curtain and Zwart, 1995) and backstepping (Krstic and Smyshlyaev, 2008a; Meurer, 2013a,b; Jadachowski et al., 2015; Baccoli and Pisano, 2015). For semi-linear systems dissipativity and matrix inequality based designs have been proposed, based on an interpretation of the semi-linear system as an interconnection of a linear and a nonlinear subsystem and performing the design step for the linear one in order to ensure adequate dissipation properties to ensure the convergence for the nonlinear system. This has been done using modal decomposition approaches for the linear subsystem (Hagen and Mezic, 2003; Hagen, 2006; Schaum et al., 2008a, 2016b, 2018), direct Lyapunov arguments (Schaum et al., 2013b), and recently also the Backstepping-based approach (Schaum and Meurer, 2020). Besides these dissipavitiy-based techniques, absolute stability based approaches (Curtain et al., 2003) as well as sliding-mode observers (Orlov, 2009) have been used. Even though these studies have established important and powerful design methods and have revealed fundamental connections, possibilities and limitations, one central issue in all approaches relies in establishing closed-loop stability, when the ̂ control law (1.7) is implemented using the observed state, denoted by x(t), i. e. ̂ u(t) = u∗ − ϖ(x(t), y(t)),

(1.9)

given that this implies the analysis of the coupled observer and control error system. This is schematically represented in Figure 1.2 with Σ̂ representing the observer.

12 | 1 Motivation and problem formulation

Figure 1.2: Closed-loop system with plant Σ, observer Σ̂ and feedback control u = u∗ − ϖ(x,̂ y).

For linear systems it results that if both, the state-feedback control (1.7) and the observer converge (exponentially) then the desired state x ∗ is an (exponentially stable) attractor for the closed-loop system with the observer-based feedback control (1.9). This result is known as separation principle (Kailath, 1980; Ogata, 1995) and forms the basis of many design approaches for linear systems. Unfortunately, the separation principle does not hold for nonlinear systems and the closed-loop stability thus has to be analyzed depending on the properties of the system and the proposed observer and control scheme. The associated problem of nonlinear finite-dimensional system closed-loop stability has been addressed by many different authors from different backgrounds and here it should suffice to mention but a few of the most essential approaches in the literature. Beside general purpose analysis exploiting input-to-state stability concepts (Arcak et al., 2002; Angeli et al., 2004) and exploiting the cascade-structure of state and observation error in combination with stabilizability and observability properties (Teel and Praly, 1994, 1995), high-gain observer-based approaches (Atassi and Khalil, 2000; Khalil, 2017), polynomial control functions (Ebenbauer et al., 2005), dissipation inequalities in form of matrix inequalities (Moreno, 2006) and the circle-criterion (Praly and Arcak, 2004; Arcak, 2005) have been exploited. Between these results, the one closest to the results that will be obtained here is (Moreno, 2006). Note that for the purpose at hand the most important aspect in Moreno (2006) is that the highly structural convergence conditions for the observer in terms of dissipativity inequalities imply the existence of quadratic Lyapunov functions and thus can be exploited to perform the analysis of the closed-loop dynamics. Even though the above studies have revealed important insights, the question remains open whether it is possible to obtain a simpler separation property if the controller is also designed explicitly on the basis of a dissipation inequality. This gives the

1.3 Main contributions |

13

closed-loop system even more structure than considered in the above studies and thus one could suppose that it should allow to obtain less conservative closed-loop stability conditions. This is one of the main questions which are addressed in the present study (see Section 3.4) in particular in view of applications for some classes of infinitedimensional systems.

1.3 Main contributions Having the previously discussed state of the art as point of departure, the main contributions of the present work can be summarized as follows: – Further extension of the dissipativity-based observer design approach (Moreno, 2005, 2004; Schaum and Moreno, 2006; Schaum et al., 2008c) to feedback control and combination of both with general closed-loop exponential stability criteria. – Presentation of new results on connections between dissipativity-theory and stability theory and their exploitation for control and observer design purposes. – Application of the results to several application examples which have not been addressed within the framework of dissipativity-theory in previous studies, according to the authors knowledge, and provide existence conditions or explicit tuning approaches. – Putting common approaches in control and observer design into the dissipativity framework and extending them to more general system structures. – Consideration of in-domain measurements and actuation for the design of distributed parameter monitoring and control systems with explicit sensor and actuator location criteria following and extending recent results of the author (e. g., Schaum et al., 2015b, 2016b, 2018, 2020; Schaum and Meurer, 2019). – Connection of the recently proposed point-wise measurement injection approach for observer design of semi-linear parabolic distributed parameter systems with the dissipativity-based observer design theory.

|

Part II: Theoretical foundations

2 Stability, dissipativity and some system-theoretic concepts The purpose of this chapter is to provide the basis of concepts from stability, dissipativity and systems theory which will be used throughout the subsequent chapters. The design approach discussed in this work is based on the notion of dissipativity as introduced by Willems (1972a,b), Hill and Moylan (1976, 1980), Moylan (1974), Byrnes et al. (1991) and since then exploited by several authors (see e. g. Brogliato et al., 2007; Sepulchre et al., 1997; Moreno, 2005 and references therein). For the stability assessment in the dissipativity-based approach the close relation to Lyapunov theory and in particular to the direct method of Lyapunov are exploited. Thus, first the basic notions and results from Lyapunov’s stability theory (Lyapunov, 1992; Zubov, 1964; LaSalle and Lefschetz, 1961; Khalil, 1996) are recalled.

2.1 A brief review of stability theory 2.1.1 Lyapunov stability Consider a system described by an autonomous differential equation for the state x(t) in the state space 𝕏 with norm ‖ ⋅ ‖. Let this differential equation be given by ẋ = f (x),

t > 0,

x(0) = x 0 ,

(2.1)

which is supposed to have a unique solution x : [0, ∞) → 𝕏 so that x(t) ∈ 𝕏 for all t ≥ 0, x 0 ∈ 𝒟 ⊆ 𝕏. Particular examples of such systems with 𝕏 = ℝn or 𝕏 = L2 (0, 1) are given in Section 1.1. For the purpose at hand it is useful to recall the basic notions on stability in the sense of Lyapunov which in the subsequent discussions will be used and put in relation to the dissipative systems approach. The following definitions go along standard ones in the literature (see e. g. LaSalle and Lefschetz, 1961; Zubov, 1964; Khalil, 1996), with a particular discussion for infinitedimensional systems e. g. in Zubov (1964). Note that in order to highlight the dependency of the evolution of the state on the initial condition sometimes the notation x(t; x 0 ) is used. Definitions 2.1.1. An element x ∗ of the state space 𝕏 is called an equilibrium solution or steady state for (2.1) if f (x ∗ ) = 0. In the sequel the focus is on stability properties of equilibrium solutions and the different relevant notions of stability are defined. Definition 2.1.2. An equilibrium x ∗ of (2.1) is said to be stable, if for any ϵ > 0 there exists a constant δ= δ(ϵ) > 0 such that for any initial deviation from equilibrium within https://doi.org/10.1515/9783110677942-002

18 | 2 Stability, dissipativity and some system-theoretic concepts a δ-neighborhood, the trajectory is contained in the associated ϵ-neighborhood, i. e. 󵄩 󵄩 󵄩 󵄩 ∀x 0 ∈ 𝕏 : 󵄩󵄩󵄩x 0 − x ∗ 󵄩󵄩󵄩 ≤ δ ⇒ 󵄩󵄩󵄩x(t; x 0 ) − x ∗ 󵄩󵄩󵄩 ≤ ϵ,

∀t ≥ 0.

(2.2)

If x ∗ is not stable, it is called unstable. Stability implies that the solutions stay arbitrarily close to the equilibrium whenever the initial condition is chosen sufficiently close to the equilibrium. Note that stability implies boundedness of solutions, at least in a neighborhood of x ∗ , but not that these converge to the equilibrium x ∗ . Convergence in turn is ensured by the concept of attractivity. Definition 2.1.3. The equilibrium x ∗ is called an attractor for the set 𝒟⊆ 𝕏, if 󵄩 󵄩 ∀x 0 ∈ 𝒟 : lim 󵄩󵄩󵄩x(t; x 0 ) − x ∗ 󵄩󵄩󵄩 = 0.

(2.3)

t→∞

It is called attractive if there exists a set 𝒟 ⊂ 𝕏 such that (2.3) holds true. The set 𝒟 is called the domain of attraction. Note that an equilibrium may be attractive without being stable, i. e., trajectories always have a large transient, so that for small ϵ no trajectory will stay for all times within the ϵ-neighborhood, but will return to it and converge to the equilibrium. An example of such a behavior is given by Vinograd’s system (Vinograd, 1957). Definition 2.1.4. An equilibrium x ∗ of (2.1) is said to be asymptotically stable if it is stable and attractive. The asymptotic stability does not state anything about convergence speed. It only establishes that after an infinite time the state x(t), if starting in the domain of attraction, i. e., x(0) = x 0 ∈ 𝒟, will approach x ∗ asymptotically. A concept which allows to overcome this issue is the one of exponential stability. Definition 2.1.5. An equilibrium point x ∗ of (2.1) is said to be exponentially stable in a set 𝒟, if it is stable, and there are constants a, λ > 0 so that 󵄩 󵄩 󵄩 󵄩 ∀x 0 ∈ 𝒟 : 󵄩󵄩󵄩x(t; x 0 ) − x ∗ 󵄩󵄩󵄩 ≤ a󵄩󵄩󵄩x 0 − x ∗ 󵄩󵄩󵄩e−λt ,

∀t ≥ 0.

(2.4)

The constant a is called the amplitude and λ the convergence rate. Note that when the set 𝒟 in Definition 2.1.5 is not a priori defined, the equilibrium x ∗ is simply called exponentially stable. Clearly, exponential stability implies asymptotic stability. It is quite noteworthy that, eventhough the convergence is still asymptotic, for exponentially stable equilibria it is possible to determine exactly the time needed to approach the equilibrium up to a given distance. To illustrate this,

2.1 A brief review of stability theory | 19

consider the time required to ensure an 98.5 % convergence.1 For this suppose that the bound (2.4) holds with a = 1 and is exact, i. e. it holds with equality. In this case the notion of characteristic time tc = λ−1

(2.5)

is useful. It follows from (2.4) that the norm behaves like a linear first-order system d‖x(t; x 0 ) − x ∗ ‖ 󵄩 󵄩 = −λ󵄩󵄩󵄩x(t; x 0 ) − x ∗ 󵄩󵄩󵄩. dt Note that the characteristic time tc corresponds to the inverse slope of the time response of this linear system at t = 0. Denote by ts the settling time, i. e. the time required for 98.5 % convergence. It follows that ‖x(ts ; x 0 ) − x ∗ ‖ = e−λts = 0.015 ‖x 0 − x ∗ ‖

⇔

ts =

ln(0.015) 4 ≈ = 4tc , λ λ

(2.6)

meaning that (practical) convergence is obtained after approximately four characteristic times. It should be noted that in the present treatise the consideration of bounded deviations due to bounded perturbations will not be taken into account. Nevertheless, in most practical applications considerable perturbations can be at play stemming from either exogenous sources or model mismatches and parameter errors. For such applications the notion of practical stability introduced by LaSalle and Lefschetz (1961) or the notion of input-to-state stability introduced by Sontag (1989, 1995) can be employed. Note that, e. g., for linear time-invariant systems the exponential stability property ensures the input-to-state stability for many cases of interest (Sontag, 1989, 1995; Karafyllis and Krstic, 2018). 2.1.2 The direct method of Lyapunov Dissipativity theory is based on the notion of energy balances. The same notion motivated the studies of Aleksandr Mikhailovich Lyapunov who, based on the previous studies of Evangelista Torricelli, Joseph-Louis Lagrange and Joseph Liouville on stability problems in astronomy and fluids (Fuller, 1992), generalized the considerations from energy preservation and dissipation to functions which are positive for any nonzero argument. To recall the main results of his works and subsequent generalizations of it, first some definitions are in order. 1 This value is motivated by noticing that two curves which are identical up to 98.5 % are hardly distinguishable with the naked eye, in particular if they are measured with a certain measurement uncertainty.

20 | 2 Stability, dissipativity and some system-theoretic concepts Definition 2.1.6. A continuous function V : 𝒟 ⊆ 𝕏 → ℝ with 0 ∈ int(𝒟) is called – positive semi-definite in 𝒟 if ∀x ∈ 𝒟 : V(x) ≥ 0. – positive definite in 𝒟 if ∀0 ≠ x ∈ 𝒟 : V(x) > 0 and V(x) = 0 only for x = 0. – negative semi-definite in 𝒟 if ∀x ∈ 𝒟 : V(x) ≤ 0. – negative definite in 𝒟 if ∀0 ≠ x ∈ 𝒟 : V(x) < 0 and V(x) = 0 only for x = 0. For the system (2.1) the rate of change of V at time t ≥ 0 along the solution x: [0, ∞) → 𝕏 can be defined as follows dV 1 (ξ ): = lim (V(x(τ; ξ )) − V(ξ )) τ→0 τ dt

(2.7)

where x(τ; ξ ) denotes the solution at time τ ≥ 0 of (2.1) starting at ξ ∈ 𝕏, i. e. x(0) = ξ . With this notion at hand the following results are obtained which will be frequently used throughout the subsequent chapters. In particular, for finite-dimensional systems the following results are known and stated here without proof (see e. g. Khalil, 1996; Zubov, 1964; Sepulchre et al., 1997 for more details) Theorem 2.1.1. Let V∈ 𝒞 1 (𝒟 ⊆ 𝕏 ⊆ ℝn , ℝ) be positive definite in 𝒟. If ∀x ∈ 𝒟 :

dV (x) ≤ 0, dt

(2.8)

then x = 0 is stable in the sense of Lyapunov. In the case that V(x) is positive definite and satisfies (2.8) it is called a Lyapunov functional . The reason why this results holds true in a finite-dimensional state space is that the level sets of V, i. e. the curves Γc = {x ∈ ℝn | V(x) = c}

(2.9)

are the boundaries of compact2 subsets of the state space, at least for a sufficiently small neighborhood about the origin. The property dV < 0 ensures that these sets are dt positively invariant, and thus, for a given ϵ > 0 choosing the set Γδ with sufficiently small δ always ensures that the trajectories remain within the ϵ-neighborhood. Thus, the condition for a stable equilibrium in the sense of Lyapunov given in Definition 2.1.2 is satisfied. Remark 2.1.1. In an infinite-dimensional set-up, the compactness of the level sets is not always guaranteed and thus special care has to be taken when considering infinitedimensional systems (see e. g. Luo et al., 1999). 2 In a metric space a set M is compact if it is sequentially compact, i. e. if every sequence contained in M contains a convergent subsequence. In ℝn this is ensured if the set is bounded and closed.

2.1 A brief review of stability theory | 21

Introducing the set 𝒳0 of states for which the right-hand side in (2.8) vanishes, i. e. n

𝒳0 = {x ∈ 𝕏⊆ ℝ |

dV (x) = 0} dt

(2.10)

one can apply the following result going to back to Nikolay Nikolayevich Krasovsky and Joseph Pierre LaSalle, which is frequently called the invariance theorem. Theorem 2.1.2 (Krasovsky–LaSalle). Let D ⊆ 𝕏= ℝn be a positively invariant compact set and V: 𝒟 → ℝ) a positive definite functional satisfying (2.8) for all x ∈ D. Then the state x(t) converges to the largest positively invariant set ℳ ⊆ 𝒳0 with 𝒳0 defined in (2.10). Corollary 2.1.1. If the conditions of Theorem 2.1.2 are satisfied and it holds that ℳ = {0}, then the origin is asymptotically stable. The analysis of the asymptotic stability of the origin using Theorem 2.1.2 and Corollary 2.1.1 implies some additional effort on the study of the dynamics (2.1) restricted to the set 𝒳0 . Alternatively, if one can find a positive definite function V such that its value strictly decreases over time, i. e. dV is negative definite in x, then the dt next result can be applied. Theorem 2.1.3. Let V : 𝒟 ⊆ 𝕏=ℝn → ℝ be positive definite. If ∀x ∈ 𝒟 : x = 0 is locally asymptotically stable in 𝒟.

dV (x) dt

< 0, then

As mentioned above special conditions to ensure the compactness of level sets must hold true in order to be able to be extend these results to the infinite-dimensional set-up. Nevertheless, by introducing some additional conditions, a result on the exponential stability can be obtained which also applies in a general metric state space as e. g. L2 (0, 1). Theorem 2.1.4. Let 𝕏 be a metric space with norm ‖ ⋅ ‖ and V∈ 𝒞 1 (𝕏, ℝ) be a positive definite functional. If there exist constants α, β, γ > 0 so that for all x ∈ 𝕏 (i) α‖x‖2 ≤ V(x) ≤ β‖x‖2

(2.11a)

(ii)

(2.11b)

dV (x) ≤ −γ‖x‖2 dt

then x = 0 is exponentially stable and (2.4) holds with a = √β/α and λ = γ/(2β). The proof is given here for completeness and follows standard arguments (see e. g. LaSalle and Lefschetz, 1961; Zubov, 1964; Khalil, 1996). Proof. In virtue of (2.11) it holds that γ dV (x) ≤ − V(x). dt β

22 | 2 Stability, dissipativity and some system-theoretic concepts From the comparison principle (Khalil, 1996) it follows that V(x(t)) ≤ V(x 0 )e

γ

−βt

,

∀t ≥ 0.

From (2.11a) this implies that γ β −γt 1 − t 󵄩2 1 󵄩󵄩 2 󵄩󵄩x(t)󵄩󵄩󵄩 ≤ V(x(t)) ≤ V(x 0 )e β ≤ e β ‖x 0 ‖ , α α α

∀t ≥ 0

and finally γ β − t 󵄩󵄩 󵄩 󵄩󵄩x(t)󵄩󵄩󵄩 ≤ √ ‖x 0 ‖e 2β , α

∀t ≥ 0.

Thus the origin x = 0 is exponentially stable as defined in (2.4) with a and λ stated in Theorem 2.1.4.

2.2 Dissipative systems The notion of dissipativity is based on a generalization of energy concepts in mechanics and thermodynamics and has been introduced by Jan Camiel Willems in the early 70’s in the seminal papers (Willems, 1972a,b) and later on intensively discussed by Peter Moylan and David Hill (Hill and Moylan, 1980). The concept of dissipativity is a generalization of the concept of passivity which had already been exploited by this time and whose implications on stability have been widely studied in the literature (Kalman, 1964, 1963; Moylan, 1974; Hill and Moylan, 1976; Byrnes et al., 1991). A very nice treatment of recent results from passivity theory is provided e. g. in Brogliato et al. (2007). In the following the concept of dissipativity is reviewed, some important results from the theory of dissipative systems are discussed and some new ones derived that will be used frequently in the subsequent considerations.

2.2.1 Definitions and general aspects Dissipativity and in particular passivity is a notion on the relations between the inputs and outputs of systems. It thus applies on static or memory-less systems of the form y = φ(u)

(2.12)

with a vector-valued function φ : 𝒰 → 𝒴 mapping from the (metric) input space 𝒰 to the (metric) output space 𝒴 , as well as to systems governed by ẋ = f (x, u),

t > 0,

x(0) = x 0

(2.13a)

2.2 Dissipative systems | 23

y = h(x),

t≥0

(2.13b)

with the state x(t) at time t ≥ 0 in the metric space 𝕏, f : 𝒟 ⊆ 𝕏 × 𝒰 → 𝕏 with its domain 𝒟 ⊆ X, and h: 𝕏 → 𝒴 . In the case that such a state-space representation is given the question arises on how these input-output relations are reflected in the structure of this representation. This question can be addressed by viewing the dissipativity in terms of the supply rate ω: 𝒰 × 𝒴 → ℝ and the storage functional 𝒮 : 𝕏 → ℝ, which represents a generalization of the concept of energy stored in the system in dependence of its state x(t) ∈ 𝕏 at time t. Definition 2.2.1. System (2.13) is called dissipative with respect to the supply rate ω: 𝒰 × 𝒴 → ℝ, if there exists a positive semi-definite storage functional 𝒮 : 𝕏 → ℝ such that for all admissable initial states x 0 and inputs u ∈ 𝒰 it holds that t

∀0 ≤ t :

𝒮 (x(t)) − 𝒮 (x 0 ) ≤ ∫ ω(u(τ), y(τ))dτ.

(2.14)

0

Note that static maps, i. e. memory-less systems, do not have states and thus no energy storage. In the case that (2.14) holds with equality, the system is called lossless. For the control design approaches discussed in this study, loss-less systems do not play an important role. If the net energy flux is into the system, the supply rate is always non-negative, i. e. ω(u, y) ≥ 0 for all u ∈ 𝒰 , y ∈ 𝒴 . In many cases a quantitative bound for the energy dissipation, i. e. the difference between the left and right hand sides of inequality (2.14) plays a crucial role. To establish important cases of energy dissipation bounds a differential version of the dissipation inequality is advantageous. For this purpose the rate of change in the stored energy with respect to time along the solution x : [0, ∞) → 𝕏 is defined similarly to (2.7) by considering the solution trajectory x(τ; ξ , u) starting at ξ and evolving with τ ≥ 0 in function of the admissable input u ∈ 𝒰 so that x(0; ξ , u) = ξ as 1 d𝒮 (ξ ): = lim [𝒮 (x(τ; ξ , u)) − 𝒮 (ξ )]. τ→0 dt τ

(2.15)

With this definition at hand the concept of strict state dissipativity is introduced. Definition 2.2.2. System (2.13) is called strictly state dissipative with dissipation rate κ > 0 and supply rate ω: 𝒰 × 𝒴 → ℝ, if there exists a positive semi-definite storage functional 𝒮 : 𝕏 → ℝ such that for all x ∈ 𝕏 and admissable inputs u ∈ 𝒰 it holds that d𝒮 (x) ≤ ω(u, y) − κ‖x‖2 . dt

(2.16)

For the purpose of designing feedback loops in a dissipation-based framework the consideration of particular supply rates is an important step. In the subsequent chapters two particular cases are considered.

24 | 2 Stability, dissipativity and some system-theoretic concepts Definition 2.2.3. Let 𝒰 , 𝒴 be equipped with inner products ⟨⋅, ⋅⟩𝒰 , ⟨⋅, ⋅⟩𝒴 , respectively, u ∈ 𝒰 , y ∈ 𝒴 . System (2.13) is called (i) passive, if 𝒰 = 𝒴 and it is dissipative with respect to the supply rate ω(u, y) = ⟨u, y⟩𝒴 , (ii) (Q, S, R)-dissipative, if it is dissipative with respect to the quadratic supply rate y Q ω(u, y) = ⟨[ ] , [ ∗ u S

S y ] [ ]⟩ := ⟨y, Qy⟩𝒴 + ⟨y, Su⟩𝒴 + ⟨u, S∗ y⟩𝒰 + ⟨u, Ru⟩𝒰 R u (2.17)

where3 Q : 𝒴 → 𝒴 , S : 𝒰 → 𝒴 , S∗ : 𝒴 → 𝒰 and R : 𝒰 → 𝒰 , (iii) (Q, S, R)-strictly state dissipative with dissipation rate κ > 0, if it is strictly state dissipative with dissipation rate κ and the supply rate (2.17). The particular evaluation of the supply rate depends on the spaces 𝒰 and 𝒴 with their associated inner products. Example 2.2.1. For the finite-dimensional case, i. e. 𝕏=ℝn and 𝒰 = ℝp , 𝒴 = ℝm , the supply rate (2.17) is given by T

y Q ω(u, y) = [ ] [ T u S with Q ∈ ℝm×m , S ∈ ℝm×p and R ∈ ℝp×p .

S y ][ ] R u

(2.18)

⬦

Example 2.2.2. For 𝒰 = (L2 (0, 1))p and 𝒴 = (L2 (0, 1))m the supply rate (2.17) is evaluated as 1

ω(u, y) = ∫ [ 0

T

y(z) Q ] [ ∗ u(z) S

with Q : 𝒴 → 𝒴 , S : 𝒰 → 𝒴 and R : 𝒰 → 𝒰 .

S y(z) ][ ] dz R u(z)

(2.19)

⬦

Having in mind the direct method of Lyapunov for stability assessment discussed in Section 2.1.2 it becomes clear from (2.16) that there is a close connection between dissipativity and stability. This is highlighted in the following example. Example 2.2.3. Consider the case of a finite-dimensional passive system with a positive definite storage function 𝒮 (x) > 0, x ∈ 𝕏=ℝn i. e. it holds that d𝒮 (x) ≤ uT y. dt

(2.20)

3 In the case of finite-dimensional systems Q, S, R are matrices and the adjoint S∗ is given by ST .

2.2 Dissipative systems | 25

Accordingly, choosing u = −Ky,

K = KT > 0

(2.21)

one obtains that d𝒮 (x) ≤ −y T Ky ≤ 0. dt By virtue of Theorem 2.1.1 it follows that x = 0 is stable in the sense of Lyapunov as defined in Definition 2.1.2. Furthermore, from Theorem 2.1.2 it follows that the trajectories converge to the set 𝒳0 = {x ∈ 𝕏 | y = 0}.

The largest positively invariant subset ℳ ⊂ 𝒳0 can be defined as ℳ = {x ∈ 𝕏 | y = 0, y

(k)

= 0, k ∈ ℕ},

(2.22)

where y (k) denotes the k-th time derivative of y. In the case of a linear system Σ(A, B, C) with u = 0 the requirement that y and all its time derivatives are zero can be written in virtue of the Cayleigh–Hamilton Theorem (Callier and Desoer, 1991) as (see also Appendix B) 𝒦o x = 0

(2.23)

where C CA .. . n−1 CA [

[ [ 𝒦o = [ [ [

] ] ] ] ] ]

is the Kalman observability matrix. Clearly, if the system is completely observable, equality (2.23) implies that x = 0, meaning that the unique trajectory contained in the set ℳ defined in (2.22) is x = 0 and thus, in virtue of Corollary 2.1.1 the origin is asymptotically stable. Accordingly, if a linear system is completely observable and passive with a positive definite storage function, the origin can be asymptotically stabilized by the output-feedback (2.21). For nonlinear finite-dimensional systems the same reasoning applies with the difference, that from the observability of the subspace in which y and its time-derivatives are zero it is not possible to conclude the observability of the complete state space. This motivates the introduction of the following concept. ⬦ Definition 2.2.4. A system is called zero-state observable, if for u = 0 the restriction y ≡ 0 implies that x(t) = 0 for all t ≥ 0. It is called zero-state detectable, if the same restriction implies that limt→∞ x(t) = 0.

26 | 2 Stability, dissipativity and some system-theoretic concepts With this definition at hand the following result is obtained. Lemma 2.2.1. Consider a passive system Σ with state x(t) ∈ ℝn , t ≥ 0 and positive definite storage functional 𝒮 : ℝn → ℝ. If Σ is zero-state observable then the origin is asymptotically stabilizable with the output-feedback controller (2.21). Proof. Let 𝒮 denote the positive definite storage function. Due to the passivity property the inequality (2.20) holds. Thus, with the controller (2.21) it follows from the Krasovsky–LaSalle invariance Theorem 2.1.2 that x(t) converges to the set ℳ defined in (2.22) for t → ∞. The zero-state observability implies that ℳ = {0} and hence, in virtue of Corollary 2.1.1 the origin x = 0 is asymptotically stable. Passivity of systems turns out to be preserved under parallel and negative feedback interconnections, implying that complex passive systems can be build up from passive subsystems, and the stability properties of the subsystems implies the stability properties of the over-all complex interconnection. Due to these nice properties the applications of passivity in control theory are quite vast. For further details the reader is referred to e. g. Brogliato et al. (2007), van der Schaft (2017), Sepulchre et al. (1997). In the remainder of this chapter and most of this work, the focus will be on the general quadratic supply rate (2.17). The concepts and results discussed in this section form the basis for the subsequent analysis and design methods. In particular it remains to clarify under which conditions dynamical systems and static maps satisfy the kind of dissipativity conditions introduced and discussed in this section, and for the design purposes which dissipativity properties of system interconnections ensure the exponential stability of a desired solution. These questions are addressed in the following sections.

2.2.2 Finite-dimensional systems: necessary and sufficient conditions Consider a nonlinear input affine system in the following form ẋ = f (x) + G(x)u, y = h(x),

t > 0, t≥0

x(0) = x 0

(2.24)

with the state x(t) ∈ 𝕏 =ℝn , f : ℝn → ℝn a smooth vector field, G : ℝn → ℝn×p a matrix valued function whose columns g i : ℝn → ℝn , i = 1, . . . , p are smooth vector fields, u ∈ ℬ([0, ∞), ℝp ), and h : ℝn → ℝm . The following result is a connection between the concept of (Q, S, R)-strict state dissipativity and the Kalman–Yakubovich–Popov lemma (Kalman, 1963; Popov, 1959, 1964; Yakubovich, 1962a,b; Brogliato et al., 2007) and has been reported first by Hill and Moylan (1976). For completeness, the proof of the Lemma is included here. Note that in contrast to the original proof in Hill and Moylan (1976) using the notion of available storage, the version of Sepulchre et al. (1997) for passive systems is adapted for the purpose at hand. To state the result recall

2.2 Dissipative systems | 27

the notion of Lie derivative Lkf 𝒮 , k ∈ ℕ0 of a function 𝒮 : 𝕏 → ℝ with respect to the vector field f : 𝕏 → 𝕏, given by L0f 𝒮 (x) = 𝒮 (x),

L1f 𝒮 (x) =: Lf 𝒮 (x) =

𝜕𝒮 (x) f (x), 𝜕x

For G : 𝕏 → ℝn×p introduce accordingly LG 𝒮 (x) :=

Lkf 𝒮 (x) =

𝜕𝒮 (x)G(x) 𝜕x

𝜕Lk−1 f 𝒮 (x) 𝜕x

f (x).

∈ ℝ1×p for all x ∈ 𝕏.

Lemma 2.2.2. Let Q ∈ ℝm×m , S ∈ ℝm×p , R ∈ ℝp×p . The system (2.24) is (Q, S, R)-dissipative with storage functional 𝒮 : 𝕏 = ℝn → ℝ if and only if there exist functions q: ℝn → ℝn and W : ℝn → ℝn×p such that 1 Lf 𝒮 (x) = hT (x)Qh(x) − qT (x)q(x) 2 LG 𝒮 (x) = 2hT (x)S − qT (x)W(x) T

2R = W (x)W(x).

(2.25a) (2.25b) (2.25c)

Proof. Sufficiency: Consider that the conditions (2.25) hold and write the difference between the supplied energy according to (2.18) and the actual change in the stored energy ω(y, u) −

d𝒮 (x) = y T Qy + 2y T Su + uT Ru − Lf 𝒮 (x) − LG 𝒮 (x)u dt = hT (x)Qh(x) + 2hT (x)Su + uT Ru − Lf 𝒮 (x) − LG 𝒮 (x)u = hT (x)Qh(x) + 2hT (x)Su + uT Ru

1 − (hT (x)Qh(x) − qT (x)q(x)) − (2hT (x)S − qT (x)W(x))u 2 1 = qT (x)q(x) + uT Ru + qT (x)W(x)u 2 1 1 = qT (x)q(x) + uT W T (x)W(x)u + qT (x)W(x)u 2 2 1 T = (q(x) + W(x)u) (q(x) + W(x)u)≥ 0. 2

It follows that T

T

T

̇ 𝒮 (x) ≤ ω(y, u) = y Qy + 2y Su + u Ru. meaning that (2.24) is (Q, S, R)-dissipative. Necessity: Suppose that (2.24) is (Q, S, R)-dissipative, then ̇ 0 ≤ y T Qy + 2y T Su + uT Ru − 𝒮 (x)

= y T Qy + 2y T Su + uT Ru − Lf 𝒮 (x) − LG 𝒮 (x)u

= hT (x)Qh(x) + 2hT (x)Su + uT Ru − Lf 𝒮 (x) − LG 𝒮 (x)u

=: d(x, u).

(2.26)

28 | 2 Stability, dissipativity and some system-theoretic concepts Given that d(x, u) is quadratic in u and non-negative in u, x it follows that there must exist a vector-valued function q(x) and a matrix-valued function W(x) such that the scalar function d(x, u) can be expressed in terms of the quadratic form4 1 T d(x, u) = [q(x) + W(x)u] [q(x) + W(x)u] ≥ 0. 2

(2.27)

Expanding this quadratic form yields 1 1 d(x, u) = qT (x)q(x) + uT W T (x)q(x) + uT W T (x)W(x)u. 2 2 Comparing coefficients with the definition of d(x, u) in (2.26) yields the expressions (2.25). 2.2.3 Linear systems with quadratic storage functions Consider the linear time-invariant system Σ(A, B, C) given by the dynamics ẋ = Ax + Bu, y = Cx,

t > 0, t≥0

x(0) = x 0

(2.28)

with state x(t)∈ 𝕏 at time t ≥ 0, input u(t) ∈ 𝒰 , output y(t) ∈ 𝒴 , operator A : 𝒟(A) ⊆ 𝕏 → 𝕏 with domain 𝒟(A), B : 𝒰 → 𝕏 and C: 𝕏 → 𝒴 . The (Q, S, R)-strict state dissipativity of (2.28) can be characterized in terms of a linear operator inequality (LOI), as stated in the following Lemma (cp. Pandolfi, 1998). Lemma 2.2.3. The linear system (2.28) with state x(t) ∈ 𝕏, t ≥ 0, input u(t) ∈ 𝒰 , output y(t) ∈ 𝒴 is (Q, S, R)-strictly state dissipative with rate κ > 0 for Q : 𝒴 → 𝒴 , S : 𝒰 → 𝒴 , R : 𝒰 → 𝒰 and quadratic storage functional 𝒮 : 𝕏 → ℝ with 𝒮 (x) = ⟨x, Px⟩> 0 with self adjoint operator P = P ∗ : 𝕏 → 𝕏 if and only if the linear operator inequality PA + A∗ P + κI ⟨χ, ([ B∗ P

PB C ∗ QC ]−[ ∗ 0 S C

C∗ S ]) χ⟩ ≤ 0, R

∀0 ≠ χ ∈ 𝕏 × 𝒰 .

(2.29)

holds true. Proof. Taking the time derivative of 𝒮 along solutions of (2.28) yields d𝒮 (x) = ⟨x, P(Ax + Bu)⟩ + ⟨(Ax + Bu), Px⟩ dt = ⟨x, [A∗ P + PA]x⟩ + ⟨x, PBu⟩ + ⟨u, B∗ Px⟩ x PA + A∗ P = ⟨[ ] , [ u B∗ P

PB x ] [ ]⟩ . 0 u

4 This actually corresponds to a factorization of d(x, u) ≥ 0 similar to taking the square root of a scalar.

2.2 Dissipative systems | 29

Sufficiency: Using (2.29) it follows immediately that x C ∗ QC d𝒮 (x) ≤ ⟨[ ] , [ ∗ u S C dt y Q = ⟨[ ] , [ ∗ u S

C∗ S x ] [ ]⟩ − κ⟨x, x⟩ R u

S y ] [ ]⟩ − κ‖x‖2 R u

showing that (2.28) is (Q, S, R)-strictly state dissipative with dissipation rate κ. Necessity: Suppose that (2.28) is (Q, S, R)-strictly state dissipative with the given storage function 𝒮 . Then the result follows by substituting the definition of (Q, S, R)-strict . state dissipativity into d𝒮(x) dt Note that this result holds for the finite-dimensional as well as the infinitedimensional set-up. For the case of passivity, i. e. Q = 0, S = 21 I, R = 0 this result goes at hand with the celebrated Kalman–Yakubovich–Popov lemma (Kalman, 1963; Popov, 1959, 1964; Yakubovich, 1962a,a; Brogliato et al., 2007) summarizing the statespace properties of minimal realizations of positive real transfer functions. In the finite dimensional case the preceding result can also be shown using Lemma 2.2.2 as follows. According to Lemma 2.2.2 the system (2.28) is (Q, S, R)-dissipative if and only if there exist q(x) and W(x) such that 1 Lf 𝒮 (x) = x T (AT P + PA)x = x T C T QCx − qT (x)q(x) 2 Lg 𝒮 (x) = 2x T PB = 2x T C T S − qT (x)W(x) 2R = W T (x)W(x)

Substituting these expressions into the quadratic form (2.29) (with κ = 0) yields 1 − (qT (x)q(x) + 2qT (x)W(x)u + uT W T (x)W(x)u) 2 1 T = − [q(x) + W(x)u] [q(x) + W(x)u] ≤ 0. 2 In the case that κ > 0 the strict inequality is obtained. This shows the necessity ̇ part. Furthermore, the LMI (2.29) implies that d(x, u) := −𝒮 (x) + ω(Cx, u) ≥ 0 and is quadratic in u. Thus d can be factorized as in (2.27) showing that q(x) and W(x) exist satisfying (2.25), and thus (2.29) is also sufficient. At this place, for the finite-dimensional setup it is convenient to introduce for later reference the case that the matrix B depends on additional variables, summarized in a possibly time-varying parameter vector p(t) ∈ 𝒫 ⊂ ℝnp for all t ≥ 0, i. e. dx = A(p)x + B(p)u, dt y = Cx,

t > 0, x(0) = x 0 ,

(2.30a)

t ≥ 0.

(2.30b)

30 | 2 Stability, dissipativity and some system-theoretic concepts The consideration of such a parameter varying linear system has several application areas in control engineering (see e. g. Mohammadpour and Scherer, 2012) and offers interesting approaches to address also the control and observation of nonlinear systems. For these kind of systems the following concept of dissipativity turns out to be useful. Definition 2.2.5. The linear parameter varying system (2.30) is called uniformly (Q, S, R)-strictly state dissipative with dissipation rate κ > 0 if (2.29) holds uniformly for all p ∈ 𝒫 . Note that in the theory of linear parameter-varying systems the output operator C is also possibly parameter-varying. Nevertheless, for the present study it is sufficient to consider the simpler case of constant output operators.

2.2.4 Exponential stability and dissipativity As a central concern in exploiting dissipativity theory for design purposes consists in establishing conditions for which a desired dissipativity property is achieved, it is helpful to have relations at hand between the stability properties of open systems with zero input and the dissipativity properties with non-zero input. This holds in particular, because there are many approaches and methods known for observer and feedback control design which ensure exponential stability. When interconnecting such exponentially stable systems with others the question always arises if the stability is preserved in the interconnections. Passivity is known to preserve such nice properties and for the subsequent analysis it will be crucial to have criteria for which the more general (Q, S, R)-dissipativity properties ensure the same. As a first step to approach this question, connections between the exponential stability of the zero-solution of a system with zero input and its dissipativity for non-zero input are established here. Lemma 2.2.4. Consider the linear system ẋ = Ax + Gu, y = Hx,

t > 0,

t ≥ 0,

x(0) = x 0

(2.31a) (2.31b)

with state x(t) ∈ 𝕏 at time t ≥ 0, input u(t) ∈ 𝒰 , output y(t) ∈ 𝒴 , A : 𝒟(A) ⊆ 𝕏 → 𝕏, G : 𝒰 → 𝕏, and H : 𝕏 → 𝒴 . Let Q : 𝒴 → 𝒴 , S : 𝒰 → 𝒴 , R : 𝒰 → 𝒰 and A : 𝒟(A) ⊆ 𝕏 → 𝕏 be the infinitesimal generator of an exponentially stable C0 semi-group of contractions (T(t))t≥0 with growth bound ω = −γ < 0, i. e., for all t ≥ 0 it holds true that ‖T(t)‖O ≤ e−γt . If it holds that for all χ ∈ 𝕏 × 𝒰 (2γ − κ)I − H ∗ RH ⟨χ, [ SH − G∗

H ∗ S∗ − G ] χ⟩ ≥ 0, −Q

(2.32)

2.2 Dissipative systems | 31

then the linear system Σ(A, G, H) defined in (2.31) is (−R, S∗ , −Q)-strictly state dissipative with dissipation rate κ > 0. Proof. Let A be as stated the infinitesimal generator of an exponentially stable contraction semi-group with growth bound ω = −γ < 0. Thus, for all admissable x ∈ 𝕏 and t, τ ≥ 0 it holds true that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 −γτ 󵄩󵄩x(t + τ; x 0 )󵄩󵄩󵄩 = 󵄩󵄩󵄩T(τ)x(t; x 0 )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩x(t; x 0 )󵄩󵄩󵄩e ,

∀t, τ ≥ 0.

(2.33)

Define the positive definite storage functional 𝒮 : 𝕏 → 𝕏 as 2

𝒮 (x) = ‖x‖ = ⟨x, x⟩.

For u = 0 it follows that the rate of change of 𝒮 along solutions x can be expressed on the one side in function of A, and on the other side can be bounded by using (2.33), i. e. for all x ∈ 𝕏 it holds true that d𝒮 (x) = (⟨Ax, x⟩ + ⟨x, Ax⟩) dt 1 󵄩 󵄩2 󵄩 󵄩2 = lim (󵄩󵄩󵄩x(t + τ)󵄩󵄩󵄩 − 󵄩󵄩󵄩x(t)󵄩󵄩󵄩 ) τ→0 τ 1 󵄩 󵄩2 󵄩 󵄩2 ≤ lim (󵄩󵄩󵄩x(t)󵄩󵄩󵄩 e−2γτ − 󵄩󵄩󵄩x(t)󵄩󵄩󵄩 ) τ→0 τ (e−2γτ − 1) 󵄩󵄩 󵄩2 = lim 󵄩󵄩x(t)󵄩󵄩󵄩 τ→0 τ 󵄩 󵄩2 = −2γ 󵄩󵄩󵄩x(t)󵄩󵄩󵄩 where the last step is obtained by applying the rule of L’Hôpital–Bernoulli (Strang, 1991). For the input u ≠ 0, we thus have that d𝒮 (x) = (⟨Ax + Gu, x⟩ + ⟨x, Ax + Gu⟩) dt = (⟨Ax, x⟩ + ⟨x, Ax⟩) + (⟨Gu, x⟩ + ⟨x, Gu⟩) ≤ −2γ‖x‖2 + (⟨Gu, x⟩ + ⟨x, Gu⟩)

or equivalently x −2γI d𝒮 (x) ≤ ⟨[ ] , [ ∗ u G dt

G x ] [ ]⟩ 0 u

(2.34)

In virtue of Lemma 2.2.3 the (−R, S∗ , −Q)-strict state dissipativity is thus ensured if for all [x T uT ]T ∈ 𝕏 × 𝒰 it holds true that x −2γI ⟨[ ] , [ ∗ u G

G x x −κI − H ∗ RH ] [ ]⟩ ≤ ⟨[ ] , [ 0 u u SH

or equivalently, if (2.32) holds true.

H ∗ S∗ x ] [ ]⟩ −Q u

32 | 2 Stability, dissipativity and some system-theoretic concepts From this result in a general metric space 𝕏 the following corollary for the finitedimensional case is obtained. Corollary 2.2.5. Let 𝕏 = ℝn , A ∈ ℝn×n , G ∈ ℝn×p , H ∈ ℝm×n , Q ∈ ℝm×m , S ∈ ℝm×p , R ∈ ℝp×p with Q < 0 and T(t) = eAt with ‖T(t)‖O ≤ e−γt for t ≥ 0 so that γ>

1 max λ, 2 λ∈σ(ℳ)

T

T T

T

ℳ = κI + H RH − (G − H S )Q (G − SH). −1

(2.35)

Then the linear system Σ(A, G, H) defined in (2.31) is (−R, ST , −Q)-strictly state dissipative with dissipation rate κ > 0. Proof. Let (2.35) hold true, implying that (2γ − κ)I − H T RH + (H T ST − G)Q−1 (SH − GT ) > 0.

(2.36)

Given that Q < 0 by assumption, it must hold that (2γ − κ)I − H T RH > 0.

(2.37)

Note that the left hand side of (2.36) is the Schur complement (see Appendix A.2) of the matrix (2γ − κ)I − H T RH K0 = [ SH − GT

H T ST − G ] −Q

so that from the condition (2.36) it follows that K0 > 0. The (−R, ST , −Q)-strict state dissipativity with dissipation rate κ of the linear system Σ(A, G, H) follows from Lemma 2.2.4. The importance of these results will become evident in the following discussions, but at this moment it should be highlighted, that for linear systems there are many different well-established approaches for ensuring that a given exponential convergence rate is satisfied, as long as an according detectability or stabilizability property is ensured for the linear system (Kailath, 1980; Curtain and Zwart, 1995). The result in Corollary 2.2.5 can be extended to the case of nonlinear systems in 𝕏 ⊆ ℝn which can be transformed into the particular form ẋ = Ax + G(x)ν,

σ = h(x)

ν = −φ(σ)

x(0) = x 0

(2.38a) (2.38b) (2.38c)

where the matrix-valued function G(x) has entries gij (x) which are smooth in x and are bounded, so that there is a constant ḡ > 0 such that ‖G(x)‖ ≤ ḡ on 𝕏, and h is Lipschitz-continuous with Lipschitz constant Lh . A special situation where this kind

2.2 Dissipative systems | 33

of interconnection of a partially linear dynamical subsystem interconnected with a nonlinear static map occurs is given when parts of a nonlinear system can be exactly linearized by state-feedback control and the remaining nonlinear parts which are not compensated are treated as a possibly destabilizing nonlinear feedback loop (see Section 3.3 for more details). For this kind of systems the following result is useful. Lemma 2.2.6. Let Q ≤ −qI with q > 0, R ≤ 0 and ‖G(x)‖ ≤ ḡ for all x ∈ 𝕏 = ℝn . Then the subsystem (2.38a)–(2.38b) with state space 𝕏 ⊆ ℝn is (−R, ST , −Q)-strictly state dissipative with dissipation rate κ > 0 if σ(A) ⊂ ℂ− and μ > κ + ‖R‖L2h +

(ḡ + ‖S‖Lh )2 , q

μ=−

max

λ∈σ(A+AT )

λ,

(2.39)

where μ is the modulus of the maximum eigenvalue of the symmetric matrix A + AT and Lh the Lipschitz constant of the output map h. Proof. Consider the storage function 𝒮 (x) = ⟨x, x⟩ > 0

and its rate of change over time d T 𝒮 (x) = ⟨x, (A + A )x⟩ + 2⟨x, G(x)ν⟩. dt By the Courant–Fischer theorem (Dym, 2007) it holds that λ− ⟨x, x⟩ ≤ ⟨x, (A + AT )x⟩ ≤ λ+ ⟨x, x⟩,

λ− =

min

λ∈σ(A+AT )

λ,

λ+ =

max

λ∈σ(A+AT )

λ.

Taking into account that σ(A) ⊂ ℂ− it follows that ⟨x, (A + AT )x⟩ ≤ −μ⟨x, x⟩,

μ=−

max

λ∈σ(A+AT )

λ.

On the other hand, from the above considerations it is clear that the system is (−R, ST , −Q)-strictly state dissipative with dissipation rate κ > 0 if σ −R d 𝒮 (x) ≤ ⟨[ ] , [ ν S dt

ST σ ] [ ]⟩ − κ⟨x, x⟩ = −⟨σ, Rσ⟩ + 2⟨σ, ST ν⟩ − ⟨ν, Qν⟩ − κ⟨x, x⟩. −Q ν

Summarizing, the dissipativity is ensured if ⟨x, (A + AT )x⟩ + 2⟨x, G(x)ν⟩ + ⟨σ, Rσ⟩ − 2⟨σ, ST ν⟩ + ⟨ν, Qν⟩ + κ⟨x, x⟩

≤ −μ⟨x, x⟩ + 2⟨x, G(x)ν⟩ + ⟨h(x), Rh(x)⟩ − 2⟨h(x), ST ν⟩ + ⟨ν, Qν⟩ + κ⟨x, x⟩ ≤ 0.

(2.40)

34 | 2 Stability, dissipativity and some system-theoretic concepts By assumption it holds that 󵄩 󵄩󵄩 ̄ 󵄩󵄩G(x)ν 󵄩󵄩󵄩 ≤ g‖ν‖,

󵄩 󵄩󵄩 󵄩󵄩h(x)󵄩󵄩󵄩 ≤ Lh ‖x‖,

ν T Qν ≤ −q‖ν‖2

with q > 0. Accordingly, (2.40) is fulfilled if (−μ + κ + ‖R‖L2h )‖x‖2 + (2ḡ + 2‖S‖Lh )‖x‖‖ν‖ − q‖ν‖2 ≤ 0. This is a quadratic form and thus the inequality is fulfilled if −μ + κ + ‖R‖L2h [ ḡ + ‖S‖Lh

ḡ + ‖S‖Lh ] < 0. −q

Using the Schur complement (Lemma A.2.1), the preceding condition is ensured if (i)

μ − κ − ‖R‖L2h > 0,

(ii) μ − κ − ‖R‖L2h −

(ḡ + ‖S‖Lh )2 > 0. q

Given that q > 0 the condition (ii) is stronger than condition (i). Thus, condition (ii) is sufficient for establishing the (−R, ST , −Q)-strict state dissipativity with dissipation rate κ. Further it can be written as in (2.39), proofing the statement. Note that in the preceding proof instead of the Schur complement in this twodimensional set-up one can argue equivalently using the trace and the determinant of the matrix.

2.2.5 Static (memory-less) subsystems and sector nonlinearities In the following consider nonlinear static maps of the form y = φ(u)

(2.41)

with φ : 𝒰 → 𝒴 and u ∈ 𝒰 , the space of input functions. For the static map (2.41) the actual output depends only on the actual input and not on the past, so that these maps are called memory-less. Accordingly, the state space is void and the internal storage 𝒮 is zero. For this reason the dissipativity condition (2.14) just reads5 ω(u, y) ≥ 0

∀u ∈ 𝒰 .

A particular class of static maps are sector nonlinearities. 5 See also the discussion right after Definition 2.2.1.

2.2 Dissipative systems | 35

Example 2.2.4. For the simple case example y = sin(u),

(2.42)

u ∈ ℝ,

the graph of the map is illustrated in Figure 2.1. It can be seen that the graph is completely contained in the sector6 [k1 , k2 ] = [−0.22, 1] so that (k2 u − φ(u))(φ(u) − k1 u) = (k2 u − y)(y − k1 u) ≥ 0,

∀u ∈ ℝ.

(2.43)

Expanding terms leads to the particular (Q, S, R)-dissipativity condition T

y −1 − y2 − (k1 + k2 )yu − k1 k2 u2 = [ ] [ 1 u (k + k2 ) 2 1

1 (k 2 1

+ k2 ) y ] [ ] = ω(u, y) ≥ 0, u −k1 k2 (2.44)

∀u ∈ ℝ. ⬦

Figure 2.1: Sector condition for the static map y = sin(u) in (2.42).

According to the preceding example, scalar sector nonlinearities are generically (Q, S, R)-dissipative with Q = −1,

1 S = (k1 + k2 ), 2

R = −k1 k2 .

(2.45)

It should be noted that passive nonlinearities are described by maps for which y u = φ(u)u ≥ 0,

∀u ∈ ℝ

6 The slopes of the sector bounds can be determined by the maximum and minimum slope of the nonlinear function φ. In the case of (2.42) the maximum slope is +1, and the minimum slope of the sector is determined by the point of intersection of the straight line y = mu with the map y = sin(u), defined by the condition sin(u) = cos(u)u, stemming from the requirement that the slope of sin(u) equals the slope of mu at the intersection point.

36 | 2 Stability, dissipativity and some system-theoretic concepts and which are thus completely contained in the first-third quadrant pair. Such maps are (Q, S, R)-dissipative with Q = 0,

1 S= , 2

R = 0.

Clearly, the concepts of output-feedback passivity (OFP) with φ(u)u − ρφ2 (u) ≥ 0,

∀u ∈ ℝ

for some ρ ≠ 0, or input-feedforward passivity (IFP) with φ(u)u − γu2 ≥ 0,

∀u ∈ ℝ

for some γ ≠ 0 perfectly fit into the framework of (Q, S, R)-dissipativity with Q = −ρ, Q = 0,

1 S= , R=0 2 1 S = , R = −γ 2

(OFP) (IFP).

These particular concepts will not find special use in the following considerations and are thus only included for completeness here. The reader who is interested in these and further particular passivity concepts and their application is referred to the books (Brogliato et al., 2007; van der Schaft, 2017; Khalil, 1996; Sepulchre et al., 1997; Bao and Lee, 2007). Sometimes it is useful to write sector conditions as quadratic inequalities of the form u(k1 u − y) ≤ 0,

u(y − k2 u) ≤ 0,

∀u ∈ ℝ

(2.46)

or simply, after separating the mixed term uy as k1 u2 ≤ uy ≤ k2 u2 ,

∀u ∈ ℝ.

The geometrical interpretation of this relation is depicted in Figure 2.2.

Figure 2.2: Sector condition (2.47) for the nonlinear map y = sin(u) with [k1 , k2 ] = [−0.22, 1].

(2.47)

2.2 Dissipative systems | 37

Summarizing the above, the following useful result is obtained. Lemma 2.2.7. Let y = φ(u) be contained in the sector [k1 , k2 ]. The following assertions are equivalent: 1. The inequality (k2 u − y)(y − k1 u) ≥ 0 is satisfied for all u ∈ ℝ 2. It holds that k1 u2 ≤ uy ≤ k2 u2 for all u ∈ ℝ 3. The map y = φ(u) is (Q, S, R)-dissipative with Q = −1, S = 21 (k1 + k2 ), R = −k1 k2 . Proof. The equivalence between properties 1 and 3 has already been shown above. Thus it remains to show that property 2 and 1 are equivalent. For this purpose let property 2 hold. It follows that for all u ∈ ℝ k1 u2 − yu ≤ 0 ≤ k2 u2 − yu

(2.48a)

⇒ (k1 u2 − yu)(k2 u2 − yu) ≤ 0

(2.48b)

⇔ u2 (k1 u − y)(k2 u − y) ≤ 0

(2.48c)

⇔ (y − k1 u)(k2 u − y) ≥ 0

(2.48d)

showing that property 1 also holds true. The reverse direction follows directly up to inequality (2.48b). By assumption it holds that k2 > k1 and thus (2.48a) follows. Even though it becomes geometrically intractable for high-dimensional systems, the sector concept can be extended to the case where u ∈ 𝒰 = ℝp and y ∈ 𝒴 = ℝ by considering (k T2 u − φ(u))(φ(u) − k T1 u) = (uT k 2 − φ(u))(φ(u) − k T1 u) ≥ 0,

u ∈ ℝp .

(2.49)

Substituting y = φ(u) and re-arranging into a quadratic form, this relation defines the supply rate T

−1 y ω(u, y) = [ ] [ 1 u (k + k 2 ) 2 1

1 T (k 2 1

+ k T2 )

−k 2 k T1

y ] [ ] ≥ 0, u

∀u ∈ ℝp ,

(2.50)

showing that in this case the nonlinear map is (Q, S, R)-dissipative with Q = −1,

1 S = (k T1 + k T2 ), 2

R = −k 2 k T1 .

In the case that φ is a vector-valued function whose entries φi (u), i = 1, . . . , q satisfy sector conditions of the form (2.49), or equivalently which are associated to supply rates ωi (u, φi (u)) of the form (2.50), a dissipativity property for φ(u) can be obtained e. g. by building the weighted sum of the supply rates ωi (u, φi (u)) with positive weights

38 | 2 Stability, dissipativity and some system-theoretic concepts θi ≥ 0, i = 1, . . . , q, i. e. q

q

i=1

i=1

ω(u, φ(u)) = ∑ θi ωi (u, yi ) = ∑ θi (−φi (u)2 + φi (u)(k Ti1 + k Ti2 )u − uT k i2 k Ti1 u) θ1 (k T11 + k T12 ) q ] [ .. ] u − uT (∑ θi k i2 k T )u = −φT (u)Θφ(u) + φT (u) [ i1 . ] [ i=1 T T θ (k + k ) q2 ] [ q q1 or summarizing T −Θ y ω(u, φ(u)) = [ ] [ 1 u (K + K2 ) 2 1

1 (K T 2 1

+ K2T )

−K2 Θ

−1

K1T

y ] [ ] ≥ 0, u

∀u ∈ ℝp

(2.51)

T

with matrices Θ = diag(θi ) and Ki = [θ1 k 1i ⋅ ⋅ ⋅ θq k qi ] ∈ ℝp×q , i = 1, 2. A particular class of dissipativity conditions are those determined by Lipschitz constants. In the present context this becomes particularly interesting when considering the case that φ(u) = α(ν + u) − α(ν)

(2.52)

for some vector valued function α : 𝒰 → ℝq which is Lipschitz continuous with local Lipschitz constant Lαν (Ω) over the domain Ω, i. e. it holds 󵄩󵄩 󵄩 α 󵄩󵄩α(ν 2 ) − α(ν 1 )󵄩󵄩󵄩 ≤ Lν ‖ν 2 − ν 1 ‖,

∀ν 1 , ν 2 ∈ Ω ⊂ 𝒰 .

(2.53)

This inequality can be rewritten as a dissipativity condition for the function φ(u) defined in (2.52) by setting u = ν 2 − ν 1 and rewriting (2.53) as T

φ(u) −I 2 0 ≤ (Lαν ) uT u − φ(u)T φ(u) = [ ] [ u 0

0 φ(u) ][ ], (Lαν )2 I u

∀u ∈ ℝp

showing that φ as defined in (2.52) is (Q, S, R)-dissipative with Q = −I,

S = 0,

2

R = (Lαν ) I.

(2.54)

The Lipschitz condition is one of the most direct approaches for determining dissipativity conditions for nonlinearities depending on more than one variable. For example, alternatively to the sector in the introducing example (2.42) the Lipschitz sector [−1, 1] could be chosen. Comparing both sectors in this example already shows that the dissipativity condition derived from the Lipschitz continuity is more conservative. Example 2.2.5. For the determination of a Lipschitz-based dissipativity condition for a more complex example, consider the non-isotonic Langmuir–Hinshelwood reaction rate r(c, T) =

kc e−γ/T (1 + σc)2

(2.55)

2.2 Dissipative systems | 39

occurring in the description of catalytic reactions in chemical process engineering (Tronci et al., 2011; Vasanth Kumar et al., 2008). Here c denotes the concentration of the reactant and T the temperature of the reaction medium at the catalytic surface. Concentrations can be normalized into the unit interval c ∈ [0, 1] where c = 1 refers to pure reactant. The temperature is limited by the energy conservation into an interval T ∈ [T − , T + ], so that for practical purposes it is sufficient to determine a dissipativity condition over the compact domain D = [0, 1] × [T − , T + ] ⊂ 𝕏 =ℝ2 . It holds that ∀[c, T]T ∈ D :

󵄩󵄩 󵄩 󵄨 󵄨 󵄩󵄩r(c + ec , T + eT ) − r(c, T)󵄩󵄩󵄩 ≤ max󵄨󵄨󵄨∇r(c, T)e󵄨󵄨󵄨, D

e e = [ c]. eT

Furthermore, it holds that 󵄨 󵄨 󵄩 󵄩 max󵄨󵄨󵄨∇r(c, T)e󵄨󵄨󵄨 ≤ max󵄩󵄩󵄩∇r(c, T)󵄩󵄩󵄩‖e‖ D

D

and 󵄩 󵄩 max󵄩󵄩󵄩∇r(c, T)󵄩󵄩󵄩 = max √(𝜕c r)2 + (𝜕T r)2 D

D

= k max[e−γ/T √( D

+

≤ ke−γ/(T ) √1 +

2

2

γc 1 − σc ) +( 2 )] (1 + σc)3 T (1 + σc)2

γ2 . (T − )2

Introducing the constant Lrc,T := ke−γ/(T ) √1 + +

γ2 (T − )2

(2.56)

it follows from the preceding calculations that Lrc,T is a (local) Lipschitz constant for r(c, T) so that ∀[c, T]T ∈ D :

󵄩󵄩 󵄩 r 󵄩󵄩r(c + ec , T + eT ) − r(c, T)󵄩󵄩󵄩 ≤ Lc,T ‖e‖

and in consequence the function φ(ec , eT ; c, T) = r(c +ec , T +eT )−r(c, T) is (Q, S, R)-dissipative with Q = −1,

S = 0,

2

R = (Lrc,T ) = k 2 e−2γ/(T ) (1 + +

γ2 ). (T − )2

⬦

(2.57)

Using local Lipschitz constants local dissipativity properties can be established, which can then be further exploited in the design of observers and control mechanisms as discussed in the considerations in the subsequent chapters. Anyway, as mentioned before, Lipschitz sectors can be rather conservative.

40 | 2 Stability, dissipativity and some system-theoretic concepts Example 2.2.6. Consider the nonlinear function φ(u) = u sin(ku),

(2.58)

u ∈ ℝ,

which actually is not Lipschitz continuous over the real axis but only over compact subintervals, because φ󸀠 (u) = sin(ku) + ku cos(ku),

󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨φ (u)󵄨󵄨 ≤ 1 + ku,

∀u ∈ ℝ.

Nevertheless, the map satisfies the (sharp) inequality 󵄨󵄨 󵄨 󵄨󵄨φ(u)󵄨󵄨󵄨 ≤ |u|,

∀u ∈ ℝ

and is thus completely contained in the linear sector [−1, 1] as can be seen in Figure 2.3. Accordingly (2.58) is (Q, S, R)-dissipative with (Q, S, R) = (−1, 0, 1).

Figure 2.3: Sector condition for the function φ(u) = u sin(ku) with k = 10, included in the (nonLipschitz) sector [−1, 1].

Considering in this case the Lipschitz sector over the compact interval u ∈ [−π, π], i. e. Lφ = 1 + kπ would lead to the conservative bound 󵄨󵄨 󵄨 󵄨󵄨φ(u)󵄨󵄨󵄨 ≤ (1 + kπ)|u|,

∀u ∈ ℝ.

For k = 10 (as in Figure 2.3) this means that the Lipschitz bound is about 32.4 times more conservative than the sharp one. ⬦ Even though they might be conservative, in many applications Lipschitz sector conditions provide a good first estimate. Furthermore, notice the following relation (see also Schaum and Meurer, 2019a). Lemma 2.2.8. A scalar static map φ: ℝ → ℝ which is (Q, S, R)-dissipative with Q < 0 and R > −S2 /|Q|2 is contained in the sector [k1 , k2 ] with k1 =

1 (S − √S2 + R|Q|), |Q|

k2 =

1 (S + √S2 + R|Q|). |Q|

(2.59)

2.2 Dissipative systems | 41

Proof. By assumption φ(u) is (Q, S, R)-dissipative with Q < 0. Thus in virtue of Definition 2.2.3 it holds that −φ(u)2 +

R 2 2S uφ(u) + u ≥ 0, |Q| |Q|

∀u ∈ ℝ.

This condition can be identified with the sector condition (k2 u − φ(u))(φ(u) − k1 u)

= −φ2 + (k1 + k2 )φu − k1 k2 u2 ≥ 0,

∀u ∈ ℝ

setting k1 + k2 =

2S , |Q|

−k1 k2 =

R |Q|

or equivalently k12 −

2S R k − = 0, |Q| 1 |Q|

k2 =

2S − k1 . |Q|

Given that by assumption R > −S2 /|Q| holds true it follows that the solutions for k1 , k2 of the preceding equation set are given by (2.59). In some applications it might also be useful to have the following result at hand: Lemma 2.2.9. Let φ: 𝒰 → 𝒴 be (Q, S, R)-dissipative. Then, ψ = −φ is (Q, −S, R)-dissipative. Proof. By assumption φ is (Q, S, R)-dissipative, i. e. the inequality φ Q ⟨[ ] , [ ∗ u S

S φ ] [ ]⟩ ≥ 0, R u

∀u ∈ 𝒰

holds true. It follows that ψ Q ⟨[ ] , [ ∗ u −S

−S ψ ] [ ]⟩ = ⟨ψ, Qψ⟩ + 2⟨ψ, −Su⟩ + ⟨u, Ru⟩ R u = ⟨−φ, Q(−φ)⟩ + 2⟨−φ, −Su⟩ + ⟨u, Ru⟩ = ⟨φ, Qφ⟩ + 2⟨φ, Su⟩ + ⟨u, Ru⟩ φ Q = ⟨[ ] , [ ∗ u S

S φ ] [ ]⟩ ≥ 0, R u

∀u ∈ 𝒰 .

This shows that ψ = −φ is (Q, −S, R)-dissipative. To finish this section, consider the particular case of a linear translation of a nonlinear map in the form φ(u) = α(u) + Mu

(2.60)

42 | 2 Stability, dissipativity and some system-theoretic concepts with some matrix M ∈ ℝp×p . Such situations classically occur when either a system of the form ẋ = α(x) is rewritten for analysis or design purposes as ẋ = −Mx + φ(x),

φ(x) = α(x) + Mx

or when an input-feedforward component is connected to a nonlinear static map (see Figure 2.4). For this case the following Lemma applies.

Figure 2.4: Input-Feedforward configuration of a nonlinear map α(u) and a constant matrix M.

Lemma 2.2.10. Let α: 𝒰 = ℝp → 𝒴 = ℝq be (Q, S, R)-dissipative. Then φ(u) = α(u)+Mu with M ∈ ℝq×p is (Q󸀠 , S󸀠 , R󸀠 )-dissipative with Q󸀠 = Q,

S󸀠 = S − QM,

R󸀠 = R − M T (S − QM) − (S − QM)T M − M T QM.

(2.61)

Proof. By assumption it holds that T

α Q ωα (u, α(u)) = [ ] [ T u S

S α ] [ ] = αT Qα + 2αT Su + uT Ru ≥ 0, R u

∀u ∈ 𝒰

with Q = QT and R = RT . A simple calculation shows that with (Q󸀠 , S󸀠 , R󸀠 ) given in (2.61) it holds that T

φ Q󸀠 [ ] [ 󸀠T u (S )

S󸀠 φ ][ ] R󸀠 u T

α + Mu Q =[ ] [ u (S − QM)T

S − QM α + Mu ][ ] T T R − M (S − QM) − (S − QM) M − M QM u T

= (α + Mu)T Q(α + Mu) + 2(α + Mu)T (S − QM)u

+ uT [R − M T (S − QM) − (S − QM)T M − M T QM]u

2.2 Dissipative systems | 43

= αT Qα + 2αT QMu + uT M T QMu + 2αT (S − QM)u + 2uT M T (S − QM)u + uT [R − 2M T (S − QM) − M T QM]u = αT Qα + 2αT (QM + S − QM)u + uT (R + M T QM + 2M T (S − QM) − 2M T (S − QM) − M T QM)u = αT Qα + 2αT Su + uT Ru = ωα (u, α(u)) ≥ 0,

∀u ∈ 𝒰 .

Thus, φ satisfies the dissipativity condition with Q󸀠 , S󸀠 , R󸀠 given in (2.61), i. e. with respect to the quadratic supply rate for φ defined by T

φ Q󸀠 ωφ (u, φ(u)) = [ ] [ 󸀠 T u (S )

S󸀠 φ ] [ ] ≥ 0, R󸀠 u

∀u ∈ 𝒰

with Q󸀠 = (Q󸀠 )T and R󸀠 = (R󸀠 )T . This completes the proof. 2.2.6 Interconnections of dynamic and static subsystems For many finite-and infinite-dimensional systems it is known how to design feedback controllers and observers to exponentially stabilize the origin of the associated error dynamics. Nevertheless, when interconnecting such systems, the exponential stability for each separate dynamics does not necessarily imply the asymptotic stability of the origin for the interconnected system. Dissipativity concepts on the other hand may be employed to derive sufficient conditions for exponential stability in system interconnections as will be shown in the sequel. In particular, in the following the focus is put on feedback interconnections of systems with static nonlinear maps like schematically illustrated in Figure 2.5.

Figure 2.5: Negative feedback interconnection (2.62) of nonlinear dynamic and static systems.

44 | 2 Stability, dissipativity and some system-theoretic concepts These system interconnections can be written in the so-called Lur’e form (Lur’e and Postnikov, 1944; Kalman, 1963; Pandolfi, 1998) ẋ = f (x, u),

t > 0,

u = −φ(σ),

t≥0

σ = h(x),

t≥0

x(0) = x 0

(2.62a) (2.62b) (2.62c)

with x(t) ∈ 𝕏 at t ≥ 0, f : 𝕏 × 𝒰 → 𝕏, h : 𝕏 → ℋ, φ : ℋ → 𝒰 . This kind of feedback interconnection of a linear dynamical and a nonlinear static system are typically studied in the framework of absolute stability (see e. g. Aizerman and Gantmacher, 1964; Khalil, 1996). For this system the following stability result is a direct consequence of the preceding discussions. Lemma 2.2.11. Consider the feedback interconnection (2.62) shown in Figure 2.5 with x ∈ 𝕏=ℝn and let φ(σ) be (Q, S, R)–dissipative. If the open-loop system (2.62a)–(2.62b) is (−R, S∗ , −Q)–state strictly dissipative with dissipation rate κ > 0 and a positive definite storage functional 𝒮 : 𝕏 → ℝ, then the origin x = 0 is asymptotically stable. Proof. By assumption and according to Definition 2.2.3 it holds that along solutions of (2.62) σ −R d𝒮 (x) ≤ ⟨[ ] , [ u S dt = − ⟨[

ST σ σ −R ] [ ]⟩ − κ‖x‖2 = ⟨[ ],[ −Q u −φ(σ) S

φ(σ) Q ],[ T σ S

ST σ ][ ]⟩ − κ‖x‖2 −Q −φ(σ)

S φ(σ) ][ ]⟩ − κ‖x‖2 R σ

In virtue of the (Q, S, R)-dissipativity of the static map φ it follows that d𝒮 (x) ≤ −κ‖x‖2 < 0. dt The asymptotic stability of the origin follows from Theorem 2.1.3. In the case that the storage functional 𝒮 (x) is quadratically bounded, i. e. there exist positive constants α, β > 0 such that α‖x‖2 ≤ 𝒮 (x) ≤ β‖x‖2

(2.63)

the preceding result can be extended to conclude the exponential stability by employing Theorem 2.1.4 and thus to provide an estimate of the convergence velocity. Corollary 2.2.1. Let the assumption of Lemma 2.2.11 hold with 𝕏 being a Hilbert space, and further let the storage functional 𝒮 : 𝕏 → ℝ be quadratically bounded as in (2.63). β Then the origin is exponentially stable with amplitude a = √ and convergence rate γ=

κ , 2β

i. e. ‖x(t)‖ ≤ a‖x 0 ‖e−γt .

α

2.2 Dissipative systems | 45

In addition to this result, which is just an adaptation of classical results presented e. g. in Hill and Moylan (1976), Moreno (2005) the following result can be used in cases where the (−R, S∗ , −Q)-strict dissipativity of the dynamical subsystem is not easy to establish, but it is known that it has an exponentially stable solution for u = 0. This result is also valid in infinite-dimensional state spaces. Lemma 2.2.12. Consider the Lur’e interconnection of the linear system Σ(A, G, H) defined in (2.31) with (nonlinear) feedback u = −φ that is Lipschitz continuous with Lipschitz constant Lφ and let A be the infinitesimal generator of an exponentially stable C0 semi-group (T(t))t≥0 , i. e. such that ‖T(t)‖O ≤ Me−γt for all t ≥ 0 with M ≥ 1 and γ > 0. If the auxiliary system χ̇ = −γχ + gH Mν,

t > 0,

σχ = χ,

χ(0) = M‖x 0 ‖,

t≥0

gH = ‖G‖O ‖H‖O

(2.64) (2.65)

with input ν and output σχ is (0, Lφ , 1)-state strictly dissipative with dissipation rate κ > 0, then the origin x = 0 is exponentially stable. Proof. The general (implicit) solution of the system ẋ = Ax − Gφ(σ),

t > 0,

σ = Hx,

t≥0

x(0) = x 0

can be written as t

x(t) = T(t)x 0 − ∫ T(t − τ)Gφ(σ(τ))dτ.

(2.66)

0

Taking norms on both sides and recalling the exponential stability of the semigroup (T(t))t≥0 one obtains t

󵄩󵄩 󵄩 󵄩 󵄩 −γt −γ(t−τ) ‖G‖O 󵄩󵄩󵄩φ(σ(τ))󵄩󵄩󵄩dτ 󵄩󵄩x(t)󵄩󵄩󵄩 ≤ Me ‖x 0 ‖ + ∫ Me 0

t

󵄩 󵄩 = e−γt (M‖x 0 ‖ + ∫ Meγτ ‖G‖O 󵄩󵄩󵄩φ(σ(τ))󵄩󵄩󵄩dτ). 0

Introducing the variable t

ℵ(t) = e

−γt

󵄩 󵄩 (M‖x 0 ‖ + ∫ Meγτ ‖G‖O 󵄩󵄩󵄩φ(σ(τ))󵄩󵄩󵄩dτ) 0

with ℵ(0) = M‖x 0 ‖,

(2.67)

46 | 2 Stability, dissipativity and some system-theoretic concepts it follows that ∀t ≥ 0 :

(2.68)

󵄩 󵄩󵄩 󵄩󵄩x(t)󵄩󵄩󵄩 ≤ ℵ(t)

and ̇ = −γℵ(t) + M‖G‖O 󵄩󵄩󵄩φ(σ(t))󵄩󵄩󵄩, ℵ(t) 󵄩 󵄩

t > 0,

ℵ(0) = M‖x 0 ‖.

(2.69)

Now, consider the Lipschitz constant Lφ of φ, so that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩φ(σ(t))󵄩󵄩󵄩 = 󵄩󵄩󵄩φ(σ(t)) − φ(0)󵄩󵄩󵄩 ≤ Lφ 󵄩󵄩󵄩σ(t)󵄩󵄩󵄩 ≤ Lφ ‖H‖O 󵄩󵄩󵄩x(t)󵄩󵄩󵄩 ≤ Lφ ‖H‖O ℵ(t).

(2.70)

Accordingly, (2.69) is bounded as ̇ ≤ −γℵ(t) + M‖G‖O ‖H‖O Lφ ℵ(t) = −(γ − gH Lφ M)ℵ(t) ℵ(t)

(2.71)

with gH = ‖G‖O ‖H‖O . Seeking for a characterization in form of a quadratic dissipation bound,7 the exponential stability can be established using the auxiliary system χ̇ = −γχ + gH Mϖ(χ),

σχ = χ,

t > 0, t≥0

χ(0) = χ0 = ℵ0 = M‖x 0 ‖

which for χ(0) = ℵ(0) corresponds to the limiting dynamics in (2.71) with ϖ(χ) = Lφ χ, and is equivalent to (2.64) with the particular feedback ν = ϖ(χ). Accordingly, it holds that ℵ(t) ≤ χ(t) for all t ≥ 0. The function ϖ(χ) = Lφ χ satisfies the sector condition ((Lφ + ϵ)χ − ϖ(χ))(ϖ(χ) − (Lφ − ϵ)χ) ≥ 0,

∀ϵ > 0, χ ∈ ℝ,

or equivalently −ϖ2 + 2Lφ ϖχ − (L2φ − ϵ2 )χ 2 ≥ 0,

∀χ ∈ ℝ.

Thus ϖ is (Q, S, R)-dissipative with q = −1,

s = Lφ ,

r = L2φ − ϵ2 .

Taking the particular case that ϵ = Lφ it follows that ϖ is (−1, Lφ , 0)-dissipative. According to Corollary 2.2.1, if the linear system Σ(−γ, gH M, I) is (0, Lφ , 1)-state strictly dissipative with some dissipation rate κ > 0, the closed-loop system is exponentially stable (taking S(χ) = 21 χ 2 as storage function). 7 Note that at this stage it is also possible to directly conclude that if γ > gH Lφ M the bounding function ℵ(t) exponentially converges to zero with amplitude a = 1 and rate γφ = γ − gH Lφ M, i. e. 󵄩󵄩 󵄩 󵄩󵄩x(t)󵄩󵄩󵄩 ≤ ℵ(t) ≤ ℵ(0)e−γφ t = M‖x 0 ‖e−(γ−gH Lφ M)t , 󵄩 󵄩

∀t ≥ 0.

(2.72)

2.3 System theoretic concepts | 47

Exploiting the preceding relation the following exponential stability condition can be directly drawn. Corollary 2.2.2. Let the assumptions of Lemma 2.2.12 hold and additionally 2γ − κ − (gH M − Lφ )2 > 0,

gH = ‖G‖O ‖H‖O .

(2.73)

Then the origin x = 0 is an exponentially stable equilibrium solution of the Lur’e feedback interconnection ẋ = Ax + Gν,

t > 0,

ν = −φ(x),

t ≥ 0.

σ = Hx,

t≥0

x(0) = x 0

Proof. According to Lemma 2.2.12 the exponential stability is ensured if the auxiliary system Σ(−γ, gH M, I) is (0, L2φ , 1)-state strictly dissipative with a positive dissipation rate κ. According to Lemma 2.2.3 this condition is equivalent to the inequality [

−2γ + κ gH M − Lφ

gH M − Lφ ] ≤ 0. −1

(2.74)

In terms of the Schur complement8 (see Appendix A.2) this holds if (i)

− 2γ + κ < 0,

(ii)

2γ − κ − (gH M − Lφ )2 > 0.

(2.75)

Given that condition (ii) is stronger than condition (i) the dissipativity property is ensured if (ii), or equivalently (2.73) holds true. With χ and ℵ defined in the proof of Lemma 2.2.12 this property implies the exponential convergence of χ to 0, which in turn implies the exponential convergence of ℵ to 0 and thus of x to 0.

2.3 System theoretic concepts For later use in the design of observers and feedback control schemes, in this section the basic notions from systems theory concerning observability, detectability and stabilizability are introduced. It should be noticed that this is a short introduction to the basic concepts and only those concepts will be discussed that will be used in the subsequent analysis. A deep treatise on the underlying concepts goes beyond the scope of the present text and the reader is referred to some of the seminal books (Åström and Murray, 2008; Sontag, 1995; Nijmeier and van der Schaft, 1990; Isidori, 1995, 1999; van der Schaft, 2017; Slotine and Li, 1991; Curtain and Zwart, 1995; Tucsnak and Weiss, 2009; Banks, 1983). 8 Note that in this two-dimensional setup the same conditions are obtained requiring a negative trace and a positive determinant.

48 | 2 Stability, dissipativity and some system-theoretic concepts 2.3.1 Observability and detectability In this section the notions underlying the observer design problem are discussed. An observer is a dynamical system that reconstructs the state of a system on the basis of a mathematical model and the known inputs and outputs, i. e. the actuator and sensor signals, respectively. As the reconstructability of state trajectories is underline by the possibility to distinguish between different trajectories through the state space, the notion of indistinguishability must be clarified first. Definition 2.3.1. Consider the system ẋ = f (x) + G(x)u, y = h(x),

t > 0, t≥0

x(0) = x 0

(2.76)

with state x(t) ∈ 𝕏 at time t ≥ 0, f : 𝕏 → 𝕏, G : 𝕏 → ℒ(𝒰 , 𝕏), h : 𝕏 → ℝm , and u(t) ∈ ℝp . Two initial conditions9 x 01 , x 02 are called indistinguishable for a given admissable input u∈ 𝒰 if they lead to the same output y(t) for all t ≥ 0, i. e. if it holds that y(t) = h(x(t; x 01 , u)) ≡ h(x(t; x 02 , u)),

∀t ≥ 0.

It should be noted that identical initial states (and thus identical trajectories) are always indistinguishable. The importance of this concept relies in the possibility that a system can have different state trajectories for the same input-output behavior. Thus, for the reconstructability of state trajectories the indistinguishability is a key property and the absence of indistinguishable trajectories definitely enables to reconstruct (in a way to be clarified later) the system state from the knowledge of inputs and outputs. Given that this uniqueness feature is a key property for the existence of an observer, the following notions are fundamental for the subsequent discussions. Definition 2.3.2. The system (2.76) is called – observable if the indistinguishability of x 01 and x 02 implies that x 01 = x 02 – detectable if for for a given admissable input u ∈ 𝒰 any two indistinguishable initial conditions x 01 , x 02 satisfy limt→∞ ‖x(t; x 01 , u) − x(t; x 02 , u)‖ = 0. The problem in analyzing detectability stems from the fact that (i) indistinguishability of initial conditions x 01 , x 02 must be characterized for the particular dynamics and output operators at hand, and (ii) the asymptotic stability of the difference ̃ = x(⋅; x 01 ) − x(⋅; x 02 ) needs to be established. For nonlinear systems this may lead x(⋅) to rather complex analysis as can be seen e. g. in Ibarra-Rojas et al. (2004), Schaum 9 Sometimes it is convinient to define the indistinguishability for the associated solution trajectories x i (t; x 0i , u(⋅)), i = 1, 2 instead.

2.3 System theoretic concepts | 49

et al. (2005), Schaum and Moreno (2007), Moreno and Dochain (2008), Moreno et al. (2014), Schaum and Bernal Jaquez (2016), but for linear systems is rather well characterized by the Kalman decomposition. For convenience the derivation of the Kalman decomposition is recalled in equation (B.17) in Appendix B. Accordingly, a linear timeinvariant system is detectable if and only if it can be transformed by an invertible transformation ξ = Tx into the following form: Ã ξ ̇ = [ ̃ 1,1 A2,1 y = [C̃ 1

B̃ 0 ] ξ + [ ̃ 1 ] u, ̃ B2 A2,2 0] ξ ,

t > 0, ξ (0) = ξ 0

(2.77)

t ≥ 0,

(2.78)

with à 1,1 ∈ ℝr×r , à 2,1 ∈ ℝ(n−r)×r , à 2,2 ∈ ℝ(n−r)×(n−r) , B̃ 1 ∈ ℝr×p , B̃ 2 ∈ ℝ(n−r)×p , C̃ 1 ∈ ℝr×m and the pair (à 1,1 , C̃ 1 ) being completely observable and the matrix à 2,2 having all eigenvalues in the open left-half complex plane. It should be noted that this separation can also be performed in an infinitedimensional set-up if the linear operator A has a discrete eigenvalue spectrum (see e. g. Curtain and Zwart, 1995). The easiest way to see this is by means of a spectral (Fourier) decomposition and separating observable and unobservable modes. For more details on this, the interested reader is referred at this place to the seminal book of Curtain and Zwart (1995) or an application of these concepts for the observer design of a semi-linear parabolic diffusion-convection-reaction system in Section 5.1 (see also Schaum et al., 2008a, 2016b, 2018; Schaum, 2009). Similar to the linear case, a nonlinear system ẋ = f (x) + G(x)u, y = h(x),

t > 0, t≥0

x(0) = x 0

(2.79a) (2.79b)

with x ∈ 𝕏=ℝn which is locally not completely observable can be rewritten as an interconnection of an observable with an unobservable subsystem. To illustrate this consider the case that the observability map (for more details see Appendix B.5 or e. g. Isidori, 1995, 1999; Nijmeier and van der Schaft, 1990) for (2.79) given by h(x) [L h(x)] [ f ] ] Φ(x) = [ [ .. ] [ . ] n [Lf h(x)] has only no < n independent entries i. e. for all x ∈ 𝕏 the matrix 𝒪(x) =

𝜕Φ(x) 𝜕x

50 | 2 Stability, dissipativity and some system-theoretic concepts has at least rank n0 . Then the state can be transformed by a diffeomorphic map into the form (see e. g. Isidori, 1995, 1999; Nijmeier and van der Schaft, 1990 for more details) ξ̇ = [

f 1 (ξ 1 ) G̃ (ξ ) ] + [ ̃ 1 ] u, f 2 (ξ 1 , ξ 2 ) G2 (ξ )

̃ ), y = h(ξ 1

t > 0,

ξ (0) = ξ 0

t≥0

(2.80) (2.81)

so that the associated observability map for the pair (y, ξ 1 ), i. e. the matrix

𝒪̃ (ξ 1 ) =

̃ 𝜕Φ(ξ 1) , 𝜕ξ 1

̃ ) h(ξ 1 [ ̃ )] ] [ Lf h(ξ 1 ] [ 1 ̃ Φ(ξ ] . 1) = [ ] [ .. ] [ no ̃ L h(ξ ) 1 ] [ f1

has rank n0 . The system’s local detectability is ensured in this set-up if the locally nonobservable part ξ 2 asymptotically converges to a unique attractor. The reason for this is that even though one can not directly reconstruct ξ 2 from the given input-output behavior, the state itself converges to a unique attractor, and thus any other, possibly indistinguishable trajectory will also converge to the same attractor, thus ensuring the detectability property. A particularly useful notion in the context of detectability analysis which ensures a fixed convergence rate for the non-observable part, consists in the concept of δ-detectability. Definition 2.3.3. A system in the form (2.77) or (2.80) is called δ-detectable, if for any two trajectories ξ a , ξ b with ξ 1,a ≡ ξ 1,b (indistinguishability) it holds that 󵄩󵄩 󵄩 −δt 󵄩󵄩ξ 2,a (t) − ξ 2,b (t)󵄩󵄩󵄩 ≤ M‖ξ 2,a,0 − ξ 2,b,0 ‖e ,

∀t ≥ 0

with some constant M ≥ 1. The notion of δ-detectability used here is kind of parallel to the notion of δ-exponential detectability introduced in Curtain and Zwart (1995) (more precisely Definition 5.2.1 there) for linear systems but the one presented here is more adapted to the purpose at hand. To simplify the notation further the next Definition is introduced. Definition 2.3.4. The pair (A, C) of a linear system Σ(A, B, C) is called δ-detectable if Σ is δ-detectable. Note that for the linear (finite-dimensional) system (2.77) δ-detectability is ensured if the largest eigenvalue of the matrix à 22 satisfies −δ ≤ max λ. λ∈σ(à 2,2 )

(2.82)

2.3 System theoretic concepts | 51

For infinite-dimensional systems the growth bound of the semigroup generated by the operator A2,2 has to be considered (see Curtain and Zwart, 1995). The importance of the δ-detectability concepts becomes evident by the following reasoning. For the observable part the convergence rate can be assigned using some established method while for the unobservable part the convergence rate is fixed by the system itself. Thus, the ensured convergence rate is naturally limited by δ. In consequence, in the case that the δ-detectability can be established, an observer can be designed10 such that the observation error exponentially converges to zero with rate δ as will be seen in the subsequent chapters.

2.3.2 Stabilizability and relative degree For control purposes the notions corresponding to observability and detectability are the controllability (or reachability) and stabilizability, respectively (Sontag, 1995; Åström and Murray, 2008; Isidori, 1995; Curtain and Zwart, 1995). The underlying system dynamics for the subsequent discussion is given by p

ẋ = f (x) + ∑ bi (x)ui ,

t > 0,

y = h(x),

t≥0

i=1

x(0) = x 0

(2.83a) (2.83b)

with state x ∈ 𝕏 ⊆ ℝn , inputs ui , i = 1, . . . , p, outputs yi , i = 1, . . . , m, smooth vector fields f , bi and output function h. Given that for the purpose at hand the stabilizability property is the essential one, the following definition is introduced. Definition 2.3.5. The origin x = 0 of the system (2.83) is called locally stabilizable (by state feedback control) in D ⊂ 𝕏 if there exists a feedback law u = ν(x) such that for all x 0 ∈ D ⊂ 𝕏 it is an asymptotically stable attractor for the closed-loop dynamics p

ẋ = f (x) + ∑ bi (x)νi (x), i=1

t > 0, x(0) = x 0 .

If this holds for D = 𝕏 than the origin is called (globally) stabilizable in 𝕏. 10 For the interested reader, the existence of the observer which ensures that the observation error exponentially converges with rate δ is actually the basic statement of Definition 5.2.1 in Curtain and Zwart (1995) mentioned above.

52 | 2 Stability, dissipativity and some system-theoretic concepts Clearly, this notion presents a necessary condition for the functioning of a feedback control for the stabilization of the origin. If the origin is (locally) open-loop asymptotically stable, the (local) stabilizability is trivially given. The question on stabilizability is an essential one and can be assessed in a similar manner as discussed for the detectability by an adequate Kalman state decomposition into controllable and uncontrollable parts. The details about this can bee found e. g. in the classical books on linear or nonlinear control theory (Sontag, 1995; Isidori, 1995). For the subsequent discussions it will turn out to be more important to introduce the following concepts which are central in the question on how to design stabilizing controllers. Definition 2.3.6. The (SISO) system (2.83) with p = m = 1 has relative degree r at x ∗ ∈ D ⊆ 𝕏 if ∀x ∈ D 1. Lb Lkf h(x) = 0, k = 0, 1, . . . , r − 2 and 2. Lb Lr−1 f h(x) ≠ 0. A relative degree r = 0 corresponds to a direct feed-through of the input to the output, i. e. the case that y = h(x, u). As this case is not considered here, it holds in the sequel that r ≥ 1. It should be noted that for a system with (well-defined) relative degree r ≤ n the state transformation h(x) ] [ [ Lf h(x) ] ] [ ] [ .. ] [ . ] [ ] [ r−1 z = Φ(x) = [Lf h(x)] ] [ [ Φr+1 (x) ] ] [ ] [ .. ] [ . ] [ Φ (x) ] [ n brings the system into the so-called Byrnes–Isidori normal form (Isidori, 1995) z2 z1 ] [ . ] [ .. ] [ . ] [ . ] [ . ] [ ] ] [ [ ] [α(z) + β(z)u] z d [ r ], ]=[ [ ż = ] ] [ dt [ [zr+1 ] [ φ1 (z, u) ] ] [ . ] [ .. ] [ . ] [ ] [ . ] [ . φ (z, u) z ] [ n ] [ n−r with α(z) = Lrf h(x)|x=Φ−1 (z) ,

β(z) = Lb Lr−1 f h(x)|x=Φ−1 (z) ,

φk (z, u) = (Lf Φr+k (x) + Lb Φr+k (x)u)|x=Φ−1 (z) ,

k = 1, . . . , n − r.

(2.84)

2.3 System theoretic concepts | 53

Actually, Φr+k k = 1, . . . , n − r can be chosen so that φk is not a function of u by considering the conditions Lb Φr+k (x) = 0,

k = 1, . . . , n − r.

(2.85)

From this consideration it is immediately clear, that the dynamics of zr+1 to zn can not be influenced by the control input u, and thus the (locally) controllable subspace is r-dimensional. Furthermore, as can be seen from the dynamics in Byrnes–Isidori normal form, the relative degree determines how many times the output of the system must be differentiated with respect to time in order to obtain a relation to the input. In the case of a linear system (or a system linearization about x̄ ∗ ) the relative degree thus corresponds to the order of the (proper) transfer function as long as the observable subspace is at least r-dimensional.11 Due to these reasons the dynamics of zr+1 to zn are referred to as the internal dynamics and often a new state vector is introduced zr+1 [ . ] ] η=[ [ .. ] , [ zn ]

η̇ = φ(z)

(2.86)

where it has been assumed that the conditions (2.85) are satisfied. Given that accordingly the internal dynamics can not be influenced by means of feedback control, the stability of the origin η = 0 is an important condition which is necessary for stabilizability. An important concept related to the internal dynamics is the so-called zero dynamics given by η̇ = φ(0, η) = φ0 (η),

η(0) = η0 .

(2.87)

A system whose zero dynamics are asymptotically stable is said to have the minimum phase property, in accordance with the fact that for linear systems the internal dynamics determines the roots of the numerator of the transfer function. The concept of relative degree can also be extended to the case of MIMO systems. This is recalled here for the case that p = m, i. e. the number of inputs and outputs is the same. Definition 2.3.7. The system (2.83) with p = m has vector relative degree r = {r1 , . . . , rp } in x ∗ ∈ D ⊆ 𝕏 if 11 To see this relation recall that the order of the transfer function corresponds to the dimension of the observable and controllable subspace.

54 | 2 Stability, dissipativity and some system-theoretic concepts 1. 2.

Lbi Lkf hj (x) = 0, for all k = 1, . . . , rj − 2, i, j = 1 . . . , m and x ∈ D and the decoupling matrix r −1

Lb1 Lf1 h1 (x) [ .. ℬx (x) = [ . [ rm −1 L L [ b1 f hm (x)

⋅⋅⋅ .. . ⋅⋅⋅

r −1

Lbm Lf1 h1 (x) ] .. ] . ] rm −1 Lbm Lf hm (x)]

has full rank for all x ∈ D. The Byrnes–Isidori normal form can then be derived in a way similar to the SISO case presented above. At this place this is left to the interested reader and will be discussed with more detail in Section 3.2.3 where the exact input-state linearization is discussed in the context of the dissipativity-based design framework.

2.4 Summary In this chapter the basic notions of stability, dissipativity and the most essential system theoretic concepts that will be used in this study have been introduced. Important connections have been drawn between exponential stability and dissipativity, some of which have not been reported in the literature as far as the author knows. Besides classical results, like Lemma 2.2.2 which basically resembles the famous Kalman– Yakubovich–Popov Lemma (Kalman, 1963; Popov, 1959, 1964; Yakubovich, 1962a,a; Brogliato et al., 2007), particular and new connections between exponential stability of a system and its strict state dissipativity have been presented in Lemmas 2.2.4 and 2.2.6 and explicit conditions for exponential stability of the interconnection of a dynamical system with a nonlinear static feedback map in Lemma 2.2.11 and Corollary 2.2.1. On the basis of these concepts and results, explicit design approaches can be developed for exponentially stabilizing controllers and exponentially convergent observers, for which additionally separation-principle-like closed-loop stability properties can be ensured. This is the subject of the following chapter.

3 Dissipativity-based observer and feedback control design In this chapter the proposed design approaches for nonlinear observers and controllers on the basis of dissipative systems theory are presented. The results are derived on the basis of a general set-up, so that most of them are applicable to finiteand infinite-dimensional systems. In the cases that results are particularly restricted to finite-dimensional systems it is highlighted to prevent confusion.

3.1 Dissipative observers In this section dissipativity-based observer design approaches are discussed. Observers have important applications in modern autonomous system applications. In particular, they play the key element in the model-based evaluation of on-line and off-line sensor and actuator data for the purpose of: – Process monitoring in the sense that the actual system state is reconstructed from the sensor and actuator data in a computational unit on the basis of a mathematical model of the system behavior. – Fault detection for system self-monitoring and functioning evaluation in the light of comparison with certain threshold behavior to infer from the input-output behavior on possible sensor or actuator faults, leakages, damages, etc. – Behavior prediction for the purpose of evaluation of possible process outputs, security risks, cost versus gain estimates on the basis of the actual state estimate (see first point above) in combination with the model-based prediction capabilities, which are much more precise than pure simulation models. – Data evaluation for (on-line or off-line) system identification, exploiting the identifiability properties of dynamical input-output systems. These main use cases of observers in modern automation theory and application can clearly be combined with the use in modern state-feedback control approaches in the understanding that an appropriate state estimate together with the knowledge about possible actuator failures is essential to ensure a desired closed-loop functioning. The attention here is placed on the first and most classical use case, namely the state reconstruction from input-output data in combination with the system model. First, the classical result on dissipative observers for linear dynamic subsystems with nonlinear static feedback interconnection which is known from the literature (Moreno, 2004, 2005; Schaum and Moreno, 2006; Schaum et al., 2007, 2008c) is recalled and formulated in the context of the present text. Then a discussion follows for systems which are only partially observable and finally a connection with the extended Luenberger observer, i. e. the Luenberger observer with state-dependent correction gains is drawn. https://doi.org/10.1515/9783110677942-003

56 | 3 Dissipativity-based observer and feedback control design 3.1.1 Linear dynamic subsystems with linear correction scheme Consider the system interconnection ẋ = Ax + Bu + Gφ(σ) + ψ(y, u),

σ = Hx, y = Cx,

t > 0,

t≥0

x(0) = x 0

t≥0

(3.1a) (3.1b) (3.1c)

with the state x(t) ∈ 𝕏, t ≥ 0, input u(t) ∈ 𝒰 , output y(t) ∈ 𝒴 = ℝm , linear operator A: 𝒟(A) ⊆ 𝕏 → 𝕏 with domain 𝒟(A), B: 𝒰 → 𝕏, H : 𝕏 → ℋ, φ : ℋ → ℱ , G: ℱ → 𝕏, ψ : 𝒴 × 𝒰 → 𝒳 , and output (measurement) operator C : 𝕏 → 𝒴 . Examples of systems with this structure and explicit spaces 𝕏, 𝒰 , 𝒴 , ℋ, ℱ and operators A, B, G, H, C are provided in Section 1.1 (see system (1.4) and the subsequent examples) and in Chapters 4 and 5. The particular structure of (3.1) is motivated by the problem of Lur’e (Lur’e and Postnikov, 1944; Kalman, 1963; Brogliato et al., 2007; Pandolfi, 1998) as will be specified below. The following nonlinear Luenberger observer1 is set dx̂ = Ax̂ + Bu + Gφ(σ)̂ + ψ(y, u) − L(C x̂ − y), dt σ̂ = H x,̂

t > 0,

̂ x(0) = x̂ 0

t≥0

(3.2a) (3.2b)

with the correction gain matrix L which is the design degree of freedom. Consider the errors x̃ = x̂ − x,

σ̃ = σ̂ − σ

(3.3)

with the associated dynamics dx̃ = (A − LC)x̃ + G[φ(σ + σ)̃ − φ(σ)], dt σ̃ = H x,̃

t > 0,

̃ x(0) = x̃ 0

t≥0

(3.4a) (3.4b)

1 Note that in the original treatments on the dissipative observer design in finite dimensions (Moreno, 2004, 2005; Schaum and Moreno, 2006; Schaum et al., 2007, 2008c) an additional correction mechanism is considered in the feedback loop resulting in the observer dx̂ = Ax̂ + Bu + Gφ(σ̂ − N(C x̂ − y)) + ψ(y, u) − L(C x̂ − y), dt σ̂ = H x,̂

t > 0,

̂ x(0) = x̂ 0

t ≥ 0.

This additional degree of freedom is not considered in the sequel in order to simplify the presentation and focus on the essential structure, but can in some cases improve the performance.

3.1 Dissipative observers | 57

Introducing AL = A − LC,

̃ σ,̃ σ) = φ(σ) − φ(σ + σ)̃ φ(

(3.5)

the estimation error dynamics (3.4) can be written in the form of a Lur’e-type interconnection between the linear dynamical subsystem Σ(AL , G, H) and the nonlinear static map ν = −φ,̃ i. e. ẋ̃ = AL x̃ + Gν,

σ̃ = H x,̃

̃ σ,̃ σ), ν = −φ(

t > 0,

t≥0

t≥0

̃ x(0) = x̃ 0

(3.6a) (3.6b) (3.6c)

As a direct consequence of Lemma 2.2.11 and Corollary 2.2.1 the following result is obtained. Theorem 3.1.1. Consider the system (3.1) and let φ̃ in (3.5) be (Q, S, R)-dissipative uniformly in σ. If there exists an operator L: 𝒴 → 𝕏 such that Σ(AL , G, H) is (−R, S∗ , −Q))strictly state dissipative with rate κ > 0 and positive definite storage functional 𝒮 : ̃ P = P ∗ such that ∃ 0 < p− , p+ ∈ ℝ for which is holds true 𝕏 → ℝ with 𝒮 (x)̃ = ⟨x,̃ P x⟩, − ̃ 2 + that p ‖x‖ ≤ 𝒮 (x)̃ ≤ p ‖x‖̃ 2 for all x̃ ∈ 𝕏, then the estimation error x̃ exponentially converges to zero. Proof. The result follows immediately by applying Corollary 2.2.1 to the Lur’e system interconnection (3.6). Remark 3.1.1. From Theorem 3.1.1 it becomes clear that the convergence velocity depends on κ and on p+ . In general these two measures depend on each other, i. e. for design purposes the particular choice of storage function 𝒮 and the dissipation rate κ are not independent. For implementation purposes it is important to consider a tradeoff between small p+ and large κ. Further, the smaller p+ , the smaller also p− . For constant p+ the amplitude a increases with decreasing p− . This interdependency of the triplet (κ, p− , p+ ) is related to the overshooting phenomenon for large feed-back or correction gain known from different areas of feedback control and observer design and can best be overcome by either tuning or optimization. This will be further explored in Section 4.5. Besides this the reader is referred to Schaum et al. (2020). Remark 3.1.2. Note that the influence of the actual system state is only through the coupling with σ in the nonlinear function φ.̃ As by assumption the influence of this coupling is bounded, in the sense that for all σ the function φ̃ satisfies the (Q, S, R)-dissipativity property, the convergence is uniform with respect to the system state x. This fact is particularly useful in connection with feedback control, as will be detailed in Section 3.4. Remark 3.1.3. Having in mind the sufficient conditions for (−R, ST , −Q)-strict state dissipativity in Lemma 2.2.4 and 2.2.6 it is immediately clear that the existence of the observer is ensured if one can show that L can be chosen so that A−LC generates either an

58 | 3 Dissipativity-based observer and feedback control design exponentially stable C0 semigroup of contractions with sufficiently negative growth bound (or maximum eigenvalue), or if the modulus of the maximum eigenvalue of (A−LC)∗ +(A−LC) is large enough. This can be achieved in the finite-dimensional case if the pair (A, C) is completely observable, or if it is δ-detectable (cp. Definition 2.3.3) with sufficiently large δ. These conditions directly extend to the infinite-dimensional case if A − LC is a Riesz-spectral operator (see e. g. Schaum et al. (2018) and the discussion in Section 5.1). 3.1.2 Partially observable finite-dimensional systems Most observer design approaches (including the above treatment up to this point) do not consider the structure which underlies the system dynamics explicitly, but rather try to ensure that a full-state observer with measurement injection in all state variables exists and consider the question on how to choose the associated injection gain matrix in order to ensure local or non-local asymptotic or exponential convergence conditions. An interesting exception to this rule is the geometric observer design approach (Alvarez and Lopez, 1999; Fernandez et al., 2012; Frau et al., 2010; Porru et al., 2013) which introduces the concept of estimation structure. The estimation structure is referred to the choice of state variables whose dynamics are explicitly modified by the measurement injection, and (associated to this question) the number ki , i = 1, . . . , m of Lie derivatives of each measurement function hi : 𝕏 → ℝ, i = 1, . . . , m. The advantage of this approach is on one side the possibility to be implemented even if the system is not observable (and thus the associated observability distribution is not involutive) or if the observability map is badly conditioned and thus the observer will suffer a lack of robustness with respect to measurement noise and parameter uncertainties. For this reason the above dissipative observer design approach will be applied in this section in combination with an explicit consideration of a given choice of estimation structure. Consider the autonomous system2 ẋ = f (x),

y = h(x),

t > 0, t≥0

x(0) = x 0

(3.7a) (3.7b)

with the state x(t) ∈ 𝕏 = ℝn at time t ≥ 0, smooth vector field f : 𝕏 → 𝕏, output y(t) ∈ 𝒴 = ℝm and h : 𝕏 → ℝm . In the geometric observer design approach (Alvarez and Lopez, 1999; Fernandez et al., 2012; Frau et al., 2010; Porru et al., 2013) the first step consists in determining which states are corrected by the measurements and which are determined without 2 The approach can also be directly extended to the non-autonomous case, if the estimation structure is chosen such that time-derivatives of the input signal are avoided.

3.1 Dissipative observers | 59

measurement injection, i. e., in form of an open-loop simulation. This can be done by a local observability analysis and basically results in a decision about the number ki ≥ 0, i = 1, . . . , m of time derivatives of the m measured outputs that are considered in the local observability map. Let no = m + ∑m i=1 ki denote the related number of states that will be corrected by measurement injection, and nu = n − no denote the number of remaining states which will be determined without measurement injection. Introduce the state transformation [

xo ] = Φ(x), xu

dim(x 0 ) = no ,

dim(x u ) = nu

(3.8)

with a diffeomorphism Φ: 𝕏 → 𝕏 of the form h1 (x) ] [ .. ] [ ] [ . ] [ [ hm (x) ] ] [ [ L1 h (x) ] ] [ f 1 ] [ .. ] [ ] [ . ] [ Φo (x) k [ 1 Φ(x) = [ L h1 (x) ] ] = [Φ (x)] , f ] [ u ] [ .. ] [ . ] [ ] [ km [ Lf hm (x) ] ] [ ] [ ] [ ] [ ] [ Φ (x) u ] [

n = no + nu .

(3.9)

Note that the resulting system dynamics can always be written in form of the interconnection structure3 ẋ o = Ao x o + Go φo (σ o , x u ),

σ o = Ho x o ,

ẋ u = Au x u + φu (x o , x u ), y = Cx o ,

t > 0,

x o (0) = x o,0

t > 0,

x u (0) = x u,0

t≥0

t ≥ 0.

(3.10a) (3.10b) (3.10c) (3.10d)

A possible observer for these dynamics is given by ẋ̂ o = Ao x̂ o + Go φo (σ̂ o , x̂ u ) − L(C x̂ o − y),

σ̂ o = Ho x̂ o ,

t > 0,

t≥0

x̂ o (0) = x̂ o,0

(3.11a) (3.11b)

3 The simplest choice is Ao ∈ ℝno ×no , Au ∈ ℝnu ×nu , Go = I, Ho = I, φo = [𝜕x Φ(x)]1,...,no − Ao x o , φu = [𝜕x Φ(x)]no +1,...,n − Au x u .

60 | 3 Dissipativity-based observer and feedback control design ẋ̂ u = Au x̂ u + φu (x̂ o , x̂ u ),

t > 0,

x̂ x̂ = Φ−1 ([ o ]) , x̂ u

x̂ u (0) = x̂ u,0

t ≥ 0.

(3.11c) (3.11d)

The dynamics of the associated errors x̃ o = x̂ o − x o and x̃ u = x̂ u − x u read ẋ̃ o = (Ao − LC)x̃ o + Go ψo (σ̃ o , x̃ u ),

σ̃ o = Ho x̃ o , ẋ̃ u = Au x̃ u + ψu (x̃ o , x̃ u ),

t > 0,

x̂ o (0) = x̂ o,0

t > 0,

x̂ u (0) = x̂ u,0

t≥0

(3.12a) (3.12b) (3.12c)

with ψo (σ̃ 0 , x̃ u ) = φo (x o + σ̃ o , x u + x̃ u ) − φo (x o , x u ) and ψu (x̃ o , x̃ u ) = φu (x o + x̃ o , x u + x̃ u ) − φu (x o , x u ) Given that by construction the linear part of the dynamics of x̃ o is in observability normal form the eigenvalues of (Ao − LC) can always be assigned arbitrarily. Thus, if ψo (σ̃ o , x̃ u ) is (Q, S, R)-dissipative with respect to σ̃ o uniformly in x̃ u , it is possible to design L, by following the steps in Section 3.1.1 such that x̃ o will converge exponentially to zero with rate γo > 0 and amplitude ao , uniformly in x̃ u . Accordingly, in the sequel assume that 󵄩󵄩 ̃ 󵄩 −γ t 󵄩󵄩x o (t)󵄩󵄩󵄩 ≤ ao ‖x̃ o,0 ‖e 0 ,

∀t ≥ 0

(3.13)

holds true. Let there exists a constant mu ≥ 1 so that for the matrix Au it holds that 󵄩󵄩 Au t 󵄩󵄩 λ+ t 󵄩󵄩e 󵄩󵄩O ≤ mu e u ,

∀t ≥ 0

where λu+ denotes the maximum eigenvalue of the matrix Au (or more generally the growth rate of the semigroup eAu t ). Accordingly for the variable x̃ u (t) it holds that t

x̃ u (t) = eAu t x̃ u,0 + ∫ eAu (t−τ) ψu (x̃ o (τ), x̃ u (τ))dτ

(3.14)

0

and after taking norms on both sides t

󵄩󵄩 ̃ 󵄩 󵄩 A t󵄩 󵄩 A (t−τ) 󵄩󵄩󵄩󵄩 󵄩 󵄩󵄩x u (t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩e u 󵄩󵄩󵄩O ‖x̃ u,0 ‖ + ∫󵄩󵄩󵄩e u 󵄩󵄩󵄩󵄩ψu (x̃ o (τ), x̃ u (τ))󵄩󵄩󵄩dτ 0

t

≤ mu eλ t ‖x̃ u,0 ‖ + ∫ mu eλ +

0

+

(t−τ)

||ψu (x̃ o (τ), x̃ u (τ))‖dτ,

∀t ≥ 0.

3.1 Dissipative observers | 61

Under the assumption that ψu is Lipschitz continuous with respect to x̃ o and x̃ u with Lipschitz constants Loψ and Luψ , respectively, it follows that u

u

t

󵄩 󵄩󵄩 ̃ λ+ t −λ+ τ o u 󵄩󵄩x u (t)󵄩󵄩󵄩 ≤ mu e u (‖x̃ u,0 ‖ + ∫ e u (Lψu ‖x̃ o ‖ + Lψu ‖x̃ u ‖)dτ),

∀t ≥ 0

0

Defining the right hand side of the last inequality as t

χu (t) := mu eλu t (‖x̃ u,0 ‖ + ∫ e−λu τ (Loψu ||x̃ o ‖ + Luψu ‖x̃ u ‖)dτ) +

+

0

so that χu (0) = mu ‖x̃ u,0 ‖ it holds that ‖x̃ u (t)‖ ≤ χu (t) for all t ≥ 0. It follows that χu̇ = λu+ χu + mu (Loψu ‖x̃ o ‖(t) + Luψu ‖x̃ u ‖(t)) ≤ (λu+ + mu Luψu )χu + mu Loψu ‖x o ‖,

t > 0.

Introducing the variable χo (t) = ao ‖x̃ o,0 ‖e−γo t ,

χȯ = −γo χo ,

χo (0) = ao ‖x̃ o,0 ‖

(3.15)

it follows that −γo d χo [ ]≤[ mu Loψ dt χu u

λu+

0 χo u ][ ], + mu Lψ χu u

t > 0,

[

χo (0) χ ] = [ o0 ] χu0 χu (0)

(3.16a)

where the inequality has to be interpreted element-wise. Accordingly, if the condition λu+ < −mu Luψu

(3.17)

holds, it follows that 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩 χo (t) 󵄩󵄩󵄩 󵄩 χ (0) 󵄩󵄩 󵄩󵄩[ 󵄩󵄩 ≤ e−γt 󵄩󵄩󵄩[ o ]󵄩󵄩󵄩 , ] 󵄩󵄩 χu (t) 󵄩󵄩 󵄩󵄩 χu (0) 󵄩󵄩 󵄩 󵄩 󵄩 󵄩

γ = max{−γo , λu+ + Luψu },

∀t ≥ 0

(3.18)

and thus 󵄩󵄩 ̃ 󵄩 󵄩󵄩 ̃ 󵄩󵄩 󵄩󵄩 x o (t) 󵄩󵄩󵄩 󵄩 x 󵄩 󵄩󵄩[ 󵄩󵄩 ≤ ae−γt 󵄩󵄩󵄩[ o,0 ]󵄩󵄩󵄩 , ] 󵄩󵄩 x̃ u (t) 󵄩󵄩 󵄩󵄩 x̃ u,0 󵄩󵄩 󵄩 󵄩 󵄩 󵄩

a = max{ao , mu },

∀t ≥ 0,

(3.19)

i. e. the observation error exponentially converges to zero with rate γ. This is summarized in the following theorem. Theorem 3.1.2. Consider the system (3.7) with 𝕏=ℝn and let no in (3.9) be such that Φ: 𝕏 → 𝕏 is a diffeomorphism. Let further ψ̃ o in (3.12) be (Q, S, R)-dissipative uniformly in x̃ u and ψ̃ u be Lipschitz continuous with respect to x̃ o and x̃ u with Lipschitz constants Loψ and Luψ , respectively. Then the observer (3.11) exponentially converges to zero with u u convergence rate γ given in (3.18) if

62 | 3 Dissipativity-based observer and feedback control design 1. 2.

The correction gain matrix L is chosen so that Σ(Ao −LC, I, Ho ) is (−R, ST , −Q)-strictly state dissipative with dissipation rate κ > 0 and The maximum eigenvalue λu+ of Au satisfies (3.17).

Note that according to this analysis the convergence velocity is limited by the constant λu+ + Luψ and thus the observer convergence can not be made arbitrarily fast. This u is due to the fact that the state x̃ u converges with its natural velocity to zero. This velocity is largely due to the detectability of the system (3.10), which is ensured if the convergence condition (3.17) is satisfied. This result allows to see also how the δ-detectability (see Definition 2.3.3) can be assessed for a nonlinear system and what its implications are. In the present set-up the δ-detectability is ensured if −δ ≤ λu+ + Luψu . Accordingly, under this assumptions it is possible to design an observer with convergence rate −δ following the above steps. Rewriting the observer (3.11) in original coordinates leads to an interesting observation. To derive this it is convenient to introduce the variable x̂ ξ̂ = [ o] , x̂ u

̇ ξ̂ = f ̄ (ξ ̂ ) − L(̄ C̄ ξ ̂ − y)

with f ̄ , L̄ and C̄ given according to (3.11) by A x̂ + Go φo (Ho x̂ o , x̂ u ) ], f ̄(ξ ̂ ) = [ o o Au x̂ u + φu (x̂ o , x̂ u )

L L̄ = [ ] , 0

C̄ = [C

0] .

In original coordinates x̂ = Φ−1 (ξ ̂ ) the observer (3.11) can be written as4 𝜕Φ−1 (ξ ̂ ) ̄ ̂ ̂ ẋ̂ = [f (ξ ) − L(̄ C̄ ξ ̂ − y)] = f (x)̂ − Le (x)(h( x)̂ − y), 𝜕ξ ̂

t > 0,

̂ x(0) = x̂ 0 ,

4 Recall from the inverse function theorem (see e. g. Strang, 1991) the following result: Let ξ = Φ(x). For any x it holds that 1 = 𝜕x [Φ−1 (Φ(x))] = 𝜕ξ Φ−1 (ξ )𝜕x Φ(x). In consequence it follows that −1

𝜕ξ Φ−1 (Φ(x)) = (𝜕x Φ(x)) . In particular this implies that f (x) = 𝜕ξ Φ−1 (ξ )f ̄ (ξ ).

3.1 Dissipative observers | 63

where Le (x)̂ =

−1 𝜕Φ−1 (ξ ̂ ) ̄ 𝜕Φ(x)̂ L=[ ] L̄ 𝜕x̂ 𝜕ξ ̂

is the state-dependent correction gain matrix. This shows that by following this approach a dissipativity-based observer design for a partially observable and detectable system can be carried out using a state-dependent gain matrix, putting the dissipativity-based approach in perspective with the geometric observer (Alvarez, 1997, 1999; Alvarez and Lopez, 1999; Alvarez and Fernandez, 2009; Tronci et al., 2005), the extended Luenberger observer (Zeitz, 1987; Birk and Zeitz, 1988, 1983), and the extended Kalman Filter (Gelb, 1978) (without explicit measurement and process noises at this stage). Note that such a state-dependent gain enables in principle an on-line adaptation of the gains, in particular their sign and value, in order to account for changes in the direction of the vector field. In contrast to the extended Luenberger observer the approach presented here does not require (local) observability of the system but only detectability, just as in the assumptions for the existence of the geometric observer.

3.1.3 Further results for state-dependent gain matrices in finite dimensional systems In addition to the result presented in the preceding section on the design of dissipativity-based observers for nonlinear detectable and only partially observable systems with state-dependent gains, here an alternative approach is used for the general finitedimensional nonlinear system of the form ẋ = f (x, u), y = h(x),

t > 0,

t ≥ 0.

x(0) = x 0

(3.20a) (3.20b)

Consider the nonlinear observer ẋ̂ = f (x,̂ u) − l(x,̂ u)(h(x)̂ − y),

t > 0,

̂ x(0) = x̂ 0

(3.21)

with state dependent gain l(x,̂ u) evaluated at the estimated state x.̂ In terms of the estimation error x̃ = x̂ − x,

x = x̂ − x̃

(3.22)

the actual system dynamics (3.20) can be rewritten by its Taylor series expansion about x̂

64 | 3 Dissipativity-based observer and feedback control design ̃ ẋ = f (x̂ − x,̃ u) = f (x,̂ u) − J(x,̂ u)x̃ + ϕ∗ (x),

t > 0,

̃ y = h(x̂ − x),

x(0) = x 0

t ≥ 0,

(3.23a) (3.23b)

with J(x,̂ u) =

𝜕f (x,̂ u) 𝜕x

being the Jacobian evaluated at the pair (x,̂ u) and ϕ∗ being the sum of second and higher order terms from the Taylor series expansion which clearly satisfies ϕ∗ (0) = 0. The associated dynamics of the estimation error are given by ẋ̃ = ẋ̂ − ẋ = f (x,̂ u) − l(x,̂ u)(h(x)̂ − y) − f (x,̂ u) + J(x,̂ u)x̃ − ϕ∗ (x)̃ ̃ = J(x,̂ u)x̃ − l(x,̂ u)(h(x)̂ − y) − ϕ∗ (x),

t > 0,

̃ x(0) = x̃ 0 .

In virtue of the mean value theorem the term h(x)̂ − y can be written as h(x)̂ − y = h(x)̂ − h(x) = h(x)̂ − h(x̂ − x)̃ =

𝜕h (x̂ − ηx)̃ x,̃ 𝜕x

η ∈ (0, 1).

Substituting this equation into the preceding development it turns out that the dynamics of the estimation error can be written as 𝜕h ̃ x̃ − ϕ∗ (x), ̃ ẋ̃ = (J(x,̂ u) − l(x,̂ u) (x̂ − ηx)) 𝜕x

η ∈ (0, 1),

t > 0,

̃ x(0) = x̃ 0 .

In the most simple case where h(x) = Cx this reduces to ̃ ẋ̃ = (J(x,̂ u) − l(x,̂ u)C)x̃ − ϕ∗ (x),

t > 0,

̃ x(0) = x̃ 0 .

(3.24)

Of particular interest is of course the case where the linearization about every point x̂ is completely observable and by appropriately defining a diffeomorphism on the basis of the linear Kalman observability matrix (evaluated at x)̂ the system can be brought into a permutated observer canonical form (see Appendix B.3 for a derivation for the SISO case). To illustrate this idea consider the case of a SISO system given in the form5 so that 0 [ [ [1 [ [ ̂ J(x, u) = [0 [ [. [. [. [0

⋅⋅⋅ .. . .. .

−a0 (x,̂ u)

⋅⋅⋅ .. ..

. .

] ] ] ] ] ], ] ] ] −an−2 (x,̂ u)] −an−1 (x,̂ u)] −a1 (x,̂ u) .. .

0 1

C = [0

⋅⋅⋅

0

1] .

(3.25)

5 If the system dynamics is not given in this form it can be transformed into it by a linear equivalence transformation (see Appendix B.3) as long as its linear approximation is completely observable at any state x̂ ∈ 𝕏.

3.1 Dissipative observers | 65

Let li (x,̂ u), i = 1, . . . , n denote the entries of the vector-valued gain function l(x,̂ u). Then it follows that 0 [ [ [1 [ [ J(x,̂ u) − l(x,̂ u)C = [0 [ [. [. [.

⋅⋅⋅ .. . .. .

[0

−a0 (x,̂ u) − l1 (x,̂ u)

⋅⋅⋅ .. ..

. .

] ] ] ] ] ] ] ] ] −an−2 (x,̂ u) − ln−1 (x,̂ u)] −an−1 (x,̂ u) − ln (x,̂ u) ] −a1 (x,̂ u) − l2 (x,̂ u) .. .

0 1

and by choosing li (x,̂ u) = −ai−1 (x,̂ u) + ā i−1

(3.26)

it turns out that 0 [ [ [1 [ [ ̄ A := J(x,̂ u) − l(x,̂ u)C = [0 [ [. [. [. [0

⋅⋅⋅ .. . .. .

−ā 0

⋅⋅⋅ .. ..

. .

] ] ] ] ] ] ] ] ] −ā n−2 ] −ā n−1 ] −ā 1 .. .

0 1

is a constant matrix with the ā k , k = 0, . . . , n−1 being the coefficients of the characteristic polynomial. This basically corresponds to the classical result using the Ackermann formula for eigenvalue assignment (Ackermann, 1972; Kailath, 1980). By this fact, the estimation error dynamics can again be written in the form of an interconnection of a linear dynamic and a nonlinear static subsystem ẋ̃ = Ā x̃ + ν,

σ̃ = x,̃

̃ ν = −ϕ (σ), ∗

t > 0,

t≥0

t ≥ 0,

̃ x(0) = x̃ 0

(3.27a) (3.27b) (3.27c)

which fits perfectly into the above considerations on the dissipative observer design (cp. Equation (3.6)). In particular, according to Proposition 3.1.1, if ϕ∗ is (Q, S, R)-dissipative and Σ(A,̄ I, I) is (−R, ST , −Q)-strictly state dissipative with dissipation rate κ > 0, the exponential convergence of the observation error to zero is ensured. To analyze the dissipativity properties of ϕ∗ methods from Section 2.2.5 can be employed, in particular in combination with the result of Lemma 2.2.10. Furthermore, this approach can be used to design local observers and thus goes along typical approaches for nonlinear observer design (Krener and Isidori, 1983; Birk and Zeitz, 1983; Krener and Respondek, 1985; Birk and Zeitz, 1988; Rudolph and Zeitz, 1994; Tami et al., 2013, 2016).

66 | 3 Dissipativity-based observer and feedback control design

3.2 State-feedback control design In this section the dissipativity-based approach to state-feedback control is discussed and some new results derived, extending classical design methods to certain classes of nonlinear systems.

3.2.1 Dissipation through feedback control in finite dimensions As an introductory consideration to this section, consider the nonlinear SISO control system ẋ = f (x) + g(x)u, t > 0, y = h(x),

x(0) = x 0

t≥0

(3.28) (3.29)

with the state x(t) ∈ 𝕏 = ℝn at time t ≥ 0, smooth vector fields f : 𝕏 → 𝕏, g : 𝕏 → 𝕏, input u(t) ∈ 𝒰 = ℝ, output y(t) ∈ 𝒴 = ℝ and h : 𝕏 → ℝ. If the relative degree between the input u and output z = h(x) is equal to one (see Section 2.3.2), i. e., Lg h(x) ≠ 0 at least locally for x ∈ D ⊆ 𝕏 with 0 ∈ int(D), it is a convenient approach to consider a passivity-based control (Byrnes et al., 1991; Sepulchre et al., 1997). For this purpose consider the input-output relation ż = Lf h(x) + Lg h(x)u,

y = z,

t > 0,

t ≥ 0.

z(0) = z0

As Lg h(x) ≠ 0 one can introduce a new input ν together with the controller u=

ν − Lf h(x) Lg h(x)

.

(3.30)

Accordingly, in closed-loop one obtains a simple integrator dynamics ż = ν,

y = z,

t > 0,

t ≥ 0.

z(0) = z(0)

Considering the storage function 𝒮 (z) = 21 z 2 one directly obtains d𝒮 (z) = νz = νy dt so that the passivity follows immediately. This classical approach has been used e. g. in Gonzalez and Alvarez (2005), Castellanos-Sahagun and Alvarez (2006), DiazSalgado et al. (2007, 2012), Schaum et al. (2007, 2008c, 2009, 2010, 2012a, 2013a) to design (saturated) state-feedback controllers for chemical and biological reactors.

3.2 State-feedback control design |

67

In particular it has been shown e. g. in Gonzalez and Alvarez (2005), CastellanosSahagun and Alvarez (2006), Schaum et al. (2010, 2012a, 2013a), Diaz-Salgado et al. (2012) that in combination with an appropriately designed observer the resulting observer-based output-feedback controller is equivalent to a (saturated) PI controller (with observer-based anti-windup protection scheme). A direct generalization of the passivity-based approach is obtained by considering a general storage function candidate 𝒮 (x) and its rate of change over time d𝒮 (x) = Lf 𝒮 (x) + Lg 𝒮 (x)u, dt

t > 0.

(3.31)

Dissipativity with respect to a new (virtual) input ν and the associated supply rate ω(y, ν) is ensured if Lf 𝒮 (x) + Lg 𝒮 (x)u ≤ ω(y, ν) = ω(h(x), ν) In the case that Lg 𝒮 (x) ≠ 0 a particular control which achieves this inequality is given by u = Lg 𝒮 (x)−1 (ω(h(x), ν) − Lf 𝒮 (x)) for which (3.31) evaluates to d𝒮 (x) = ω(h(x), ν). dt

(3.32)

Note that the condition Lg 𝒮 (x) ≠ 0 implies that the (actual) input u has relative degree r = 1 with respect to the virtual output defined by 𝒮 (x). If there is a direct correspondence between 𝒮 (x) and the output y = h(x) like for the quadratic (positive semi-definite) storage function 𝒮 (x) =

1 2 1 2 y = h (x) ≥ 0 2 2

then this property corresponds again to the relative degree one property of the original input-output pair (u, y) exploited above. Besides this direct assessment providing dissipativity with respect to a new input ν, the main question which is addressed in this section relates to the system p

ẋ = f (x) + ∑ bi (x)ui + G(x)φ(σ), t > 0, i=1

σ = h(x),

t ≥ 0,

x(0) = x 0

(3.33a) (3.33b)

with the state x(t) ∈ 𝕏 = ℝn at time t ≥ 0, smooth vector fields f : 𝕏 → 𝕏, bi : 𝕏 → 𝕏, i = 1, . . . , p, inputs ui (t) ∈ ℝ, i = 1, . . . , p, G : 𝕏 → ℝn×r with entries gij ∈ 𝒞 1 (𝕏, ℝ), φ : ℋ = ℝq → ℱ = ℝr and h : 𝕏 → ℋ.

68 | 3 Dissipativity-based observer and feedback control design This system can be viewed as the two-subsystem interconnection p

ẋ = f (x) + ∑ bi (x)ui + G(x)ν,

t > 0,

σ = h(x),

t≥0

i=1

ν = φ(σ),

x(0) = x 0

t ≥ 0.

(3.34a) (3.34b)

(3.34c)

On the basis of the stability assessments in Lemma 2.2.11 and Corollary 2.2.1 for these kind of system interconnections the important question consists in how to assign desired dissipativity properties to the dynamical subsystem (3.34a)–(3.34b). In comparison to the above analysis of how to design the controller for the input u in order to achieve a desired dissipativity with respect to a new input, the question here consists in how to design the controller for u in order to establish a desired dissipativity with respect to the given second input ν. To answer this question the structure of the system consisting of the vector fields f , b, the matrix-valued function G and the output map h have to be taken into account and appropriately exploited. 3.2.2 Linear feedback control for linear subsystems Consider the linear system ẋ = Ax + Bu + Gν,

σ = Hx,

t > 0,

t ≥ 0,

x(0) = x 0

(3.35a)

(3.35b)

which is identical in its structure to (3.1) with ψ = 0. In view of Lemma 2.2.11 and Corollary 2.2.1 the interesting dissipativity property required for (3.35) is the (−R, S∗ , −Q)-strict state dissipativity (see Definition 2.2.3). By considering a linear state feedback control of the form u = −Kx

(3.36)

with the operator K: 𝕏 → 𝒰 the closed-loop dynamics can be written as ẋ = Ac x + Gν,

σ = Hx,

t > 0, t ≥ 0,

x(0) = x 0 ,

Ac = (A − BK)

(3.37a)

(3.37b)

and sufficient conditions for the closed-loop system to be (−R, S∗ , −Q)-strictly state dissipative with a positive dissipation rate can be directly concluded from Lemma 2.2.3. Lemma 3.2.1. The linear system (3.35) with the state-feedback controller (3.36) is (−R, S∗ , −Q)-strictly state dissipative with dissipation rate κ > 0 and positive definite storage functional 𝒮 : 𝕏 → ℝ with 𝒮 (x) = ⟨x, Px⟩ with P = P ∗ if ⟨x, PAc x⟩ + ⟨Ac x, Px⟩ + ⟨x, (κI + H ∗ RH)x⟩ + 2⟨x, (PG − H ∗ S∗ )ν⟩ + ⟨ν, Qν⟩ ≤ 0. (3.38)

3.2 State-feedback control design |

69

Note that for a finite-dimensional linear system Σ(A, B, G, H) the dissipativity condition (3.38) simplifies to the matrix inequality PA + ATc P + κI + H T RH [ c GT P − SH

PG − H T ST ]≤0 Q

(3.39)

and several methods are known to find solutions (if they exist) for these kind of problems. In particular, if the pair (A, B) is completely controllable, the eigenvalues of the matrix Ac can be assigned arbitrarily and thus one possible solution (not necessarily the best) is given by choosing P = I and assigning the eigenvalues so that the matrix [

Ac + ATc + κI + H T RH GT − SH

G − H T ST ] < 0. Q

Provided that Q < 0 this can be ensured if the first entry and its Schur complement (Lemma A.2.1) are negative definite, i. e. (i) Ac + ATc + κI + H T RH < 0,

(ii) Ac + ATc + κI + H T RH − (G − H T ST )Q−1 (GT − SH) < 0 or (iii) Q − (GT − SH)(Ac + ATc + κI + H T RH) (G − H T ST ) < 0 −1

By the Courant–Fischer theorem (Dym, 2007) it holds that μl x T x ≤ x T [Ac + ATc ]x ≤ μu x T x,

μl =

min

λ∈σ(Ac +ATc )

λ, μu =

max

λ∈σ(Ac +ATc )

λ

with μl and μu being the minimum and maximum eigenvalues of the matrix Ac + ATc . Accordingly it holds that Ac + ATc + κI + H T RH < 0

μu + κ + ‖H‖2 ‖R|| < 0

if

so that a first condition consists in assigning μu so that 󵄩2 󵄩 μu < −(κ + ‖H 󵄩󵄩󵄩 ||R󵄩󵄩󵄩). Using this bound in condition (ii) yields μu + κ + ‖H‖2 ‖R‖ +

‖G − H T ST ‖2 < 0, qm

with qm < 0 : νQν ≤ qm ν 2 , ∀ν

what leads to the final condition on μu given by T T 2 󵄩2 󵄩 ‖G − H S ‖ μu < −(κ + ‖H 󵄩󵄩󵄩 ||R󵄩󵄩󵄩 + ). qm

This condition goes at hand with the one stated in Lemma 2.2.6.

(3.40)

70 | 3 Dissipativity-based observer and feedback control design 3.2.3 Nonlinear finite-dimensional systems Consider the following nonlinear multiple-input-multiple-output (MIMO) system p

ẋ = f (x) + ∑ bj (x)uj + Gν,

t > 0,

σ = Hx,

t≥0

j=1

y = h(x),

x(0) = x 0

(3.41a) (3.41b)

t≥0

(3.41c)

with the state x(t) ∈ 𝕏 = ℝn at time t ≥ 0, smooth vector fields f : 𝕏 → 𝕏, bj : 𝕏 → 𝕏, j = 1, . . . , p, inputs uj (t) ∈ ℝ, j = 1, . . . , p and ν(t) ∈ ℝr , G ∈ ℝn×r , H ∈ ℝq×n , output y(t) ∈ 𝒴 = ℝm and h : 𝕏 → ℝm . In the following consider that m = p, i. e. the number of inputs and outputs is the same and assume that the vector relative degree r = {r1 , . . . , rm } (see Definition 2.3.7) satisfies m

∑ rk = n

(3.42)

k=1

so that the decoupling matrix introduced in Definition 2.3.7 r −1

Lb1 Lf1 h1 (x) [ .. Bx (x) = [ . [ rm −1 [Lb1 Lf hm (x)

⋅⋅⋅ .. . ⋅⋅⋅

r −1

Lbm Lf1 h1 (x) ] .. ] . ] rm −1 Lbm Lf hm (x)]

(3.43)

has full rank, i. e. rank(B(x)) = m for all x ∈ 𝕏. In this case the diffeomorphic state transformation h1 (x) [ L h (x) ] ] [ f 1 ] [ .. ] [ ] [ . ] [ r −1 [ L 1 h (x) ] ] [ f 1 ] [ y ] [ 1 ] [ ] [ ẏ ] [ ] [ 1 ] [ h2 (x) ] [ . ] [ ] [ . ] [ .. ] [ . ] [ ] [ (r1 −1) ] [ . ] z = Φ(x) = [ y1 ] [ Lr2 −1 h (x) ] = [ ] 2 ] [ [ f [ ] [ y2 ] [ ] [ . ] [ ] [ . ] [ ] [ . ] [ . ] . ] [ . (r −1) ] [ ] [ymm ] [ ] [ ] [ ] [ h (x) m ] [ ] [ .. ] [ . ] [ rm −1 L h (x) m ] [ f

(3.44)

3.2 State-feedback control design

| 71

brings the system into the generalized Byrnes–Isidori normal form ż =

󵄨󵄨 p 󵄨󵄨 𝜕Φ(x) (f (x) + ∑ bj (x)uj + Gν)󵄨󵄨󵄨 󵄨󵄨 𝜕x j=1 󵄨x=Φ−1 (z)

̄ = Az + B(B(z)u + α(z)) + G(z)ν,

t>0

σ = hσ (z),

t≥0

y = Cz,

t ≥ 0,

(3.45a) (3.45b) (3.45c)

with initial condition z(0) = z 0 = Φ(x 0 ) and A1 [ [0 [ A=[ [ .. [. [0

0 A2 .. . ⋅⋅⋅

⋅⋅⋅ .. . .. . 0

0 [. [. [. [ [ Ai = [ ... [ [. [. [.

0 .. ] . ] ] ], ] 0] Am ]

1,

B = {bij }i=1,...,n,j=1,...,p ,

bij = {

0,

1 ..

[0

.

0 .. . .. .

⋅⋅⋅

⋅⋅⋅

(i, j) = (rj , j) else,

⋅⋅⋅ .. . .. . .. . ⋅⋅⋅

0 .. ] ] .] ] ] 0] ] ] ] 1] 0]

B(z) = Bx (Φ−1 (z)),

(3.45d)

with Bx given in (3.43) and r

αk (z) = Lfk hk (Φ−1 (z)), k = 1, . . . , p,

𝜕Φ −1 ̄ G(z) = (Φ (z))G, 𝜕x

hσ (z) = H T Φ−1 (z) (3.45e)

Given that Φ(x) is a diffeomorphic map and G, H are constant matrices, it follows that over any compact subset D ⊆ Φ(𝕏) there exist constants ḡm , Lh , such that 󵄩󵄩 ̄ 󵄩󵄩 󵄩󵄩G(z)󵄩󵄩O ≤ ḡm ,

‖hσ (z)|| ≤ Lh ‖z‖,

(3.46)

i. e. Lh is the Lipschitz constant of the function h. Furthermore, given that Bx (x) has full rank for all x ∈ 𝕏, B(z) has full rank for all z ∈ Φ(𝕏), and thus its inverse exists. Accordingly, the control u = −B−1 (z)(α(z) + Kz)

(3.47)

achieves the closed-loop dynamics ̄ ż = Ac z + G(z)ν, t>0

σ = hσ (z),

t ≥ 0.

z(0) = z 0 ,

Ac = A − BK

(3.48a) (3.48b)

This system has exactly the form (2.38), so that sufficient conditions to conclude (−R, ST , −Q)-strict state dissipativity can be drawn directly from Lemma 2.2.6 and are stated next.

72 | 3 Dissipativity-based observer and feedback control design Lemma 3.2.2. Let Q ≤ −qI with q > 0, R ≤ 0 and ḡm , Lh be constants such that (3.46) holds for all z ∈ D. The dynamical subsystem (3.48) resulting from applying the control (3.47) to the system (3.41) is (−R, ST , −Q)-strictly state dissipative with dissipation rate κ > 0 in D if μ > (κ + ‖R‖O L2h +

(ḡ + ‖S‖O Lh )2 ), q

μ=−

max

λ∈σ(Ac +ATc )

λ,

(3.49)

where μ denotes the modulus of the maximum eigenvalue of Ac + ATc . A comparison with the condition (3.40) derived for the case of the linear system (3.35) (i. e., with G and hσ being constant matrices) shows that quite similar results are obtained in terms of the maximum eigenvalue of the closed-loop matrix Ac .

3.3 Dissipativity-based stabilization of dynamic-static system interconnections A classical approach of dissipative control theory is for the purpose of stabilization of a linear system in presence of a possibly destabilizing nonlinear source. For this purpose consider the system ẋ = Ax + Bu + Gφ(σ),

σ = Hx,

t > 0, t≥0

x(0) = x 0

(3.50a) (3.50b)

which is identical in its structure to (3.1) with ψ = 0. By considering a linear state feedback control of the form u = −Kx

(3.51)

with the linear operator K, the closed-loop dynamics can be written as ẋ = Ac x + Gν, t > 0,

σ = Hx,

ν = −η(σ),

η = −φ(σ),

t≥0

t≥0

t≥0

x(0) = x 0 ,

Ac = A − BK

(3.52a) (3.52b) (3.52c) (3.52d)

Note that (3.52a)–(3.52c) represent a negative feedback interconnection of a linear dynamic system Σ(A − BK, G, H) with a nonlinear static map η = −φ(σ). Based on the general considerations of Section 2.2 and in particular Lemma 2.2.11 and Corollary 2.2.1 one obtains the following stability result for the closed-loop dynamics (3.52). Theorem 3.3.1. Consider system (3.52). Let η(σ) in (3.52c) be (Q, S, R)-dissipative and let K: 𝕏 → 𝒰 be such that the system Σ(Ac , G, H) is (−R, S∗ , −Q)-strictly state dissipative

3.3 Dissipativity-based stabilization of dynamic-static system interconnections | 73

with rate κ > 0 and positive definite storage functional 𝒮 : 𝕏 → ℝ with 𝒮 (x) = ⟨x, Px⟩ with P = P ∗ such that there exist 0 < p− , p+ ∈ ℝ so that p− ‖x‖2 ≤ 𝒮 (x) ≤ p+ ‖x‖2 . Then the origin x = 0 is exponentially stable with rate γ = κ/(2p+ ) and amplitude a = √p+ /p− . Proof. The result follows directly from Corollary 2.2.1 applied to system (3.52). For the finite-dimensional nonlinear system (3.41) interconnected with a static nonlinear map in the form p

ẋ = f (x) + ∑ bj (x)uj + Gν,

t > 0,

σ = Hx,

t≥0

(3.53b)

t≥0

(3.53d)

j=1

y = h(x),

x(0) = x 0

(3.53a)

t≥0

ν = −η(σ),

(3.53c)

the following result is obtained. Theorem 3.3.2. Consider the feedback interconnection (3.53) and let η be (Q, S, R)-dissipative. Let the vector relative degree satisfy ∑pi=1 ri = n. Consider the controller u = −B−1 x (x)(α(Φ(x)) + KΦ(x)),

(3.54)

with Φ: 𝕏 → 𝕏 given in (3.44) and Bx (x) being the decoupling matrix defined in (3.43). The origin x = 0 is exponentially stable for the closed-loop system with (3.54) if the gain matrix K is chosen so that μ > (κ + ‖R‖L2h +

(ḡ + ‖S‖Lh )2 ), q

μ=−

max

λ∈σ(Ac +ATc )

λ

(3.55)

with Lh , ḡ given in (3.46) and Ac as defined in (3.52) with A given in (3.45d). Proof. Consider the state transformation z = Φ(x) that brings the system into the normal form (3.45). Recall from 3.2.2 (or Lemma 2.2.6) that under the condition (3.55) the dynamic subsystem is (−R, ST , −Q)-strictly state dissipative with dissipation rate κ > 0 with the storage function 𝒮 : 𝕏 → ℝ with 𝒮 (z) = z T z. Remember from Lemma 2.2.11 and Corollary 2.2.1 that the interconnection of the static subsystem which is (Q, S, R)-dissipative and the dynamic one which is (−R, ST , −Q)-strictly state dissipative with positive dissipation rate achieves an exponentially stable origin z = 0, i. e. there exist a, γ > 0 so that 󵄩󵄩 󵄩 −γt 󵄩󵄩z(t)󵄩󵄩󵄩 ≤ a‖z 0 ‖e ,

∀t ≥ 0.

From x = Φ−1 (z) and the fact that for bounded x and z there exist constants m, M > 0 so that Φ(x) ≤ m and Φ−1 (z) ≤ M it follows that 󵄩󵄩 󵄩 󵄩 −1 󵄩 −γt −γt 󵄩󵄩x(t)󵄩󵄩󵄩 = 󵄩󵄩󵄩Φ (z)󵄩󵄩󵄩 ≤ M||z(t)‖ ≤ Ma‖z 0 ‖e ≤ Mam||x 0 ‖e ,

∀t ≥ 0

74 | 3 Dissipativity-based observer and feedback control design or summarizing 󵄩 󵄩󵄩 ̄ 0 ‖e−γt , 󵄩󵄩x(t)󵄩󵄩󵄩 ≤ a||x

ā = Mam ≥ 1,

∀t ≥ 0.

This completes the proof.

3.4 Observer-based feedback control In this section the problem of analyzing the closed-loop dynamics with an observerbased output-feedback control scheme is considered. For this purpose, first the linear observer-controller structures from Sections 3.1.1 and 3.2.2 are combined and an explicit closed-loop exponential stability result is established which resembles the separation principle known for linear systems. For this purpose it is shown that the combined control and observation error reflects a triangular dependency which is exploited, in a way similar to other nonlinear separation results in the literature (see e. g. Teel and Praly, 1994, 1995) to proof the closed-loop stability. Then the exact feedback linearizing control from Section 3.2.3 is combined with a dissipative observer and the closed-loop exponential stability is characterized following a similar approach. 3.4.1 Linear correction and control structure Consider the system ẋ = Ax + Bu + Gφ(σ), σ = Hx, y = Cx,

t > 0, t≥0 t≥0

(0) = x 0

(3.56a) (3.56b) (3.56c)

which in its structure is equivalent to (3.1) with ψ = 0, together with the dissipative observer (see Section 3.1) 𝜕t x̂ = Ax̂ + Bu + Gφ(σ)̂ − L(C x̂ − y), Bx̂ = ub , σ̂ = H x,̂

t > 0, t≥0 t≥0

̂ x(0) = x̂ 0

(3.57a) (3.57b) (3.57c)

where L : 𝒴 → 𝕏 is a linear operator. Consider further the observer-based feedback controller u = −K x̂

(3.58)

with the linear operator K: 𝕏 → 𝒰 . In terms of the estimation error x̃ = x̂ − x the associated closed-loop dynamics can be written as ẋ = Ac x + Gφ(σ) − BK x,̃ t > 0, ẋ̃ = AL x̃ + G(φ(σ + σ)̃ − φ(σ)), t > 0,

x(0) = x 0 ̃ x(0) = x̃ 0

(3.59a) (3.59b)

3.4 Observer-based feedback control | 75

σ = Hx,

t≥0

σ̃ = H x,̃

t≥0

(3.59c) (3.59d)

with the operators Ac = A − BK,

AL = A − LC.

These dynamics can be written in the form of two interconnected Lur’e systems ẋ = Ac x + Gω − BK x,̃

σ = Hx,

t > 0,

t≥0

ω = −η(σ),

t≥0

ẋ̃ = AL x̃ + Gν,

t > 0,

σ̃ = H x,̃

ν = −ψ(σ,̃ σ),

x(0) = x 0

(3.60a) (3.60b) (3.60c)

̃ x(0) = x̃ 0

t≥0

t ≥ 0,

(3.60d) (3.60e) (3.60f)

with η(σ) = −φ(σ)

(3.61a)

ψ(σ,̃ σ) = −(φ(σ + σ)̃ − φ(σ)).

(3.61b)

The structure of this interconnection is shown in Fig. 3.1. As can be seen, the dependency of the observation error dynamics on the control error is only through the nonlinearity ψ(σ,̃ σ). Again, consider the case that the associated static subsystem ν = −ψ uniformly satisfies a dissipativity condition, e. g. in form of a sector condition. Accordingly, by ensuring the appropriate strict dissipativity of the linear dynamic subsystem Σ(AL , G, H) following Lemma 2.2.11 and Corollary 2.2.1 the observer converges independent of the control error. The control error itself depends directly on the observation error through the term BK x.̃ This implies a triangular structure in the linear dependencies equivalent to the one obtained in the linear case which can be exploited to proof the following result. Theorem 3.4.1. Consider the closed-loop dynamics (3.60)–(3.61) depicted in Fig. 3.1 and let ‖BK‖O ≤ bK̄ ∈ ℝ. Let η be (Qc , Sc , Rc )-dissipative and ψ be uniformly (Qo , So , Ro )-dissipative. If the linear system (3.60a)–(3.60b) is (−Rc , Sc∗ , −Qc )-strictly state dissipative with dissipation rate κc > 0 and positive definite storage functional 𝒮c : 𝕏 → ℝ with 𝒮c (x) = ⟨x, Pc x⟩, Pc = Pc∗ and the linear system (3.60d)–(3.60e) is (−Ro , So∗ , −Qo )-strictly state dissipative with dissipation rate κo > 0 and positive definite storage functional ̃ Po = Po∗ satisfying 𝒮o : 𝕏 → ℝ with 𝒮o (x)̃ = ⟨x,̃ Po x⟩, αc− ‖x‖ ≤ Sc (x) ≤ αc+ ‖x‖,

αo− ‖x‖̃ ≤ So (x)̃ ≤ αo+ ‖x‖̃

for some constants αc− , αc+ , αo− , αo+ > 0. Then the origin [x T , x̃ T ]T = 0 is exponentially κ κ stable with convergence rate given by γ = 21 min{ α+o , α+c }. o

c

76 | 3 Dissipativity-based observer and feedback control design

Figure 3.1: Interconnected control and observation error dynamics (3.60) in Lur’e configuration.

Proof. The proof exploits the cascade structure of the closed-loop dynamics (3.60)– (3.61). Introducing the closed-loop operators Ac = A − BK,

Ao = A − LC

the solutions of the control and observation error can be written as x(t) = e

Ac t

t

x0 + ∫ e

Ac (t−τ)

t

̃ Gν c (τ)dτ − ∫ eAc (t−τ) BKx(τ)dτ

0

t

0

̃ = eAo t x̃ 0 + ∫ eAo (t−τ) Gν o (τ)dτ. x(t) 0

Taking norms on both sides and applying the triangle inequality yields t 󵄩󵄩 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 A t A (t−τ) 󵄩󵄩 ̃ Gν c (τ)dτ󵄩󵄩󵄩 + 󵄩󵄩󵄩∫ eAc (t−τ) BKx(τ)dτ 󵄩󵄩x(t)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩e c x 0 + ∫ e c 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 󵄩󵄩 󵄩 󵄩 0 0

t 󵄩󵄩 󵄩󵄩 t 󵄩󵄩 Ac t 󵄩󵄩 󵄩 󵄩 󵄩 ̃ 󵄩󵄩 Ac (t−τ) 󵄩 ≤ 󵄩󵄩e x 0 + ∫ e Gν c (τ)dτ󵄩󵄩󵄩 + ∫󵄩󵄩󵄩eAc (t−τ) 󵄩󵄩󵄩O ‖BK‖O 󵄩󵄩󵄩x(τ) 󵄩󵄩dτ 󵄩󵄩 󵄩󵄩 󵄩 󵄩 0 0 t

󵄩󵄩 ̃ 󵄩󵄩 A t A (t−τ) Gν o (τ)dτ‖. 󵄩󵄩x(t)󵄩󵄩 = ||e o x̃ 0 + ∫ e o 0

3.4 Observer-based feedback control | 77

According to the strict dissipativity properties assumed for the linear systems Σ(Ac , G, H) and Σ(Ao , G, H) it follows from Theorems 3.1.1 and 3.3.1 that 󵄩󵄩 ̃ 󵄩󵄩 −γ t 󵄩󵄩x(t)󵄩󵄩 ≤ ao ‖x̃ 0 ‖e o ,

γo = −

κo , 2αo+

ao = √

αo+ , αo−

∀t ≥ 0

󵄩󵄩 󵄩 −γ t 󵄩󵄩x(t)󵄩󵄩󵄩 ≤ ac ||x 0 ‖e c ,

γc = −

κc , 2αc+

ac = √

αc+ , αc−

∀t ≥ 0.

󵄩󵄩 Ac (t−τ) 󵄩󵄩 −γ t 󵄩󵄩e 󵄩󵄩O ≤ ac e c ,

∀t ≥ 0.

and for x̃ = 0

Furthermore it holds that

Accordingly, one obtains that t

̃ ‖x(t)‖≤ac ‖x 0 ‖e−γc t + ∫ ac e−γc (t−τ) ‖BK‖O ‖x(τ)‖dτ −γo t ̃ ‖x(t)‖≤a , o ‖x̃ 0 ||e

0

∀t ≥ 0.

Recall that ‖BK‖O ≤ bK̄ and introduce the bounding functions σc (t) = e

−γc t

t

󵄩 ̃ 󵄩󵄩 (ac ‖x 0 ‖ + ∫ ac eγc τ bK̄ 󵄩󵄩󵄩x(τ) 󵄩󵄩dτ),

σo (t) = ao ‖x̃ 0 ‖e−γo t

0

with σc (0) = ac ‖x 0 || and σo (0) = ao ‖x̃ 0 ‖. It holds that σ̇ c ≤ −γc σc + bK̄ σo

σ̇ o ≤ −γo σo , t > 0. Accordingly, σc , σo are in a classical cascade structure. Introducing the bounding states χ χ = [ 1] , χ2

σ (0) χ0 = [ c ] σo (0)

with dynamics χ̇ = [

−γc 0

bK̄ ] χ, −γo

t > 0,

χ(0) = χ 0

it follows that 󵄩󵄩 󵄩 −γt 󵄩󵄩χ(t)󵄩󵄩󵄩 ≤ e ‖χ 0 ‖,

γ = min{γc , γo },

∀t ≥ 0

78 | 3 Dissipativity-based observer and feedback control design and χi (t) ≤ e−γt χi0 ,

i = 1, 2,

∀t ≥ 0.

Thus it holds that 󵄩 󵄩󵄩 −γt −γt 󵄩󵄩x(t)󵄩󵄩󵄩 ≤ σc (t) ≤ σc (0)e = ac ‖x 0 ‖e 󵄩󵄩 ̃ 󵄩󵄩 −γt −γt 󵄩󵄩x(t)󵄩󵄩 ≤ σo (t) ≤ σo (0)e = ao ‖x̃ 0 ‖e ,

∀t ≥ 0,

implying the exponential convergence of the observation and control error to zero with κ κ convergence rate γ = 21 min{ α+o , α+c } as stated in the Theorem. o

c

Besides this direct result, the dissipation properties in form of matrix inequalities can be exploited to obtain a closed-loop stability condition in terms of a quadratic Lyapunov function as stated in the following theorem. Note that in contrast to the preceding direct result, here (i) the result is constrained to finite-dimensions, and (ii) additional conditions show up, which have to be fulfilled. Thus the following result is more restrictive in nature, but possibly more useful in the context of extensions to larger classes of interconnected systems. Theorem 3.4.2. Consider the closed-loop dynamics (3.60)–(3.61) with 𝕏 = ℝn , depicted in Fig. 3.1 and let the assumptions of Theorem 3.4.1 be satisfied. Let κc , κo are sufficiently large, so that there exists a γ > 0 satisfying (γ − κc )(γ − κo )I > K T BT Pc2 BK.

(3.62)

Then the origin [x T , x̃ T ]T = 0 is exponentially stable. Proof. By assumption it holds that x AT P + P A ⟨[ ] , [ c c T c c ω G Pc

Pc G x ] [ ]⟩ 0 ω

x −κ I − H T Rc H ≤ ⟨[ ] , [ c ω Sc H

H T ScT x ] [ ]⟩ −Qc ω

(3.63)

and x̃ AT P + P A ⟨[ ] , [ L o T o L ν G Po

Po G x̃ ] [ ]⟩ 0 ν

x̃ −κ I − H T Ro H ≤ ⟨[ ] , [ o ν So H

H T SoT x̃ ] [ ]⟩ −Qo ν

Taking 𝒮 = 𝒮c + 𝒮o as storage function it follows from (3.60) that x ATc Pc + Pc Ac [ x̃ ] [ −K T BT P [ ] [ c 𝒮 ̇ = ⟨[ ] , [ [ω] [ GT Pc 0 [ν] [

−Pc BK ATL Po + Po AL 0 G T Po

Pc G 0 0 0

0 x [ ] Po G ] ] [ x̃ ] ] [ ]⟩ 0 ] [ω] 0 ][ν]

(3.64)

3.4 Observer-based feedback control | 79

With (3.63) and (3.64) it turns out that in virtue of the dissipativity properties of the individual subsystems it holds that x −κc I − H T Rc H [ x̃ ] [ −K T BT P [ ] [ c 𝒮 ̇ ≤ ⟨[ ] , [ [ω] [ Sc H 0 [ν] [

−Pc BK −κo I − H T Ro H 0 So H

H T ScT 0 −Qc 0

x 0 [ x̃ ] H T SoT ] ][ ] ] [ ]⟩ 0 ] [ω] −Qo ] [ ν ]

x −κ I = ⟨[ ] , [ T cT x̃ −K B Pc

−Pc BK x x −H T Rc H ] [ ]⟩ + ⟨[ ] , [ −κo I x̃ ω Sc H

x −κ I = ⟨[ ] , [ T cT x̃ −K B Pc

−Pc BK x σ −R ] [ ]⟩ + ⟨[ ] , [ c −κo I x̃ ω Sc

x̃ −H T Ro H + ⟨[ ] , [ ν So H

σ̃ −R + ⟨[ ] , [ o ν So

σ̃ SoT ] [ ]⟩ −Qo ν

x −κ I = ⟨[ ] , [ T cT x̃ −K B Pc ψ Q − ⟨[ ] , [ To So σ̃

H T SoT x̃ ] [ ]⟩ ν −Qo

−Pc BK x η Q ] [ ]⟩ − ⟨[ ] , [ Tc Sc −κo I x̃ σ

H T ScT x ] [ ]⟩ −Qc ω

x ScT ] [ ]⟩ −Qc ω

Sc η ] [ ]⟩ Rc σ

So ψ ] [ ]⟩ Ro σ̃

x −κ I ≤ ⟨[ ] , [ T cT x̃ −K B Pc

−Pc BK x ] [ ]⟩ −κo I x̃

where the (Qc , Sc , Rc ) dissipativity of η and the uniform (Qo , So , Ro ) dissipativity of ψ have been exploited. Accordingly, there exists a constant γ > 0 for which 𝒮 ̇ ≤ −γ 𝒮 holds true if [

−κc I −K T BT Pc

−Pc BK ] ≤ −γI < 0 −κo I

or equivalently [

−(γ − κc )I K T BT Pc

Pc BK ] < 0. −(γ − κo )I

In virtue of the Schur complement (cp. Appendix A.2) this inequality is satisfied if (γ − κo )I −

K T BT Pc2 BK >0 (γ − κc )

what is equivalent to (3.62). Note that the dissipation κc depends on the control gain vector K and that with larger κc the necessary K becomes larger too. Thus, the possibility to satisfy the condition (3.62) of Theorem 3.4.2 amount in first place in designing the observer so that

80 | 3 Dissipativity-based observer and feedback control design it converges sufficiently fast, given that the closed-loop stability does not depend explicitly on the observer gain L. Clearly, the larger L the higher the noise amplification, presenting an additional constraint and implying the need for a trade-off in practical applications.

3.4.2 Observer-based feedback subsystem-linearization in finite dimensions Recall from Section 3.2.3 the exact feedback linearizing controller (3.47) u = −B−1 (z)(α(z) + Kz) = μ(z)

(3.65)

for the nonlinear system (3.41) in Byrnes–Isidori normal form ̄ ż = Az + B(B(z)u + α(z)) + G(z)ν,

t > 0,

ν = −φ(σ),

t≥0

σ = hσ (z), y = Cz,

t≥0

z(0) = z 0

(3.66a) (3.66b) (3.66c)

t≥0

(3.66d)

with A, B, α, β, Ḡ defined in (3.45)–(3.45d). The main issue about implementing the state-feedback controller (3.65) resides in the requirement to have access to the complete state z. On the basis of the Byrnes–Isidori normal form (3.66) the following nonlinear observer scheme is proposed ̂ + α(z)) ̂ + G(̄ z)̂ ν̂ − L(Cẑ − y), ż̂ = Aẑ + B(B(z)u

̂ σ̂ = hσ (z),

̂ ν̂ = −φ(σ),

t > 0,

t≥0

̂ z(0) = ẑ 0

t≥0

(3.67a) (3.67b) (3.67c)

with the correction gain L. Denoting by z̃ = ẑ − z the observation error, the observer error dynamics can be written as6 ż̃ = (A − LC)z̃ + ϖ,

σ̃ = z,̃

ϖ = −ψO (σ;̃ z) = −ψO (z;̃ z),

t > 0, t≥0 t≥0

̃ z(0) = z̃ 0

(3.68a) (3.68b) (3.68c)

with ̄ + z)φ(σ ̄ ̃ ψO (z;̃ z) = −(B([B(z + z)̃ − B(z)]u + α(z + z)̃ − α(z)) − G(z + σ)̃ + G(z)φ(σ)) 6 Note that this representation does not take into account any structure but puts together the different effects of the nonlinearities in the single function ψO . Alternatively, one could reformulate the problem considering several non-linearity’s in an extended set-up. Accordingly, the presented convergence conditions are sufficient but possibly conservative.

3.4 Observer-based feedback control | 81

Note that it holds for all u ∈ 𝒰 , z ∈ D ⊆ ℝn that ψ(0; z) = 0. Thus, if the function ψO (z;̃ z) is (Q, S, R)-dissipative, and the correction gain is chosen so that the dynamic subsystem Σ(A − LC, I, I) given by (3.68a)–(3.68c) is (−R, ST , −Q)strictly state dissipative with dissipation rate κ > 0, then the observation error exponentially converges to zero. Conditions for this property can be found above in Theorem 3.1.1. The observer-based implementation of the feedback control (3.65) reads ̂ u = −B−1 (z)(α( z)̂ + Kz)̂ = μ(z)̂ = μ(z) + [μ(z + z)̃ − μ(z)]

(3.69)

and the closed-loop dynamics are given by ̄ ż = (A − BK)z + Bν c + G(z)ν,

t > 0,

σ = hσ (z),

t≥0

(3.70b)

ν = −φ(z),

t≥0

(3.70c)

t≥0

(3.70d)

t≥0

(3.70e)

ν c = −ψc (z;̃ z), y = Cz,

z(0) = z 0

(3.70a)

with the observer-induced error function ψc (z;̃ z) = −B(z)(μ(z + z)̃ − μ(z)),

ψc (0; z) = 0.

(3.71)

Note that the control design can be performed as outlined in Lemma 3.2.2 in order to achieve exponential stability for the case that ν c = 0 and then verifying the exponential stability of the complete closed-loop system interconnection ̄ ż = (A − BK)z + Bν c + G(z)ν,

t > 0,

z(0) = z 0

(3.72a)

σ = hσ (z),

t≥0

ż̃ = (A − LC)z̃ + ϖ,

t>0

σ̃ = z,̃

t≥0

(3.72d)

ν = −φ(z),

t≥0

(3.72e)

ν c = −ψc (z;̃ z),

t≥0

(3.72f)

ϖ = −ψO (σ;̃ z),

t ≥ 0.

(3.72g)

(3.72b) ̃ z(0) = z̃ 0

(3.72c)

Note that, similar to the linear case discussed above in Section 3.4.1 the observer convergence is independent of the controller convergence. This result is subject to the

82 | 3 Dissipativity-based observer and feedback control design assumption that the function ψO is (Q, S, R)-dissipative uniformly with respect to u and z. In the linear case only uniform dissipativity with respect to z was necessary. Provided this uniform dissipativity property of the nonlinear subsystem ψO is given, the dynamic subsystems of the combined error dynamics show a cascade structure, so that the closed-loop exponential stability can be concluded if the appropriate dissipation inequalities of the controller and observer are fulfilled. This leads to the following separation result, which is similar to the one presented in Theorem 3.4.1. Theorem 3.4.3. Consider the closed-loop dynamics (3.72). Let φ be (Qc , Sc , Rc )-dissipative, ψO be (Qo , So , Ro )-dissipative and ψc be Lipschitz continuous in z,̃ uniformly in z ψ with Lipschitz constant Lc , i. e. 󵄩󵄩 󵄩 ψ ̃ 󵄩󵄩ψc (z;̃ z)󵄩󵄩󵄩 ≤ Lc ‖z‖. If the dynamic subsystem (3.72a)–(3.72b) is (−Rc , ScT , −Qc )-strictly state dissipative with dissipation rate κc > 0 and quadratically bounded storage function 𝒮c : 𝕏 → ℝ and the dynamic subsystem (3.72c)–(3.72d) is (−Ro , SoT , −Qo )-strictly state dissipative with dissipation rate κo > 0 and quadratically bounded storage function 𝒮o : 𝕏 → ℝ, than the origin [z T , z̃ T ]T = 0 is exponentially stable. Proof. The proof follows the reasoning of the proof of Theorem 3.4.1. Recall the definition of the closed-loop matrices Ac = A − BK,

Ao = A − LC

and write the solutions of the control and observation error as t

t

0

0

̄ ̃ z(t) = eAc t z 0 + ∫ eAc (t−τ) G(z)ν(τ)dτ − ∫ eAc (t−τ) Bψc (z(τ); z(τ))dτ t

̃ = eAo t z̃ 0 + ∫ eAo (t−τ) ϖ(τ)dτ. z(t) 0

Taking norms on both sides and applying the triangle and Cauchy–Schwarz inequalities yields t 󵄩󵄩 󵄩󵄩󵄩 t 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 Ac t 󵄩 󵄩 󵄩 󵄩 󵄩 Ac (t−τ) ̄ ̃ G(z)ν c (τ)dτ󵄩󵄩󵄩 + ∫󵄩󵄩󵄩eAc (t−τ) 󵄩󵄩󵄩O ‖B‖O 󵄩󵄩󵄩ψc (z(τ); z(τ))󵄩󵄩󵄩dτ 󵄩󵄩z(t)󵄩󵄩 ≤ 󵄩󵄩e z 0 + ∫ e 󵄩󵄩 󵄩󵄩 󵄩 󵄩 0 0

t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ̃ 󵄩󵄩 󵄩󵄩󵄩 Ao t ̃ Ao (t−τ) 󵄩󵄩. z(t) = e z + e ϖ(τ)dτ ∫ 󵄩󵄩 󵄩󵄩 󵄩󵄩 0 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 0

3.4 Observer-based feedback control | 83

̄ According to the strict dissipativity properties assumed for the systems Σ(Ac , G(z), hσ (z)) and Σ(Ao , I, I) it follows that 󵄩󵄩 ̃ 󵄩󵄩 −γ t 󵄩󵄩z(t)󵄩󵄩 ≤ ao ‖z̃ 0 ‖e o ,

1 γo = − κo , 2

∀t ≥ 0

󵄩 󵄩󵄩 −γ t 󵄩󵄩z(t)󵄩󵄩󵄩 ≤ ac ||z 0 ‖e c ,

1 γc = − κc , 2

∀t ≥ 0.

and for z̃ = 0

Taking into account the uniform Lipschitz continuity of the function ψc it follows that t

̃ ‖z(t)‖ = ac ‖z 0 ‖e−γc t + ∫ ac e−γc (t−τ) ‖B‖Lψ c ‖z(τ)‖dτ 󵄩󵄩 ̃ 󵄩󵄩 −γ t 󵄩󵄩z(t)󵄩󵄩 = ao ‖z̃ 0 ‖e o .

0

Using the bounding functions t

󵄩󵄩 ̃ 󵄩󵄩 σc (t) = ac ‖z 0 ‖e−γc t + ∫ ac e−γc (t−τ) ‖B‖Lψ c󵄩 󵄩z(τ)󵄩󵄩dτ,

σo (t) = ao ‖z̃ 0 ‖e−γo t

0

with σc (0) = ac ‖z 0 || and σo (0) = ao ‖z̃ 0 ‖ it follows that σ̇ c ≤ −γc σc + ‖B‖Lψ c σo

σ̇ o ≤ −γo σo ,

t > 0.

Accordingly the new state χ χ = [ 1] , χ2

σ (0) χ0 = [ c ] σo (0)

with dynamics −γc 0

χ̇ = [

ψ

‖B‖Lc ] χ, −γo

t > 0,

χ(0) = χ 0

satisfies χi (t) ≤ e−γt χi0 ,

i = 1, 2,

γ = min{γc , γo },

∀t ≥ 0.

It follows that 󵄩󵄩 󵄩 −γt −γt 󵄩󵄩z(t)󵄩󵄩󵄩 ≤ σc (t) ≤ σc (0)e = ac ‖z 0 ‖e 󵄩󵄩 ̃ 󵄩󵄩 −γt −γt 󵄩󵄩z(t)󵄩󵄩 ≤ σo (t) ≤ σo (0)e = ao ‖z̃ 0 ‖e ,

∀t ≥ 0,

implying the exponential convergence of the observation and control error to zero as stated in the Theorem.

84 | 3 Dissipativity-based observer and feedback control design

3.5 Summary In this chapter general approaches for the design of dissipativity-based observers, state-feedback and observer-based output-feedback control schemes have been discussed and developed. The presented results generalize some of the well-established observer and feedback design approaches allowing to be applied on a subdynamics of the system by interpreting the complete system model as an interconnection of two parts, a dynamical subsystem and a static subsystem. Conditions for the exponential convergence, or exponential stability in the case of feedback control systems have been derived on the basis of the dissipativity properties of each subsystem. Different approaches to assign a prescribed dissipation property for the dynamical subsystem have been discussed using classical design methods, like linear observer and statefeedback, feedback linearization or the extended Luenberger observer. The exponential stability of the observer-based dissipative feedback control systems has been analyzed and sufficient conditions drawn which resemble the separation principle for linear systems. In the next part application examples will be considered that serve as a basis to illustrate and discuss the general considerations of this chapter.

|

Part III: Application examples

In this part case studies are presented in order to highlight different aspects of the dissipativity-based design approach for finite- and infinite-dimensional systems. Each case considered here illustrates a different aspect and employs a different methodological approach within the dissipativity framework. In the first chapter observer-based output-feedback control for nonlinear finitedimensional systems is considered. First, a simple academic example is studied to illustrate the main ideas and results from the preceding chapters. Then, some particular disispativity-based observer and feedback control design approaches are discussed for a nonlinear ship model, an exothermic chemical and a biological stirred tank reactor, the inverted pendulum with applied torque and an academic nonlinear MIMO control system of a three-dimensional relaxation oscillator. In the second chapter semi-linear infinite-dimensional systems are considered. First, the idea of spectral dissipation assignment for the linear subsystem is presented and applied to an academic example of a two-state coupled 1D pde model with measurements in the domain and/or on the boundary. Next, the point-wise measurement injection observer design is discussed and put into the perspective of dissipativitybased design principles. Then, a dissipativity-based (static) proportional outputfeedback control for an unstable semi-linear heat equation with in-domain actuation is designed. In these three examples the choice of appropriate in-domain sensor and actuator locations is a main question which is resolved using the theory provided in the preceding chapters. Finally, the backstepping observer and control design approach for boundary control systems is discussed in the perspective of dissipativity theory. In particular this is done for two case examples: an unstable semi-linear heat equation and a semi-linear partial integro-differential equation.

https://doi.org/10.1515/9783110677942-004

4 Finite-dimensional systems In this chapter the problem of observer and control design for some particular nonlinear finite-dimensional systems is discussed. The examples have been chosen in order to illustrate the main results developed in the preceding chapters as well as different dissipativity-based design approaches. First, a nonlinear observer-based output-feedback control is designed for a simple two-dimensional academic example of an interconnection of a linear dynamic and a nonlinear static subsystem. Then a linear output-feedback control strategy based on a nonlinear observer with linear correction mechanism is employed to stabilize different set-points for a second-order nonlinear ship course model proposed by Norrbin (1960). The combination of a nonlinear input-output linearizing state-feedback controller with a constant correction gain nonlinear observer is considered next for a simple model of a two-state continuous-stirred-tank-reactor with exothermic reaction. Afterwards, a nonlinear Luenberger observer is designed for a two-state bioreactor model with monotonic kinetics and compared with an asymptotic observer. Then, the combination of a constant gain nonlinear observer with an exact feedback linearizing controller is employed for the output-feedback stabilization of the model of an inverted pendulum with applied torque. Finally, the (partial) exact feedback linearization of a three-state nonlinear MIMO model is presented illustrating the connection of dissipativity-based and geometric control approaches.

4.1 Two-state linear-nonlinear system interconnection Consider a system which is composed by a linear dynamic subsystem ẋ = Ax + bu + gν,

t > 0,

σ = h x,

t≥0

(4.1b)

y = c x,

t≥0

(4.1c)

T

T

x(0) = x 0

(4.1a)

with x(t) ∈ ℝ2 , u(t), ν(t) ∈ ℝ and a nonlinear static feedback ν = −φ(σ),

(4.2)

where φ(σ) is given by φ(σ) =

k0 σ(1 − σ) . 1 + γσ 2

(4.3)

In the following consider the case that A=[

1 2

−1 ], −5

1 b = [ ], 0

https://doi.org/10.1515/9783110677942-005

1 g = [ ], −1

hT = [1

−1] ,

cT = [0

1] ,

90 | 4 Finite-dimensional systems k0 = 0.8,

γ = 10

so that the eigenvalues of the dynamics matrix A are given by λ1,2 = −2 ± √7 implying that the origin x = 0 is a saddle point. On the other hand the linear subsystem is completely observable and controllable according to the Kalman observability and controllability criterion, as 𝒦c = [b, Ab] = [

1

1

],

0

2

0 ]=[ c A 2

1

𝒦o = [

cT T

⇒

], −5

⇒

rank(𝒦c ) = 2 rank(𝒦o ) = 2

(4.4a) (4.4b)

and thus the eigenvalues of the linear subsystem can be assigned arbitrarily by linear feedback control. The graph of the nonlinearity (4.3) is shown in Figure 4.1.

Figure 4.1: Graph of the nonlinearity φ(σ) given in (4.3).

To construct a sector containing the graph of φ note that the maximum slope sm = k0 is attained at σ = 0, so that an upper sector bound is given by the line k0 σ. The lower sector bound has to be chosen so that the line kl σ and φ(σ) has exactly one solution besides σ = 0 at a point σl where the slope is exactly kl , i. e. the lower sector bound is tangential to φ. This is satisfied if kl σl = φ(σl ),

kl = φ󸀠 (σl ).

4.1 Two-state linear-nonlinear system interconnection

| 91

For the above example these equations have the solution σl = 1 + √

11 ≈ 2.049, 10

kl =

k0 (1 − σl ) ≈ −0.020. 1 + γσl2

Accordingly, the following sector is obtained (k0 σ − φ(σ))(φ(σ) − kl σ) ≥ 0,

∀ σ ∈ ℝ,

(4.5)

implying that the static map φ is (Q, S, R)-dissipative with Q = −1,

1 S = (k0 + kl ) ≈ 0.390, 2

R = −k0 kl ≈ 0.016.

Thus, according to Theorem 3.3.1 an exponentially stabilizing linear state-feedback controller u = −k T x

(4.6)

is obtained if the closed-loop linear subsystem Σ(Ak , g, h) with Ak = A − bk T is (−0.016, 0.390, 1)-strictly state dissipative with a positive dissipation rate κ > 0. Setting the quadratic storage functional 𝒮 : ℝ2 → ℝ with T

𝒮 (x) = x x,

i. e. with P = I and choosing κ = 1, according to Lemma 2.2.3 this property is ensured if the gain k T is chosen such that the matrix inequality A + ATk + κI + hRhT [ k g T − ShT

3.016 − 2k1 g − Sh [ ] = [ 0.984 − k2 Q [ 0.61

0.984 − k2 −8.984 −0.61

0.61 ] −0.61] ≤ 0 −1 ]

holds true. This inequality can be solved directly (e. g. using the Schur complement discussed in Appendix A.2) or a solution can be found using Lemma 2.2.4, i. e. choosing k so that the largest eigenvalue λ+ ∈ ℂ− of Ak satisfies the inequality γ>

1 max R(λ), 2 λ∈σ(ℳ)

T

−1

T

ℳ = κI + hRh − (g − Sh)Q (g − Sh) .

For the case at hand, choosing κ = 1 the matrix M is given by 1 + 0.016 + 0.612 −0.016 − 0.612

ℳ=[

−0.016 − 0.612 1.388 ]=[ 1 + 0.016 + 0.612 −0.388

with eigenvalues λ1 = 1.776,

λ2 = 1.0.

−0.388 ] 1.388

92 | 4 Finite-dimensional systems Accordingly, the closed-loop stability is ensured if it holds that the eigenvalue with maximum modulus λ+ of Ak satisfies λ+ < −0.888.

(4.7)

The feedback gain k can thus be assigned e. g. using the Ackermann formula (for n = 2) (Ackermann, 1972; Kailath, 1980) k T = w T (A2 + p1 A + p0 I),

w T = [0

1] 𝒦c−1

with p0 , p1 being the coefficients of the desired closed-loop characteristic polynomial and 𝒦c the Kalman controllability matrix given in (4.4a). Setting the desired closedloop eigenvalues of the matrix Ak as λ1,2 = −4 (and correspondingly p0 = 16, p1 = 8) for which the requirement (4.7) is obviously fulfilled, the control gain is obtained as k T = [4.0

(4.8)

−0.5] .

The resulting closed-loop behavior is presented in Figure 4.2 for the initial condition x 0 = [1, −1]T showing a smooth convergence to the origin in about 8 time units. In order to implement the feedback control the complete knowledge about the state is necessary, while only the measurement y = cT x is accessible without somehow employing the system model. Thus, an observer is required to reconstruct the state information from the knowledge of the output measurement y and the input u. Having in mind the preceding discussion, consider the nonlinear Luenberger observer ẋ̂ = Ax̂ + bu − gφ(σ)̂ − l(cT x̂ − y),

t > 0,

σ̂ = hT x,̂

t ≥ 0.

̂ x(0) = x̂ 0

(4.9a) (4.9b)

The associated observation error dynamics can be written as ẋ̃ = (A − lcT )x̃ + g ν,̃

t > 0,

σ̃ = hT x,̃

t≥0

(4.10b)

ν̃ = −ψ(σ;̃ σ),

t ≥ 0,

(4.10c)

̃ x(0) = x̃ 0

with the nonlinearity ψ(σ;̃ σ) = φ(σ + σ)̃ − φ(σ),

ψ(0; σ) = 0.

According to the mean value theorem it holds that φ(σ + σ)̃ − φ(σ) = 𝜕σ φ(σ + ησ)̃ σ,̃

η ∈ (0, 1).

(4.10a)

4.1 Two-state linear-nonlinear system interconnection

| 93

Figure 4.2: Closed-loop response of the nonlinear system (4.1)–(4.3) with the linear dissipative state-feedback control (4.6).

Note that the slope 𝜕σ φ is bounded in the interval [sl , sm ] with sl = −0.168, sm = 0.82. Accordingly, ψ is contained in the sector [sl , sm ] and is (Qo , So , Ro )-dissipative (uniformly in σ) with Q0 = −1,

1 So = (0.82 − 0.168) = 0.326, 2

R = −0.82(−0.168) = 0.138.

94 | 4 Finite-dimensional systems Thus, if the correction gain l is chosen so that the eigenvalues of the matrix Al = A − lcT = [

1 2

−(1 + l1 ) ] −(5 + l2 )

are such that the maximum modulus |λo+ | satisfies 󵄨󵄨 + 󵄨󵄨 1 max R(λ), 󵄨󵄨λo 󵄨󵄨 > 2 λ∈σ(ℳo )

T

T

−1

ℳo = κo I + hRo h − (g − So h)Qo (g − So h) ,

then the observed stated x̂ exponentially converges to the actual state x. Setting κo = 3, the matrix ℳo is given by ℳo = [

3.138 + (0.674)2 −0.138 − (0.674)2

−0.138 − (0.674)2 3.592 ]=[ 3.138 + (0.674)2 −0.592

−0.592 ] 3.592

with eigenvalues μo,1 = 4.184,

μo,2 = 3.0.

Accordingly, if the largest eigenvalue of Al is less than −μo,1 the convergence condition is satisfied. For this purpose the observer gain is determined using the Ackerman formula 2

lT = w To ((AT ) + m1 AT + m0 I),

w To = [0

1] (𝒦oT )

−1

with m0 , m1 being the coefficients of the desired characteristic polynomial of the matrix Al and 𝒦o the Kalman observability matrix given in (4.4b). Setting the desired eigenvalues of the matrix Al as λo,12 = −15 < −4.184 (and correspondingly m0 = 225, m1 = 30) the gain is obtained as lT = [127

26] .

(4.11)

The resulting behavior of the system (4.1)–(4.3) using the output-feedback controller u = −k T x̂

(4.12)

with k and l chosen according to (4.8) and (4.11), respectively, is shown in Figure 4.3 for the initial conditions x 0 = [1, −1]T , x̂ 0 = [−1, 0]T with the observed state represented by the dashed line. It can be seen that the observer converges in about 2 time units and afterwards the state converges in about 10 time units to the origin, thus recovering sufficiently well the behavior of the state-feedback controller.

4.2 Ship course control | 95

Figure 4.3: Closed-loop response of the nonlinear system (4.1)–(4.3) with the linear dissipative observer-based output-feedback control (4.12). The observed state is represented by the dashed lines.

4.2 Ship course control The design of autopilots for ship maneuvering has a long history (cp. Fossen, 2002). Several models of different complexity have been proposed and used for the analysis and design of ship course control systems. In the sequel the nonlinear Norrbin model (Norrbin, 1960; Åström and Källström, 1976; van Amerongen, 1982) is used to design a dissipativity-based output-feedback control for the stabilization of set-point changes

96 | 4 Finite-dimensional systems for the heading angle ψ(t) ∈ ℝ of the ship at time t ≥ 0. According to the Norrbin model, the dynamics of the rate of change of the heading angle ψ satisfy T ψ̈ + H(ψ)̇ = Kδ,

t>0

(4.13)

̇ with respective initial conditions ψ(0) = ψ0 , ψ(0) = vψ0 . In (4.13) T is the characteristic time constant of the ship, K is a constant gain, δ represents the rudder angle, which acts as the control input, and H(ψ)̇ is a nonlinear ship maneuvering characteristic, approximated by the polynomial H(ψ)̇ = b0 + b1 ψ̇ + b2 ψ̇ 2 + b3 ψ̇ 3 ,

t>0

(4.14)

with constant coefficients b0 , . . . , b3 . The offset b0 and the quadratic term account for the possible lack of ship symmetry and the sign of b1 indicates the course stability of the ship: for b1 < 0 the ship is course unstable and for b1 > 0 it is course stable. Course instability goes at hand with the existence of multiple steady-states with associated bi-stability and hysteresis features in the steering dynamics and thus represents an interesting task for control design purposes. In the following it is assumed that the ship is symmetric and thus b0 = 0,

b2 = 0.

The shape of the curve H(ψ)̇ is shown in Figure 4.4 for the case b1 = −1, b3 = 1.

Figure 4.4: Shape of the maneuvering characteristics H(ψ)̇ (4.14) with b0 = 0, b1 = −1, b2 = 0, b3 = 1.

Furthermore, it is assumed that the heading angle ψ is measured.1 The ship model (4.13)–(4.14) can be written in the Lur’e form as ẋ = Ax + bδ + gν, t > 0,

x(0) = x 0

(4.15a)

1 The Norrbin model is commonly referred to as a first-order model given that the output is normally taken as ψ.̇

4.2 Ship course control | 97

σ = hT x,

t≥0

(4.15b)

y = c x,

t≥0

(4.15c)

T

ν = −H(σ),

t≥0

(4.15d)

with the state x(t) ∈ ℝ2 at time t ≥ 0, A ∈ ℝ2×2 , and b, g, h, c ∈ ℝ2 given by ψ x = [ ], ψ̇

0 ψ x0 = [ 0 ] , A = [ vψ0 0

1 0

],

0 b = [K ] , T

0 0 g = [1], h = [ ], 1 T

1 c = [ ]. 0

Note that for bounded δ and large deviations in x2 (i. e., in ψ)̇ the third order term will always dominate and cause a reduction of |x2 |. Thus, x2 is naturally restricted to a finite interval x2 = σ ∈ [x2− , x2+ ] with x2− = −x2+ for the considered symmetric case. As a consequence the function H(σ) is contained in the sector spanned by two lines with slope (see Figure 4.4) s−H =

min

σ∈[−x2+ ,x2+ ]

𝜕σ H(σ) = b1 ,

s+H = H(x2+ )

(4.16)

implying that H(σ) is (Qc , Sc , Rc )-dissipative with Qc = −1,

1 Sc = (s−H + s+H ), 2

Rc = −s−H s+H .

Using a linear state feedback control of the form δ = −k T (x − x d ),

xd = [

ψd ] 0

(4.17)

with the constant ship heading set-point value ψd , the dynamics of the control error x̃ = x − x d satisfy ẋ̃ = (A − bk T )x̃ + gν,

t > 0,

σ = h x,̃

t≥0

(4.18b)

ỹ = c x,̃

t≥0

(4.18c)

T

T

ν = −H(σ),

̃ x(0) = x̃ 0

t ≥ 0.

(4.18a)

(4.18d)

Given that the pair (A, b) is completely controllable, k can always be chosen such that Σ(A − bk, g, hT ) is (−Rc , ScT , −Qc )-strictly state dissipative with a dissipation rate κc > 0, and thus, in virtue of Theorem 3.3.1 x̃ converges exponentially to zero. This result is somewhat intuitively clear, given that the cubic term in H is stabilizing and only the linear one is destabilizing. Thus, by choosing the control gain k2 (related to x2 ) large enough, one can compensate the destabilizing forces and introduce additional damping into the system. The design can be carried out by finding a matrix Pc = PcT > 0 and a control gain k T so that for a constant κc > 0 it holds that [

Pc (A − bk T ) + (AT − kbT )Pc + κc I + Rc hhT g T Pc − hT Sc

Pc g − hSc ] ≤ 0. Qc

(4.19)

98 | 4 Finite-dimensional systems In order to implement the state-feedback controller (4.17) the following observer is set up ẋ̂ = Ax̂ + bδ + g ν̂ − l(cT x̂ − y), t > 0,

σ̂ = hT x,̂

̂ x(0) = x̂ 0

t≥0

̂ ν̂ = −H(σ),

(4.20a) (4.20b)

t≥0

(4.20c)

with the correction gain vector l. The associated observation error dynamics are given by ė = (A − lcT )e + g ν,̃

t > 0,

ζ = h e,

t≥0

T

ν̂ = −φ(ζ ; σ),

e(0) = e0

(4.21a) (4.21b)

t≥0

(4.21c)

with φ(ζ ; σ) = H(σ + ζ ) − H(σ).

(4.22)

According to the mean value theorem it holds that ̄ = (b1 + 3b3 (σ)̄ 2 )ζ , φ(ζ ; σ) = 𝜕σ H(σ)ζ

σ̄ = σ + ϵζ ,

ϵ ∈ (0, 1).

It follows that over a specific interval [ζ − , ζ + ] it holds that s−φ ζ 2 ≤ ζφ(ζ ; σ) ≤ s+φ ζ 2 with s−φ = min (b1 + 3b3 (σ)̄ 2 ) = b1 , − +

(4.23)

σ∈[ζ ,ζ ]

2

2

s+φ = max (b1 + 3b3 (σ)̄ 2 ) = b1 + 3 max{(s− ) , (s+ ) }. − + σ∈[ζ ,ζ ]

(4.24)

According to Lemma 2.2.7 the map φ(ζ ; σ) is uniformly (Qo , So , Ro )-dissipative over the interval [ζ − , ζ + ] with Qo = −1,

1 So = (s−φ + s+φ ), 2

Ro = −s−φ s+φ .

Given that the pair (A, cT ) is completely observable, the gain l can always be chosen so that the system Σ(A − lcT , g, hT ) is (−Ro , SoT , −Qo )-strictly state dissipative with a dissipation rate κo > 0. This is ensured if the gain l ∈ ℝ2 is chosen so that a matrix Po = PoT > 0 and the constant κo > 0 exist for which the dissipation inequality [ is satisfied.

Po (A − lcT ) + (AT − lcT )Po + κo I + Ro hhT g T Po − hT So

Po g − hSo ]≤0 Qo

(4.25)

4.2 Ship course control | 99

Summarizing, both, the state-feedback control gain and the observer correction gain can be chosen such that both the control error and the observation error exponentially converge to zero. Given the nonlinearity of the system dynamics (4.13), it is nevertheless not immediately clear that the output-feedback controller, i. e. δ = −k T (x̂ − x d )

(4.26)

will also exponentially stabilize the zero solution x̃ = 0. This conclusion in turn can be drawn taking into account Theorem 3.4.1 provided that both dissipativity inequalities (4.19) and (4.25) are satisfied. Simulation studies for the ship model (4.13) have been carried out using the parameter set K = 0.5,

T = 31,

b1 = −1,

b3 = 0.4

which correspond to the parameters for the ship ROV Zeefakkel with a velocity of 5 ms as provided in van Amerongen (1982) and setting b1 = −1 to introduce course-instability. ∘ For these parameters, the dissipation inequality (4.19) is satisfied for x2+ = 2 s choosing κc = 0.02,

0.9481 Pc = [ 4.8818

4.8818 ], 100

k T = [0.3162

10.8411] .

(4.27)

The results for the state-feedback controller (4.17) with a constant set-point ψd = 50∘ for the heading angle and initial deviation x 0 = [−10, 0]T are shown in Figure 4.5. It can be seen, that the rudder angle δ (bottom) quickly returns to small values after a short initial correction within reasonable limits is performed,2 while the heading angle ψ (top) converges without overshoot to the desired value in about 150 s. During ∘ the transient the rate of change x2 = ψ̇ (middle) remains below x2+ = 2 s . The dissipation inequality (4.25) for the observer is satisfied by choosing κo = 0.2027,

Pc = [

38.9679 −8.0905

−8.0905 ], 41.6327

4.0244 l=[ ]. 3.0978

(4.28)

The observer-based output-feedback has been tested in simulations for a large initial deviation in the initial heading angle according to −10 x0 = [ ], 10

0 x̂ 0 = [ ] . 0

With the gains chosen as in (4.27) and (4.28) exponential convergence of the closedloop is ensured by Theorem 3.4.1. 2 The maximum rudder angle is about 30°.

100 | 4 Finite-dimensional systems

Figure 4.5: Simulation results for the closed-loop behavior of the ship model (4.13) with the statefeedback control (4.17) for a constant set point ψd = 50∘ : heading angle ψ (top) and ψ̇ (middle) and rudder angle δ (bottom).

The simulation result for the closed-loop system is presented in Figure 4.6 showing that the observer converges in about 5 s, and that the controller stabilizes the system at the desired heading angle ψd = 0 with a reasonable control effort. To further test the behavior of the closed-loop system, a series of set-point changes has been simulated with the following initial conditions and set-points:

−3 x0 = [ ] , 1

0 x̂ 0 = [ ] , 0

0, t ≤ 60 s { { { ψd = {−20, 60 s < t ≤ 300 s { { 300 s < t. {20,

The associated results are shown in Figure 4.7. It can be seen that (i) the observer converges within about 5 s, (ii) the initial offset is quickly compensated and the initial set-point ψd = 0 reached after about 1 min, and (iii) the subsequent set-points are reached after short transients of about 150 s with reasonably small rudder deflections.

4.3 Continuous stirred-tank reactor with exothermic reaction | 101

Figure 4.6: Simulation results for the closed-loop behavior of the ship model (4.13) with the observer-based output-feedback (4.26) with (4.20): heading angle ψ (top) and ψ̇ (middle) and rudder angle δ (bottom). The observed state is represented by the discontinuous line.

4.3 Continuous stirred-tank reactor with exothermic reaction Consider the continuous-stirred tank reactor (CSTR) with exothermic monotonic reaction rate ċ = d(ce − c) − ke−γ/T r(c),

t > 0,

c(0) = c0

(4.29a)

Ṫ = d(Te − T) − η(T − Tc ) + βke−γ/T r(c), t > 0,

T(0) = T0

(4.29b)

y = T, r(c) =

c k0 + c

t≥0

(4.29c) (4.29d)

with concentration c, temperature T, dilution rate d (i. e. the quotient of volumetric flow rate q over the reactor volume V), feed concentration ce and temperature Te , maximum reaction rate k, Arrhenius coefficient γ, heat transfer coefficient η between the reactor wall and the cooling jacket with temperature Tc (which acts in the following as the control input u), adiabatic temperature rise β, reaction rate function r(c) with

102 | 4 Finite-dimensional systems

Figure 4.7: Simulation results for the closed-loop behavior with the state (upper two plots, continuous line) and estimate (upper two plots, discontinuous line) and rudder angle (bottom plot).

half-saturation constant k0 , and temperature measurement y. It can be quickly verified that the interval [0, ce ] is a positively invariant set for the concentration: for c = 0 it follows that ċ > 0 implying that c increases and for c = ce it results that ċ < 0 implying that c decreases. A similar reactor model has been used in Schaum et al. (2008c, 2007, 2009), Schaum (2009), where a passivity-based control design has been carried out in combination with a dissipativity-based observer for a non-monotonic reaction rate. In contrast to these studies, here (i) a monotonic reaction rate is considered, and (ii) a dissipativity-based constant gain feedback control is used in combination with a dissipative observer. In the sequel it will be advantageous to write the dynamics (4.29) in a different coordinate system, namely using the enthalpy measure (Aris, 1969; Dochain, 2000, 2001; Schaum et al., 2008c) ζ = T + βc

(4.30)

with linear dynamics ζ ̇ = d(ζe − ζ ) − η(ζ − βc − Tc ),

t > 0,

ζ (0) = ζ0 ,

ζe = Te + βce

(4.31)

4.3 Continuous stirred-tank reactor with exothermic reaction | 103

which does no more depend on the nonlinear reaction rate. In these coordinates the complete reactor dynamics are given by ċ = d(ce − c) − ke−γ/(ζ −βc) r(c), ζ ̇ = dζ + ηT + ηβc − (d + η)ζ , e

t > 0,

t > 0,

c

y = T = ζ − βc,

t ≥ 0.

c(0) = c0

(4.32a)

ζ (0) = ζ0

(4.32b) (4.32c)

As can be easily seen, these dynamics can be written in the form of a Lur’e linear dynamic and nonlinear static system interconnection in terms of the state vector x = [c ζ ]T as follows ẋ = Ax + dx e + bu + g(y)ν, t > 0, T

σ = h x,

x(0) = x 0

(4.33a) (4.33b)

ν = −r(σ),

t≥0

t≥0

(4.33c)

y = c x,

t ≥ 0,

(4.33d)

T

with the state x(t) ∈ ℝ2 at time t ≥ 0, A ∈ ℝ2×2 , and b, g(y), h, c ∈ ℝ2 given by c x = [ ], ζ

c xe = [ e] , ζe

1 h = [ ], 0

c=[

−d A=[ ηβ

0 ], −(d + η)

0 b = [ ], η

g(y) = [

ke−γ/y ], 0

−β ]. 1

Setting the observer ẋ̂ = Ax̂ + dx e + bu + g(y)ν̂ − l(cT x̂ − y), t > 0, T

σ̂ = h x,̂

̂ ν̂ = −r(σ),

̂ x(0) = x̂ 0

(4.34a)

t≥0

(4.34b)

t≥0

(4.34c)

̃ x(0) = x̃ 0

(4.35a)

and the associated observation error x̃ = x̂ − x, the observation error dynamics read ẋ̃ = Al x̃ + g(y)ν,̃

t > 0,

σ̃ = h x,̃

t≥0

T

ν̃ = −φ(σ;̃ σ),

(4.35b)

t≥0

(4.35c)

with φ(σ;̃ σ) = r(σ + σ)̃ − r(σ),

φ(0; σ) = 0

(4.36)

104 | 4 Finite-dimensional systems and −(d + βl1 )

−l1

β(η + l2 )

−(d + η + l2 )

Al = A − lcT = [

].

Note that even though the concentration c is restricted to the interval [0, ce ] as discussed above, the estimated concentration ĉ could become negative in principle. Nevertheless, with adequate tuning the trajectories should not enter this critical region. Accordingly, for the purpose at hand focus on observer trajectories within the physically meaningful interval [0, ce ]. Extensions of the proposed observer scheme for which the positive invariance of the interval [0, ce ] can be explicitly ensured go beyond the scope of the present illustration purpose. In virtue of the mean-value theorem it holds that φ(σ;̃ σ) = 𝜕c r(σ + ρσ)̃ σ,̃

ρ ∈ (0, 1)

and consequently by introducing s−r = inf 𝜕c r(σ) = 𝜕c r(ce ) = σ∈[0,ce ]

k0 , (k0 + ce )2

s+r = sup 𝜕c r(σ) = 𝜕c r(0) = σ∈[0,ce ]

1 k0

it follows that φ is contained in the sector [s−r , s+r ] and thus it is (Q, S, R)-dissipative uniformly in σ with Qo = −1,

1 So = (s−r + s+r ), 2

Ro = −s−r s+r .

(4.37)

It follows directly from Lemma 2.2.11 and Corollary 2.2.1 that if the linear system Σ(A, g(y), hT ) is (−Ro , So , −Qo )-strictly state dissipative with dissipation rate κ > 0 uniformly in y and with a quadratic storage function, then the observation error converges exponentially to zero. This in turn is achieved if there exists a matrix P = PT > 0 such that for all y ∈ [T − , T + ] the matrix inequality Po Al + ATl Po + κI + (s−r s+r )hhT

[ [

g T (y)Po −

s−r +s+r T h 2

Po g(y) − h −1

s−r +s+r 2 ]

≤0

(4.38)

]

holds true, given that in this case a quadratic storage function is given by 𝒮 (x)̃ = x̃ T Px̃ > 0. The controller is considered by a simple linear feedback of the form Tc = −k T x

4.3 Continuous stirred-tank reactor with exothermic reaction | 105

leading to the closed-loop dynamics ẋ = (A − bk T )x + be x e + g(y)ν, t > 0, T

σ = h x,

x(0) = x 0

t≥0

ν = −r(σ),

(4.39a) (4.39b)

t ≥ 0.

(4.39c)

Due to the monotonicity feature of the reaction rate r(c), the nonlinearity r(σ) (over the compact positively-invariant interval [0, ce ]) is contained in the sector [r(ce ), 1/k0 ] and thus is (Qc , Sc , Rc ) dissipative with Qc = −1,

1 1 Sc = (r(ce ) + ) 2 k0

Rc = −

r(ce ) . k0

By the preceding analysis it is clear that the closed-loop system (4.39) is exponentially stable if the linear subsystem Σ(AK , b, hT ) is (−Rc , Sc , −Qc ) strictly state dissipative with positive dissipation rate κc > 0 uniformly in y, where AK = A − bk T . This in turn is fulfilled if the matrix inequality Pc Ak + ATk Pc + κI +

[ [

g T (y)Pc −

r(ce ) hhT k0

Pc g(y) − h

k0 r(ce )+1 T h 2k0

k0 r(ce )+1 2k0

−1

]≤0

(4.40)

]

is satisfied for all y ∈ [T − , T + ]. The closed-loop exponential stability of the reactor with observer-based outputfeedback control can be concluded from Theorem 3.4.1 if both matrix inequalities (4.38) and (4.40) are satisfied simultaneously. The observer-based output-feedback control for a dimensionless reactor model has been evaluated in simulation studies with the parameters d = 1,

ce = 1,

k = exp(25),

γ = 104 ,

Te = Tc = 400,

η = 1,

β = 50.

The control is designed in order to improve the stability of the reactor operation point c̄ 0.61 ] x̄ = [ ̄ ] = [ T 409.8 and thus the velocity with which perturbations are rejected. For this purpose the control and observer gains are adjusted such that (i) the dissipation inequalities (4.38) and (4.40) are satisfied and (ii) a reasonable control effort is ensured. A reasonably well-performed behavior is achieved for k T = [−0.1520

1.9848] ,

lT = [

−0.0567 ]. 0.8633

106 | 4 Finite-dimensional systems In Figure 4.8 the open and closed-loop responses of the reactor can be seen for an initial condition x 0 = [0.1, 380]T representing a strong initial state perturbation and thus a rather extreme case for the testing of the closed-loop behavior. It can be seen in Figure 4.8 that the open-loop trajectories (discontinuous thin lines) converge to the operation point in about 4 time units. For the closed-loop operation, the observer recovers the reactor state in about 2 time units (in the measured temperature component already after about 1 time unit) and the controller steers the reactor trajectory to the operation point x̄ in about 2 time units, i. e. about two times faster than in open-loop operation.

Figure 4.8: Simulation results for the open-loop (dotted lines) and closed-loop reactor (continuous lines) and the observer states (discontinuous lines) with concentration c (top), temperature T (center) and cooling temperature Tc (bottom).

4.4 Nonlinear Luenberger observer for bioreactor monitoring To illustrate the observer design using state-dependent gains discussed in Section 3.1.3, in this section a two-state bioreactor model is considered. Let x1 denote

4.4 Nonlinear Luenberger observer for bioreactor monitoring

| 107

the biomass and x2 the substrate, both normalized to the substrate feed concentration, with biomass growth described by a Monod kinetics μ(x2 ) ẋ1 = −dx1 + μ(x2 )x1 ,

ẋ2 = d(1 − x2 ) − γμ(x2 )x1 , y = x1 ,

t > 0,

x1 (0) = x10

t > 0,

x2 (0) = x20

t ≥ 0.

(4.41a) (4.41b) (4.41c)

Here d is the dilution rate, i. e. the quotient of feed flow rate and reactor volume, and γ is the inverse yield coefficient, i. e. the substrate to biomass conversion factor. It is considered that the biomass is measured on-line. A nonlinear Luenberger observer with state dependent gains is set according to ẋ̂1 = −dx̂1 + μ(x̂2 ) − l1 (x,̂ d)(x̂1 − y), ẋ̂2 = d(1 − x̂2 ) − γμ(x̂2 ) − l2 (x,̂ d)(x̂1 − y),

t > 0,

x̂1 (0) = x̂10

(4.42a)

t > 0,

x̂2 (0) = x̂20

(4.42b)

The pseudo-linear dynamics matrix of the error dynamics (3.24) is given by J(x,̂ d) =

−d + μ(x̂2 ) 𝜕f (x,̂ d) = [ −γμ(x̂2 ) 𝜕x

μ󸀠 (x̂2 )x̂1 ] −d − γμ󸀠 (x̂2 )x̂1

The linear Kalman observability map evaluated at the pair (x,̂ d) is given by 1 −d + μ(x̂2 )

𝒦0 (x,̂ d) = [

0 ] μ (x̂2 )x̂1 󸀠

(4.43)

which has full rank for all x̂1 , x̂2 ≠ 0. Note that x1 = 0 corresponds to the pathologic washout case, in which no more biomass is contained in the reactor and thus no biomass will be produced any more, given that in the feed no biomass is contained. As y = x1 this situation directly implies from a biological point of view that the observability is lost (see also the discussion in Schaum et al., 2005; Schaum and Moreno, 2007). Thus, the observer design can be developed for the case that3 x̂1 > 0. The transformation which brings the matrix J into the form (see Equation (B.22) in Appendix B.3) is given by T = [ŵ

̂ , J w]

0 ŵ = 𝒦o−1 (x,̂ d) [ ] 1

3 Actually it should be carefully evaluated if x̂1 > 0 holds for the situations which will be treated in praxis, given that, even though in the actual reactor model the biomass will nether become negative in the observer this may happen due to the injection of the measurement. To preclude negative estimates is an interesting subject, but it will not be treated in this illustrative example.

108 | 4 Finite-dimensional systems which evaluates to T(x,̂ d) = [

0

1

d+γμ󸀠 (x̂ )x̂ ] . − μ󸀠 (x̂ )x2 ̂ 1 2 1

1 μ󸀠 (x̂2 )x̂1

Accordingly the transformed pseudo-linear dynamics matrix is given by −γdμ󸀠 (x̂2 )x̂1 + dμ(x̂2 ) − d2 ] −γμ󸀠 (x̂2 )x̂1 + μ(x̂2 ) − 2d

0 J(̃ x,̂ d) = T −1 (x,̂ d)J(x,̂ d)T(x,̂ d) = [ 1

and with measurement injection the pseudo-linear error dynamics matrix reads 0 J(̃ x,̂ d) − l(̃ x,̂ d)C̃ = [ 1

−γdμ󸀠 (x̂2 )x̂1 + dμ(x̂2 ) − d2 − l1̃ (x,̂ d) ] −γμ󸀠 (x̂2 )x̂1 + μ(x̂2 ) − 2d − l2̃ (x,̂ d)

where l(̃ x,̂ d) = T −1 (x,̂ d)l(x,̂ d),

C(̃ x,̂ d) = CT(x,̂ d)

is the transformed injection gain and measurement vector, respectively. Accordingly, choosing γdμ󸀠 (x̂2 )x̂1 − d μ(x̂2 ) + d2 + p0 p1 ] l(̃ x,̂ d) = [ 󸀠 γμ (x̂2 )x̂1 − μ(x̂2 ) + 2d + (p0 + p1 ) the pseudo-linear part of the estimation error dynamics is governed by the constant dynamics matrix 0 Ā = [J(̃ x,̂ d) − l(̃ x,̂ d)] = [ 1

−p0 p1 ] −(p0 + p1 )

with the characteristic polynomial λ2 + (p0 + p1 )λ + p0 p1 = 0 and solutions λ1 = −p0 ,

λ2 = −p1 .

Thus, the desired eigenvalues can be directly assigned. The decision on how the gains should be chosen can be performed using an eigenvalue assignment in the light of Lemma 2.2.4 once the dissipativity condition satisfied by the nonlinear function ϕ∗ (x,̃ x,̂ d) = f (x̂ − x,̃ d) − f (x,̂ d) − J(x,̂ d)x̃ is established. Denote in the sequel φ(x;̃ x,̂ d) = f (x̂ − x,̃ d) − f (x,̂ d).

4.4 Nonlinear Luenberger observer for bioreactor monitoring

| 109

Note that by assumption f is Lipschitz continuous. Let Lfx denote the associated Lipschitz constant. Following the discussion in Section 2.2.5 it follows that φ is (Q, S, R)-dissipative with Q = −I,

S = 0,

2

R = (Lfx ) I.

The dissipativity condition of ϕ∗ can now be determined e. g. using Lemma 2.2.10 noting that ϕ∗ (x,̃ x,̂ d) = φ(x,̃ x,̂ d) − J(x,̂ d)x.̃ Following Lemma 2.2.10 it results that ϕ∗ is (Qe , Se , Re )-dissipative with Qe = Q,

̂ Se = S − QJ(x),

T

̂ ̂ − (S − QJ(x)) ̂ J(x)̂ − J T (x)QJ( ̂ ̂ R󸀠 = R − J T (x)(S − QJ(x)) x).

Accordingly, gains which ensure the observer convergence in virtue of Lemma 2.2.11 and Corollary 2.2.1 can always be found as from the above analysis it follows that the eigenvalues of the matrix (J(x,̂ d) − l(x,̂ d)C) and thus the associated dissipation property of the linear dynamic subsystem of the observation error dynamics in the transformed state ξ = T −1 (x,̂ d)x̃ ̄ + ν, ξ ̇ = Aξ

t > 0,

σ = ξ,

∗ ν = −ϕ̄ (σ, x,̂ d),

t≥0

ξ (0) = ξ 0

∗ ϕ̄ (σ) = T −1 ϕ∗ (T −1 σ), t ≥ 0,

with the above choice of l(̃ x,̂ d) can be assigned arbitrarily. Simulation studies have been performed for the parameters γ = 1.5,

k0 = 0.5,

ks = 0.2,

d = 0.2.

The observer gain has been adjusted along the observer trajectory so that observation error linearization has the pair of conjugate complex eigenvalues λ = −0.6 ± 0.125 i. For comparison purposes an asymptotic observer (Dochain et al., 1992; Dochain, 2003; Bastin and Dochain, 1990) has been implemented and tested against the proposed observation scheme. For this purpose the reaction invariant m = γb + s,

ṁ = d(se − m),

m(0) = m0

is simulated with the guess for the initial condition given by ̂ m0 = γy(0) + s(0) and the substrate estimate calculated by ̂ = m(t) − γy(t). s(t)

110 | 4 Finite-dimensional systems The obtained results are illustrated in Figure 4.9 for the initial conditions x 0 = [0.4, 0.6]T and x̂ 0 = [0.7, 0.3]T , with the reactor state (continuous line) and the observed state with the proposed observer scheme (discontinuous line), as well as the substrate estimate on the basis of the asymptotic observer (dotted line). It can be seen that with the proposed scheme (discontinuous line) the observer convergence can be significantly improved. This is by no means surprising giving that the proposed observer is based on the assumption that the growth rate is sufficiently well known to implement a model-based observation scheme. In practice the growth rate is often not well known and thus the asymptotic observer can be preferred, depending on the particular needs, if a slow convergence rate in the substrate estimate is sufficient.

Figure 4.9: Simulation results for the bioreactor example with the extended Luenberger observer (dashed lines), the actual reactor trajectory (continuous lines) and the substrate estimate of an asymptotic observer (dotted line) with biomass b (top) and substrate s (bottom).

4.5 Inverted pendulum with applied torque Consider in the following a simple inverted pendulum with applied torque M. Based on simple energy conservation arguments the following model is obtained l2 l (J + m )ϕ̈ + cp ϕ̇ + mg sin(ϕ) = M, 4 2

t>0

(4.44)

4.5 Inverted pendulum with applied torque | 111

̇ with initial condition ϕ(0) = ϕ0 , ϕ(0) = θ0 . Here, J is the moment of inertia, m the mass of the pendulum, l the distance from the articulation to the center of mass, ϕ the angle, cp the coefficient of Coulomb friction, and g the gravitation constant. Introducing the coefficients δ=

cp J+

2 m l4

γ=

,

mg 2l

J+

2 m l4

u=

,

M

2

J + m l4

,

the variables x1 = ϕ,

x2 = ϕ,̇

x10 = ϕ0 ,

x20 = θ0 ,

x1 x=[ ] x2

and the measurement y = x1 = ϕ, the pendulum dynamics can be written in the equivalent form 0

1

0 0 ] x + [ ] sin(x1 ) + [ ] u, −δ γ 1

ẋ = [

0

y = [1

x(0) = x 0

0] x

(4.45a) (4.45b)

or in compact notation ẋ = Ax + gν + bu, t > 0,

x(0) = x 0

(4.46)

σ = hT x,

t≥0

(4.47)

ν = −φ(σ),

t≥0

(4.48)

T

t ≥ 0,

(4.49)

y = c x,

with the state x(t) ∈ ℝ2 at time t ≥ 0, A ∈ ℝ2×2 , b, g, h, c ∈ ℝ2 and φ : ℝ → ℝ given by 0 A=[ 0

1 −δ

],

0 g = [ ], γ

1 h = [ ], 0

0 b = [ ], 1

φ(σ) = − sin(σ),

cT = [1

0] .

The observer is set up as ẋ̂ = Ax̂ + g ν̂ + bu − l(cT x̂ − y),

t > 0,

σ̂ = hT x,̂

t≥0

(4.50b)

̂ ν̂ = −φ(σ),

t≥0

(4.50c)

with correction gain vector l = [l1 dynamics are given by ė = (A − lcT ) x̂ + g ν̃ = [ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ :=Al

̂ x(0) = x̂ 0

(4.50a)

T

l2 ] . With the observation error e = x̂ − x, the error −l1 −l2

1 ] e + g ν,̃ t > 0, −δ

̃ x(0) = x̃ 0

(4.51a)

112 | 4 Finite-dimensional systems σ̃ = hT x,̃

t≥0

ν̃ = −ψ(σ;̃ σ) = −(φ(σ + σ)̃ − φ(σ)),

(4.51b)

t≥0

(4.51c)

On the basis of the mean value theorem for any σ it holds that ψ(σ;̃ σ) = 𝜕σ φ(σ + ησ)̃ σ,̃

η ∈ (0, 1).

With φ(σ) = − sin(σ) one finds that ψ(σ;̃ σ) = − cos(σ + ησ)̃ σ,̃

η ∈ (0, 1).

Note that cos(ξ ) ∈ [−1, 1], ξ ∈ ℝ, so that one can set k1 = 1, k2 = −1 in equation (2.45) implying that ψ is (Q, S, R)-dissipative uniformly in σ with Q = −1,

S = 0,

R = 1.

It follows from Theorem 3.1.1 that for exponential convergence of the observer with rate θ > 0 it is sufficient that the linear system Σ(Al , g, hT ) is (−R, S, −Q) = (−1, 0, 1)-strictly state dissipative with sufficiently large dissipation rate κ > 0. In virtue of Lemma 2.2.3 this is ensured if there exists a matrix P = P T > 0 such that MO := [

PAl + ATl P + κI gT P

Pg −hRhT ]−[ 0 ShT

hST ] ≤ 0, −Q

Al = A − lcT .

In difference to the preceding sections, here the search for a feasible choice for the triplet (P, l, κ) is carried out using numerical optimization tools, solving the corresponding optimization problem min ( max λ +

(P,l,κ) λ∈σ(MO )

1 + ωlT l) κ

subject to: P = PT > 0

κ>0

where ω ≥ 0 is used to constraint the size of the observer gains for noise attenuation. For the purpose of controlling the inverted pendulum the applied torque u (and thus M) can be assigned so that the nonlinearity is exactly compensated (cp. Section 3.2.3 on feedback linearization). This is achieved for u = −γ sin(σ) − k T x, or equivalently in the present notation bu = −gν − bk T x.

(4.52)

4.5 Inverted pendulum with applied torque | 113

Substituting this into (4.46) leads to the closed-loop dynamics with state-feedback control 0 ẋ = (A − bk T ) x = [ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ −k1

1 ] x, −(δ + k2 )

:=Ak

t > 0,

x(0) = x 0 .

Clearly, k can be chosen by eigenvalue assignment or any other technique leading to a desired closed-loop behavior. Implementing the controller (4.52) with the observed state value x,̂ i. e. u = −γ sin(σ)̂ − k T x̂ yields the corresponding exactly linearized closed-loop observer dynamics 1 ] x,̂ −(δ + k2 )

−l1 ẋ̂ = (Al − bk T )x̂ = [ −l2 − k1

t > 0,

̂ x(0) = x̂ 0 .

In terms of the observation error x̃ = x̂ − x introduced above, the coupled system and observation error closed-loop dynamics can be written as ẋ = Ak x − bk T x̃ − g ν,̃

t > 0,

σ = hT x, ẋ̃ = Al x̃ + g ν,̃

t≥0

σ̃ = h x,̃

t≥0

t > 0,

T

ν̃ = −ψ(σ;̃ σ) = −𝜕σ φ(σ + ησ)̃ σ,̃

x(0) = x 0

(4.53a) (4.53b)

̃ x(0) = x̃ 0

η ∈ (0, 1), t ≥ 0.

(4.53c) (4.53d) (4.53e)

These dynamics can again be written in the Lur’e form as follows ẋ e = Ae x e + g e ν,̃ σ̃ =

hTe x e ,

t > 0,

x e (0) = x e0

t≥0

ν̃ = −ψ(σ;̃ σ),

(4.54a) (4.54b)

t≥0

(4.54c)

with x xe = [ ] , x̃ hTe = [0T

−bk T ], Al

A Ae = [ k 0 hT ] = [0

0

1

0] .

0 [−γ ] −g [ ] ge = [ ] = [ ] , [0] g [γ]

Again, by Lemma 2.2.11 and Corollary 2.2.1, as ψ(σ;̃ σ) is (Q, S, R)-dissipative with (Q, S, R) = (−1, 0, 1) uniformly with respect to σ, if Σ(Ae , g e , he ) in (4.54) is (−R, ST , −Q)-, i. e. (−1, 0, 1)-strictly state dissipative with dissipation rate κ > 0 and a quadratically

114 | 4 Finite-dimensional systems bounded storage function, then the joint controller and observation error x e will exponentially converge to zero. This property holds true if there exists a matrix Pe = PeT > 0, a constant κ > 0 and gains l1 , l2 , k1 , k2 such that the following matrix inequality is satisfied: P A + ATe Pe + κI + he hTe MCL := [ e e g Te Pe

Pe eg e ] ≤ 0. Q

A solution quadruplet (P, l, k, κ) for this inequality can again be obtained e. g. using optimization tools solving the minimization problem min ( max λ + ω0

(P,l,k,κ) λ∈σ(MCL )

1 + ω1 (lT l + k T k)) κ

subject to:

(4.55a) (4.55b)

T

(4.55c)

κ > κ0

(4.55d)

P=P >0

where ω0 , ω1 > 0 and ω1 ≥ 0 can be used to reduce the size of the gains l and k. Note, that the size of the solution set can be reduced using particular forms for the matrix P = P T like e. g. a block-diagonal form. Simulation studies have been performed to test the above observer-controller structure for the parameter set cp = 0.006 kg m2 /s,

J = 0.17 kg m2 ,

m = 0.5 kg,

l = 0.45 m,

g = 9.81 m/s2

initial conditions (angles are given in radians) x 0 = [−π

T

0.2] ,

x̂ 0 = [0.1

T

2] .

The observer and controller gains have been chosen on the basis of the optimization problem (4.55) with ω0 = 10−3 , ω1 = 10−4 , κ0 = 0.05 and P set as a block-diagonal matrix, solved in Matlab using the function fmincon. The resulting gain vectors are given by l = [5.51 21.45]T , k T = [9.63 9.55]. Figure 4.10 shows the convergence in spite of a large initial error. In the left column the simulation results for the case that there is no measurement noise are presented and in the right column the results for the same scenario but with high-frequency measurement noise of 0.5 radians amplitude. These results show that the dissipative feedback control achieves a good performance even in presence of measurement noise. The good noise compensation effect is due to the assignment of the observer gains on the basis of a minimization criterion, similar to the approach in classical Kalman Filtering, with the advantage of having an explicit proof of exponential convergence.

4.6 Nonlinear state-feedback control of a MIMO system

| 115

Figure 4.10: Simulation results for the closed-loop operation of the pendulum (4.44) for the (dissipative) feedback-linearizing control (4.52) with dissipative observer (4.50) without measurement noise (left) and with measurement noise (right). Upper plots: state (continuous line) and observed state (discontinuous line). Bottom: control inputs.

4.6 Nonlinear state-feedback control of a MIMO system In this section the stabilization by state-feedback of a nonlinear MIMO system is considered in order to illustrate the design approach for a nonlinear dynamic subsystem coupled with a nonlinear static map, illustrating the general considerations in Section 3.2.3. The control system is given by −(1 + x12 )x2 0 0 [ ] [ ] [ ] ẋ = [x1 + sin(x2 )] + [1 + x32 ] u1 + [0] u2 , 3 [ x2 − x3 ] [ 0 ] [1]

t > 0,

y=[

t ≥ 0.

x1 ], (1 + x22 )x3

x(0) = x 0

(4.56a) (4.56b)

The open-loop behavior is presented in Figure 4.11, showing that the trajectories converge to a relaxation oscillation in the three-dimensional state space. This system can be written in the form of a negative feedback interconnection of a nonlinear dynamical subsystem and a nonlinear static subsystem −(1 + x12 )x2 0 0 0 [ ] [ ] [ ] [ ] ẋ = [ x1 ] + [1 + x32 ] u1 + [0] u2 + [ 1 ] ν, t > 0, 3 [ x2 − x3 ] [ 0 ] [1] [0]

x(0) = x 0

(4.57a)

116 | 4 Finite-dimensional systems

Figure 4.11: Simulation results for the open-loop dynamics (4.56), showing the convergence to a three-dimensional relaxation oscillation.

σ = [0 y=[

0] x,

t≥0

(4.57b)

x1 ], (1 + x22 )x3

t≥0

(4.57c)

t≥0

(4.57d)

1

ν = −φ(σ), with φ(σ) = − sin(σ). In compact form these dynamics are written as ẋ = f (x) + b1 (x)u1 + b2 u2 + gν,

t > 0,

σ = h x,

t≥0

T

ν = −φ(σ),

x(0) = x 0

t≥0

(4.58a) (4.58b) (4.58c)

with the state x(t) ∈ ℝ3 at time t ≥ 0, f : ℝ3 → ℝ3 , b1 : ℝ3 → ℝ3 , b2 , g, h ∈ ℝ3 , ui (t) ∈ ℝ, i = 1, 2, and φ : ℝ → ℝ. As follows from example (2.42) in Section 2.2.5, the nonlinear function φ(σ) is contained in the sector [−1, 0.22] so that the static subsystem is (Q, S, R)-dissipative with Q = −1,

S=

1 − 0.22 = 0.39, 2

R = −(−1) 0.22 = 0.22.

In accordance with Corollary 2.2.1 the origin of (4.56) is exponentially stable with rate κ > 0 if the control input u = [u1 , u2 ]T is chosen such that the (still nonlinear) subsystem (4.58a)–(4.58b) is (−R, S, −Q) = (−0.22, 0.39, 1)− strictly state dissipative with dissipation rate γ = 2κ > 0 and a positive definite storage function S(x). For this purpose note that the vector relative degree of the system is given by r = {2, 1}

(4.59)

4.6 Nonlinear state-feedback control of a MIMO system

| 117

so that by the diffeomorphic transformation h1 (x) x1 ] [ ] [ z = Φ(x) = [Lf h1 (x)] = [−(1 + x12 )x2 ] , 2 [ h2 (x) ] [ (1 + x2 )x3 ]

Lf h1 = 𝜕x h1 (x)f (x)

(4.60)

the associated dynamics in z-coordinates reads 0 0 ̄ ] + B[ ] u + g(z)ν, ż = Az + [ a(z) B(z)

t > 0,

z(0) = z 0

(4.61a)

with 0 [ A = [0 [0

1 0 0

0 ] 0] , 0]

a(z) = [

a1 (z) ], a2 (z)

a1 (z) = L2f h1 (x)|x=Φ−1 (z) ,

a2 (z) = Lf h2 (x)|x=Φ−1 (z) and 0 [ B = [1 [0

0 ] 0] , 1]

L L h (x) B(z) = [ b1 f 1 Lb1 Lf h2 (x)

Lb2 Lf h1 (x) ] , Lb2 Lf h2 (x) x=Φ−1 (z)

̄ g(z) = Lg Φ(x)|x=Φ−1 (z) .

By the relative degree property (4.59) the matrix B(z) is invertible for all z ∈ Φ(𝕏), so that the control a1 (z) ] − Kz) = ϖ(z) a2 (z)

(4.62)

k2 0

(4.63)

u = B−1 (x)|x=Φ−1 (z) (− [ with the particular4 control gain matrix k K=[ 1 0

0 ] k3

achieves the closed-loop dynamics 0 [ ̄ ż = (A − BK)z + g(z)ν = [−k1 [ 0

σ = hσ (z) = hT Φ−1 (z),

1 −k2 0

0 ] ̄ 0 ] z + g(z)ν, −k3 ]

t > 0, t ≥ 0.

z(0) = z 0 .

(4.64a) (4.64b)

4 The control gain matrix K has been chosen in this way to simplify the calculation of the closed-loop eigenvalues of the linear z-dynamics.

118 | 4 Finite-dimensional systems It should be noted that the input-output properties, and thus the dissipativity properties up to the state dissipation rate of the system in z-coordinates and the one in x-coordinates are the same. Thus, if one ensures by an appropriate choice of the matrix K that the system (4.64) is (−R, ST , −Q)-strictly state dissipative, the original one (4.58) with the feedback control u(t) = ϖ(Φ(x(t))) given in (4.62) will also be. To analyze the dissipativity property Lemma 2.2.6 can be used. For this purpose notice for the case at hand the variables in condition (2.39) are given by ‖R‖ = 0.22,

‖S‖ = 0.39,

q = 1,

󵄩 󵄩 Lh = max 󵄩󵄩󵄩𝜕z hσ (z)󵄩󵄩󵄩, D⊂Φ(𝕏)

ḡ = 1,

so that the closed-loop dynamical subsystem is (−R, S, −Q) = (−0.22, 0.39, 1)− strictly state dissipative with dissipation rate κ > 0 if 1 μ > (κ + 0.22L2h + (1 + 0.39Lh )2 ), 2

μ = − max R(λ). λ∈σ(A)

(4.65)

The characteristic equation for the matrix A − BK is given by (λ + k3 )(λ2 + k2 λ + k1 ) = 0 so that the eigenvalues are given by λ1,2 = −

k2 √ k22 ± − k1 , 2 4

λ3 = −k3 .

The closed-loop convergence with the state-feedback controller is shown in Figure 4.12 for the choice of k1 = 12,

k2 = 7,

k3 = 5,

yielding λ1 = −3,

λ2 = −4,

λ3 = −5

⇒

γ = 3.

Figure 4.12: Simulation results for the closed-loop dynamics (4.56) with the feedback controller (4.62).

4.6 Nonlinear state-feedback control of a MIMO system

| 119

It can be seen that the unstable dynamics is effectively compensated by the feedback linearizing controller while the stability introduced by the imposition of the linear exponentially stable closed-loop dynamics is robust against bounded perturbations of the nonlinearity φ(σ) = sin(σ).

5 Infinite-dimensional systems In this chapter generalizations of the preceding results are presented to systems described by semi-linear partial differential or integro-partial differential equations. First, recent results by the author on the design of spectral (Schaum et al., 2008b, 2016b, 2018) and point-wise (Schaum et al., 2015b, 2016a, 2017; Schaum, 2018) measurement injection observer schemes are presented. In particular the point-wise measurement injection approach is put in perspective with the dissipative observer design discussed in this text. Both approaches allow for measurements at the boundary or in the domain. Next, the design of a proportional output-feedback control for a semilinear heat equation with in-domain actuation is considered following and extending the results in Schaum and Meurer (2019b). All these studies involve in-domain measurement and control and provide explicit solutions for the sensor and actuator location (and shape design) problem. Additionally, first results on the design of dissipativity-based backstepping control and observer design are presented for a semi-linear unstable heat equation with Neuman boundary actuation and boundary measurement as well as a semi-linear partial integro-differential equation with Dirichlet boundary control following the approach in Schaum and Meurer (2019a, 2020).

5.1 Spectral measurement injection dissipative observer In the following a system described by coupled semi-linear parabolic pdes is considered in the form 𝜕t x = D𝜕z2 x − v𝜕z x + Kx + Gφ(σ), σ = Hx,

t > 0, t ≥ 0,

z ∈ (0, 1) z ∈ (0, 1)

(5.1a) (5.1b)

with boundary conditions B00 x(0, t) + B01 𝜕z x(0, t)= f 0 (t), B10 x(1, t) + B11 𝜕z x(1, t)= f 1 (t),

t≥0 t≥0

(5.1c) (5.1d)

and output y = Cx,

t≥0

(5.1e)

where t ≥ 0 denotes the time, z ∈ [0, 1] the space, and x(⋅, t) ∈ 𝕏 = (L2 (0, 1))n the state with initial condition x(⋅, 0) = x 0 . Accordingly, at each z ∈ [0, 1] and t ≥ 0 it holds that x(z, t) ∈ ℝn . Further, D = diag(di ) with the positive diffusion coefficients di > 0, i = 1, . . . , n, v ≥ 0 is the convection coefficient, K ∈ ℝn×n is a coupling matrix, G ∈ ℝn×p , φ ∈ 𝒞 1 (ℝq → ℝp ), and σ ∈ ℝq is a combination of states defined according https://doi.org/10.1515/9783110677942-006

5.1 Spectral measurement injection dissipative observer

| 121

to H ∈ ℝq×n in (5.1b). The exogenous boundary inputs f i ∈ 𝒞 ∞ ([0, ∞), ℝn ), i = 0, 1 are known functions of time, and the matrices Bij ∈ ℝn×n , i, j = 0, 1 are such that the boundary conditions are well defined and the existence of a unique solution is ensured for all x 0 ∈ (H 2 (0, 1))n (see e. g. Henry, 1981; Fridman and Orlov, 2009; Pazy, 1983; Schaum et al., 2013b). At each time t ≥ 0 the measurement vector is denoted by y(t) ∈ ℝm and the associated output operator C : 𝕏 → ℝm is defined by cT1 ⟨δζ1 , x(⋅, t)⟩ [ ] Cx(⋅, t) = [ ... ] T c ⟨δ , x(⋅, t)⟩ [ m ζm ]

(5.2)

with δζ being the Dirac δ-function centered at z = ζ and cTj ∈ ℝ1×n , j = 1, . . . , m. The observer is set up as 𝜕t x̂ = D𝜕z2 x̂ − v𝜕z x̂ + K x̂ + Gφ(σ)̂ − L(C x̂ − y), σ̂ = H x,̂

t > 0, t > 0,

z ∈ (0, 1) z ∈ (0, 1)

(5.3a) (5.3b)

with boundary conditions ̂ t)= f 0 (t), ̂ t) + B01 𝜕z x(0, B00 x(0, ̂ t) + B11 𝜕z x(1, ̂ t)= f 1 (t), B10 x(1,

t≥0

(5.3c)

t≥0

(5.3d)

̂ 0) = x̂ 0 . The associated estimation error x̃ = x̂ − x is governed and initial condition x(⋅, by the dynamics ̃ 𝜕t x̃ = D𝜕z2 x̃ − v𝜕z x̃ + K x̃ − LC x̃ − Gψ(σ), σ̃ = H x,̃

ψ(σ)̃ = −[φ(σ + σ)̃ − φ(σ)],

t > 0,

z ∈ (0, 1)

t > 0,

z ∈ (0, 1)

t > 0,

z ∈ (0, 1)

(5.4a) (5.4b) (5.4c)

with boundary conditions ̃ t) + B01 𝜕z x(0, ̃ t) = 0, B00 x(0,

̃ t) + B11 𝜕z x(1, ̃ t) = 0, B10 x(1,

t≥0

t≥0

(5.4d) (5.4e)

̃ 0) = x̃ 0 . The preceding dynamics can be rewritten as a Lur’eand initial condition x(⋅, type dynamic-static system interconnection ̄ ẋ̃ = AL x̃ + Gν, σ̃ = H̄ x,̃ ̃ ν = −F(σ),

t>0

t≥0 t≥0

(5.5a) (5.5b) (5.5c)

̃ ̃ with x(t) ∈ 𝕏 with initial condition x(0) = x̃ 0 , operators AL = A − LC, Ḡ : ℱ = 2 p 2 q (L (0, 1)) → 𝕏, H̄ : 𝕏 → ℋ = (L (0, 1)) , F : ℋ → ℱ defined by A = D𝜕z2 − v𝜕z + K,

̄ (Gν)(z)= Gν(z),

2

n

𝒟(A) = {x̃ ∈ (H (0, 1)) | (5.4d)–(5.4e) hold true} (5.5d)

∀ν ∈ ℱ , z ∈ (0, 1)

(5.5e)

122 | 5 Infinite-dimensional systems ̄ (Hx)(z)= Hx(z),

(F(σ))(z)= φ(σ(z)),

∀x ∈ 𝕏, z ∈ (0, 1)

(5.5f)

∀σ ∈ ℋ, z ∈ (0, 1).

(5.5g)

Based on the general results stated in Lemma 2.2.11 and Corollary 2.2.1 the observation error exponentially converges to zero if ψ is (Q, S, R)-dissipative and Σ(A, G, H) is (−R, S∗ , −Q)-strictly state dissipative with positive dissipation rate κ > 0. According to Lemma 2.2.3 this is satisfied if there exists a matrix-valued function W(z) = W T (z) > 0 so that the linear operator inequality (LOI) WAL + A∗L W + H ∗ RH + κI [ (G∗ W − SH)

(WG − H ∗ S∗ ) ]≤0 Q

(5.6)

is satisfied. Note that in (5.6) the adjoint operator A∗L is used, implying that the operator on the left-hand side can be applied only on functions which lie in the intersection1 of 𝒟(AL ) and 𝒟(A∗L ). On the other hand this inequality can be expressed in a different manner without using the adjoint operator as ⟨x, WAL x⟩ + ⟨AL x, Wx⟩ + ⟨x, H ∗ RHx⟩ + κ‖x‖2 + 2⟨x, (WG − H ∗ S∗ )ν⟩ + ⟨ν, Qν⟩ ≤ 0. (5.7) This representation is valid for all functions x which are element of the domain 𝒟(AL ). The main question in comparison to the finite-dimensional case consists in how to evaluate the LOI (5.6) (or (5.7)) and how to design L to ensure that it is satisfied. This question is addressed next on the basis of a spectral decomposition approach. Assumption 5.1.1. In the following it is assumed that K is such that A in (5.5d) is a Riesz spectral operator which has discrete real eigenvalues with algebraic equal geometric multiplicity which form a decreasing sequence i. e. λ1 ≥ λ2 ≥ . . . with limn→∞ λn = −∞ (cp. Delattre et al., 2003 for the case with n = 1). As a consequence, the number k of eigenvalues, which are larger than a given constant c is always finite (cp. Curtain and Zwart, 1995; Christofides and Daoutidis, 1997; Delattre et al., 2003) or equivalently λi ≤ c

∀i ≥ k.

Spectral dissipation assignment The subsequent analysis and design procedure summarizes the results of Schaum et al. (2008b, 2016b, 2018). According to the connection between the dominant eigenvalue λ∗ of the linear operator A and its dissipativity properties as discussed in Section 2.2.4 and summarized in Lemma 2.2.4, the LOI (5.6) or (5.7) is satisfied, if the eigenvalues of the linear operator AL lie in the open left-half complex plane sufficiently away from the imaginary axis. This in turn can be achieved using a modal observer design technique (Curtain, 1982; Curtain and Zwart, 1995) as is illustrated next. 1 Note that this is related to the fact that the properties of a quadratic form ⟨x, Ax⟩ are determined only by the symmetric part of the operator A (see e. g. Appendix A.1).

5.1 Spectral measurement injection dissipative observer

| 123

Single measurement injection Let ϕi denote the i-th eigenfunction of the operator A associated to the eigenvalue λi and ψi the i-th eigenfunction of the adjoint operator. According to Assumption 5.1.1 the eigenfunctions form a basis of the Hilbert space 𝕏 and any function x ∈ 𝕏 can be written as ∞

x(z) = ∑ ai ϕi (z), i=0

ai = ⟨x, ψi ⟩.

(5.8)

According to the spectral decomposition (5.8) the measurement at z = ζ can be written as ∞

∞

i=1

i=1

y = ∑ ai cT ⟨δ(z − ζ ), ϕi ⟩ = ∑ ai cT ϕi (ζ ). The output component of the estimation error at z = ζ is ∞

cT ⟨δζ , x⟩̃ = ∑ x̃i cT ϕi (ζ ), i=1

x̃i = ⟨x,̃ ψi ⟩

Choosing the innovation gain l as k

l = ∑ li ϕi

(5.9)

i=1

the innovated linear operator becomes ∞

k

∞

i=1

i=1

r=1

Al x̃ = ∑ λi x̃i ϕi − ∑ li ϕi ∑ x̃r cT ϕr (ζ )) k

∞

k

∞

i=k+1

i=1

r=1,r =i̸

= ∑ ϕi [λi − li cT ϕi (ζ )]x̃i + ∑ ϕi λi x̃i − ∑ ϕi li ∑ cT ϕr (ζ )x̃r i=1 k

∞

∞

i=1

r=1

i=k+1

= ∑ ϕi ∑ ϖri (li , ζ )x̃r + ∑ ϕi λi x̃i

(5.10)

with λi − li cT ϕi (ζ ),

ϖri (li , ζ ) = {

T

−li c ϕr (ζ ),

r=i

r ≠ i.

Using the subindexes s, f for slow (s) and fast (f ) modes and introducing the matrixvalued functions and vectors Φs = [ϕ1 x̃ s = [x̃1

⋅⋅⋅ ⋅⋅⋅

ϕk ] , T

x̃k ] ,

Φf = [ϕk+1 x̃ f = [x̃k+1

⋅ ⋅ ⋅] T

⋅ ⋅ ⋅]

124 | 5 Infinite-dimensional systems Λs = diag(λi )i 0 can be assigned for the linear subsystem Σ(Al , G, H) in (5.5) as long as the sensor is located accordingly. Thus, under the observability assumption (5.14) the gains li , i = 1, . . . , k can be chosen such that the LOI (5.6) or (5.7) is satisfied and consequently the observation error x̃ exponentially converges to zero. Multiple measurement injections For the multi-output case with m measurements set the innovation gain k

L = ∑ ϕi [li1 i=1

⋅⋅⋅

lim ]

(5.15)

5.1 Spectral measurement injection dissipative observer

| 125

such that it holds that ∞

k

m

∞

i=1

i=1

s=1

r=1

Λs − ℒCs 0

AL x̃ = ∑ λi x̃i ϕi − ∑ ϕi ∑ lis ∑ x̃r cTs ϕr (ζs ) = [Φs

Φf ] [

−LCf x̃ s ][ ] Λf x̃ f

(5.16)

with

ℒ = [l1s

lms ] ,

⋅⋅⋅

cTis = [cTi ϕ1 (ζi )

⋅⋅⋅

cT1s [ . ] ] Cs = [ [ .. ] , T [cms ]

cTi ϕk (ζi )] ,

cT1f [ . ] ] Cf = [ [ .. ] T [cmf ]

cTif = [cTi ϕk+1 (ζi )

⋅ ⋅ ⋅]

(5.17)

and Λs , Λf , Φs , Φf , x̃ s , x̃ f , ljs , cTjs , cTjf defined as in (5.11) and (5.13). Accordingly, if the pair (Λs , Cs ) is observable, the slow eigenvalues λ1 , . . . , λk can be rearranged so that the dominant eigenvalue λ∗ becomes λ∗ = max{λk+1 ,

max

λ∈σ(Λs −LCs )

λ}.

(5.18)

With these preliminaries the main result is stated as follows. Theorem 5.1.1. Consider the Lur’e-type interconnection (5.5) and let ψ(σ) be (Q, S, R)dissipative. If the dimension k of the observer innovation and the sensor locations ζ1 , . . . , ζm are such that the pair (Λs , Cs ) is observable with Λs ∈ ℝk×k and Cs ∈ ℝm×k defined in (5.11), (5.13) and (5.17), then the gain matrix ℒ in (5.17) can be chosen so that the dominant eigenvalue λ∗ of the linear operator AL = A − LC in (5.5) satisfies λ∗ I
0, i. e. Σ(AL , G, H) is (−R, S∗ , −Q)-state strictly dissipative with rate κ > 0 and the estimation error exponentially converges to zero with rate γ = κ/2 and amplitude a = 1. Proof. Let λ∗ be such that the inequality (5.19) holds true. By assumption the pair (Λs , Cs ) is observable and thus there exists ℒ so that the maximum eigenvalue of Λs − ℒCs satisfies (5.18). Given the cascade structure in (5.16) it follows that the maximum eigenvalue of the operator AL satisfies the condition (5.18) and hence the requirements of Lemma 2.2.4. Thus the linear system Σ(AL , G, H) is (−R, S∗ , −Q)-state strictly dissipative with rate κ > 0 and the LOI (5.6) is satisfied with W = I. The exponential stability result follows from Corollary 2.2.1.

126 | 5 Infinite-dimensional systems Remark 5.1.1. The proposed observer design does not require the approximate observability property of all modes (Curtain and Zwart, 1995), but only the observability of a finite-dimensional subset of modes, which can be easily achieved by appropriately choosing the sensor locations ζ1 , . . . , ζm on the basis of the eigenfunctions ϕ1 (z), . . ., ϕk (z). This offers a simple criterion for the location of sensors for observation purposes for semi-linear parabolic systems, in particular having in mind that most studies focus on the approximate observability (implying that all eigenvalues can be moved) which is only ensured when the sensor is located at the boundary (see e. g. Delattre et al., 2004). Remark 5.1.2. The idea of approaching the design by means of a spectral separation into slow and fast dynamics is well-known from the literature on distributedparameter systems (Curtain, 1982; Curtain and Zwart, 1995; Christofides, 2001), including studies on the effect of possible observer spill-over (Hagen and Mezic, 2003). The proposed approach using the dissipativity framework and (Q, S, R)-dissipation inequality for the nonlinearity, nevertheless, has not been exploited in the literature to the authors knowledge and does implicitly account for the spill-over by means of the dissipation bounds for the linear and nonlinear subsystems. Application example Consider the following coupled two-state semi-linear pde system with two in-domain measurements 𝜕t x1 = d1 𝜕z2 x1 + γx2 ,

t>0

(5.20a)

t>0

(5.20b)

t≥0

(5.20d)

t≥0

(5.20e)

y2 (t) = x2 (ζ2 , t), t ≥ 0

(5.20f)

𝜕t x2 = d2 𝜕z2 x2 + φ(x2 ),

x1 (0, t) = x1 (1, t) = 0,

x2 (0, t)= 0, φ(x2 ) =

t≥0

𝜕z x2 (1, t) = 0,

x2 (1 − x22 ) , 1 + x24

y1 (t) = x1 (ζ1 , t),

(5.20c)

with d1 , d2 , γ > 0 and ζ1 , ζ2 ∈ [0, 1]. The set of eigenvalue-eigenfunction pairs of the linear operator are given by ϕI (λkI , ϕIk ) = (−d1 (kπ)2 , [ k1 ]) , 0 (λkII , ϕIIk ) = (−d2 (

2

ϕII 2k − 1 π) , [ IIk1 ]) ϕk2 2

for k ∈ ℕ with ϕIk1 = √2 sin(ωIk z),

ωIk = kπ

5.1 Spectral measurement injection dissipative observer

| 127

and ϕIIk1 = BIIk1 (

sin(ωk1 )(ωk1 sin(ωIIk z) − ωIIk sin(ωk1 z)) ωIIk sin(ωk1 ) − ωk1 sin(ωIIk ))

+ sin(ωk1 z))

ϕIIk2 = BIIk2 sin(ωIIk z),

with normalization constant BIIk1 and ωk1 = √ BIIk2 = −

−λkII d1

= ωIIk √

d2 , d1

ωIIk =

2k − 1 π 2

d1 (ω2k1 − ωII2 k ) sin(ωk1 )

γ(ωIIk sin(ωk1 ) − ωk1 sin(ωIIk ))

By a direct calculation it can be seen that 𝒟(A) = 𝒟(A∗ ). The system is a cascaded interconnection between a stable heat equation and a bi-stable semi-linear diffusion equation and has three steady-state solutions (two of them being asymptotically stable and the other one a repulsor). The function ψ(σ,̃ x2 ) = φ(x2 + σ)̃ − φ(x2 ) satisfies a sector condition of the form with K1 = 1, K2 = −1.5, where the value of K2 is chosen sufficiently small to ensure that ψ is completely comprised in the sector. Hence the static subsystem (5.5c) is (Q, S, R)-dissipative with Q = −1, S = −0.25, R = 1.5. For the numerical evaluation the system and observer dynamics have been implemented in Matlab using a finite-difference discretization with N = 40 collocation points and subsequent integration of the resulting set of coupled nonlinear equations with the standard solver ode15s. The system parameters are assigned as γ = −10, d1 = 0.5, d2 = 0.2, ζ1 = ζ2 = 0.3 and the initial conditions are set as x 0 = ϕII1 and x̂ 0 = −ϕII1 . The first k = 3 eigenvalues have been moved such that the dominant eigenvalue of the linear innovated subsystem is λ∗ = λ3II ≈ −12.337 and the linear operator inequality (5.6) is satisfied. In particular, a dissipation rate κ ≈ 10 is ensured, meaning that the estimation error should have a 98.5 % convergence to zero in about 4 characteristic times tc = 1/κ ≈ 0.1 meaning in about 0.4 time units. For comparison purposes Figure 5.1 shows the convergence of the system trajectories to the upper steady-state solution (upper row) and the convergence of a pure simulation without measurement injection to the lower steady-state profile (center row), yielding a non-vanishing estimation error (lower row). The time evolution for the proposed observation scheme is illustrated in Figure 5.2. The figure shows the convergence of the observer to the upper steady-state solution (upper row) and the exponential convergence to zero of the associated estimation error (lower row). The time evolution of the approximated L2 -norms of the spatially discretized errors are shown in Figure 5.3. The convergence due to the correction scheme is clearly

128 | 5 Infinite-dimensional systems

Figure 5.1: State evolution (first row), state estimate (second row) and observation error (third row) without measurement injection.

visible. It can be seen that the estimation error converges in about 0.15 time units, i. e. faster than predicted by the dissipation rate κ ≈ 10.

5.2 Point-wise measurement injection observer Consider the semi-linear heat equation 𝜕t x = D𝜕z2 x + μ(x),

t > 0,

z ∈ (0, 1)

(5.21a)

with boundary conditions 𝜕z x(0, t) = 0,

t≥0

(5.21b)

5.2 Point-wise measurement injection observer

| 129

Figure 5.2: State estimate (first row) and observation error (second row) of the proposed observer with measurement injection in the first k = 3 modes.

Figure 5.3: Comparison of the approximated L2 -norm for the open-loop simulation (dashed line) and the proposed observer (continuous line).

𝜕z x(1, t) = u(t),

t≥0

(5.21c)

and boundary and in-domain measurement y (t) x(0, t) y(t) = [ 1 ] = [ ], y2 (t) x(ζ , t)

t ≥ 0,

(5.21d)

130 | 5 Infinite-dimensional systems where t ≥ 0 denotes the time, z ∈ [0, 1] the space, x(⋅, t) ∈ 𝕏 = L2 (0, 1) the state with initial condition x(⋅, 0) = x0 , and D > 0 the diffusion constant. The function μ : L2 (0, 1) → L2 (0, 1) has uniformly bounded slope so that there are constants sl , su ∈ ℝ such that ∀x ∈ L2 (0, 1).

sl ≤ 𝜕x μ(x) ≤ su ,

(5.22)

Following the point-wise injection approach proposed in Schaum et al. (2015b, 2016a, 2017), Schaum (2018) the observer is set by directly imposing the measurement at the measurement location in form of an algebraic constraint, i. e. ̂ 𝜕t x̂ = D𝜕z2 x̂ + μ(x),

t > 0,

z ∈ (0, 1)

(5.23a)

̂ 0) = x̂0 and with initial condition x(⋅, ̂ t) = 0 𝜕z x(0,

̂ t) = u(t) 𝜕z x(1,

̂ t) = y1 (t) x(0,

̂ , t) = y2 (t) x(ζ

t≥0

(5.23b)

t≥0

(5.23d)

t≥0

(5.23c)

t ≥ 0.

(5.23e)

The associated error x̃ = x̂ − x is governed by 𝜕t x̃ = D𝜕z2 x̃ − φ(x;̃ x),

̃ φ(x;̃ x) = μ(x) − μ(x + x),

t > 0,

t ≥ 0,

z ∈ (0, 1)

z ∈ (0, 1)

(5.24a) (5.24b)

̃ 0) = x̃0 and with initial condition x(⋅, ̃ t) = 0 𝜕z x(0,

t≥0

̃ t) = 0 x(0,

t≥0

̃ t) = 0 𝜕z x(1, ̃ , t) = 0 x(ζ

(5.24c)

t≥0

(5.24d)

t ≥ 0,

(5.24f)

(5.24e)

i. e. the error vanishes exactly (up to possibly measurement imperfections not considered here) at the measurement locations. Note that for the function φ it holds that φ(0; x) = 0 ∀x ∈ 𝕏. According to the above, in order to analyze the stability of the error dynamics with respect to the L2 norm it is sufficient to prove the exponential stability of the zero solution of the pde 𝜕t x̃ = D𝜕z2 x̃ − φ(x;̃ x),

̃ φ(x;̃ x)= μ(x) − μ(x + x),

t > 0, t ≥ 0,

z ∈ (0, 1) z ∈ (0, 1)

(5.25a) (5.25b)

5.2 Point-wise measurement injection observer |

131

̃ 0) = x̃0 and the boundary and in-domain conditions with initial condition x(⋅, ̃ t) = 0 x(0,

t≥0

̃ t) = 0 𝜕z x(1,

t ≥ 0.

̃ , t) = 0 x(ζ

(5.25c)

t≥0

(5.25d) (5.25e)

Note that this pde corresponds to (5.24) neglecting the Neuman boundary condition at z = 0. Accordingly, the set of solutions of (5.25) includes the set of solutions of (5.24) as a subset. The dynamics (5.25) can be written in the form of a Lur’e interconnection given by 𝜕t x̃ = D𝜕z2 x̃ + ν, t > 0, ν = −φ(x;̃ x),

t ≥ 0,

z ∈ (0, 1) z ∈ (0, 1)

(5.26a) (5.26b)

̃ 0) = x̃0 and with x(⋅, ̃ t) = 0 x(0,

t≥0

̃ t) = 0 𝜕z x(1,

t ≥ 0.

̃ , t) = 0 x(ζ

(5.26c)

t≥0

(5.26d) (5.26e)

The additional constraint at z = ζ actually separates the domain in two parts, so that the error dynamics can be written as x̃1 (z, t), z ∈ [0, ζ ) { { { ̃ t) = {0, x(z, z=ζ { { {x̃2 (z, t), z ∈ (ζ , 1],

t≥0

(5.27)

where x̃1 and x̃2 are the solutions of the Lur’einterconnections 𝜕t x̃1 = D𝜕z2 x̃1 + ν1 ,

t > 0,

z ∈ (0, 1)

z ∈ (0, 1)

(5.28b)

+ ν2 , t > 0,

z ∈ (0, 1)

(5.28c)

ν1 = −φ(x̃1 ; x),

𝜕t x̃2 =

D𝜕z2 x̃2

ν2 = −φ(x̃2 ; x),

t ≥ 0, t ≥ 0,

z ∈ (0, 1)

(5.28a)

(5.28d)

with initial conditions x̃i (⋅, 0) = x̃i,0 , i = 1, 2 and boundary conditions x̃1 (0, t) = 0

t≥0

(5.28e)

x̃2 (ζ , t) = 0

t≥0

(5.28g)

x̃1 (ζ , t) = 0

𝜕z x̃2 (1, t) = 0

t≥0

t ≥ 0.

(5.28f) (5.28h)

132 | 5 Infinite-dimensional systems In a similar way to the preceding sections, the dissipativity properties of each twosystems interconnection, i. e. for x̃1 and x̃2 can be analyzed using e. g. a spectral decomposition, given that the operators A1 = D𝜕z2 ,

A2 = D𝜕z2 ,

𝒟(A1 ) = {v ∈ 𝕏 | v(0) = 0, v(ζ ) = 0} 𝒟(A2 ) = {v ∈ 𝕏 | v(ζ ) = 0, 𝜕z v(1) = 0}

are Riesz-spectral operators which generate contraction semigroups with growth bound ωi , i = 1, 2 determined by the maximum eigenvalue, if ζ is chosen so that the maximum eigenvalue is negative. Actually, it is straight-forward to show that the eigenvalues satisfy λ1,n = −D(

2

nπ ), ζ

λ2,n = −Dω2n,2 ,

n≥1

(5.29a)

tan(ωn,2 ζ ) = tan(ωn,2 )

⇒

π ωn,2 ≈ (2n + 1) , 2

(5.29b)

showing that both eigenvalues are smaller than the maximum eigenvalue for the heat equation with Neuman boundary conditions, which is actually λ0 = 0. On the basis of this analysis, if ν = −φ is (Q, S, R)-dissipative, on can use Lemma 2.2.4 to establish sufficient conditions for the linear subsystems in terms of the sensor location which ensures that both linear dynamic subsystems are (−R, S∗ , −Q)-strictly state dissipative with a positive dissipation rate κi > 0 i = 1, 2. Such conditions are stated in the next Proposition. Proposition 5.2.1. Let ωζ be the solution of tan(ωζ ) = tan(ω) and let 2

π γ = min{D( ) , Dω2ζ }. ζ If ζ ∈ (0, 1) is chosen so that (1 − S)2 1 ), γ > (κ + R − 2 Q

(5.30)

then the observation error x̃ in (5.24) exponentially converges to zero in the L2 norm. Proof. According to (5.29a) the operator A = D𝜕z2 ,

𝒟(A) = {v ∈ 𝕏 | v(0) = v(ζ ) = 0, 𝜕z v(1) = 0}

generates a semigroup of contractions with growth bound 2

π −γ = max{−D( ) , −Dω2ζ }. ζ

5.2 Point-wise measurement injection observer

| 133

Note that in the dissipativity framework presented in the preceding chapter, for the particular case at hand the matrix pair (G, H) is given by (1, 1). Recall from Lemma 2.2.4 that, accordingly Σ(A, 1, 1) is (−R, S∗ , −Q)-strictly state dissipative with dissipation rate κ > 0 if (5.30) holds true. Thus 𝒮 = ⟨x,̃ x⟩̃ is a quadratic storage functional and by Corollary 2.2.1 the exponential stability of the zero solution for the interconnection (5.24) follows. In order to illustrate an alternative approach in the line of the direct Lyapunov analysis (Schaum et al., 2013b; Fridman and Orlov, 2009), which has the advantage that it can be applied also for space and time varying coefficients, consider the Lyapunov functional candidate 1

V(x)̃ = ⟨x,̃ x⟩̃ = ∫ x̃ 2 (z, t)dz 0 ζ

1

= ∫ x̃ (z, t)dz + ∫ x̃ 2 (z, t)dz 2

0

ζ

ζ

1

= ∫ x̃12 (z, t)dz + ∫ x̃22 (z, t)dz 0

(5.31)

ζ

with rate of change over time ζ

1

dV (x)̃ = 2 ∫ x̃1 (z, t)(D𝜕z2 x̃1 (z, t) + ν1 )dz + 2 ∫ x̃2 (z, t)(D𝜕z2 x̃2 (z, t) + ν2 )dz dt 0

ζ

ζ

2

ζ = 2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x̃1 (z, t)D𝜕z x̃1 (z, t)|0 −2 ∫(D(𝜕z x̃1 (z, t)) − x̃1 (z, t)ν1 )dz+ 0

=0

1

2

+ 2⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ x̃2 (z, t)D𝜕z x̃2 (z, t)|1ζ −2 ∫(D(𝜕z x̃2 (z, t)) − x̃2 (z, t)ν2 )dz ζ

=0

ζ

1

2

2

= −2 ∫(D(𝜕z x̃1 (z, t)) − x̃1 (z, t)ν1 )dz − 2 ∫(D(𝜕z x̃2 (z, t)) − x̃2 (z, t)ν2 )dz 0

ζ

̃ , t) = 0 by Wirtingers inequality (Hardy et al., 1952) it holds that Given that x(ζ ζ

2

− ∫ D(𝜕z x̃1 (z, t)) dz ≤ − 0

1

0

2

− ∫ D(𝜕z x̃2 (z, t)) dz ≤ − ζ

ζ

Dπ 2 ∫ x̃12 (z, t)dz 4ζ 2 1

Dπ 2 ∫ x̃22 (z, t)dz. 4(1 − ζ )2 ζ

134 | 5 Infinite-dimensional systems Summarizing, one arrives at 1

Dπ 2 dV Dπ 2 ̃ t)νdz. (x)̃ ≤ − min{ 2 , }V(x)̃ + 2 ∫ x(z, dt 2ζ 2(1 − ζ )2

(5.32)

0

This lead directly to the following convergence result. Proposition 5.2.2. If ζ ∈ (0, 1) is chosen so that min{

Dπ 2 (1 − S)2 Dπ 2 , } > κ + R − , Q 2ζ 2 2(1 − ζ )2

(5.33)

then the observation error x̃ in (5.24) exponentially converges to zero in the L2 norm. Proof. Consider the storage function 𝒮 (x)̃ = V(x)̃ = ⟨x,̃ x⟩̃ and the associated bound (5.32) of its rate of change over time. Accordingly, the linear system Σ(A, 1, 1) is (−R, S∗ , −Q)-strictly state dissipative if 2

2

− min{ Dπ , Dπ } + κ + R 2ζ 2 2(1−ζ )2

M=[

1−S

1−S Q

] ≤ 0.

This ensured if condition (5.33) holds true, as can be seen using either the Schur complement (see also Appendix A.2) or analyzing the trace and determinant of the matrix M. For illustration purposes the above observer (5.23) is considered for the case of the nonlinearity μ(x) =

x(1 − x 2 ) . 1 + x4

(5.34)

The derivative 𝜕x μ of μ is contained in the interval [−0.07, 1] and thus the function φ is contained in the sector [−0.07, 1] and is (Q, S, R)-dissipative with Q = −1,

S = 0.465,

R = −0.07.

With this nonlinearity and setting u = 0 the system exhibits steady-state multiplicity with three constant solutions xn (z) = −1,

xc (z) = 0,

xp (z) = 1.

It can be easily verified that the two non-zero solutions correspond to attractor profiles while the zero solution corresponds to an open-loop repulsor. The initial conditions chosen for the numerical simulations are the following x0 (z) = 0.2,

−0.2, z ∉ {0, ζ }

x̂0 (z) = {

0.2,

z ∈ {0, ζ }

5.2 Point-wise measurement injection observer |

135

̃ satisfying the condition x(z) = 0 for z = 0, ζ . As can be seen below in the simulation results, both initial conditions are in the domain of attraction of different attractor profiles. The second sensor location has been chosen as ζ = 2/3. The corresponding state profile evolution over time is shown in Figure 5.4 showing that the state converges in about 4 time units to the stationary profile xp (z) = 1.

Figure 5.4: System state profile evolution starting from the initial condition x0 (z).

Starting the simulation with the initial condition of the observer but without performing the measurement injection and just running an (open-loop) simulation yields the evolution shown in Figure 5.5. It can be seen that the state profile converges to the other attractor stationary profile xn (z) = −1.

Figure 5.5: Simulation without measurement injection starting from the initial condition x0̂ (z).

136 | 5 Infinite-dimensional systems In direct comparison, the time evolution of the observer state profile with the pointwise measurement injection is shown in Figure 5.6 showing that besides the erroneous initial condition the observed state profile converges in about 0.5 time units to the actual one. The associated observation error evolution is shown in Figure 5.7

Figure 5.6: Point-wise injection observer evolution. The dark lines indicate the sensor location.

Figure 5.7: Observation error evolution for the pointwise injection observer (5.23). The dark lines indicate the sensor location.

The time evolution of the approximated L2 -norms of the spatially discretized errors are shown in Figure 5.8, showing how the convergence is achieved using the proposed pointwise injection scheme, whereas the pure simulation without measurement injection does not achieve a convergent estimate.

5.3 In-domain output-feedback control of a semi-linear heat equation

| 137

Figure 5.8: Comparison of the approximated L2 -norms without measurement injection (dashed line) and with pointwise measurement injection (continuous line).

5.3 In-domain output-feedback control of a semi-linear heat equation Consider the semilinear heat equation with Dirichlet boundary conditions and indomain control 𝜕t x = D𝜕z2 x − gφ(σ) + bu, t > 0, σ = Hx,

t ≥ 0,

z ∈ (0, 1)

(5.35a) (5.35b)

with boundary conditions x(0, t) = x(1, t) = 0,

t≥0

(5.35c)

and initial condition x(⋅, 0) = x0 , where z represents space, t time, x the state profile, D the diffusion coefficient, g:∈ L2 (0, 1) a smooth function of z, φ∈ 𝒞 1 (ℝ, ℝ) a smooth function of σ = Hx with H: 𝕏 → ℝ defined by Hx = ⟨h, x⟩,

x ∈ 𝕏,

b∈ L2 (0, 1) and h∈ L2 (0, 1) the input (actuator) and output (sensor) shape functions, respectively, and u(t) ∈ ℝ the associated control input. In the sequel only the so called collocated case is considered, i. e. b = h. The semilinear heat equation (5.35) is written as an abstract Cauchy problem in the Hilbert space 𝕏 = L2 (0, 1) as ẋ = Ax − GF(σ) + Bu, t > 0,

x(0) = x0

(5.36a)

138 | 5 Infinite-dimensional systems σ = Hx,

t ≥ 0,

(5.36b)

with operators A : 𝕏 → 𝕏, G : ℱ = ℝ → 𝕏, B : 𝒰 = ℝ → 𝕏 and F : ℋ = ℝ → ℱ defined by A = D𝜕z2 ,

2

𝒟(A) = {v ∈ H (0, 1) | v(0) = v(1) = 0},

∀z ∈ [0, 1], ξ ∈ ℝ,

(Bu)(z) = b(z)u,

∀z ∈ [0, 1], u ∈ ℝ,

(F(σ))(z) = φ(σ),

(Gξ )(z) = g(z)ξ , ∀z ∈ [0, 1], σ ∈ ℝ.

Consider the linear output-feedback control u = −kσ.

(5.37)

In the sequel the corresponding closed-loop operator will be denoted by Ak := A − kbH,

𝒟(Ak ) = 𝒟(A).

The closed-loop dynamics can be written as a feedback interconnection of a linear dynamical and a nonlinear static subsystem in Lur’eform ẋ = Ak x + gν

σ = Hx

ν = −F(σ).

(5.38a) (5.38b) (5.38c)

Following the dissipativity-based approach outlined in this study, the stability assessment can be carried out on the basis of Lemma 2.2.11 and Corollary 2.2.1 assuming that the nonlinearity φ in (5.38c) is (Q, S, R)-dissipative. Thus, the main focus is to derive (sufficient) conditions that ensure that the linear dynamical subsystem, given by the controlled heat equation is (−R, S∗ , −Q) strictly state dissipative. This analysis is carried out following two alternative approaches: (i) a direct Lyapunov approach and (ii) a modal (or spectral) decomposition approach. Both approaches highlight different aspects of the closed-loop linear subsystem and give insight into the control design problem. Direct Lyapunov approach Consider the linear subsystem Σ(Ak , g, H) given in (5.38a)–(5.38b) with the storage function 1

V(x) = ⟨x, x⟩ = ∫ x 2 dz > 0. 0

5.3 In-domain output-feedback control of a semi-linear heat equation

| 139

With u given in (5.37) it holds that 1

1

dV (x) = 2 ∫ x𝜕t xdz = 2 ∫ x(D𝜕z2 x − bkσ + gν)dz dt 0

=

2Dx𝜕z x|10

1

0

1

2

1

− 2 ∫ D(𝜕z x) dz − 2kσ ∫ bxdz + 2 ∫ xgνdz. 0

0

0

In virtue of the Dirichlet boundary conditions and recalling Wirtinger’s inequality (Hardy et al., 1952) it follows that 1

1

1

0

0

0

dV (x) ≤ −2Dπ 2 ∫ x2 dz + 2 ∫ xgνdz − 2kσ ∫ bxdz. dt To further analyze this inequality, note that in the case of a collocated input-output pair (i. e., b = h) 1

1

∫ bxdz = ∫ hxdz = ⟨h, x⟩ = σ 0

(5.39)

0

so that 1

1

0

0

dV (x) ≤ −2Dπ 2 ∫ x2 dz + 2 ∫ xgνdz − 2kσ 2 dt = −2Dπ 2 ‖x‖2 + 2⟨x, gν⟩ − 2⟨σ, kσ⟩. In order to apply the exponential stability result for the interconnection stated in Corollary 2.2.1 the linear subsystem must be strictly (−R, S, −Q) state dissipative with positive dissipation rate κ > 0. This is ensured if the inequality − 2Dπ 2 ‖x‖2 + 2⟨x, gν⟩ − 2⟨σ, kσ⟩

≤ −κ‖x‖2 + ⟨σ, −Rσ⟩ + 2⟨σ, Sν⟩ + ⟨ν, −Qν⟩

(5.40)

holds true. Clearly, this inequality can only be satisfied if Q ≤ 0. Recall from the discussion in Section 2.2.5 on sector nonlinearities that this constraint is not restrictive from a practical point of view and thus in the sequel it is assumed that Q < 0 is satisfied. Furthermore, it can be seen from (5.40) that in the left-hand side of the inequality no coupled terms in σν appear. For S = 0, the above inequality can be rewritten as x −2Dπ 2 + κ ⟨[ ] , [ ν g

g x ] [ ]⟩ − (2k − R)σ 2 ≤ 0 Q ν

140 | 5 Infinite-dimensional systems so that for k > R/2 and a constant g the (−R, 0, −Q) strict state dissipativity with a positive dissipation rate κ is ensured if [

−2Dπ 2 + κ g

g ] < 0. Q

(5.41)

Recalling that by the above assumption Q < 0 this condition holds true if Q(−2Dπ 2 + κ) − g 2 > 0. A necessary condition for the existence of a positive κ is that (see also Schaum and Meurer, 2019b) Q (R + ). 2 Q(κ − 2Dπ 2 ) − g 2

(5.44)

This is summarized in the following Proposition. Proposition 5.3.1. Consider the semi-linear heat equation (5.35) with the output-feedg2 back control (5.37). Let g be constant, φ(σ) be (Q, S, R)-dissipative with Q < − 2Dπ 2 and the control gain satisfy inequality (5.44). Then, (i) the linear subsystem Σ(Ak , g, H) given in (5.38) is (−R, S∗ , −Q) strictly state dissipative with positive dissipation rate κ satisfying (5.43), and (ii) the solution x = 0 is exponentially stable. Proof. From the preceding analysis it follows that the linear subsystem Σ(Ak , g, H) in (5.38) is (−R, S, −Q) strictly state dissipative with positive dissipation rate κ if (5.42) holds and k satisfies (5.44). The exponential stability follows from Corollary 2.2.1. Note that the preceding result states only conditions on the system parameters (D, g), the nonlinearity (Q, S, R) and the control gain (k) but no explicit conditions on the sensor and actuator locations and shape function (h). The question about localization and shape of the sensor and actuator characteristic functions is nevertheless a crucial one. A possibility to address this question is using a modal decomposition as outlined in the following section. Modal (spectral) decomposition approach The eigenvalue problem associated to the open-loop dynamics (i. e., for u = 0) is given by D𝜕z2 ϕn = λn ϕn subject to the boundary conditions ϕn (0) = ϕn (1) = 0, n ∈ ℕ. Setting jωn = √

λn , D

ϕn (z) = αn sin(ωn z) + βn cos(ωn z)

142 | 5 Infinite-dimensional systems it follows by substitution in the boundary conditions that ωn = nπ. This implies that the eigenvalues and eigenfunctions are given by λn = −D(nπ)2 ,

ϕn (z) = √2 sin(ωn z).

(5.45)

Given that the eigenfunctions ϕn form a basis of the state space it is possible to express the state as x(t) = ∑ an (t)ϕn , n

an (t) = ⟨x(t), ϕn ⟩

and the output as σ(t) = Hx(t) = ⟨h, ∑ an (t)ϕn ⟩ = ∑ an (t)hn , n

n

hn = ⟨h, ϕn ⟩.

Accordingly, the output-feedback control can be written as u(t) = −kσ(t) = −kHx(t) = − ∑ kan (t)hn . n

The action of the closed-loop operator on the state x can thus be expressed as Ak x = (A − bkH) ∑ an ϕn n

= ∑ λn an ϕn − bk ∑ an hn n

n

Projecting the closed-loop dynamics of the linear subsystem with ν = 0 on the i-th eigenfunction ϕi , recalling ⟨ϕk , ϕi ⟩ = δik and denoting by bi = ⟨b, ϕi ⟩ yields ȧ i = ⟨𝜕t x, ϕi ⟩ = ⟨∑ λn an ϕn − bk ∑ an hn , ϕi ⟩ n

n

= ∑ λn an ⟨ϕn , ϕi ⟩ − ⟨b, ϕi ⟩k ∑ an hn n

n

= λi ai − bi k ∑ an hn n

= (λi − kbi hi )ai − kbi ∑ an hn . n=i̸

Note that for the collocated case considered here it holds that hn = bn and thus the preceding equation is equivalent to ȧ i = (λi − kh2i )ai − khi ∑ hn an , n=i̸

i ∈ ℕ.

(5.46)

5.3 In-domain output-feedback control of a semi-linear heat equation

| 143

This corresponds to the associated l2 formulation of the closed-loop linear subsystem. In terms of the storage functional V(x) = ⟨x, x⟩ = ⟨∑ an ϕn , ∑ al ϕl ⟩ = ∑ a2n n

n

l

(5.47)

with rate of change over time dV (x) = 2 ∑ an ȧ n dt n

(5.48)

equivalent statements are thus obtained using the pde, the abstract differential equation or the corresponding l2 formulation. Note that for l2 the complete infinite-set of coupled differential equations can be written using the infinite-dimensional symmetric matrix and modal coefficient vector λ1 − kh21 [ −kh1 h2 Ak = [ [ .. [ .

−kh1 h2 λ2 − kh22

⋅⋅⋅ ..

.

] ], ] ]

a1 [ ] a] a=[ [ 2] .. [.]

in the compact form ȧ = Ak a,

a(0) = a0

and the storage functional over l2 as V(x) = 𝒱 (a) = aT a, with d𝒱 (a) = aT ȧ + (a)̇ T a = aT (ATk + Ak )a = 2aT Ak a dt due to the symmetry of Ak . From Lemma 2.2.4 it is known that the dissipativity properties of the linear subsystem can be accessed through the eigenvalues of the linear operator Ak . Equivalently this can be done using the eigenvalues of the infinitedimensional matrix Ak . Note that due to the fact that the matrix Ak is not sparse, the eigenvalues cannot be directly calculated. Nevertheless, bounds for the eigenvalues can be determined using e. g. the extension of Gershgorin’s circle criterion for infinitedimensions (see e. g. Shivakumar et al., 1987; Aleksić et al., 2014 and compare also Franco de los Reyes et al., 2019b,c). Following the reasoning in Aleksić et al. (2014) and adapting the notation to better fit into the present specific framework, the infinitedimensional matrix is strictly diagonal dominant (in l2 ) if for all i ∈ ℕ there exists a sequence {wi }i∈ℕ with wi > 0 and ‖w‖ = √∑n wn2 < 1 such that it holds that (assuming k > 0) 󵄨 󵄨 wi 󵄨󵄨󵄨λi − kh2i 󵄨󵄨󵄨 ≥ ri = √∑ |khi hn |2 = k|hi |√∑ |hn |2 n=i̸

n=i̸

(5.49)

144 | 5 Infinite-dimensional systems where ri denotes the i-th spectral radius. The existence of the series {wi }i∈ℕ in turn is equivalent to the requirement that s2 = √∑( i

2

ri ) < 1. |λi − kh2i |

(5.50)

In the case that Ak is a strictly diagonal dominant matrix, an upper bound λ∗ for the (real parts of the) eigenvalues is given by λ∗ = sup{(λn − kh2n ) + rn }, n∈ℕ

(5.51)

what is a direct generalization of the circle criterion of Geršgorin (1931). In order to find conditions for the strict diagonal dominance of the matrix Ak note that ri2 = k 2 h2i ∑ h2n = k 2 h2i (∑ h2n − h2i ) = k 2 h2i (‖h‖2 − h2i ) ≤ k 2 h2i ‖h‖2 n

n=i̸

(5.52)

with finite ‖h‖2 = ⟨h, h⟩ = ∑n h2n given that h ∈ L2 (0, 1). Next, as k > 0 (5.50) can be assessed as follows: s22 = ∑( i

=∑ i

2

ri2 ri ) = ∑ 2 2 2 |λi − kh2i | i | − D(iπ) − khi |

ri2 ri2 ≤ ∑ 2 2 √ |D(iπ)2 + khi |2 i (2 kDiπ|hi |)

in virtue of the second binomial formula a2 + b2 ≥ 2ab. Furthermore it follows from (5.52) that ∑ i

ri2

(2√kDiπ|hi |)2

≤∑ i

k 2 h2i ‖h‖2 h2i k 2 ‖h‖2 k‖h‖2 1 k‖h‖2 = = ζ (2) = ∑ ∑ 4Dπ 2 i i2 4Dπ 2 (2√kDiπ|hi |)2 4kDπ 2 i i2 h2i

with the Rieman ζ -function ζ (s) = ∑ n

1 . ns

It holds that ζ (2) =

(2π)2 B 22 2

with the Bernoulli number B2 =

1 . 6

5.3 In-domain output-feedback control of a semi-linear heat equation

| 145

Summarizing, the value s2 is bounded by k‖h‖2 4π 2 k‖h‖2 = . 24D 4Dπ 2 24

s2 ≤ Thus, condition (5.50) is satisfied if

s2 ≤

k‖h‖2 0 the diffusion coefficient, μ : 𝕏 → 𝕏 a smooth function, and u(t), y(t) the control input and output at time t, respectively. The problem consists in designing state and output-feedback control schemes so that the state x exponentially converges to zero. For this purpose the system is viewed in the interconnection form 𝜕t x = D𝜕z2 x + ν, t > 0, σ = x,

ν = −φ(σ),

t≥0 t≥0

z ∈ (0, 1)

(5.60a) (5.60b) (5.60c)

150 | 5 Infinite-dimensional systems with boundary conditions 𝜕z x(0, t) = 0

(5.60d)

𝜕z x(1, t) = u(t)

(5.60e)

for t ≥ 0 and the feedback nonlinearity φ(σ) = −μ(σ),

t ≥ 0.

(5.61)

As known from the fundamental results presented in Chapters 2 and 3, in particular the exponential stability results of Lemma 2.2.12 and Corollary 2.2.2, if the nonlinearity μ(x) is linearly bounded, i. e. there exists a constant m > 0 so that |μ(x)| ≤ m|x| and the linear subsystem (5.60a)–(5.60b) is exponentially stable, then the exponential stability of (5.60) is ensured if an associated auxiliary system (to be specified later) satisfies an appropriate dissipativity condition. To address this in detail, first the statefeedback control problem is considered. A similar approach will then be employed for the observer design and coupled with the feedback control in virtue of the separation result established in Theorem 3.4.1. State-feedback control First a backstepping controller will be designed for the linear subsystem in (5.60) for the case that ν(z, t) = 0. The target dynamics for the linear dynamic subsystem is given by Krstic and Smyshlyaev (2008a) 𝜕t ξ (z, t) = D𝜕z2 ξ (z, t) − βξ ,

t > 0,

z ∈ (0, 1)

(5.62a)

with initial condition ξ (⋅, 0) = ξ0 and boundary conditions 𝜕z ξ (0, t) = 0

(5.62b)

𝜕z ξ (1, t) = 0

(5.62c)

for t ≥ 0. The backstepping transformation is given by z

ξ (z, t) = x(z, t) − ∫ k(z, ζ )x(ζ , t)dζ .

(5.63)

0

As is shown in Appendix D.1.1 the kernel satisfies the hyperbolic pde β k(z, ζ ), D βz k(z, z) = − 2D 𝜕ζ k(z, 0) = 0.

𝜕z2 k(z, ζ ) − 𝜕ζ2 k(z, ζ ) =

ζ ∈ (0, 1), z ∈ (ζ , 1)

(5.64a) (5.64b) (5.64c)

5.4 Backstepping-based dissipative control and observer design

| 151

The solution of this pde is derived in Appendix D.1.2 (see also Krstic and Smyshlyaev, 2008a) and is given by β

2 2 β I1 (√ D (z − ζ )) k(z, ζ ) = − z D √ β (z 2 − ζ 2 )

(5.65)

D

with I1 being the modified Bessel of first order and first kind. Accordingly, the control input can be determined by the equation 1

0 = 𝜕z ξ (1, t) = 𝜕z x(1, t) − k(1, 1)x(1, t) − ∫ 𝜕z k(1, ζ )x(ζ , t)dζ 1

0

= u(t) − k(1, 1)x(1, t) − ∫ 𝜕z k(1, ζ )x(ζ , t)dζ 0

and is thus given by 1

u(t) = k(1, 1)x(1, t) + ∫ 𝜕z k(1, ζ )x(ζ , t)dζ , 0

From (5.64b) it follows that2 k(1, 1) = −

β 2D

β

and, introducing α(z, ζ ) = √ D (z 2 − ζ 2 ), β I (α(z, ζ )) β I2 (α(z, ζ )) d k(z, ζ ) = − 1 − z 𝜕 α(z, ζ ) dz D α(z, ζ ) D α(z, ζ ) z with 𝜕z α(z, ζ ) =

βz β D√ D (z 2

− ζ 2)

,

so that β 2 β 2 2 2 β I1 (√ D (z − ζ )) β I2 (√ D (z − ζ )) d k(z, ζ ) = − [ + z2 ]. 2 2 dz D √ β (z 2 − ζ 2 ) D z −ζ D

2 Note that the same result is obtained taking into account that (see e. g. Arfken, 1985; Krstic and Smyshlyaev, 2008a) lim

x→0

I1 (x) 1 = , x 2

d I1 (x) I2 (x) = . dx x x

(5.66)

152 | 5 Infinite-dimensional systems In particular this implies that β β 2 2 β I1 (√ D (1 − ζ )) β I2 (√ D (1 − ζ )) d + k(1, ζ ) = − [ ], dz D √ β (1 − ζ 2 ) D 1 − ζ2 D

so that the backstepping-based state-feedback controller is given by 1 β β I1 (√ D (1 − ζ 2 )) β I2 (√ D (1 − ζ 2 )) β x(1, t) + ∫[ ]x(ζ , t)dζ ). u(t) = − ( + D 2 D 1 − ζ2 √ β (1 − ζ 2 ) 0

(5.67)

D

The backward transformation has the form z

x(z, t) = w(z, t) + ∫ l(z, ζ )w(ζ , t)dζ 0

with the kernel satisfying the pde (see Appendix D.2 or Krstic and Smyshlyaev, 2008a) β 𝜕z2 l(z, ζ ) − 𝜕ζ2 l(z, ζ ) = − l(z, ζ ), D βz l(z, z) = − 2D 𝜕ζ l(z, 0) = 0.

ζ ∈ (0, 1), z ∈ (ζ , 1)

(5.68a) (5.68b) (5.68c)

The solution of this pde is given by (see Appendix D.2 for the derivation) β 2 2 β J1 (√ D (z − ζ )) l(z, ζ ) = − z , D √ β (z 2 − ζ 2 )

(5.69)

D

with the Bessel function J1 (x) of first kind and order one. Combining the forward and backward transformation with the ideas of Lemma 2.2.12 and Corollary 2.2.2 to establish the dissipativity properties of an auxiliar system yields the following result. Proposition 5.4.1. Consider the semi-linear heat equation (5.59) and let the function μ(x) be linearly bounded, i. e. there exists a constant m > 0 such that |μ(x)| ≤ m|x|. Then the state x(z, t) converges exponentially to zero in the L2 -norm with convergence rate κ if the controller is given by (5.67) with β≥

2κ + Mc2 + m2 − 2Dπ 2 2

(5.70)

where Mc = (1+max |k(z, ζ )|)(1+max |l(z, ζ )|) with k(z, ζ ) given in (5.65) and l(z, ζ ) given in (5.69).

5.4 Backstepping-based dissipative control and observer design

| 153

Proof. Write the target dynamics in abstract notation over the Hilbert space 𝕏 dξ (t) = Aξ (t), dt

ξ (0) = ξ0

where A is given by A = D𝜕z2 − βI,

𝒟(A) = {ξ (z) ∈ 𝕏 | ξ (0) = ξ (1) = 0}.

The operator A is the infinitesimal generator of a contraction semigroup Tξ (t) satisfying 󵄩󵄩 󵄩 −γ t 󵄩󵄩Tξ (t)󵄩󵄩󵄩 ≤ e c ,

∀t ≥ 0

where γc = |λ1 | = Dπ 2 + β

(5.71)

is the modulus of the largest eigenvalue λ1 of A. Thus it holds that 󵄩󵄩 󵄩 󵄩 󵄩 −γ t 󵄩󵄩ξ (t)󵄩󵄩󵄩 = 󵄩󵄩󵄩Tξ (t)ξ0 󵄩󵄩󵄩 ≤ ‖ξ0 ‖e c ,

∀t ≥ 0.

Given that the kernel of the forward and backward transformations are bounded it holds that there exist constants M1 and M2 with 󵄨 󵄨 M1 = max 󵄨󵄨󵄨k(z, ζ )󵄨󵄨󵄨, z,ζ ∈[0,1]

󵄨 󵄨 M2 = max 󵄨󵄨󵄨l(z, ζ )󵄨󵄨󵄨 z,ζ ∈[0,1]

such that (using the Cauchy–Schwarz inequality) z z 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩x(z, t)󵄩󵄩󵄩 = 󵄩󵄩󵄩ξ (z, t) + ∫ l(z, ζ )ξ (ζ , t)dζ 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩ξ (z, t)󵄩󵄩󵄩 + ∫󵄩󵄩󵄩l(z, ζ )󵄩󵄩󵄩󵄩󵄩󵄩ξ (ζ , t)󵄩󵄩󵄩dζ 󵄩󵄩󵄩 󵄩󵄩󵄩 0 0 1

1

󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩ξ (z, t)󵄩󵄩󵄩 + ∫󵄩󵄩󵄩l(z, ζ )󵄩󵄩󵄩󵄩󵄩󵄩ξ (z, t)󵄩󵄩󵄩dζ ≤ 󵄩󵄩󵄩ξ (z, t)󵄩󵄩󵄩(1 + ∫󵄩󵄩󵄩l(z, ζ )󵄩󵄩󵄩dζ ) 0

󵄩 󵄩 󵄩 󵄩 ≤ (1 + M1 )󵄩󵄩󵄩ξ (z, t)󵄩󵄩󵄩 ≤ (1 + M1 )󵄩󵄩󵄩ξ0 (z)󵄩󵄩󵄩e−γc t z 󵄩󵄩󵄩 󵄩󵄩󵄩 󵄩 󵄩 ≤ (1 + M1 )󵄩󵄩󵄩x0 (z) − ∫ k(z, ζ )x0 (ζ )dζ 󵄩󵄩󵄩e−γc t 󵄩󵄩 󵄩󵄩 󵄩 󵄩 0

0

z

󵄩 󵄩 󵄩 󵄩󵄩 󵄩 ≤ (1 + M1 )(󵄩󵄩󵄩x0 (z)󵄩󵄩󵄩 + ∫󵄩󵄩󵄩k(z, ζ )󵄩󵄩󵄩󵄩󵄩󵄩x0 (ζ )󵄩󵄩󵄩dζ )e−γc t z

0

󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ (1 + M1 )(1 + ∫󵄩󵄩󵄩k(z, ζ )󵄩󵄩󵄩dζ )󵄩󵄩󵄩x0 (z)󵄩󵄩󵄩e−γc t ≤ (1 + M1 )(1 + M2 )󵄩󵄩󵄩x0 (z)󵄩󵄩󵄩e−γc t , ∀t ≥ 0

0

154 | 5 Infinite-dimensional systems or summarizing 󵄩 −γ t 󵄩 󵄩 󵄩󵄩 󵄩󵄩x(z, t)󵄩󵄩󵄩 ≤ Mc 󵄩󵄩󵄩x0 (z)󵄩󵄩󵄩e c ,

Mc = (1 + M1 )(1 + M2 ),

∀t ≥ 0.

(5.72)

Note that, on the other hand, x(z, t) is the solution of the abstract differential equation ẋ = Ac x + ν, ν = 0,

t > 0,

t ≥ 0,

x(0) = x0

with the operator Ac = D𝜕z2 ,

1

𝒟(Ac ) = {x ∈ 𝕏 | 𝜕z x(0) = 0, 𝜕z x(1) = k(1, 1)x(1) + ∫ 𝜕z k(1, ζ )x(ζ )dζ } 0

(5.73)

that according to the above considerations generates a C0 semigroup Tc (t) that satisfies 󵄩󵄩 󵄩 −γ t 󵄩󵄩Tc (t)󵄩󵄩󵄩 ≤ Mc e c ,

∀t ≥ 0

(5.74)

with Mc ≥ 1 defined in (5.72) and γc given in (5.71). In consequence, according to Lemma 2.2.12 the exponential stability of x = 0 follows if the auxiliary system χ̇ = −γc χ + Mc ϖ, t > 0,

σχ = χ,

t≥0

χ(0) = Mc ‖x0 ‖

(5.75a) (5.75b)

is (0, m, 1)-strictly state dissipative with dissipation rate κc > 0. According to Corollary 2.2.2 this is satisfied if [

−2γc + κc Mc − m

Mc − m ]≤0 −1

(5.76)

or equivalently, if γc >

κc + (Mc − m)2 . 2

(5.77)

Taking into account the relation of γc and β given in (5.71), this implies that the control design degree of freedom β must satisfy (5.70) with κ = κc /2. Note that the proof establishes sufficient conditions for the exponential convergence of the semi-linear system (5.59) with the linear backstepping control (5.67). The fact that Mc and γc are related in a non-trivial fashion with the value of β implies that the condition (5.70) is eventually conservative. Anyway, the present consideration shows that it is possible to achieve closed-loop stability for the semi-linear dynamics

5.4 Backstepping-based dissipative control and observer design

| 155

by means of a linear feedback controller and connects the backstepping control approach with the explicit dissipation assignment in the framework of closed-loop stabilization of Lur’e systems, similar to results in the line of reasoning of absolute stability theory. Note further that following the steps of the proof of Lemma 2.2.12 the input-tostate-stability property (Sontag, 1995; Karafyllis and Krstic, 2018) of the closed-loop linear subsystem can be easily assessed. To see this, consider the general solution t

x(z, t) = Tc (t)x0 (z) + ∫ Tc (t − τ)ν(z, τ)dτ.

(5.78)

0

Taking norms on both sides and developing the right-hand side yields t t 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ‖x‖(t) = 󵄩󵄩Tc (t)x0 (z) + ∫ T(t − τ)ν(z, τ)dτ󵄩󵄩 ≤ ‖Tc ‖O (t)‖x0 ‖ + ∫ ‖T‖O (t − τ)‖ν‖(τ)dτ 󵄩󵄩 󵄩󵄩 󵄩 󵄩 0 0 t

≤ Mc e−γc t ‖x0 ‖ + ∫ Mc e−γc (t−τ) ‖ν‖(τ)dτ. 0

Defining the variable ℵ(t) = Mc e

−γc t

t

‖x0 ‖ + ∫ Mc e−γc (t−τ) ‖ν‖(τ)dτ

(5.79)

0

it follows that ̇ = −γc ℵ(t) + Mc ‖ν‖(t), ℵ(t)

t > 0,

ℵ(0) = Mc ‖x0 ‖.

(5.80)

It follows that ‖x‖(t) ≤ ℵ(t) = Mc (e−γc t ‖x0 ‖ + (1 − e−γc t )

‖ν‖∞ ), γc

‖ν‖∞ = sup ‖ν‖(t), t∈ℝ

∀t ≥ 0.

Simulation results for the state-feedback controller For the purpose of illustrating the above control design, numerical simulation studies have been carried out for D = 2,

μ(x) =

x(1 − x 2 ) 1 + x4

(5.81)

implying that the nonlinearity μ(x) has a Lipschitz constant of Lμ = 1 and thus is linearly bounded according to 󵄨󵄨 󵄨 󵄨󵄨μ(x)󵄨󵄨󵄨 ≤ Lμ |x| = |x|.

156 | 5 Infinite-dimensional systems Accordingly, μ is (−1, 1, 0)-dissipative implying that, in virtue of Lemma 2.2.9 the map φ(x) = −μ(x) is (Q, S, R)-dissipative with Q = −1,

S = −1,

R = 0.

(5.82)

For the simulation of the open-loop and closed-loop system behavior the parabolic equation (5.59) has been spatially discretized using the finite-difference scheme and implemented with N = 50 collocation points. The integral term for the evaluation of the backstepping controller has been implemented by means of the trapezoidal rule. The resulting set of ordinary differential equations has been solved in Matlab using the standard solver ode15s. The simulation results for the unstable open-loop system are shown in Figure 5.13, showing the convergence of the state profile to a non-zero equilibrium profile in about 4 time units. The behavior of the closed-loop system with the proposed controller (5.67) with β = 3 is presented in Figure 5.14, showing that the controller stabilizes the system and it converges to the zero profile in about 1 time unit.

Figure 5.13: Open-loop behavior of the solutions of the semi-linear heat equation (5.59) with (5.81).

Figure 5.14: Closed-loop behavior of the solutions of the semi-linear heat equation (5.59) with (5.81) and the backstepping controller (5.67) with β = 3.

5.4 Backstepping-based dissipative control and observer design

| 157

Backstepping-based observer design Next, a backstepping-based observer is designed for the semi-linear unstable heat equation (5.59) and the exponential convergence is assessed on the basis of the dissipativity approach discussed in the previous sections. For this purpose, consider the observer 𝜕t x̂ = D𝜕z2 x̂ + μ(x)̂ − l(ŷ − y),

t > 0,

̂ ̂ t), y(t)= x(0,

t≥0

z ∈ (0, 1)

(5.83a) (5.83b)

̂ 0) = x̂0 and boundary conditions with initial condition x(⋅, ̂ − y(t)) ̂ t) = −l0 (y(t) 𝜕z x(0,

(5.83c)

̂ t) = u(t) 𝜕z x(1,

(5.83d)

for t ≥ 0. Introducing the associated observation error variables x̃ = x̂ − x,

̃ x)̃ = μ(x + x)̃ − μ(x) μ(x,

(5.84)

̃ 0) = 0 and the error dynamics with μ(x, ̃ x)̃ − lx(0, ̃ t), 𝜕t x̃ = D𝜕z2 x̃ + μ(x,

t > 0,

z ∈ (0, 1)

(5.85a)

̃ 0) = x̃0 and boundary conditions with initial condition x(⋅, ̃ t) ̃ t) = −l0 x(0, 𝜕z x(0, ̃ t) = 0, 𝜕z x(1,

(5.85b) (5.85c)

for t ≥ 0, the estimation error can be written in the interconnection form ̃ t) + ν,̃ 𝜕t x̃ = D𝜕z2 x̃ − lx(0, σ̃ = x̃

ν = −φ(x,σ)̃

t > 0, z ∈ (0, 1)

t ≥ 0, z ∈ (0, 1) t ≥ 0, z ∈ (0, 1)

(5.86a) (5.86b) (5.86c)

with boundary conditions ̃ t) = −l0 x(0, ̃ t) 𝜕z x(0, ̃ t) = 0 𝜕z x(1,

(5.86d) (5.86e)

for t ≥ 0 with ̃ σ)̃ φ(x,σ)̃ = −μ(x,

t ≥ 0,

z ∈ (0, 1).

(5.86f)

According to Lemma 2.2.12, if μ̃ is contained in a linear sector and the linear subsystem in (5.86) of the observation error is exponentially stable with sufficiently large

158 | 5 Infinite-dimensional systems convergence rate, than the dissipativity of the associated auxiliary system can be ensured according to Corollary 2.2.2 and thus the exponential convergence of the observer. For this purpose a backstepping-based observer is designed in the first step for achieving an assignable exponential convergence rate for the linear subsystem (i. e. with ν̃ = 0). Consider the target dynamics for the observation error 𝜕t ξ ̃ = D𝜕z2 ξ ̃ − βo ξ ̃

t ≥ 0,

z ∈ (0, 1)

(5.87a)

with initial condition ξ ̃ (⋅, 0) = ξ0̃ and boundary conditions 𝜕z ξ ̃ (0, t) = 0 𝜕 ξ ̃ (1, t) = 0

(5.87b) (5.87c)

z

for t ≥ 0. The solutions of these dynamics converge exponentially to zero with rate κ = βo , as can be verified by calculating the maximum eigenvalue of the operator à = D𝜕z2 − βo ,

𝒟(A)̃ = {x ∈ 𝕏 | 𝜕z x(0) = 𝜕z x(1) = 0}.

The transformation of (5.86) with ν = 0 into (5.87) is given by z

̃ t) = ξ ̃ (z, t) − ∫ p(z, ζ )ξ ̃ (ζ , t)dζ . x(z,

(5.88)

0

In Appendix D.3 it is shown that the kernel p(z, ζ ) has to satisfy the pde βo p(z, ζ ), D β d p(z, z) = o dz 2D 𝜕z p(1, ζ ) = 0,

𝜕z2 p(z, ζ ) − 𝜕ζ2 p(z, ζ ) = −

ζ ∈ (0, 1), z ∈ (ζ , 1)

(5.89a) (5.89b) (5.89c)

and that the observer gains are given by l(z) = D𝜕ζ p(z, 0),

l0 = p(0, 0).

(5.90)

The solution of this pde is derived in Appendix D.3 and is given by p(z, ζ ) = −

β

I1 (√ Do (z − ζ )(2 − (z + ζ ))) βo (1 − ζ ) . D √ βo (z − ζ )(2 − (z + ζ ))

(5.91)

D

Just as in the case of the state-feedback control addressed above, taking into account (5.66), the boundary observer gains l0 and l(z) can now be directly determined. For the boundary gain it follows that l0 = −

βo . 2D

(5.92)

| 159

5.4 Backstepping-based dissipative control and observer design

β

For the in-domain gain l(z) redefine α(z, ζ ) = √ Do (z − ζ )(2 − (z + ζ )) and note that 𝜕ζ

I1 (α(z, ζ )) I2 (α(z, ζ ) = 𝜕 α(z, ζ ), α(z, ζ ) α(z, ζ ) ζ

𝜕ζ α(z, ζ ) =

β

−βo (1 − ζ )

D√ Do (z − ζ )(2 − (z + ζ ))

.

Accordingly, one obtains from (5.90) that l(z) = βo [

β

I1 (√ Do (z − ζ )(2 − (z + ζ ))) √ βo (z D

− ζ )(2 − (z + ζ ))

√ βo (z − ζ )(2 − (z + ζ )))

I( 2 2

+ (1 − ζ )

D

(z − ζ )(2 − (z + ζ ))

].

(5.93)

The forward transformation is given by z

̃ t) + ∫ w(z, ζ )x(ζ ̃ , t)dζ ξ ̃ (z, t) = x(z,

(5.94)

0

and can be determined following the same procedure as above (see also Krstic and Smyshlyaev, 2008a). For this reason the explicit determination of the forward transformation is not addressed here, and in the sequel it will be directly assumed that it exists and it is bounded. Following the reasoning of the proof of Proposition 5.4.1 the following convergence result is obtained. Proposition 5.4.2. Consider the semi-linear heat equation (5.59) and let μ(̃ σ)̃ be con̂ t) be the solution of the observer dynamics (5.83), tained in the sector [m̃ − , m̃ + ]. Let x(z, ̂ t) converges with the gains l0 and l(z) given by (5.92) and (5.93), respectively. Then x(z, exponentially to x(z, t) in the L2 -norm if there exists γo > 0 so that βo ≥

̃ 2 γo + (Mo − m) , 2

󵄨 󵄨󵄨 󵄨 m̃ = max{󵄨󵄨󵄨m̃ − 󵄨󵄨󵄨, 󵄨󵄨󵄨m̃ + 󵄨󵄨󵄨}, Mo = (1 + max |p(z, ζ ))(1 + max(w(z, ζ ))

(5.95)

with p(z, ζ ) and w(z, ζ ) being the kernels of the backward and forward transformations (5.88) and (5.94), respectively. Proof. The proof follows exactly the steps of the proof of Proposition 5.4.1, with the differences that (i) the maximum eigenvalue of the semigroup To (t) generated by the observation target operator Ao = D𝜕z2 − βo ,

𝒟(Ao ) = {x ∈ 𝕏 | 𝜕z x(0) = 𝜕z x(1) = 0}

is given by λ0 = −βo and thus Tξ ̃ (t) is a contraction semigroup with growth bound ̃ ≤ m|̃ σ|. ̃ Accordingly, the auxiliary system in −βo , and (ii) the function μ̃ satisfies |μ(̃ σ)| Lemma 2.2.12 is given by ℵ̇ = −βo ℵ + Mo ϖ, χ=ℵ

t > 0,

ℵ(0) = Mo ‖x̃0 ‖

160 | 5 Infinite-dimensional systems and the exponential convergence of the observer is ensured if this system is (0, m,̃ 1)strictly state dissipative with dissipation rate γo > 0. According to Corollary 2.2.2 this is ensured when the inequality (5.95) holds true. Simulation results for the backstepping observer Simulations of the backstepping-based observer for the semi-linear heat equation (5.59) in open-loop operation have been carried out for the application example with D, μ defined in (5.81). The function μ̃ is Lipschitz continuous with Lipschitz constant Lμ = 1, so that m̃ = Lμ can be chosen and the feedback nonlinearity −φ(σ) is uniformly (Q, S, R)-dissipative with Q, S, R given in (5.82). Figure 5.15 shows the convergence of ̂ t) in about 2 time units to the actual state profile x(z, t) the observer state profile x(z, for the choice of βo = 10. Note that the initial profile x̂0 (z) = −0.5 belongs to the ̄ domain of attraction of the lower equilibrium profile x(z) = −1 and thus the observer convergence is only due to the correction mechanism.

Figure 5.15: State estimation (left) and observation error (right) of the backstepping observer for the semi-linear heat equation (5.59) with (5.81) in open-loop.

Output-feedback control The problem of output-feedback control is addressed next. For this purpose, the backstepping-based observer (5.83) with (5.92), (5.93) and feedback controller (5.67) are combined, leading to the output-feedback control law 1 β β I1 (√ D (1 − ζ 2 )) I2 (√ D (1 − ζ 2 )) ̂ t) β x(1, ̂ , t)dζ ) u(t) = − ( + ∫[ + ]x(ζ D 2 1 − ζ2 √ β (1 − ζ 2 ) 0

(5.96)

D

̂ t) is the solution of (5.83) with (5.92), (5.93). Note that, given the linearity of where x(z, the feedback operator, the controller can be written as ̂ t) = −Kx(z, t) − K x(z, ̃ t), u(t) = −K x(z,

(5.97)

5.4 Backstepping-based dissipative control and observer design

| 161

with Kx(z, t) =

1 β β I1 (√ D (1 − ζ 2 )) I2 (√ D (1 − ζ 2 )) β x(1, t) + ( + ∫[ ]x(ζ , t)dζ ). D 2 1 − ζ2 √ β (1 − ζ 2 ) 0

D

Accordingly, the closed-loop system can be written in the form (3.60) and by Theõ t)]T = 0 is exponentially stable. rem 3.4.1 it follows that the combined state [x(z, t), x(z, This is summarized in the next proposition. Proposition 5.4.3. Consider the semi-linear heat equation (5.59) with the backstepping controller (5.67) and the observer (5.83) with (5.92), (5.93). Let the conditions of Propositions 5.4.1 and 5.4.2 be satisfied. Then the zero profile x(z) = 0 of the closed-loop system with the observer-based output-feedback controller is exponentially stable. Simulation results for the observer-based output-feedback loop Simulation results for the backstepping-based observer (5.83) with (5.92), (5.93) for the case example (5.81) in closed-loop with the output-feedback controller (5.96) are shown in Figure 5.16, showing that the state profile of the system and the observer exponentially converge in about 1.5 time units, in accordance with Theorem 3.4.1.

Figure 5.16: State profile evolution (left) and state estimation (right) of the closed-loop system (5.59) with parameters (5.81) in closed-loop with the observer-based implementation of the feedback control (5.67) with βo = 10 and β = 3.

5.4.2 Control of a semi-linear partial integro-differential equation The study of dynamical systems described by partial integro-differential equations (pides) has gained increased focus during the last two decades motivated by analyzing particle size distribution in suspension reactors (Alvarez et al., 1994), cell population balances (Villadsen, 1999; Srienc, 1999; Mantzaris et al., 2002; Henson,

162 | 5 Infinite-dimensional systems 2003), evolution of option prices (Cont and Voltchkova, 2005), crystal growth processes (Motz et al., 2003; Vollmer and Raisch, 2002) or fish populations (Thompson and Cauley, 1999; Hernandez and Gasca-Leyva, 2003). Control approaches have been employed on different levels of complexity ranging from approximation-based finitedimensional (Mantzaris et al., 2002; Henson, 2003; Motz et al., 2003; Vollmer and Raisch, 2002) to distributed-parameter and optimal control approaches (Krstic and Smyshlyaev, 2008a,b; Khurshudyan, 2014; Frerik et al., 2014; Bribiesca-Argomedo and Krstic, 2015; Kumar et al., 2016; Meurer, 2016). For linear boundary control systems the backstepping control method proposed e. g. in Krstic and Smyshlyaev (2008a), Bribiesca-Argomedo and Krstic (2015) ensures the exponential convergence to zero of the control error. Nevertheless, as mentioned earlier in this text, only few results are reported for the application of backstepping-based controllers to semi-linear systems in general and for semi-linear pides in particular no previous studies are known to the author besides the recent work presented in Schaum and Meurer (2019a, 2020) which is put here in the perspective of the presented general framework. Problem statement Consider the following class of semi-linear partial integro-differential equation systems z

𝜕t x = 𝜕z x + ∫ f x dζ − μ(x),

t > 0,

z ∈ (0, 1)

(5.98a)

0

with initial condition x(⋅, 0) = x0 and boundary condition x(1, t) = u(t),

t ≥ 0,

(5.98b)

where t is time, z is space, x(⋅, t) ∈ 𝕏 = L2 (0, 1) is the state at time t ≥ 0, f : [0, 1] × [0, 1] → ℝ is a weighting function, μ: ℝ → ℝ is a smooth non-linear map, and u(t) ∈ ℝ is the control input acting on the right boundary at time t ≥ 0. In the sequel the considered problem consists in designing the control u in such a way that the zero solution x = 0 becomes exponentially stable. For this purpose the system is decomposed into a linear dynamic subsystem in feedback interconnection with a nonlinear static map according to z

𝜕t x = 𝜕z x + ∫ f x dζ + ν,

t > 0,

z ∈ (0, 1)

(5.99a)

t ≥ 0,

z ∈ (0, 1)

(5.99b)

0

σ = x,

ν = −μ(σ),

t ≥ 0,

z ∈ (0, 1)

(5.99c)

with boundary condition x(1, t) = u(t),

t ≥ 0.

(5.99d)

5.4 Backstepping-based dissipative control and observer design

| 163

This form fits directly into the dissipativity-based framework developed above and it can be already asserted that if the nonlinear map μ is (Q, S, R)-dissipative and the control u(t) can be designed such that the linear subsystem is (−R, S∗ , −Q)-strict state dissipative with a positive dissipation rate κ > 0, then the exponential stability of the solution x = 0 is ensured. Having in mind the relations between exponential stability and dissipativity which have been exploited in the design procedures in the previous chapters it is known that, if the linear subsystem is exponentially stable for ν = 0 with a sufficiently large convergence rate γ, than the required dissipativity property is established. Following the discussion in Krstic and Smyshlyaev (2008b,a), for ν(z, t) = 0 the linear system can be transformed into the target system 𝜕t ξ = 𝜕z ξ , t > 0,

ξ (1, t) = 0,

t≥0

z ∈ (0, 1)

(5.100a) (5.100b)

where ξ (z, t) is given by the Volterra integral transformation z

ξ (z, t) = x(z, t) − ∫ k(z, ζ )x(ζ , t)dζ ,

∀t ≥ 0, z ∈ [0, 1].

(5.101)

0

The solutions of (5.100) converge to zero in the finite time t∗ = 1. Before discussing the associated pde for the kernel and deriving the control input, note that the target pde is a first-oder hyperbolic equation and that the solutions move along characteristic curves (in this case actually straight anti-diagonal lines from right to left) in the (z, t) plane. The associated differential operator has no discrete spectrum and thus, in comparison with the parabolic case discussed above, no explicit bound for the exponential stability assessment. From the finite time convergence it can be seen that one can find a family of pairs (M, γ) so that (cp. Figure 5.17) 󵄩󵄩 󵄩 󵄩 󵄩 −γt 󵄩󵄩ξ (z, t)󵄩󵄩󵄩 ≤ M 󵄩󵄩󵄩ξ0 (z)󵄩󵄩󵄩e ,

∀t ≥ 0,

󵄩 󵄩 󵄩 󵄩 M 󵄩󵄩󵄩ξ0 (z)󵄩󵄩󵄩e−γt∗ = 󵄩󵄩󵄩ξ0 (z)󵄩󵄩󵄩

t∗ =1

⇔

M = eγ (5.102)

meaning that γ and M will always be coupled in the sense that the larger γ is chosen the larger the amplitude M will be. This fact makes it difficult to assess the problem at hand with the above discussed results. To achieve a more suitable exponential stability bound consider the weighted L2 norm for the target dynamics 1

1 2 V(ξ ) = ∫ w(z)[ξ (z, t)] dz, 2

w(z) > 0 ∀z ∈ [0, 1].

0

It holds that w− w+ ‖x‖2 ≤ V(ξ ) ≤ ‖x‖2 2 2

164 | 5 Infinite-dimensional systems

Figure 5.17: Illustration of the relation between finite-time convergence in norm and exponential stability according to (5.102).

with w− = min w(z), z∈[0,1]

w+ = max w(z). z∈[0,1]

Calculating the associated rate of change in time yields 1

dV (ξ ) = ∫ w(z)ξ (z, t)𝜕z ξ (z, t)dz dt 0

2

1

1

2

= w(z)[ξ (z, t)] |10 − ∫ 𝜕z w(z)[ξ (z, t)] dz − ∫ w(z)𝜕z ξ (z, t)ξ (z, t)dz 0 0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ = dV (ξ ) dt

so that 1

1 dV 2 2 (ξ ) = (w(z)[ξ (z, t)] |10 − ∫ 𝜕z w(z)[ξ (z, t)] dz) dt 2 0

1

1 2 2 2 = (w(1)[ξ (1, t)] − w(0)[ξ (0, t)] − ∫ 𝜕z w(z)[ξ (z, t)] dz) 2 1

0

1 2 2 ≤ (w(1)[ξ (1, t)] − ∫ 𝜕z w(z)[ξ (z, t)] dz) 2 1

0

1 2 ≤ − ∫ 𝜕z w(z)[ξ (z, t)] dz 2 0

where in the last step the boundary condition (5.100b) has been employed. For weights w(z) > 0 satisfying 𝜕z w(z) ≥ μ > 0

5.4 Backstepping-based dissipative control and observer design

| 165

it follows directly that μ μ dV (ξ ) ≤ − ‖ξ ‖2 ≤ − + V(ξ ) dt 2 w implying that μ μ 2 w+ 2 󵄩2 󵄩󵄩 − t 2 − t 󵄩󵄩ξ (z, t)󵄩󵄩󵄩 ≤ − V(ξ ) ≤ − V(ξ0 )e w+ ≤ − ‖ξ0 ‖ e w+ , w w w

∀t ≥ 0.

This in turn implies the exponential stability in the L2 norm of the zero profile according to3 󵄩󵄩 󵄩 󵄩 −γt 󵄩 󵄩󵄩ξ (z, t)󵄩󵄩󵄩 ≤ Me 󵄩󵄩󵄩ξ0 (z)󵄩󵄩󵄩,

M=√

w+ , w−

γ=

μ , 2w+

⇔

w+ = w(0)eμ

∀t ≥ 0.

3 Note that if the weight satisfies 𝜕z w(z) = μw(z),

w(z) = w(0)eμz

⇔

and w(0) = 1 is chosen then it holds that dV (ξ ) ≤ −μV(ξ ) dt and the exponential stability follows with M=√

w+ = eμ/2 , w−

γ=

μ . 2

This is actually the same as has been derived above using a simple geometric argumentation. Furthermore, this Lyapunov function has been used e. g. in Bastin and Coron (2016). Accordingly, with this choice for large μ > 0 a fast convergence with rate γ but also a large amplitude M is obtained. On the other hand, using the linear weighting function w(z) = w(0) + μz,

⇒

M=√

w(0) + μ w(1) =√ , w(0) w(0)

lim M = 1

w(0)→∞

and with w(0) = 1 M=√

1+μ , 1

γ=

μ 2(1 + μ)

yields the result μt w(0) + μ − 2(w(0)+μ) 󵄩󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩ξ (z, t)󵄩󵄩󵄩 ≤ √ 󵄩󵄩x0 (z)󵄩󵄩󵄩. e 󵄩 󵄩 󵄩 󵄩 w(0)

In comparison to the exponential weighting function this choice yields a slower convergence assessment but with a smaller amplitude.

166 | 5 Infinite-dimensional systems To ensure that the transformed state profile w(z, t) satisfies (5.100) the kernel k(z, ζ ) has to satisfy the pde (see Appendix D.4 for the derivation) z

𝜕ζ 𝜕z k(z, ζ ) + 𝜕ζ2 k(z, ζ ) = −k(z, ζ )f (ζ , ζ ) + ∫ k(z, s)𝜕ζ f (s, ζ )ds − 𝜕ζ f (z, ζ )

(5.103a)

ζ

for z ∈ (0, 1), ζ ∈ (0, z) and k(z, 0) = 0

(5.103b) z

k(z, z) = − ∫ f (ζ , ζ )dζ .

(5.103c)

0

The associated control input u(t) that achieves the desired finite-time convergent behavior for ν = 0 is calculated from the transformation (5.101) and the boundary condition (5.100b) yielding 1

u(t) = ∫ k(1, ζ )x(ζ , t)dζ ,

t ≥ 0.

(5.104)

0

Application example In order to illustrate the general idea presented above, consider the particular integral kernel and nonlinearity f (z, ζ ) = aeb(z−ζ ) ,

μ(x) =

x . 1 + x2

(5.105)

According to Appendix D.4 the kernel of the backstepping transformation (5.101) is given by k(z, ζ ) = −aζ

I1 (2√az(z − ζ )) b(z−ζ ) e , √az(z − ζ )

where I1 is the modified Bessel function of first kind and order 1. The state-feedback controller is accordingly given by 1

u(t) = −a ∫ ζ 0

I1 (2√a(1 − ζ )) b(1−ζ ) e x(ζ , t)dζ . √a(1 − ζ )

(5.106)

As shown in Appendix D.5 the inverse backstepping transformation is given by z

x(z, t) = ξ (z, t) + ∫ l(z, ζ )ξ (ζ , t)dζ 0

5.4 Backstepping-based dissipative control and observer design

| 167

with the kernel l(z, ζ ) = −az

J1 (2√aζ (z − ζ )) b(z−ζ ) e , √aζ (z − ζ )

with the Bessel function J1 or order 1. Given that both, the forward and backward transformation are bounded, and that the exponential stability is ensured according to the above considerations, Lemma 2.2.12 and Corollary 2.2.2 can be employed to derive sufficient conditions depending on the convergence rate γ and the amplitude M. Taking into account that μ is linearly bounded with a constant slope m = 1 a direct application of Corollary 2.2.2 yields the condition for (−1, 0, 1) strict state dissipativity of the associated auxiliary system ℵ̇ = −γℵ + Mν, χ = ℵ,

t > 0,

t≥0

ℵ(0) = ℵ0

with dissipation rate κ κ < 2γ − 1 − M 2 . In order to illustrate the performance of the backstepping-based controller for the semi-linear partial integro-differential equation system (5.98) numerical simulations have been carried out on the basis of a finite-difference discretization with N = 50 collocation points and using standard trapezoidal quadrature rule for the case a = 1.6,

b = 1.2.

(5.107)

The resulting unstable open-loop state profile evolution is shown in Figure 5.18. Note that the nonlinearity introduces additional instability into the system, but that

Figure 5.18: Open-loop unstable state profile evolution of (5.98) with (5.105), (5.107).

168 | 5 Infinite-dimensional systems the zero profile is already unstable for the linear partial integro-differential subsystem (5.99a) with ν = 0. The solution of the closed-loop system is shown in Figure 5.19 illustrating the exponential stability of the zero solution x = 0. The associated control u is shown separately in Figure 5.20.

Figure 5.19: Closed-loop state profile evolution of (5.98) with (5.105), (5.107) with the state-feedback controller 5.106.

Figure 5.20: Time behavior of the control input for the simulation example.

6 Conclusions and outlook In the preceding chapters it has been shown how the framework of dissipativity theory can be used in designing observers and state- and output-feedback controllers for nonlinear finite- and infinite-dimensional systems. The main theoretical tool which has been employed in all the different application examples is the direct method of Lyapunov which allows to establish constructive bounds on the convergence behavior. The usefulness of this framework resides in particular in the fact that complex interconnections of linear and nonlinear systems can be handled with simple methods based on conventional design approaches when some dissipativity properties of the system components are known. In particular this has been exploited for the case when the system can be viewed as an interconnection of a static nonlinear map with a quadratic dissipativity property and a linear or nonlinear system for which geometric or constructive control approaches can be employed. For example, in the case of interconnections of linear dynamical and nonlinear static systems satisfying a Lipschitz condition or, additionally a sector bound, it has been shown that it is sufficient to assign the dominant eigenvalue of the linear system accordingly. When the dynamic component is a nonlinear finite-dimensional system, then the design can be based e. g. on an input-output linearization when the internal dynamics are exponentially stable with sufficiently fast convergence rate. In particular, the usefulness of the presented approach becomes clear in the context of infinite-dimensional semi-linear systems, where most design approaches, like spectral (eigenvalue) decomposition or (linear) backstepping, are applicable only to linear systems. In these cases the known approaches can be used to assign sufficiently fast convergence rates for the semi-group generated by the linear closed-loop operator, so that the interconnection with a nonlinear static map remains exponentially stable. Given that the nonlinear map is treated in this context as a perturbation, with potentially destabilizing effects, the results presented can be interpreted as absolute stability results, i. e. answers to the question under which circumstance the linear systems stability is sufficient to conclude the stability of the linear-nonlinear system interconnection. Nevertheless, in many cases nonlinearities are quite beneficial for the closed-loop stability. In such cases the results presented here should be adapted in such a way that the beneficial effects are adequately accounted for. The framework of (Q, S, R)-dissipativity used throughout this study includes such situations up to some extend. Besides the named advantages for the design of observers and state-feedback controllers, a particularly useful property resulting from this approach is the closed-loop stability of observer-based output-feedback controllers in a way similar to the separation principle for linear systems. Additional to the presented results, which do not consider the case of model uncertainties, exogenous perturbations, unknown inputs or stochastic processes the https://doi.org/10.1515/9783110677942-007

170 | 6 Conclusions and outlook framework can be extended to consider such problems. In particular at this stage the model-free design approach should be mentioned which has been shown in several places to be able to directly connect to the constructive (i. e. passivity-based) control design. Some illustrative case studies where this connection has been exploited are distillation columns (Gonzalez and Alvarez, 2005; Castellanos-Sahagun and Alvarez, 2006), exothermic chemical reactors (Diaz-Salgado et al., 2007, 2012; Schaum et al., 2008c), and biological reactors (Schaum et al., 2012a,b, 2013a, 2015a; Garcia Sandoval et al., 2016). From the preceding considerations it is clear that there are still several open questions and fields where the proposed framework should be extended and applied to. One interesting class of systems are networked control systems. It is known that the (correct) interconnection of passive systems remains passive, so that the stability properties of passive systems are inherited to the network of systems. Dissipative systems present a direct generalization to this concept and thus the preceding results should be clearly possible to extend to the network case. Some interesting questions hereby are how relevant the network topology is in this context and which topologies may be more suitable to obtain dissipativity-based stability results. The question on the robustness properties of the presented approaches or adaptations of them to account for model uncertainties, stochastic perturbations, etc. also provide important and interesting research areas. In the realm of distributed parameter systems there are also still many open questions to be answered. For example a class of systems which have not been considered so far are semi-linear wave equations. Quasi-linear hyperbolic systems, as those described by the Burger’s equation for example should also be addressed. Coupled odepde systems and higher-dimensional problems will also provide a rich area for future studies. Besides these areas, new system classes should be considered. In particular, there are recent extensions of the dissipative observer design approach to discrete-time systems (Aviles and Moreno, 2018) and consequently extensions of the observer-based feedback control approaches should appear within a short time horizon. Furthermore, it will be interesting to study extensions to impulsive and hybrid systems (Haddad et al., 2006, 2001a,b) and delay equations (Chellaboina et al., 2005; Kharitonov, 2013; Fridman, 2014). Another area of interesting extensions could be opened using recent concepts from dominance analysis (Forni and Sepulchre, 2019) for studying stability of nonequilibrium solutions. The main messages are that the dissipativity framework is quite flexible with respect to the particular design approaches and that the design problem for complex system interconnections can be carried out using a kind of divide and conquer approach when the appropriate interconnection topology and the right stability and input-output properties are given or can be assigned for each of the components.

A Lemmata on quadratic forms In this chapter some useful results from linear algebra with respect to the analysis of quadratic forms are recalled and proven that are frequently used in the main text.

A.1 Quadratic forms Consider the quadratic form associated to a matrix A given by ⟨x, Ax⟩ = x T Ax.

(A.1)

Note further that A can always be written as 1 1 A = (A + AT ) + (A − AT ) 2 2 and thus the quadratic form (A.1) is equivalent to 1 1 x T Ax = x T (A + AT )x + x T (A − AT )x. 2 2 Further it holds that x T (A − AT )x = x T Ax − x T AT x = x T (Ax) − (Ax)T x = x T (Ax) − [x T (Ax)]

T

= x T (Ax) − x T (Ax) = 0 given that x T (Ax) ∈ ℝ and thus x T (Ax) = [x T (Ax)]T . Accordingly, it holds that 1 ⟨x, Ax⟩ = x T Ax = x T (A + AT )x. 2

(A.2)

Thus, in the sequel it is assumed, that the matrix A is symmetric, because, if not, the quadratic form is identical to the one of the symmetric complement 21 (A + AT ). Note that a quadratic form is positive (or negative) definite (see Definition 2.1.6) if and only if the symmetric matrix A (or its symmetric complement 21 (A + AT ) when A is not symmetric) has only positive (or negative) eigenvalues. The proof of this statement relies on the fact that symmetric n × n matrices have only real eigenvalues and are diagonalizable. Thus there exists a basis of eigenvectors which span ℝn . Expressing the vector x in the eigenvector basis yields the result after some manipulation (see e. g. Dym 2007). It must be noted that for a non-symmetric matrix A with only positive eigenvalues, the associated quadratic form is not necessarily positive definite.1 1 A counterexample is given by the matrix 1 0

A=[

−4 ], 1

σ(A) = {1, 1},

1 1 1 1 ⟨[ ] , A [ ]⟩ = ⟨[ ] , [ 1 1 1 −2

−2 1 ] [ ]⟩ = −2 < 0, 1 1

showing that, even though σ(A) ⊂ ℝ+ the quadratic form can achieve negative values. https://doi.org/10.1515/9783110677942-008

172 | A Lemmata on quadratic forms Note that a matrix A which is diagonalizable and whose eigenvectors are mutually orthogonal satisfies the property that if all eigenvalues are positive (or negative) the associated quadratic form is positive (or negative) definite. This results from the fact, that for a diagonalizable matrix there exists a basis of ℝn of mutually orthogonal eigenvectors, and thus the same idea can be exploited that is mentioned above. In particular, there exists an orthogonal matrix V with inverse V T = V −1 so that the matrix VAV T = Λ is diagonal and thus symmetric, and the quadratic form can be written as x T Ax = x T V T VAV T Vx = (Vx)T VAV T (Vx) = ξ T Λξ with ξ = Vx. Accordingly, if λ has only positive (or negative) entries than for all 0 ≠ ξ ∈ ℝn the quadratic form ξ T Λξ is positive (or negative). By the invertability of V this implies that for all 0 ≠ x ∈ ℝn the original quadratic form is positive (or negative) as well.2 For the purpose of control or observer design discussed in the preceding chapters, this implies that as long as the eigenvalues of the closed-loop matrices AL = A − LC or AK = A − BK are chosen differently with mutually orthogonal eigenvectors, the associated matrices are diagonalizable and thus the definiteness properties directly relate to the sign of the eigenvalues.

A.2 Schur complement The Schur complement is a very useful construct in the analysis of the definiteness of quadratic forms and used several times in the main text. For this reason some major results for the Schur complement are summarized next. The interested reader is also referred to, e. g. Dym (2007). Lemma A.2.1. Consider the block matrix M M = [ 11 M21

M12 ] M22

with M11 ∈ ℝn1 ×n1 , M12 ∈ ℝn1 ×n2 , M21 ∈ ℝn2 ×n1 , M22 ∈ ℝn2 ×n2 , n1 + n2 = n. Then the following holds: −1 i. If det(M11 ) ≠ 0, then det(M) = det(M11 ) det(Δ11 ) with Δ11 = M22 − M21 M11 M12 . FurT T thermore, if M = M then M > 0 if and only if M11 = M11 > 0 and Δ11 > 0, −1 ii. If det(M22 ) ≠ 0, then det(M) = det(M22 ) det(Δ22 ) with Δ22 = M11 − M12 M22 M21 . T T Furthermore, if M = M then M > 0 if and only if M22 = M22 > 0 and Δ22 > 0, where Δii is the Schur complement of Mii , i = 1, 2. 2 Note that in the counterexample given in footnote 1, the matrix A is not diagonalizable because the geometric multiplicity of the eigenvalue λ = 1 is one while its algebraic multiplicity is two.

A.2 Schur complement | 173

Proof. A simple calculation shows that if det(M11 ) ≠ 0 the matrix M can be decomposed as −1 In ×n 0 In ×n M11 M12 M 0 ][ 1 1 M = [ 1 1−1 ] [ 11 ], 0 Δ11 M21 M11 In2 ×n2 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 0 In2 ×n2 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ =:N1

=:U1

(A.3)

=:V1

−1 with Δ11 = M22 − M21 M11 M12 and In×n denotes the (n × n) identity matrix. Given the triangular structure of U1 and V1 their determinants are equal to one. Consequently,

det(M) ≤ det(U1 ) det(N1 ) det(V1 ) = det(N1 ). On the other hand the matrices U and V are invertible with U1−1 = [

In1 ×n1

0

−1 −M21 M11

In2 ×n2

],

In1 ×n1

V1−1 = [

−1 −M11 M12

0

In2 ×n2

]

so that det(U1−1 ) = det(V1−1 ) = 1 holds true. Writing N1 = U1−1 MV1−1 it follows that det(N1 ) ≤ det(U1−1 ) det(M) det(V1−1 ) = det(M) which in combination with the previously shown fact that det(M) ≤ det(N1 ) implies that det(M) = det(N1 ) = det(M11 ) det(Δ11 ). To see the assertion about the definiteness, let M = M T > 0 and consider the quadratic form x T Mx = [x T1

x x T2 ] M [ 1 ] = x T1 M11 x 1 + 2x T1 M12 x 2 + x T2 M22 x 2 > 0 x2

∀ x ∈ ℝn

with x 1 ∈ ℝn1 and x 2 ∈ ℝn2 . Accordingly, the minimum value attained over x 1 is also positive. This minimum is achieved for T T 𝜕x 1 (x T Mx) = x T1 (M11 + M11 ) + 2x T2 M12 = 0.

Now, let M = M T > 0. Taking the particular case that x = [x 1 0]T , it follows that T > 0, showing that M11 = M11 > 0 is necessary. Consequently, M11 is invertible and the minimum of the quadratic form is attained for

x T1 M11 x 1

−1 x 1 = x ∗1 (x 2 ) = −M11 M12 x 2 .

174 | A Lemmata on quadratic forms −1 T −1 Taking into account that M11 = (M11 ) the value at the minimum is calculated as ∗ ∗T T x ∗T 1 M11 x 1 + 2x 1 M12 x 2 + x 2 M22 x 2 T −1 −1 T −1 = x T2 (M12 M11 )M11 (M11 M12 x 2 ) − 2x T2 (M12 M11 )M12 x 2 + x T2 M22 x 2 T −1 = x T2 (M22 − M12 M11 M12 )x 2

= x T2 Δ11 x 2 > 0, T showing that Δ11 > 0 is also necessary. Thus, M = M T > 0 implies M11 = M11 > 0 and Δ11 > 0. T For the sufficiency, let M11 = M11 > 0 and Δ11 > 0, so that U1 , V1 ∈ ℝn×n exist so that M can be written in the form of the decomposition (A.3). Consider the quadratic form

x T Mx = x T U1 N1 V1 x with U1 , N1 , V1 defined in (A.3). Then, because M11 > 0 and Δ11 > 0, it holds that x T Mx = [x T1 = [ξ T1

In1 ×n1 −1 M21 M11

x T2 ] [

M x T2 ] [ 11 0

0

M ] [ 11 In2 ×n2 0

I 0 ] [ n1 ×n1 0 Δ11

ξ 0 ] [ 1] > 0 Δ11 x 2

−1 M11 M12 x 1 ][ ] In2 ×n2 x2

−1 where ξ 1 = x 1 + M11 M12 x 2 ∈ ℝn1 . Thus x T Mx > 0 for arbitrary x ∈ ℝn what implies that M > 0. This completes the proof of assertion i). The proof of assertion ii) goes along the same lines considering that if det(M22 ) ≠ 0 the matrix M can be written as −1 I 0 I M12 M22 Δ 0 ] [ n−11 ×n1 ] M = [ n1 ×n1 ] [ 22 M M I 0 I 0 M n2 ×n2 n2 ×n2 22 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 22 21 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ =:U2

=:N2

(A.4)

=:V2

−1 with Δ22 = M11 − M12 M22 M21 . By the same arguments as above it follows that

det(M) = det(N2 ) = det(M22 ) det(Δ22 ). T For the proof of the definiteness, note that M = M T > 0 implies that M22 = M22 > 0. T Taking the minimum of the quadratic form x Mx > 0 with respect to x 2 it follows that the minimum value is given by x T1 Δ22 x 1 > 0, showing that Δ22 > 0 holds true. The T sufficiency is shown by considering M22 = M22 > 0, Δ22 > 0 and using the decomposition (A.4) to show that

x T Mx = [x T1

Δ ξ T2 ] [ 22 0

0 x ] [ 1] > 0 M22 ξ 2

−1 with ξ 2 = M22 M21 x 1 + x 2 ∈ ℝn2 . This implies that M > 0.

B Kalman decomposition for observer design B.1 The Kalmann observability criterion Consider a linear time-invariant finite-dimensional system governed by the dynamics ẋ = Ax + Bu,

t > 0,

y = Cx,

t≥0

x(0) = x 0

(B.1) (B.2)

with solution t

x(t) = S(t)x 0 + ∫ S(t − τ)Bu(τ)dτ,

t≥0

0

and S(t) = exp(At) being the semi-group generated by A. In the case that rank(C) = m < n, we have m equations y(t) = Cx(t) so that for determining x(t) (and thus x 0 ) we need n−m additional equations. Knowing the signal y(t) over the time interval [0, T], we know, in principle, its derivatives and can establish the set of equations y(t) = Cx(t) ̇ = C x(t) ̇ = CAx(t) + CBu(t) y(t) ̈y(t) = CA2 x(t) + CABu(t) + CBu(t) ̇ .. . n−2−i y (n−1) (t) = CAn−1 x(t) + ∑n−2 Bu(i) (t). i=0 CA

(B.3)

mk = CAk−1 B

(B.4)

The coefficient

is also known as k-th Markov parameter and weights the influence of the i-th derivative of u in the (k + i)-th derivative of the output y. Rearranging equation (B.3) so that on the left known terms and on the right the unknown ones appear, one obtains the equation 𝒴(t) − 𝒰 (t) = 𝒦o x(t)

(B.5)

with the known vectors y(t) ̇ y(t) .. . (n−1) y (t) [

[ [ 𝒴(t) = [ [ [

] ] ], ] ] ]

https://doi.org/10.1515/9783110677942-009

0 m1 u(t) .. . n−2 (n−2−i) m (t) ∑ [ i=0 n−1−i u

[ [ 𝒰 (t) = [ [ [

] ] ] ] ] ]

(B.6)

176 | B Kalman decomposition for observer design and the known matrix C CA .. . n−1 CA [

[ [ 𝒦o (A, C) = [ [ [

] ] ] ] ]

∈ ℝmn×n .

(B.7)

]

This matrix is called Kalman observability matrix. The system of mn equations (B.5) has a unique solution x(t) (and thus a unique solution for x 0 ) if and only if 𝒦o has rank n, i. e. rank(𝒦o ) = n.

(B.8)

Given that the matrix 𝒦o is independent of time, it follows that the observability property for linear time invariant systems is independent of the final time T. This is summarized in the following theorem. Theoren B.1.1. The system (B.1) is completely observable in time T > 0 if and only if the Kalman observability matrix 𝒦o (B.7) has rank(𝒦o ) = n. Moreover, in this case it is observable for any T > 0. To be more specific, let the system be completely observable and consider the quadratic matrix1 𝒦O,m consisting of (arbitrary) n linearly independent rows of the nm × n matrix 𝒦o . Then −1 x(t) = 𝒦O,m (𝒴(t) − 𝒰 (t))

and thus with the semigroup S(t) = eAt it follows that t

x0 =

−1 S(−t)(𝒦O,m (𝒴 (t)

− 𝒰 (t)) − ∫ S(t − τ)Bu(τ)dτ) 0

The rank of the Kalman observability matrix is independent of the choice of the basis of the state space, i. e. for any regular transformation T it follows that ̃ Ã = TAT −1 , C̃ = CT −1 ⇒ rank(𝒦o (A, C)) = rank(𝒦O (A,̃ C)).

(B.9)

B.2 Detectability Clearly, observability is a property which depends on the inherent interaction mechanisms between the system states xi and the measurement (or output) y, and thus 1 Alternatively, the pseudo-inverse could be used here.

B.2 Detectability | 177

depends as well as on the matrix C as on the structure of A. Given that observability represents a central property in system monitoring and feedback control problems, the choice of an adequate sensor for a given system has to be carefully analyzed. In many cases the system is not observable, and there do not exist sensors which whould enhance the observability properties of the system. For these cases, the associated weaker concept of detectability (as introduced in Definition 2.3.2) is important. For the purpose at hand consider the following definition which in the case of a linear system is equivalent to Definition 2.3.2. Definition B.2.1. The pair (A, C) is called detectable if (with u = 0) 󵄩 󵄩 y ≡ 0 ⇒ lim 󵄩󵄩󵄩x(t)󵄩󵄩󵄩 = 0. t→∞

(B.10)

Note that the first condition (y ≡ 0) implies that y and its derivatives are zero, i. e. 𝒴(t) = 0 with 𝒴 defined in (B.6). In the case that the system is completely observable this implies that x(t) ≡ 0, or equivalently x 0 = 0 when u(t) = u for all t ≥ 0. In the case that the system is not completely observable the observability matrix has rank(𝒦o ) = r < n and it follows from (B.5) (with u ≡ 0) that y ≡ 0 implies that x(t) lies within the nullspace 𝒩O of the observability matrix 𝒦o , or equivalently of 𝒦o,m . The complement of 𝒩O in ℝn is called the observable subspace, and we will denote it by 𝒪 ⊆ ℝn . Clearly, if the system is completely observable, the nullspace 𝒩O = {0} and 𝒪 = ℝn . Now, consider the map defined by the transposed of the observability matrix 𝒦o , i. e. T

n×mn

𝒦o ∈ ℝ

: Y → ℝn

where Y denotes a generalized space of output functions y : ℝ → ℝm with their first n − 1 successive time derivatives, in the sense of the vector 𝒴 in (B.5). Clearly, 𝒴 ∈ Y. It follows that 𝒪 is the image of the map 𝒦oT , i. e. T

(B.11)

𝒪 = ℛ(𝒦o )

with dimension equal to the number of independent rows of 𝒦o . By the same reasoning it follows that 𝒩o = ker(𝒦o ). Note that AT 𝒦oT = [AT C T

AT (CA)T

= [(CA)T

(CA2 )T

⋅⋅⋅ ⋅⋅⋅

AT (CAn−1 )T ] (CAn )T ]

and, in virtue of the Theorem of Cayleigh-Hamilton (Dym, 2007), it follows that T

T

T

ℛ(A 𝒦o ) ⊆ ℛ(𝒦o ) = 𝒪,

(B.12)

178 | B Kalman decomposition for observer design meaning that the observable subspace 𝒪 is AT -invariant, i. e. for all x ∗ ∈ 𝒪 it follows that AT x ∗ ∈ 𝒪. The AT -invariance of 𝒪 is important in the following reasoning. Given that dim(𝒪) = rank(𝒦oT ) = r, there are r linearly independent column vectors ρi , i = 1, . . . , r of the matrix 𝒦oT that form a basis of the r-dimensional subspace 𝒪 ⊂ ℝn , with the orthogonal complement 𝒩O = ℝn ∖ 𝒪. Now, let {κi , i = 1, . . . , n − r} be a basis of this complement 𝒩O . Define the transformation T = [ρ1

ρr

⋅⋅⋅

κ1

κn−r ] ,

⋅⋅⋅

ξ = T −1 x

(B.13)

and the associated matrices à = T −1 AT,

C̃ = CT.

Denote by 𝒪̃ = T −1 𝒪, 𝒩̃ O = T −1 𝒩O , and note that this transformation implies that the first r components of the vector ξ = T −1 x correspond to components in the r-dimensional observable subspace, while the remaining n − r components correspond to the non-observable subspace, i. e. ξ ξ o = [ 1 ] ∈ 𝒪̃ , 0

0 ξ no = [ ] ∈ 𝒩̃ O ξ2

(B.14)

with ξ 1 ∈ ℝr , ξ 2 ∈ ℝn−r . Thus, for any ξ ∈ T −1 𝒪 it holds that à T ξ ∈ 𝒪̃

⇔

à T ξ o = [

à T1,1 à T

à T2,1 ξ 1 AT1,1 ξ 1 ] [ ] = [ ] à T 0 AT ξ

1,2

1,2 1

2,2

but à T ξ o ∈ 𝒪̃ and thus AT1,2 = 0. This implies that the matrix à is of the form à 1,1 à = [ à 2,1

à T1,1

0 ], Ã 2,2

à T = [

0

à T2,1 ] à T 2,2

(B.15)

Further, as the image of C T (or C̃ T ) lies in 𝒪 (or 𝒪̃ ), it follows that C̃ = [c̃1

⋅⋅⋅

c̃r

0

⋅⋅⋅

0] .

(B.16)

The matrix B̃ does not have a particular structure in the general case. This shows, that any linear time-invariant system of the form (B.1) can be brought by a regular state transformation into a form in which the observable part is decoupled from the unobservable one. In terms of this decomposition, the detectability property can now be interpreted easily. ̃ ≡ 0. Having in mind the form of the matrices Consider y = Cx ≡ 0, and thus ỹ = Cξ Ã and C,̃ it follows that ξ 1 , . . . , ξ r ≡ 0. Thus, the remaining (possibly non-zero) state is

B.3 A permutated observer canonical form | 179

of the form ξ no in (B.14), and can be written as 0 ξ =[ ] ξ2 with ξ 2 having the dynamics ξ ̇ 2 = à 2,2 ξ 2 . Accordingly, the system is detectable in the sense of Definition B.2.1 if and only if the dynamics of the unobservable part are asymptotically stable, i. e. limt→∞ ||ξ 2 (t)|| = 0. These results are summarized in the following theorem. Theorem B.2.1. Any system of the form (B.1) with rank(𝒦o ) = r ≤ n can be decomposed by a regular transformation into the form à d ξ = [ ̃ 1,1 A2,1 dt ỹ = [C̃ 1

0 B̃ ] ξ + [ ̃ 1 ] u, ̃ A2,2 B2 0] ξ ,

t > 0,

ξ (0) = ξ 0

t≥0

(B.17) (B.18)

with à 1,1 ∈ ℝr×r , à 2,1 ∈ ℝ(n−r)×r , à 2,2 ∈ ℝ(n−r)×(n−r) , B̃ 1 ∈ ℝr×p , B̃ 2 ∈ ℝ(n−r)×p , C̃ 1 ∈ ℝr×m and with the pair (à 1,1 , C̃ 1 ) being completely observable. The system is detectable if and only if à 2,2 is Hurwitz. In virtue of Definition 2.3.1 the subset of ℝn of indistinguishable states associated to the representation (B.17), is independently of the input given by n

ℐ (x 0 ) = {x ∈ ℝ | x = x 0 + v, v ∈ span{κ1 , . . . , κn−r }}.

(B.19)

This defines a separation of the state space into parallel slides passing through the initial state x 0 .

B.3 A permutated observer canonical form Besides the descomposition normal form (B.17) which exists for any system, there is an important particular intrinsic structure of any completely observable system that will be analyzed next. Suppose that we want to transform the system into such coordinates that the measured output corresponds to the last state z n , i. e. y = c̃ T z = [0

⋅⋅⋅

0

1] z

(B.20)

If the system is completely observable, then rang(𝒦o ) = n, and there exists a unique solution to the equation 0 [.] [ .. ] ] 𝒦O ŵ = c̃ = [ [ ] [0] [1]

(B.21)

180 | B Kalman decomposition for observer design given by 0 [.] [ .. ] ] ŵ = 𝒦o−1 [ [ ]. [0] [1] On the other hand, equation (B.21) can be written as 0 T cT ŵ ŵ c [.] [ ] [ ] [ .] .. .. ]=[ ] = T T c = c̃ = [ . ] 𝒦o ŵ = [ . . [ ] [ ] [ ] [0] T n−1 T T n−1 [c A ŵ ] [ŵ (A ) c] [1] with the matrix

T = [ŵ

Aŵ

⋅⋅⋅

̂ , An−1 w]

0 [.] [ .. ] ] ŵ = 𝒦o−1 [ [ ]. [0] [1]

(B.22)

Denote by sTi , i = 0, . . . , n − 1 the rows of the inverse matrix T −1 . The following two identities hold: sT0 Aŵ .. . T s [ n−1 Aŵ

[ T −1 AT = [ [

⋅⋅⋅ ⋅⋅⋅

sT0 An ŵ ] .. ], . ] sTn−1 An ŵ ]

sT0 ŵ [ . T −1 T = [ [ .. T [sn−1 ŵ

⋅⋅⋅ ⋅⋅⋅

sT0 An−1 ŵ ] .. ] = I. . ] sTn−1 An−1 ŵ ]

Comparing both products it results that for i = 1, . . . , n − 1 the i-th column of T −1 AT is identical to the (i + 1)-th column of the identity matrix. This in turn implies that the matrix T −1 AT has a sub-diagonal with unit entries. Thus the elements of the last column are exactly the negative coefficients of the characteristic polynomial, i. e. for the characteristic polynomial λn + an−1 λn−1 + ⋅ ⋅ ⋅ + a1 λ + ao we have (T −1 AT)in = −ai−1 . Given that the characteristic polynomial is invariant with respect to regular transformations, these coefficients are exactly the ones of the characteristic polynomial associated to the original matrix A. Thus the transformed dynamics are given by ż = AO z + BO u,

t > 0,

y=

t ≥ 0,

cTO z,

z(0) = z 0

(B.23) (B.24)

| 181

B.4 The linear Luenberger observer

with 0 [1 [ [ [ [0 [ AO = [ . [ .. [ [. [. [. [0

⋅⋅⋅ 0 1

..

.

⋅⋅⋅

.

..

.

0 0 .. . .. .

. ⋅⋅⋅

1 0

0 1

⋅⋅⋅ .. .. ..

.

⋅⋅⋅

−a0 −a1 .. . .. .

] ] ] ] ] ] ], ] ] ] ] −an−2 ] −an−1 ]

BO = T −1 B,

cTO = [0

⋅⋅⋅

0

1]

This particular structure is a permutated version of the observer canonical form. It can be easily verified that y = cTO z = cT Tz = cT TT −1 x = cT x. Given that the only condition for the transformability is that the Kalman observability matrix 𝒦o has rank n, the following result is obtained. Corollary B.3.1. The LTI system (B.1) can be transformed into observability normal form (B.23) if and only if it is completely observable. Proof. The sufficiency has allready been shown above. For the necessity, note that if the system is not completely observable, then the transformation matrix T (B.22) does not exist, because 𝒦o is not invertible. The observability normal form is particularly useful in the design of a state observer, as will be analyzed in the next section.

B.4 The linear Luenberger observer David Luenberger (Luenberger 1964, 1971) proposed the following simple observer scheme (called the Luenberger observer) for the system (B.1) ẋ̂ = Ax̂ + Bu − L(C x̂ − y),

t > 0,

̂ x(0) = x̂ 0 .

(B.25)

The observer can be viewed as consisting of two parts: – A copy of the system model (B.1) Σ(A, B, C) for the state prediction on the basis of the considered initial value x̂ 0 . – An innovation (or correction) scheme for model adaptation in dependence of the measured output y(t). Clearly, if the initial value x̂ 0 and the simulation model Σ̂ coincides with the actual ones x 0 and Σ, respectively, the simulation part of the observer would predict the correct state x̂ at any time. In most cases, nevertheless, the initial value is not exactly

182 | B Kalman decomposition for observer design known, so that the simulation part will not be able to correctly predict the state evolution. In this case, the adaptation mechanism tries to steer the predicted state towards the actual one, through the information contained in the measurement. Thus, it becomes clear that the underlying notion of observability and detectability are crucial for the performance of the observer. To analyze how observability properties determine the observer functioning consider the observation error x̃ = x̂ − x,

(B.26)

with dynamics ẋ̃ = ẋ̂ − ẋ = Ax̂ + Bu − LC(x̂ − x) − Ax − Bu = (A − LC)x,̃

t > 0,

̃ x(0) = x̃ 0 .

(B.27)

To ensure convergence of the observer the spectrum of the matrix A − LC has to be contained in the open left-half plane of the complex numbers, i. e. σ(A − LC) ∈ ℂ− Let us analyze this matrix with more attention, and focus on the SISO case with a completely observable system. In this case the dynamics can be brought into observability normal form (B.23) by the regular transformation T defined in (B.22). When applying the Luenberger observer structure (B.25) in these coordinates, the following particular form is obtained 0 [ [ [1 [ [ ̃ ̃ A + LC = [0 [ [. [. [. [0

⋅⋅⋅ .. . .. . .. . ⋅⋅⋅

.

0 .. . .. .

. 0

0 1

⋅⋅⋅ .. ..

0 −a0 [ .. ] ] [ . ] [ ] [ .. ] − [ [ . ] ] [ ] .. ] [ [ . ] [ −an−1 ] [0

⋅⋅⋅ .. . .. . .. . ⋅⋅⋅

l1

0 ] [ ] [ ] [1 ] [ ] [ ] = [0 ] [ ] [. ] [. ] [.

ln ]

[0

⋅⋅⋅ .. . .. . .. . ⋅⋅⋅

.

0 .. . .. .

. 0

0 1

⋅⋅⋅ .. ..

−a0 − l1 ] .. ] ] . ] ] .. ] . ] ] .. ] . ] −an−1 − ln ] (B.28)

and the characteristic polynomial is given by n

̃ 𝒫 [Ã − LC](λ) = λ + (an−1 + ln )λ

n−1

+ ⋅ ⋅ ⋅ + (a1 + l2 )λ + (a0 + l1 ).

(B.29)

This shows that an arbitrary characteristic polynomial can be assigned to the observer dynamics by choosing the coefficients li . In particular, this can be done using the Ackerman formula. In Matlab or Octave this can be implemented using the place and acker commands for the dual system. This subtle fact implies that in the case of a completely observable system the observer convergence can be ensured by adequately choosing the observer gains li .

B.5 Nonlinear systems | 183

Moreover, this does not rely on the system stability. Finally, it should be noted that the fact that an arbitrary characteristic polynomial can be imposed implies that the observer can be made, in principle,2 arbitrarily fast. Now, consider the case that the system is not completely observable. Then, according to Theorem B.2.1, by a regular transformation the system can be brought into the form (B.17) with the pair (à 1,1 , C̃ 1 ) being completely observable. The application of the observer structure (B.25) in this case yields à à − L̃ C̃ = [ ̃ 1,1 A2,1

0 L̃ C̃ ] − [ ̃ 1 ̃1 ̃ L2 C2 A2,2

à − L̃ 1 C̃ 1 0 ] = [ ̃ 1,1 A2,1 − L̃ 2 C̃ 1 0

0 ] Ã 2,2

(B.30)

showing that the spectrum of the matrix à − L̃ C̃ is given by σ(à − L̃ C)̃ = σ(à 1,1 − L̃ 1 C̃ 1 ) ∪ σ(à 2,2 ).

(B.31)

This means that, besides the fact that the completely observable part (à 1,1 , C̃ 1 ) can, in principle, be made to converge arbitrarily fast as shown before, the observer convergence completely relies on the spectrum of the unobservable part associated to à 2,2 . Thus, it is necessary for the existence of an asymptotically convergent observer that σ(à 2,2 ) ∈ ℂ− , i. e. that the system is detectable according to Theorem B.2.1. This is summarized in the following theorem. Theorem B.4.1. There exists a matrix L ∈ ℝn×m such that the dynamics (B.25) is an observer for (B.1) if and only if (B.1) is detectable. If (B.1) is completely observable the convergence rate can be assigned arbitrarily.

B.5 Nonlinear systems In this chapter we will consider the observation problem for input affine nonlinear systems of the form p

ẋ = f (x) + ∑ g i (x)ui ,

t > 0,

y = h(x))

t ≥ 0,

i=1

x(0) = x 0

(B.32a) (B.32b)

with x(t) ∈ ℝn , f : ℝn → ℝn , g i : ℝ → ℝn , ui : [0, ∞) → ℝ, i = 1, . . . , p, h : ℝn → ℝm , and f , g being sufficiently smooth. In the sequel, given the smoothness of the vector fields, it is assumed that a unique solution exists over the time interval t ∈ [0, T] for some T > 0, so that for t ≥ 0 x(t) = x(t; x 0 , u1 , . . . , up ). 2 The restriction comes from the fact that in presence of measurement noise this will be amplified proportional to the observer gains.

184 | B Kalman decomposition for observer design B.5.1 Local observability In the case of linear systems considered above it turned out that the set of indistinguishable states is independent of the input vector u. This is no longer the case for nonlinear systems. Already in the bi-linear case observability will depend on the input. To see this, consider the following example Example B.5.1. ẋ1 = −x1 + x2 (1 − u), ẋ2 = −x2 , y = x1

t > 0, t > 0, t ≥ 0.

x1 (0) = x10 x2 (0) = x20

Clearly, taking y and ẏ (as in the linear case) the following relation can be defined x1 = y,

x2 =

ẏ + y , u ≠ 1 1−u

(B.33)

but for u = 1 it is impossible to determine the x2 component from the signal y.̇ Clearly, the same holds for any higher derivative of the output y, and thus the system is not observable for u = 1, while for any u ≠ 1 it is observable. ⬦ In the sequel consider the case that u = 0. Using the notion of Lie-derivative of the function h along the vector field f , according to (see also Section 2.2.2) L0f h(x) = h(x),

Lf h(x) =

𝜕h(x) f (x), 𝜕x

Lkf h(x) = Lf (Lk−1 f h(x))

the sequence of the first n time derivatives of y can be written as y(t) = h(x(t)) ̇ = Lf h(x(t)) y(t) .. .

y (n−1) (t) = Ln−1 f h(x(t)) Now, introducing the observability map h(x) [ L h(x) ] [ f ] ] O(x) = [ .. [ ] [ ] . n−1 L h(x) [ f ]

(B.34)

B.5 Nonlinear systems | 185

the first n − 1 time derivatives above can be rewritten in a similar way as in the linear case y(t) ] [ y(t) ] [ ̇ ] = O(x(t)) 𝒴(t) = [ .. ] [ ] [ . (n−1) (t)] [y

(B.35)

Thus, if O is invertible this equation can be uniquely solved for x(t). Note that in the nonlinear case this will not enable to determine the initial condition x 0 , but as long as uniqueness of the solutions is ensured, the initial condition is uniquely associated to the trajectory (for given inputs ui (⋅), i = 1, . . . , p). To be more specific, for ui = 0, i= 1, . . . , p, and invertible O the equivalence of outputs reads O(x 1 (t)) = O(x 2 (t)),

∀ t ∈ [0, T]

and this in turn implies that x 1 (t) = x 2 (t), and thus x 01 = x 02 . Hence, the system states can be distinguished from each other, and thus the system is observable. When the input is not zero the observability map O depends also on the input and the respective time-derivatives of the input, and accordingly, as already stated above, the observability properties will also depend on the input. At this place it should be noted that, in contrast to linear time invariant (LTI) systems, for nonlinear systems special care has to be taken with such invertibility questions. In the LTI case the full rank condition of the Kalman observability matrix ensured that the observability property, if true for some value x and time t, holds for any x and t. In the nonlinear case, the map O depends on the state x and thus may be globally or only locally invertible, if invertible at all. This important question has to be analyzed carefully for every application case. Furthermore, assume that over some set 𝒟 ⊂ Rn , O is invertible. Then this does not necessarily imply observability for any initial condition x 0 ∈ 𝒟 and time T > 0, because the solution x may leave 𝒟 with time. This is actually where the time invariance comes into play, because the observability map (B.34) does not depend on time, and thus small time intervals are sufficient to ensure that x(t) is uniquely determined. Nevertheless, denoting by 𝜕𝒟 the boundary of the set 𝒟, then for initial conditions x 0 ∈ 𝜕𝒟, there may not be any such time interval over which the state x moves within 𝒟 and thus the invertibility may get lost over any time interval. This is a somekind subtle question which has to be analyzed carefully. If, in contrast, the set 𝒟 is open, then for any x 0 ∈ 𝒟 there exists a (sufficiently small) time T(x 0 ) such that x(t) ∈ 𝒟 ∀ t ∈ [0, T(x 0 )] and thus the invertibility ensures that x(t) is uniquely determined before leaving 𝒟. A property that ensures that for compact sets 𝒟 the state trajectory will stay in 𝒟 as long as x 0 ∈ 𝒟 is the positive invariance of the set 𝒟, that has to be analyzed separately.

186 | B Kalman decomposition for observer design These considerations motivate to introduce the concept of local observability at x ∗ as observability in an open set 𝒟 ⊂ Rn containing x ∗ such that O is invertible for all x ∈ 𝒟. Notice that according to the inverse function theorem it holds that a smooth map φ is a local diffeomorphim at a point x ∗ (i. e., smooth with unique smooth inverse in some neighbourhood of x ∗ ) if its Jacobian has full rank n at x ∗ . Thus, if the Jacobian 𝜕x O has rank n at x ∗ , then there exists a neighbourhood 𝒟 of x ∗ so that the restriction of O on the associated subset of ℝmn is a diffeomorphism on X and thus the system is locally observable. This is summarized in the following theorem. Theorem B.1. The system (B.32a) with ui = 0, i = 1, . . . , p is locally observable at x ∗ if [ [ rank(𝜕x ∗ O) = rank ([ [

𝜕h(x ∗ ) 𝜕x1

.. .

𝜕Ln−1 f h(x ∗ ) [ 𝜕x1

⋅⋅⋅ .. . ⋅⋅⋅

𝜕h(x ∗ ) 𝜕xn

.. .

] ] ]) = n. ]

𝜕Ln−1 f h(x ∗ ) 𝜕xn ]

(B.36)

It should be recalled here that in the linear case the Theorem of Cayleigh-Hamilton ensures that n − 1 derivatives of the output imply the maximum of independent directions which can be combined to obtain the complete ℝn . In the nonlinear case, there is no such theorem, making sure that further derivatives do not introduce further information. B.5.2 Local Kalman decomposition The following considerations are carried out for the case that ui = 0, i = 1, . . . , p. In the preceding discussion we analyzed the Jacobi matrix of the observability map O defined in (B.34). Actually, the co-vectors, defined by 𝜕Lkf hi 𝜕x

(x ∗ ) = [

𝜕Lkf hi (x ∗ ) 𝜕x1

⋅⋅⋅

𝜕Lkf hi (x ∗ )] 𝜕xn

(B.37)

span locally the observable subspace, and if there are n linearly independent covectors of this kind, the system is locally observable at x ∗ . If this holds true for all x ∗ ∈ ℝn this implies that the (locally) observable subspace corresponds to ℝn and in conclusions all trajectories can be locally distinguised from each other by means of the output (and input) signals. In the case, that the rank of the matrix 𝜕x O at x ∗ is r < n, then there are only r such linearly independent co-vectors, and the system is not locally observable. Accordingly, let there be a maximum of r linearly independent co-vectors τi , i = 1, . . . , r of the form (B.37) spanning the observable subspace 𝒪 ⊂ ℝn , i. e. 𝒪 = span{τ 1 , . . . , τ r },

i = 1, . . . , r.

τ i ∈ span{

𝜕Lkf hj 𝜕x

(x ∗ ), k = 0, . . . , n − 1, j = 1, . . . , m}, (B.38)

B.5 Nonlinear systems | 187

In the same way as for the linear case, when constructing the local indistinguishable (or unobservable) subspace 𝒩O we are interested in a space spanned locally by co-vectors of the form (B.37) which is invariant with respect to the flow of the system, i. e. a state in 𝒩O should remain in 𝒩O . For nonlinear systems this invariance property is related to the Lie-bracket [α(x), β(x)] :=

𝜕β(x) 𝜕α(x) α(x) − β(x) 𝜕x 𝜕x

(B.39)

which can be interpreted as the difference between moving along the vector field α for some time ta and than along β for some time tb , from moving along β for some time tb and than along α for some time ta , i. e. by denoting the flow along α by Φαt and along β β by Φt β

β

[α, β] = Φαta (Φtb ) − Φtb (Φαta ). The vector fields α and β are said to commute if [α, β] = 0, i. e. if it does not matter if moving first along α or β and than along β or α. For linear vector fields α(x) = Ax and β(x) = Bx this means [Ax, Bx] = BAx − ABx = (BA − AB)x and this difference is zero if and only if A and B commute. Thus, the Lie-bracket is also called the commutator of the vector fields α and β. The property of interest here, is the invariance property. A distribution 𝒟 = span{τ 1 , . . . , τ r }

(B.40)

of r linearly independent functions τi , i = 1, . . . , r is called invariant under the vector field f , if ∀ τ ∈ 𝒟 ⇒ [f , τ] ∈ 𝒟. One correspondingly writes [f , 𝒟] ∈ 𝒟. Furthermore, 𝒟 is called involutive if ∀g 1 , g 2 ∈ 𝒟 ⇒ [g 1 , g 2 ] ∈ 𝒟.

(B.41)

For convenience of notation, a distribution spanned by co-vectors is called a codistribution 𝒟⊥ . Given that the row vector entries of the Jacobian of the observability map are co-vectors, we are interested in the observability co-distribution Δ⊥ . Comming back to the original question, on how to represent the fact that states in the indistinguishable subspace 𝒩O should remain indistinguishable along time. The indistinguishability leads to the requirement that, as in the linear case, the statespace can be separated into slides which represent the indistinguishable states, and the flow carrying indistinguishable slides into indistinguishable slides. This means

188 | B Kalman decomposition for observer design that [f , Δ⊥ ] ⊂ Δ⊥ , i. e. Δ⊥ is f -invariant. Furthermore, the arbitrary combination of vector fields within the indistinguishable space has to be contained in the indistinguishable subspace, i. e. Δ⊥ has to be involutive. Next, according to (B.38), let τ 1 , . . . , τ r be a local basis for 𝒪, or in other words, that 𝒪 coincides at x ∗ with the co-distribution spanned by the co-vector fields τ i , i = 1, . . . , r. First, note that, given that the basis co-vectors are given by differentials of the time-derivatives of the output y, independently whether one follows the direction spanned by τ i and then the one spanned by τ j for i ≠ j, or vice versa, the state has to remain in 𝒪, meaning that 𝒪 has to be involutive. According to Frobenius’ theorem (see e. g. Isidori 1995) an involutive distribution has a (perpendicular at x ∗ ) co-distribution (its annihilator) which is spanned by exact differentials. Now, consider the distribution spanned by {τ Ti , i = 1, . . . , r} which is involutive under the above assumptions, and thus its co-distribution (or annihilator) 𝜕λ is spanned by n − r exact differentials of the form μj = 𝜕xj such that μj τ Ti =

𝜕λj 𝜕x

τ Ti = 0,

∀i = 1, . . . , r, j = 1, . . . , n − r.

(B.42)

Now, let k

τi (x ∗ ) =

𝜕Lf i hli

(x ∗ ) 𝜕x for some indexes ki , li , so that the τi are linearly independent, and take the diffeomorphism defined by the map k

Lf 1 hl1 (x) .. . ( ) ( kr ) (Lf hlr (x)) Ψ(x) = ( ) ( λ (x) ) ( 1 ) .. . ( λn−r (x) ) Any vector field ϕ ∈ Δ⊥ is now represented by a vector 𝜕Ψ(x) ̄ ϕ(z) =[ ϕ(x)] 𝜕x x=Ψ−1 (z) k

𝜕Lf 1 hl1 (x)

[ ] 𝜕x [ ] .. [ ] [( ) ] . [( kr ) ] [( 𝜕Lf hlr (x) ) ] [( ] ) = [( 𝜕x ) ϕ(x)] [( 𝜕λ1 (x) ) ] [( ] ) [( 𝜕x ) ] [ ] .. [ ] . [ ] [(

𝜕λn−r (x) 𝜕x

)

]x=Ψ−1 (z)

τ 1 (x) [ ] .. [ ] . [ ] [( ) ] [( τ r (x) ) ] ( ) ϕ(x)] =[ [( μ (x) ) ] [( 1 ) ] [ ] .. [ ] [ ] . μ (x) [( n−r ) ]x=Ψ−1 (z)

(B.43)

B.5 Nonlinear systems | 189 k

𝜕L r hl (x)

and as the f 𝜕xr , i = 1, . . . , n − r span the orthogonal complemente Δ of Δ⊥ , it follows that the first r components of ϕ̄ are zero, i. e. ϕ ∈ Δ⊥ ⇒ ϕ̄ = [0

0

⋅⋅⋅

ϕ̄ 1

⋅⋅⋅

T ϕ̄ r ] .

The dynamics (B.32a) in the new coordinates are given by ż = [

𝜕Ψ(x) f (x)] = f ̄ (z) 𝜕x x=Ψ−1 (z)

T

Let z = [z 1 z 2 ] with z 1 having dimension r and z 2 having dimension n − r. Then it follows that k

[ [ ż 1 = [ [( [ [

𝜕Lf 1 hl1 (x) 𝜕x k

.. .

𝜕Lf r hlr (x) 𝜕x

] ] ) f (x)] ] ] ]x=Ψ−1 (z)

τ 1 (x) [ ] . ] =[ = f ̄ 1 (z) [( .. ) f (x)] [ τ r (x) ]x=Ψ−1 (z)

Note that f 2 (z) can be constructed similarly so that ż 2 = f 2 (z). By construction z 2 ∈ 𝒩O and every vector field of the form ψi1 (z) .. ] ] . ] [ψin (z)]

[ ψi (z) = [ [ with

0

k ≠ i

1

k=i

ψik (z) = {

belongs to Δ⊥ for any r + 1 ≤ i ≤ n. Now, assume that f ̄ 1 (z) would depend on z 2 , i. e. ̄ f ̄ (z) = f ̄ (z , z ). Then 𝜕f 1 (z) ≠ 0 for some k ∈ {1, . . . , n − r}. But with ψ ∈ Δ⊥ as defined 1

1

1

2

above it follows that

i

𝜕zr+k

𝜕f ̄ (z)

1 𝜕f ̄ 𝜕f ̄ [ 𝜕zi ] [f ̄ (z), ψi ] = − ψi = − = . 𝜕f ̄ 2 (z) 𝜕z 𝜕zi [ 𝜕zi ]

Given that Δ⊥ is f -invariant, it must be f ̄ -invariant by construction, and thus [f ̄ (z), ψi ] ∈ Δ⊥ implying that 𝜕f ̄ 1 = 0, 𝜕zi

i = r + 1, . . . , n,

190 | B Kalman decomposition for observer design or equivalently, that the dynamics in the new coordinates are given in the decomposition form ż 1 = f ̄ 1 (z 1 ), ż 2 = f ̄ 1 (z 1 , z 2 ), ̄ ), y = h(z

t > 0, t > 0, t ≥ 0,

1

z 1 (0) = z 01

(B.44a)

z 2 (0) = z 02

(B.44b) (B.44c)

with the dynamics of z 1 being locally observable, and the set of indistinguishable trajectories z z2

n

n−r

ℐ (z 0 ) = {z = [ 1 ] ∈ ℝ | z 1 = z 01 and z 2 ∈ ℝ

}.

(B.45)

B.5.3 Detectability analysis Based on the representation (B.44), for the case of ui = 0, i = 1, . . . , p, the indistinguishable states are given by the set (B.45). Two states will be indistinguishable if, in T

the transformed z-representation, they have initial conditions z 0 and z 0 + [0 eT20 ] , and thus the detectability analysis amounts to analyzing the partial stability of the e2 -component of the dynamics ż 1 = f ̄ 1 (z 1 ), ż 2 = f ̄ 1 (z 1 , z 2 ),

t > 0, t > 0,

ė 2 = f 2 (z 1 , z 2 + e2 ) − f 2 (z 1 , z 2 ),

t > 0,

z 1 (0) = z 01

z 2 (0) = z 02

e2 (0) = e20 .

(B.46a) (B.46b) (B.46c)

Detectability in the sense of Definition 2.3.2 thus corresponds to the asymptotic convergence condition lim ||e2 (t)|| = 0,

t→∞

associated to the dynamics (B.46). The detectability analysis can be significantly simplified in many cases explicitely accounting for the algebraic constraints introduced by the indistinguishability conditions dk h(x + e) dk h(x) ≡ , dt k dt k

k ∈ ℕ.

Clearly, when the system is observable the stability analysis is trivial. In the nonobservable case these algebraic constraints lead to the requirement of analyzing the partial stability properties of an differential-algebraic equation (DAE) set. Particular case studies where this has been employed can be found in Ibarra-Rojas et al. (2004), Schaum and Moreno (2007), Moreno and Dochain (2008), Moreno et al. (2014), Jerono et al. (2018).

C The algebraic Riccati equation, optimality and dissipativity The relationship between dissipative and optimal systems has been discussed at different places in the literature starting from the results of Kalman (1964) up to the welldeveloped inverse optimality approach for finite-dimensional systems (Brogliato et al. 2007, Sepulchre et al. 1997, Moylan 1974). The purpose of this section consists in discussing some interesting connections between optimality and (Q, S, R)-dissipativity for systems with a quadratic storage function. Consider the linear time-invariant system Σ(A, B, C), i. e. ẋ = Ax + Bu,

t > 0,

y = Cx,

t ≥ 0,

x(0) = x 0

(C.1) (C.2)

with finite-dimensional state x ∈ ℝn , input u ∈ ℝp and output y ∈ ℝm . Also consider for a given state x(t) and input function u(⋅) ∈ 𝒰 the cost functional ∞

J(x(t)) = ∫ (x(τ)T Mx(τ) − 2x(τ)T Wu(τ) + u(τ)T Nu(τ))dτ

(C.3)

t

with ℝp×p ∋ N > 0 which is evaluated along the solution x(t; x 0 , u(⋅)) of (C.1) with initial value x 0 . The typical infinite-horizon optimal control problem consists in minimizing J(x(t)) over the set of all admissable inputs u ∈ 𝒰 , i. e. min J(x(t; u(⋅))) s. t. (C.1) holds true.

(C.4)

u∈𝒰

According to Bellman’s principle of optimality, taking an infinitesimal time step dt the optimal cost-to-go function from a starting point x(t) satisfies t+dt

V(x(t)) = min{V(x(t + dt) + ∫ (x(τ)T Mx(τ) − 2x(τ)T Wu(τ) + u(τ)T Nu(τ))} (C.5) u∈𝒰

t

Substituting V(x(t + dt)) with its Taylor-series expansion V(x(t + dt)) = V(x(t)) +

𝜕V(x(t)) ̇ x(t)dt + 𝒪2 (dt) 𝜕x 2

where 𝒪2 (dt) denotes higher order terms in dt such that limdt→0 𝒪 dt(dt) = 0. Substracting V(x(t)) on both sides of the preceding expression, dividing by dt and taking the limit dt → 0 yields 0 = min{ u∈𝒰

𝜕V(x(t)) ̇ + x(t)T Mx(t) − 2x(t)T Wu(t) + u(t)T Nu(t)}. x(t) 𝜕x

https://doi.org/10.1515/9783110677942-010

(C.6)

192 | C The algebraic Riccati equation, optimality and dissipativity This is the associated Hamilton-Jacobi-Bellman equation. Considering a quadratic cost-to-go function V(x) = x T Px, substituting the dynamics (C.1), writing the crossproducts in their symmetric form and neglecting the explicit time-dependency yields 0 = min{x T P ẋ + ẋ T Px + x T Mx − 2x T Wu + uT Nu} u∈𝒰

= min{x T P(Ax + Bu) + (Ax + Bu)T Px + x T Mx − x T Wu − uT W T x + uT Nu} u∈𝒰

or equivalently T

x AT P + PA + M 0 = min {[ ] [ T u∈𝒰 u B P − WT

PB − W x ] [ ]} N u

(C.7)

The minimum value is obtained with the optimal input u∗ satisfying 2x T (PB − W) + 2u∗T (x)N = 0. Given that det(N) ≠ 0 it follows that u∗ (x) = −Kx,

K = N −1 (BT P − W T ).

(C.8)

Substituting this into (C.7) yields 0 = x T (AT P + PA + M − 2(PB − W)K + K T NK)x = x T (AT P + PA + M − 2(PB − W)N −1 (BT P − W T ) + (PB − W)N −1 NN −1 (BT P − W T ))x. After summarizing and noticing that the equality holds for all x ∈ 𝕏 this yields the algebraic Riccati equation associated to the optimal control problem AT P + PA + M − (PB − W)N −1 (BT P − W T ) = 0

(C.9)

Now, consider a linear system Σ(A, B, C) which is (Q, S, R)-strictly state dissipative with dissipation rate κ > 0 and R > 0, i. e. such that it holds that T

x AT P + PA − C T QC + κI [ ] [ u BT P − C T ST

PB − SC x ][ ] ≤ 0 −R u

or equivalently, by employing the Schur complement of R (see Lemma A.2.1), AT P + PA − (PB − SC)R−1 (BT P − C T ST ) ≤ C T QC − κI. For the special case that the preceding inequality is satisfied identically, i. e. AT P + PA − (PB − SC)R−1 (BT P − C T ST ) = C T QC − κI,

(C.10)

C The algebraic Riccati equation, optimality and dissipativity | 193

comparing this relation with the algebraic Riccati equation (C.9) leads to the conclusion that (C.10) is the algebraic Riccati equation associated to the optimal control problem (C.3) with M = −C T QC + κI,

W = SC,

N = R.

(C.11)

This means that the (Q, S, R)-dissipativity of the open-loop linear system Σ(A, B, C) ensures the existence of the optimal feedback control u = −Kx which minimizes the cost functional (C.3) with (M, W, N) given in (C.11), and, in turn, that the solvability of the optimal control problem ensures the (Q, S, R)-strict state dissipativity of Σ(A, B, C). Remark C.1. The existence of a solution P for the algebraic Riccati equation is ensured if the pair (A, B) is controllable. For the case of an uncontrollable but stabilizable system the choice of the matrices (Q, S, R)-is substantial in the solvability question. For further information see e. g. Lancaster and Rodman (1995), Brogliato et al. (2007) and references therein.

D Kernel derivations for the backstepping approach In this chapter the transformations for the backstepping approach in section 5.4 are derived. Particular care is taken to include the most important intermediate steps in order to simplify their application and possible extensions to new situations for the reader. It should be mentioned that the cases treated here are already covered in the literature (see e. g. Krstic and Smyshlyaev 2008a,b, Meurer 2013a), but are included here in order to (i) enable a quick reference during the lecture of this book, and (ii) to provide a notation adapted to the one used in the previous sections.

D.1 Forward transformation from Subsection 5.4.1 In this section the derivation of the kernel pde for the unstable heat equation (5.59) is carried out and its solution is derived by following the steps from Krstic and Smyshlyaev (2008a) and specifying the more general calculations presented there for the particular case at hand. D.1.1 Derivation of the kernel pde Consider the Volterra integral transformation z

ξ (z, t) = x(z, t) − ∫ k(z, ζ )x(ζ , t)dζ 0

to bring the dynamics of the linear subsystem in (5.60) into the form (5.62) by adequately choosing the control input u(t). For this purpose consider the time derivative z

𝜕t ξ (z, t) = 𝜕t x(z, t) − ∫ k(z, ζ )𝜕t x(ζ , t)dζ 0

z

= D𝜕z2 x(z, t) − D ∫ k(z, ζ )𝜕ζ2 x(ζ , t)dζ 0

z

= D𝜕z2 x(z, t) − Dk(z, ζ )𝜕ζ x(ζ , t)|z0 + D ∫ 𝜕ζ k(z, ζ )𝜕ζ x(ζ , t)dζ 0

=

D𝜕z2 x(z, t)

− D[k(z, ζ )𝜕ζ x(ζ , t) −

z 𝜕ζ k(z, ζ )x(ζ , t)]0

z

− D ∫ 𝜕ζ2 k(z, ζ )x(ζ , t)dζ . 0

On the other hand it holds that 𝜕t ξ (z, t) = D𝜕z2 ξ (z, t) − βξ (z, t) =

D𝜕z2 x(z, t)

−

z 2 D𝜕z (∫ k(z, ζ )x(ζ , t)dζ ) 0

https://doi.org/10.1515/9783110677942-011

z

− βx(z, t) + ∫ βk(z, ζ )x(ζ , t)dζ 0

196 | D Kernel derivations for the backstepping approach with z 2 𝜕z (∫ k(z, ζ )x(ζ , t)dζ )

z

= 𝜕z (k(z, ζ )x(ζ , t)|ζ =z + ∫ 𝜕z k(z, ζ )x(ζ , t)dζ )

0

=

0

z

d k(z, z)x(z, t) + k(z, z)𝜕z x(z, t) + 𝜕z k(z, z)x(z, t) + ∫ 𝜕z2 k(z, ζ )x(ζ , t)dζ dz 0

with d k(z, z) = 𝜕z k(z, z) + 𝜕ζ k(z, z). dz Summarizing, it holds that 0 = − 𝜕t ξ (z, t) + D𝜕z2 ξ (z, t) − βξ (z, t) = −

(D𝜕z2 x(z, t)

− D[k(z, ζ )𝜕ζ x(ζ , t) −

z 𝜕ζ k(z, ζ )x(ζ , t)]0

z

− D ∫ 𝜕ζ2 k(z, ζ )x(ζ , t)dζ ) 0

d + D𝜕z2 x(z, t) − D k(z, z)x(z, t) − Dk(z, z)𝜕z x(z, t) − D𝜕z k(z, z)x(z, t) dz z

z

0

0

− D ∫ 𝜕z2 k(z, ζ )x(ζ , t)dζ − βx(z, t) + ∫ βk(z, ζ )x(ζ , t)dζ z

= − k(z, 0)𝜕z x(0, t) + 𝜕ζ k(z, 0)x(0, t) + ∫[D𝜕ζ2 k(z, ζ ) − D𝜕z2 k(z, ζ ) + βk(z, ζ )]x(ζ , t)dζ 0

d − D k(z, z)x(z, t) − D𝜕z k(z, z)x(z, t) − D𝜕ζ k(z, z)x(z, t) − βx(z, t). dz Taking into account the boundary condition 𝜕z x(0, t) = 0 it follows that the preceding equation can be satisfied for all solutions x(z, t) if and only if β k(z, ζ ) D β d k(z, z) = − dz 2D 𝜕ζ k(z, 0) = 0.

𝜕z2 k(z, ζ ) − 𝜕ζ2 k(z, ζ ) =

(D.1a) (D.1b) (D.1c)

The second equation can be written equivalently as k(z, z) = k(0, 0) −

β z. 2D

The value k(0, 0) can be determined by considering the first derivative of ξ (z, t) at z = 0 and the boundary conditions 𝜕z ξ (0, t) = 0, 𝜕z x(0, t) = 0 as follows: 𝜕z ξ (0, t) = 𝜕z x(0, t) − k(0, 0)x(0, t) = 0

D.1 Forward transformation from Subsection 5.4.1

| 197

implying that this holds for all x(0, t) only if k(0, 0) = 0. This implies that the pde for the kernel is given by β k(z, ζ ) D βz k(z, z) = − 2D 𝜕ζ k(z, 0) = 0.

𝜕z2 k(z, ζ ) − 𝜕ζ2 k(z, ζ ) =

(D.2a) (D.2b) (D.2c)

D.1.2 Solution of the kernel equation (D.2) To solve the pde (D.2) consider the coordinate change χ = z + ζ,

η = z − ζ,

̄ η) k(z, ζ ) = k(χ,

with 𝜕z χ = 1,

𝜕ζ χ = 1,

𝜕z η = 1,

𝜕ζ η = −1.

It holds that ̄ η)𝜕 χ + 𝜕 k(χ, ̄ η)𝜕 η 𝜕z k(z, ζ ) = 𝜕χ k(χ, z η z ̄ η) + 𝜕 k(χ, ̄ η) = 𝜕 k(χ, χ

η

̄ η)𝜕 χ + 𝜕 k(χ, ̄ η)𝜕 η 𝜕ζ k(z, ζ ) = 𝜕χ k(χ, ζ η ζ ̄ ̄ = 𝜕 k(χ, η) − 𝜕 k(χ, η) 𝜕z2 k(z, ζ ) 𝜕ζ2 k(z, ζ )

= =

χ ̄ η) 𝜕χ2 k(χ, ̄ η) 𝜕χ2 k(χ,

η

̄ η) + 𝜕2 k(χ, ̄ η) + 2𝜕χ 𝜕η k(χ, η

̄ η) + 𝜕2 k(χ, ̄ η) − 2𝜕χ 𝜕η k(χ, η

so that the pde (D.2a) can be written as ̄ η) = 𝜕z2 k(z, ζ ) − 𝜕ζ2 k(z, ζ ) = 4𝜕χ 𝜕η k(χ,

β ̄ k(χ, η). D

Note that z=

χ+η , 2

ζ =

χ−η , 2

(D.3)

so that for ζ = z it holds that η = 0 and the boundary condition (D.2b) can be written as ̄ 0) = − k(z, z) = k(χ,

βz βχ =− . 2D 4D

198 | D Kernel derivations for the backstepping approach On the other side, the second boundary condition (D.2c) leads to ̄ χ) − 𝜕 k(χ, ̄ χ) = 0 𝜕ζ k(z, 0) = 𝜕χ k(χ, η or equivalently ̄ χ) = 𝜕 k(χ, ̄ χ). 𝜕χ k(χ, η Thus, the pde (D.2) is equivalent to ̄ η) ̄ η) = β k(χ, 𝜕χ 𝜕η k(χ, 4D ̄ χ) = 𝜕 k(χ, ̄ χ) 𝜕χ k(χ, η ̄ 0) = − βχ . k(χ, 4D

(D.4a) (D.4b) (D.4c)

Integrating the first equation with respect to η from 0 to η yields η

̄ η) = 𝜕 k(χ, ̄ 0) + ∫ β k(χ, ̄ τ)dτ. 𝜕χ k(χ, χ 4D 0

̄ 0) is obtained from deriving (D.4c) with respect to χ, i. e. The constant 𝜕χ k(χ, ̄ 0) = − 𝜕χ k(χ,

β 4D

implying that η

̄ η) = − 𝜕χ k(χ,

β ̄ β +∫ k(χ, τ)dτ. 4D 4D

(D.5)

0

Next, integration with respect to χ from η to χ yields χ η

̄ η) = k(η, ̄ η) − β (χ − η) + ∫ ∫ β k(s, ̄ τ)dτds. k(χ, 4D 4D

(D.6)

η 0

̄ η) = k(χ, ̄ χ) notice that in virtue of (D.4b) it holds that To determine the constant k(η, d ̄ ̄ χ) + 𝜕 k(χ, ̄ χ) = 2𝜕 k(χ, ̄ χ). k(χ, χ) = 𝜕χ k(χ, η χ dχ On the other hand, from (D.5) it follows that χ

̄ χ) = − 𝜕χ k(χ,

β β ̄ +∫ k(χ, τ)dτ 4D 4D 0

D.1 Forward transformation from Subsection 5.4.1

| 199

so that χ

β β ̄ d ̄ k(χ, χ) = − +∫ k(χ, τ)dτ. dχ 2D 2D 0

Integrating this equation with respect to χ from 0 to χ yields χ s

̄ χ) = − β χ + ∫ ∫ β k(s, ̄ τ)dτds k(χ, 2D 2D 0 0

and thus the desired quantity η s

̄ η) = − β η + ∫ ∫ β k(s, ̄ τ)dτds. k(η, 2D 2D 0 0

Summarizing, it holds that η s

χ η

̄ η) = − β η + ∫ ∫ β k(s, ̄ τ)dτds − β (χ − η) + ∫ ∫ β k(s, ̄ τ)dτds k(χ, 2D 2D 4D 4D 0 0

=−

η s

χ η

0 0

η 0

η 0

β ̄ β β ̄ (χ + η) + ∫ ∫ k(s, τ)dτds + ∫ ∫ k(s, τ)dτds. 4D 2D 4D

This can be written equivalently in the compact notation ̄ η) = − β (χ + η) + F[k](χ, ̄ k(χ, η) 4D

(D.7)

with η s

χ η

0 0

η 0

β ̄ β ̄ ̄ F[k](χ, η) = ∫ ∫ k(s, τ)dτds + ∫ ∫ k(s, τ)dτds. 2D 4D This integral equation can be solved using the series approximation ∞

̄ η) = ∑ k̄ (χ, η), k(χ, i i=0

β k̄0 (χ, η) = − (χ + η), 4D

k̄n+1 (χ, η) = F[k̄n ](χ, η), n ∈ ℕ.

(D.8)

Next, it is shown by complete induction that the solution for k̄n (χ, η) is given by the series n+1

β k̄n (χ, η) = −( ) 4D

1 (χ + η)χ n ηn . n!(n + 1)!

(D.9)

200 | D Kernel derivations for the backstepping approach Note that for n = 0 this is satisfied according to (D.8). Assuming the solution (D.9) for n ∈ ℕ it follows that k̄n+1 (χ, η) = F[k̄n ](χ, η)

η s

n+1

β = −( ) 4D

n+2

= −(

β ) 4D

χ η

β β 1 (∫ ∫ (s + τ)sn τn dτds + ∫ ∫ (s + τ)sn τn dτds) n!(n + 1)! 2D 4D η 0

0 0 η s

χ η

0 0

η 0

1 (∫ ∫ 2(s + τ)sn τn dτds + ∫ ∫(s + τ)sn τn dτds). n!(n + 1)!

It holds that η s

η s

n n

∫ ∫ 2(s + τ)s τ dτds = ∫ ∫ 2(sn+1 τn + sn τn+1 )dτds 0 0

0 0 η

= ∫ 2 (sn+1 0 η

= ∫ 2( 0

τ=s 1 n+1 1 n+2 󵄨󵄨󵄨󵄨 τ + sn τ )󵄨󵄨 ds 󵄨󵄨τ=0 n+1 n+2

1 1 + )s2n+2 ds n+1 n+2

2(2n + 3) η2n+3 (n + 1)(n + 2)(2n + 3) 2 = η2n+3 (n + 1)(n + 2) =

and χ η

χ η

∫ ∫(s + τ)s τ dτds = ∫ ∫(sn+1 τn + sn τn+1 )dτds n n

η 0 χ

η 0

= ∫(sn+1 η

1 n+1 1 n+2 η + sn η )ds n+1 n+2

1 s=χ [sn+2 ηn+1 + sn+1 ηn+2 ]s=η (n + 1)(n + 2) 1 = (χ n+2 ηn+1 + χ n+1 ηn+2 − 2η2n+3 ). (n + 1)(n + 2) =

Summarizing, it holds that n+2

β k̄n+1 (χ, η) = −( ) 4D

n+2

= −(

β ) 4D

1 1 (χ n+2 ηn+1 + χ n+1 ηn+2 ) n!(n + 1)! (n + 1)(n + 2) 1 (χ + η)χ n+1 ηn+1 . (n + 1)!(n + 2)!

D.2 Backward transformation in Subsection 5.4.1

| 201

This completes the proof by induction and shows that k̄n (χ, η) is given by (D.9). Accordingly, the solution to the pde (D.4) is given by β

∞ ( χη)i ̄ η) = − β (χ + η) ∑ 4D , k(χ, 4D i!(i + 1)! i=0

and thus, recalling (D.3) the solution of the original pde (D.2) by k(z, ζ ) = −

β

2 2 i β ∞ ( 4D (z − ζ )) z∑ 2D i=0 i!(i + 1)!

or equivalently k(z, ζ ) = −

β

√ (z 2 − ζ 2 ) 2i β ∞ 1 z∑ ( D ) 2D i=0 i!(i + 1)! 2

(D.10)

Recalling the definition of the modified Bessel function of first kind and first order (Dettman 1988, Abramowitz and Stegun 1964) I1 (x) =

x 2i x ∞ (2) ∑ 2 i=0 i!(i + 1)!

it follows by a direct comparison that β 2 2 β I1 (√ D (z − ζ )) k(z, ζ ) = − z . D √ β (z 2 − ζ 2 )

(D.11)

D

D.2 Backward transformation in Subsection 5.4.1 The backward transformation is defined by z

x(z, t) = ξ (z, t) + ∫ l(z, ζ )ξ (ζ , t)dζ 0

Taking into account the target dynamics (5.62) and original dynamics (5.59) it holds that z

𝜕t x(z, t) = D𝜕z2 x(z, t) = D𝜕z2 (ξ (z, t) + ∫ l(z, ζ )ξ (ζ , t)dζ ) 0

=

D𝜕z2 ξ (z, t)

z

+ D𝜕z [l(z, z)ξ (z, t) + ∫ 𝜕z l(z, ζ )ξ (ζ , t)dζ ] 0

202 | D Kernel derivations for the backstepping approach = D𝜕z2 ξ (z, t) + D𝜕z l(z, z)ξ (z, t) + D𝜕ζ l(z, z)ξ (z, t) + Dl(z, z)𝜕z ξ (z, t) z

+ D𝜕z l(z, z)ξ (z, t) + ∫ D𝜕z2 l(z, ζ )ξ (ζ , t)dζ . 0

At the same time it holds that z

𝜕t x(z, t) = 𝜕t ξ (z, t) + ∫ l(z, ζ )𝜕t ξ (ζ , t)dζ 0

= =

=

D𝜕z2 ξ (z, t)

z

− βξ (z, t) + ∫ l(z, ζ )[D𝜕ζ2 ξ (ζ , t) − βξ (ζ , t)]dζ

0 2 D𝜕z ξ (z, t) − βξ (z, t) + [Dl(z, ζ )𝜕ζ ξ (ζ , t) z + ∫[D𝜕ζ2 l(z, ζ ) − βl(z, ζ )]ξ (ζ , t)dζ 0 D𝜕z2 ξ (z, t) − βξ (z, t) + Dl(z, z)𝜕z ξ (z, t) z + ∫[D𝜕ζ2 l(z, ζ ) − βl(z, ζ )]ξ (ζ , t)dζ

z

− D𝜕ζ l(z, ζ )ξ (ζ , t)]0

− D𝜕ζ l(z, z)ξ (z, t) + D𝜕ζ l(z, 0)ξ (0, t)

0

Equalling the preceding expressions for 𝜕t x(z, t) yields that l(z, ζ ) must satisfy β 𝜕z2 l(z, ζ ) − 𝜕ζ2 l(z, ζ ) = − l(z, ζ ) D β d l(z, z) = − dz 2D 𝜕ζ l(z, 0) = 0. Note that this equation set corresponds to (D.1) after substituting l(z, ζ ) = −k(z, ζ ) and β with −β, and thus the solution for the kernel of the backward transformation is given by −β 2 β 2 2 2 β I1 (j√ D (z − ζ )) β I1 (√ D (z − ζ )) =− z . l(z, ζ ) = − z D √ −β (z 2 − ζ 2 ) D j√ β (z 2 − ζ 2 ) D

D

In virtue of the property In (jx) = jn Jn (x),

j = √−1

with the Bessel function of first kind and order n denoted by Jn (x), it follows that β 2 2 β J1 (√ D (z − ζ )) l(z, ζ ) = − z . D √ β (z 2 − ζ 2 ) D

D.3 Observation error forward transformation from subsection 5.4.1

| 203

D.3 Observation error forward transformation from subsection 5.4.1 Recall the linear subsystem dynamics (5.86a)-(5.86c) of the observation error and the associated target dynamics (5.87) (for ν = 0) and set the backward transformation as z

̃ t) = ξ ̃ (z, t) − ∫ p(z, ζ )x(ζ ̃ , t)dζ . x(z, 0

It follows that ̃ t) = 𝜕t x(z,

=

D𝜕z2 (ξ ̃ (z, t) D𝜕z2 ξ ̃ (z, t)

z

̃ t) − ∫ p(z, ζ )ξ ̃ (ζ , t)dζ ) − l(z)x(0, 0

z

− D𝜕z (p(z, z)ξ ̃ (z, t) + ∫ 𝜕z p(z, ζ )ξ ̃ (ζ , t)dζ ) − l(z)ξ ̃ (0, t) 0

d = D𝜕z2 ξ ̃ (z, t) − D p(z, z)ξ ̃ (z, t) − Dp(z, z)𝜕z ξ ̃ (z, t) − D𝜕z p(z, z)ξ ̃ (z, t) dz z

− ∫ D𝜕z2 p(z, ζ )ξ ̃ (ζ , t)dζ − l(z)ξ ̃ (0, t) 0

z

= 𝜕 ξ ̃ (z, t) − ∫ p(z, ζ )𝜕 ξ ̃ (ζ , t)dζ t

t

0

z

= D𝜕z2 ξ ̃ (z, t) − βo ξ ̃ (z, t) − ∫ p(z, ζ )[D𝜕ζ2 ξ ̃ (z, t) − βo ξ ̃ (z, t)]dζ =

=

0 2 ̃ ̃ D𝜕z ξ (z, t) − βo ξ (z, t) − [Dp(z, ζ )𝜕z ξ ̃ (ζ , t) z − ∫[D𝜕ζ2 p(z, ζ ) − βo p(z, ζ )]ξ ̃ (z, t)dζ 0 D𝜕z2 ξ ̃ (z, t) − βo ξ ̃ (z, t) − Dp(z, z)𝜕z ξ ̃ (z, t) z − ∫[D𝜕ζ2 p(z, ζ ) − βo p(z, ζ )]ξ ̃ (z, t)dζ .

z − D𝜕ζ p(z, ζ )ξ ̃ (ζ , t)]0

+ D𝜕ζ p(z, z)ξ ̃ (z, t) − D𝜕ζ (z, 0)ξ ̃ (0, t)

0

Thus, it has to hold that D𝜕z2 p(z, ζ ) − D𝜕ζ2 p(z, ζ ) + βo p(z, ζ ) = 0 2D

d p(z, z) = βo dz l(z) = D𝜕ζ p(z, 0).

(D.12a) (D.12b) (D.12c)

204 | D Kernel derivations for the backstepping approach Additionally, in order to satisfy the left boundary condition it must hold that z

̃ t) = 𝜕z (ξ ̃ (z, t) − ∫ p(z, ζ )ξ ̃ (ζ , t)dζ )|z=0 = −p(0, 0)ξ ̃ (0, t) = −l0 ξ ̃ (0, t) 𝜕z x(0, 0

or equivalently l0 = p(0, 0). For the right boundary condition to hold it is required that 1

̃ t) = 𝜕z ξ ̃ (1, t) − p(1, 1)ξ ̃ (1, t) − ∫ 𝜕z p(1, ζ )ξ ̃ (ζ , t)dζ = 0. 𝜕z x(1, 0

From (D.12b) it follows that p(z, z) = C +

βo z, 2D

C ∈ ℝ.

With C = −βo /(2D) it follows that p(z, z) =

βo (z − 1) ⇒ p(1, 1) = 0 2D

and thus the additional condition on p(z, ζ ) reads 𝜕z p(1, ζ ) = 0. Summarizing, the kernel pde for the backward transformation is given by D𝜕z2 p(z, ζ ) − D𝜕ζ2 p(z, ζ ) + βo p(z, ζ ) = 0

βo (z − 1) 2D 𝜕z p(1, ζ ) = 0, p(z, z) =

and the observer gains are determined by l(z) = 𝜕z p(1, ζ ) = 0,

l0 = p(0, 0).

Introducing the change of variables z̄ = 1 − ζ ,

ζ ̄ = 1 − z,

p(̄ z,̄ ζ ̄ ) = P(z, ζ )

so that 𝜕ζ ̄ = −𝜕z ,

𝜕z̄ = −𝜕ζ

(D.13a) (D.13b) (D.13c)

D.4 Transformation from Subsection 5.4.2

| 205

and (D.13a) becomes D𝜕ζ2̄ p(̄ z,̄ ζ ̄ ) − D𝜕z2̄ p(̄ z,̄ ζ ̄ ) = −βo p(̄ z,̄ ζ ̄ ) one obtains that β 𝜕z2̄ p(̄ z,̄ ζ ̄ ) − D𝜕ζ2̄ p(̄ z,̄ ζ ̄ ) = o p(̄ z,̄ ζ ̄ ) D β p(̄ z,̄ z)̄ = − o z̄ 2D 𝜕ζ ̄ p(̄ z,̄ 0) = 0. This pde is equivalent to (D.2) with solution βo 2 ̄2 β I1 (√ D (z̄ − ζ )) p(̄ z,̄ ζ ̄ ) = − o z̄ . D √ βo (z̄2 − ζ ̄ 2 ) D

In original coordinates this becomes β I1 (√ Do ((1 − ζ )2 − (1 − z)2 )) βo p(z, ζ ) = − (1 − ζ ) D √ βo ((1 − ζ )2 − (1 − z)2 ) D

or equivalently β I1 (√ Do (z − ζ )(2 − (z + ζ ))) βo p(z, ζ ) = − (1 − ζ ) . D √ βo (z − ζ )(2 − (z + ζ )) D

D.4 Transformation from Subsection 5.4.2 Consider the original dynamics z

𝜕t x(z, t) = 𝜕z x(z, t) + ∫ f (z, ζ )x(ζ , t)dζ

(D.14a)

0

x(1, t) = u(t)

(D.14b)

and the target dynamics 𝜕t ξ (z, t) = 𝜕z ξ (z, t) ξ (0, t) = 0.

(D.15a) (D.15b)

The associated backstepping transformation is given by z

ξ (z, t) = x(z, t) − ∫ k(zζ )x(ζ , t)dζ . 0

(D.16)

206 | D Kernel derivations for the backstepping approach Derivation of the preceding transformation with respect to time yields z

z

𝜕t ξ (z, t) = 𝜕t (x(z, t) − ∫ k(zζ )x(ζ , t)dζ ) = 𝜕t x(z, t) − ∫ k(z, ζ )𝜕t x(ζ , t)dζ 0

0

z

ζ

z

= 𝜕z x(z, t) + ∫ f (z, ζ )x(ζ , t)dζ − ∫ k(z, ζ )(𝜕ζ x(ζ , t) + ∫ f (ζ , s)x(s, t)ds)dζ 0 z

0 z

0

= 𝜕z x(z, t) + ∫ f (z, ζ )x(ζ , t)dζ − ∫ k(z, ζ )𝜕ζ x(ζ , t)dζ 0

0 ζ

z

− ∫ k(z, ζ ) ∫ f (ζ , s)x(s, t)dsdζ . 0

0

It holds that z

− ∫ k(z, ζ )𝜕ζ x(ζ , t)dζ = − 0

ζ =z k(z, ζ )x(ζ , t)|ζ =0

z

+ ∫ 𝜕ζ k(z, ζ )x(ζ , t)dζ 0

z

= − k(z, z)x(z, t) + k(z, 0)x(0, t) + ∫ 𝜕ζ k(z, ζ )x(ζ , t)dζ 0

and ζ

z

z

z

− ∫ k(z, ζ ) ∫ f (ζ , s)x(s, t)dsdζ = − ∫(∫ k(z, s)f (s, ζ )ds)x(ζ , t)dζ 0

0

0

ζ

so that z

𝜕t ξ (z, t) = 𝜕z x(z, t) + ∫ f (z, ζ )x(ζ , t)dζ − k(z, z)x(z, t) + k(z, 0)x(0, t) z

0

z

z

+ ∫ 𝜕ζ k(z, ζ )x(ζ , t)dζ − ∫(∫ k(z, s)f (s, ζ )ds)x(ζ , t)dζ . 0

0

ζ

On the other hand z

𝜕t ξ (z, t) = 𝜕z ξ (z, t) = 𝜕z (x(z, t) − ∫ k(z, ζ )x(ζ , t)dζ ) 0

z

= 𝜕z x(z, t) − k(z, z)x(z, t) − ∫ 𝜕z k(z, ζ )x(ζ , t)dζ . 0

D.4 Transformation from Subsection 5.4.2

| 207

Equalling the preceeding both relations for 𝜕t ξ (z, t) yields z

𝜕z x(z, t) − k(z, z)x(z, t) − ∫ 𝜕z k(z, ζ )x(ζ , t)dζ 0

z

= 𝜕z x(z, t) + ∫ f (z, ζ )x(ζ , t)dζ − k(z, z)x(z, t) + k(z, 0)x(0, t) z

0

z

z

+ ∫ 𝜕ζ k(z, ζ )x(ζ , t)dζ − ∫(∫ k(z, s)f (s, ζ )ds)x(ζ , t)dζ 0

0

ζ

which is satisfied for all x(z, t) if and only if z

𝜕z k(z, ζ ) + 𝜕ζ k(z, ζ ) = ∫ k(z, s)f (s, ζ )ds − f (z, ζ )

(D.17a)

ζ

k(z, 0) = 0.

(D.17b)

Deriving (D.17a) with respect to ζ yields z

𝜕ζ 𝜕z k(z, ζ ) + 𝜕ζ2 k(z, ζ ) = −k(z, ζ )f (ζ , ζ ) + ∫ k(z, s)𝜕ζ f (s, ζ )ds − 𝜕ζ f (z, ζ ) ζ

Given that this has increased the order of the differential equation, a new boundary condition is required. This can be obtained by setting ζ = z in (D.17a) yielding dz k(z, z) = (𝜕z k(z, ζ ) + 𝜕ζ k(z, ζ ))|ζ =z = −f (z, z) or equivalently z

z

k(z, z) = k(0, 0) + ∫ −f (ζ , ζ )dζ = − ∫ f (ζ , ζ )dζ 0

0

given that k(0, 0) = 0 according to (D.17b). Summarizing, the differential equation set (D.17) is equivalent to z

𝜕ζ 𝜕z k(z, ζ ) + 𝜕ζ2 k(z, ζ ) = −k(z, ζ )f (ζ , ζ ) + ∫ k(z, s)𝜕ζ f (s, ζ )ds − 𝜕ζ f (z, ζ )

(D.18a)

ζ

k(z, 0) = 0

(D.18b) z

k(z, z) = − ∫ f (ζ , ζ )dζ . 0

(D.18c)

208 | D Kernel derivations for the backstepping approach For the particular case that f (z, ζ ) = aeb(z−ζ ) this yields z

𝜕ζ 𝜕z k(z, ζ ) + 𝜕ζ2 k(z, ζ ) = −ak(z, ζ ) − ∫ k(z, s)abeb(s−ζ ) dζ + abeb(z−ζ ) ζ z

= −ak(z, ζ ) − b(∫ k(z, s)f (s, ζ )ds − f (z, ζ )) ζ

Substituting the right-hand side of (D.17a) into this equation yields 𝜕ζ 𝜕z k(z, ζ ) + 𝜕ζ2 k(z, ζ ) = −ak(z, ζ ) − b(𝜕z k(z, ζ ) + 𝜕ζ k(z, ζ ))

(D.19a)

k(z, 0) = 0

(D.19b)

k(z, z) = −az.

(D.19c)

Note that at this stage no more integral term appears so that the variable s can be redefined. Introduce the change of variables k(z, ζ ) = p(s, ζ )eb(s−ζ )/2 ,

s = 2z − ζ

(D.20)

with 𝜕z s = 2,

𝜕ζ s = −1.

It follows that b 𝜕z k(z, ζ ) = (𝜕s p(s, ζ ) + p(s, ζ ) )eb(s−ζ )/2 𝜕z s 2 b = 2(𝜕s p(s, ζ ) + p(s, ζ ))eb(s−ζ )/2 2 b b 𝜕ζ k(z, ζ ) = [(𝜕s p(s, ζ ) + p(s, ζ ) )𝜕ζ s + 𝜕ζ p(s, ζ ) − p(s, ζ )]eb(s−ζ )/2 2 2 = (−𝜕s p(s, ζ ) + 𝜕ζ p(s, ζ ) − bp(s, ζ ))eb(s−ζ )/2

b p(s, ζ ))eb(s−ζ )/2 ] 2 b + 𝜕s [2(𝜕s p(s, ζ ) + p(s, ζ ))eb(s−ζ )/2 ]𝜕ζ s 2

𝜕ζ 𝜕z k(z, ζ ) = 𝜕ζ [2(𝜕s p(s, ζ ) +

= [2𝜕ζ 𝜕s p(s, ζ ) + b𝜕ζ p(s, ζ ) − b𝜕s p(s, ζ ) − − [2𝜕s2 p(s, ζ ) + b𝜕s p(s, ζ ) + b𝜕s p(s, ζ ) +

b2 p(s, ζ )]eb(s−ζ )/2 2

b2 p(s, ζ )]eb(s−ζ )/2 2

= (2𝜕ζ 𝜕s p(s, ζ ) − 2𝜕s2 p(s, ζ ) + b𝜕ζ p(s, ζ ) − 3b𝜕s p(s, ζ ) − b2 p(s, ζ ))eb(s−ζ )/2

𝜕ζ2 k(z, ζ ) = 𝜕ζ [(−𝜕s p(s, ζ ) + 𝜕ζ p(s, ζ ) − bp(s, ζ ))eb(s−ζ )/2 ]

+ 𝜕s [(−𝜕s p(s, ζ ) + 𝜕ζ p(s, ζ ) − bp(s, ζ ))eb(s−ζ )/2 ]𝜕ζ s

D.4 Transformation from Subsection 5.4.2

= [−𝜕ζ 𝜕s p + 𝜕ζ2 p − b𝜕ζ p +

| 209

b b2 b 𝜕s p − 𝜕ζ p + p]eb(s−ζ )/2 2 2 2

b b b2 𝜕s p + 𝜕ζ p − p]eb(s−ζ )/2 2 2 2

− [−𝜕s2 p + 𝜕s 𝜕ζ p − b𝜕s p −

= (𝜕s2 p − 2𝜕ζ 𝜕s p + 𝜕ζ2 p − 2b𝜕ζ p + 2b𝜕s p + b2 p)eb(s−ζ )/2 so that the differential equation (D.19) for p(s, ζ ) is given by

𝜕ζ 𝜕z k(z, ζ ) + 𝜕ζ2 k(z, ζ ) + b(𝜕z k(z, ζ ) + 𝜕ζ k(z, ζ )) = 𝜕ζ2 p(s, ζ ) − 𝜕s2 (s, ζ ) = −ap(s, ζ )

with boundary conditions p(s, 0) = 0 p(s, s) = −as. Summarizing the above yields 𝜕s2 p(s, ζ ) − 𝜕ζ2 p(s, ζ ) = ap(s, ζ )

(D.21a)

p(s, 0) = 0

(D.21b)

p(s, s) = −as.

(D.21c)

The solution of this pde is established in the following lemma. Lemma D.4.1. Let p(s, ζ ) satisfy the hyperbolic pde 𝜕s2 p(s, ζ ) − 𝜕ζ2 p(s, ζ ) = γp(s, ζ )

(D.22a)

p(s, s) = αs

(D.22b)

p(s, 0) = 0.

(D.22c)

Then p(s, ζ ) is given by p(s, ζ ) = 2αζ

I1 (√γ(s2 − ζ 2 ))

(D.23)

√γ(s2 − ζ 2 )

with I1 being the modified Bessel function of first kind and order one. Proof. To solve the pde (D.22) consider the change of coordinates η = s + ζ,

χ = s − ζ,

⇔

s=

η+χ , 2

ζ =

with partial derivatives 𝜕s η = 1,

𝜕ζ η = 1,

𝜕s χ = 1,

𝜕ζ χ = −1

η−χ , 2

(D.24)

210 | D Kernel derivations for the backstepping approach and the function ̄ χ) = p(s, ζ ) = p( p(η,

η+χ η−χ , ). 2 2

Accordingly ̄ χ) ̄ χ) + 𝜕χ p(η, ̄ χ)𝜕s χ = 𝜕η p(η, ̄ χ)𝜕s η + 𝜕χ p(η, 𝜕s p(s, ζ ) = 𝜕η p(η,

̄ χ) + 𝜕χ2 p(η, ̄ χ) + 2𝜕η 𝜕χ p(η, ̄ χ) 𝜕s2 p(s, ζ ) = 𝜕η2 p(η,

̄ χ)𝜕ζ η + 𝜕χ p(η, ̄ χ)𝜕ζ χ) = 𝜕η p(η, ̄ χ) − 𝜕χ p(η, ̄ χ) 𝜕ζ p(s, ζ ) = (𝜕η p(η,

̄ χ) − 2𝜕η 𝜕χ p(η, ̄ χ) + 𝜕χ2 p(η, ̄ χ). 𝜕ζ2 p(s, ζ ) = 𝜕η2 p(η, Furthermore, for ζ = s it holds that η = 2s so that ̄ 0) = p(s, s) = p(η,

α η. 2

Summarizing, it follows that ̄ χ) = γ p(η, ̄ χ) 𝜕s2 p(s, ζ ) − 𝜕ζ2 p(s, ζ ) = 4𝜕η 𝜕χ p(η, ̄ η) = p(χ, ̄ χ) = 0 p(s, 0) = p(η, α ̄ 0) = η p(s, s) = p(η, 2

(D.25a) (D.25b) (D.25c)

Integrating this equation with respect to χ from 0 to χ and substituting (D.25c) yields χ

χ

0

0

γ α γ ̄ χ) = 𝜕η p(η, ̄ 0) + ∫ p(η, ̄ q)dq = + ∫ p(η, ̄ q)dq, 𝜕η p(η, 4 2 4 and after integration with respect to η from χ to η, it can be seen that p̄ satisfies the integral equation η

χ

η χ

χ

0

χ 0

γ α α γ ̄ χ) = ∫( + ∫ p(r, ̄ q)dq)dr = (η − χ) + ∫ ∫ p(r, ̄ q)dqdr. p(η, 2 4 2 4

(D.26)

This integral equation can be solved using the method of successive approximation (or Picard iteration) Teschl (2012), Picard (1893). For this purpose, introduce the series ∞

̄ χ) = ∑ p̄ n (η, χ) p(η, n=0

and substitute it into the integral equation (D.26). This yields ∞

∑ p̄ n (η, χ) =

n=0

η χ

∞ γ α (η − χ) + ∑ ∫ ∫ p̄ n (r, q)dqdr. 2 4 n=0 χ 0

(D.27)

D.4 Transformation from Subsection 5.4.2

| 211

This equation can be solved by setting p̄ 0 (η, χ) =

η χ

α (η − χ), 2

γ ∫ ∫ p̄ n−1 (r, q)dqdr, n ≥ 1. 4

p̄ n (η, χ) =

(D.28)

χ 0

In the sequel it is shown by induction that the solution p̄ n of this recursive relation is given by p̄ n (η, χ) =

n

α γ (η − χ)ηn χ n ( ) . 2 4 n!(n + 1)!

(D.29)

For the proof, first note that for n = 0 the hypothesis is satisfied given that p̄ 0 (η, χ) =

0

α γ (η − χ)η0 χ 0 α ( ) = (η − χ). 2 4 0! 1! 2

Next, consider that the hypothesis holds for an arbitrary n ∈ ℕ. By (D.28) it follows that η χ

p̄ n+1 (η, χ) =

n

γ α γ (r − q)r n qn dqdr ∫∫ ( ) 4 2 4 n!(n + 1)! χ 0

n+1

α γ = ( ) 2 4

n+1

=

α γ ( ) 2 4

η χ

1 ∫ ∫(r n+1 qn − r n qn+1 )dqdr n!(n + 1)! χ 0 η

χ n+1 χ n+2 1 − rn )dr ∫(r n+1 n!(n + 1)! n+1 n+2 χ

n+1

=

α γ ( ) 2 4

=

α γ ( ) 2 4

n+1

ηn+2 χ n+1 ηn+1 χ n+2 1 ( − ) n!(n + 1)! n + 2 n + 1 n + 1 n + 2 (η − χ)ηn+1 χ n+1 . (n + 1)!(n + 2)!

This proofs hypothesis (D.29). From (D.29) it follows that n n 󵄨 󵄨󵄨 󵄨n 󵄨󵄨 ̄ 󵄨 󵄨󵄨 α 󵄨󵄨󵄨󵄨 γ 󵄨󵄨 |η − χ|η χ 󵄨󵄨pn (η, χ)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨󵄨 n! 󵄨󵄨 2 󵄨󵄨󵄨󵄨 4 󵄨󵄨

so that ∞

󵄨󵄨 ̄ 󵄨 󵄨 󵄨 󵄨󵄨p(η, χ)󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨p̄ n (η, χ)󵄨󵄨󵄨 n=0 ∞ 󵄨󵄨

󵄨 |γ/4|n ηn χ n 󵄨 α 󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨 󵄨󵄨󵄨|η − χ| 󵄨2󵄨 n! n=0󵄨 󵄨

212 | D Kernel derivations for the backstepping approach 󵄨󵄨 α 󵄨󵄨 |γ|ηχ 󵄨 󵄨 ) = 󵄨󵄨󵄨 󵄨󵄨󵄨|η − χ| exp( 󵄨󵄨 2 󵄨󵄨 4 󵄨󵄨 α 󵄨󵄨 |γ|ηχ 󵄨 󵄨 ≤ 2󵄨󵄨󵄨 󵄨󵄨󵄨 exp( ), 󵄨󵄨 2 󵄨󵄨 4 showing that the series (D.27) converges absolutely. Thus the solution of (D.26) is given by ̄ χ) = p(η,

∞ (γ/4)n (ηχ)n α (η − χ) ∑ . 2 n!(n + 1)! n=0

(D.30)

Recall that the modified Bessel function of first order I1 is given by the series Dettman (1988), Abramowitz and Stegun (1964) I1 (σ) =

σ ∞ (σ/2)2n . ∑ 2 n=0 n!(n + 1)!

(D.31)

Substituting into (D.31) σ = √γηχ and comparing the result with (D.30) shows that ̄ χ) = α(η − χ) p(η,

I1 (√γηχ) . √γηχ

(D.32)

After transforming the variables (η, χ) back into (s, ζ )-coordinates according to (D.24) this implies that the solution is given by (D.23). Note that the pde (D.21) is equivalent to (D.22) in Lemma D.4.1 substituting γ = a,

α = −a

and thus the solution is given by p(s, ζ ) = −2aζ

I1 (√a(s2 − ζ 2 )) √a(s2 − ζ 2 )

.

(D.33)

According to (D.20), after substitution of s = 2z − ζ this implies that the kernel k(z, ζ ) of the backstepping transformation (D.16) is given by k(z, ζ ) = −2aζ

= −2aζ

I1 (√a((2z − ζ )2 − ζ 2 )) √a((2z −

ζ )2

−

I1 (2√az(z − ζ )) 2√az(z − ζ )

ζ 2)

eb((2z−ζ )−ζ )/2

eb(z−ζ )

or summarizing k(z, ζ ) = −aζ

I1 (2√az(z − ζ )) b(z−ζ ) e . √az(z − ζ )

(D.34)

D.5 Inverse transformation from Section 5.4.2

|

213

D.5 Inverse transformation from Section 5.4.2 Consider the inverse transformation to bring the system (D.15) into the original system (D.14) z

x(z, t) = ξ (z, t) + ∫ l(z, ζ )ξ (ζ , t)dζ .

(D.35)

0

Accordingly, on one side it has to holds that z

𝜕t x(z, t) = 𝜕t ξ (z, t) + ∫ l(z, ζ )𝜕t ξ (ζ , t)dζ 0 z

= 𝜕z ξ (z, t) + ∫ l(z, ζ )𝜕ζ ξ (ζ , t)dζ 0

= 𝜕z ξ (z, t) +

ζ =z l(z, ζ )ξ (ζ , t)|ζ =0

z

− ∫ 𝜕ζ l(z, ζ )ξ (ζ , t)dζ 0

and on the other side that z

𝜕t x(z, t) = 𝜕z x(z, t) + ∫ f (z, ζ )x(ζ , t)dζ 0 z

z

ζ

= 𝜕z (ξ (z, t) + ∫ l(z, ζ )ξ (ζ , t)dζ ) + ∫ f (z, ζ )(ξ (ζ , t) + ∫ l(ζ , s)ξ (s, t)ds)dζ 0

0

z

0

= 𝜕z ξ (z, t) + l(z, z)ξ (z, t) + ∫ 𝜕z l(z, ζ )ξ (ζ , t)dζ 0 z

ζ

z

+ ∫ f (z, ζ )ξ (ζ , t)dζ + ∫ f (z, ζ ) ∫ l(ζ , s)ξ (s, t)dsdζ . 0

0

0

It holds that z

ζ

z

z

∫ f (z, ζ ) ∫ l(ζ , s)ξ (s, t)dsdζ = ∫(∫ f (z, s)l(s, ζ ))ds)ξ (ζ , t)dζ . 0

0

0

ζ

Equalling both expressions for 𝜕t x(z, t) yields that z

𝜕z l(z, ζ ) + 𝜕ζ l(z, ζ ) = − ∫ f (z, s)l(s, ζ )ds − f (z, ζ ) ζ

(D.36a)

214 | D Kernel derivations for the backstepping approach l(z, 0) = 0.

(D.36b)

Taking the derivative of (D.36a) with respect to z yields z

𝜕z2 l(z, ζ ) + 𝜕z 𝜕ζ l(z, ζ ) = −f (z, z)l(z, ζ ) − ∫ 𝜕z f (z, s)l(s, ζ )ds − 𝜕z f (z, ζ ) ζ

and, given that by this last step the order of the differential equation has increased by one an additional boundary condition is required. This is obtained by substituting z = ζ in (D.36a), yielding dz l(z, z) = (𝜕z l(z, ζ ) + 𝜕ζ l(z, ζ ))ζ =z = −f (z, z).

(D.37)

For the particular case that f (z, ζ ) = aeb(z−ζ ) it holds that 𝜕z f (z, ζ ) = bf (z, ζ ) so that the above becomes z

𝜕z2 l(z, ζ ) + 𝜕z 𝜕ζ l(z, ζ ) = −al(z, ζ ) − b(∫ f (z, s)l(s, ζ )ds + f (z, ζ ))

(D.38)

0

= −al(z, ζ ) + b(𝜕z l(z, ζ ) + 𝜕ζ l(z, ζ ))

(D.39)

where the last step follows from substituting equation (D.36a). The boundary condition (D.37) becomes l(z, z) = −az.

(D.40)

Summarizing, the pde for the kernel of the inverse tranformation reads 𝜕z2 l(z, ζ ) + 𝜕z 𝜕ζ l(z, ζ ) = −al(z, ζ ) + b(𝜕z l(z, ζ ) + 𝜕ζ l(z, ζ ))

(D.41a)

l(z, 0) = 0

(D.41b)

l(z, z) = −az.

(D.41c)

Denoting the solution of this equation set by l(z, ζ ; a, b) and comparing the pde (D.41) with the equation set (D.19) for the kernel k(z, ζ ; a, b) of the forward transformation, it turns out that l(z, ζ ; a, b) = k(ζ , z; a, −b). Accordingly, (D.34) implies that l(z, ζ ) = −az

I1 (2√aζ (ζ − z)) −b(ζ −z) e . √aζ (ζ − z)

(D.42)

D.5 Inverse transformation from Section 5.4.2

|

215

Given that ζ ∈ [0, z], the difference ζ − z < 0 and thus one has to take into account the evaluation of the modified Bessel function with complex-valued arguments. Fortunately, it holds that (see e. g. Arfken 1985, Abramowitz and Stegun 1964) In (ix) = in Jn (x) where i = √−1 and Jn is the Bessel function of order n. Having this in mind, one arrives at l(z, ζ ) = −az

J1 (2√aζ (z − ζ )) b(z−ζ ) e . √aζ (z − ζ )

(D.43)

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Index δ-detectability 50 admissable initial condition 6 admissable input 7 Amplitude (exponential stability) 18 Asymptotic stability 18 Attractivity 18 Backstepping control 148 Backstepping observer design 148 Bioreactor 106 Byrnes–Isidori normal form 52, 80 Characteristic time (convergence rate) 19 Continuous-stirred tank reactor (CSTR) 101 Convergence rate (exponential stability) 18 Definiteness 20 Detectability 48 Dissipativity 23 Equilibrium solution 17 Exact feedback linearization 80 Exponential stability 18 Indistinguishability 48 Internal dynamics 53 Invariance theorem (Krasovsky–LaSalle) 21 Kalman decomposition 49 Kalman–Yakubovich–Popov lemma 26 Luenberger observer 181

Lur’e 44 Lyapunov functional 20 Lyapunov stability 17 Observability 48 Observability map 49 Observer canonical form (permutated) 181 Optimality 191 Passivity 24 Pendulum 110 Point-wise measurement injection observer 128 (Q, S, R) dissipativity 24 Quadratic form 171 Relative degree 52 Riccati equation 191 Schur complement 172 Sector condition 34 Ship course control 95 Spectral measurement injection observer 120 Stabilizability 51 Storage functional 23 Strict state dissipativity 23 Supply rate 23 Vector relative degree 53 Zero dynamics 53 Zero-state observability 25