Differentially Flat Systems [1 ed.] 9781315214658, 9781351830546, 9781351821858, 9781482276640, 9781420030396, 9780824754709

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Differentially Flat Systems [1 ed.]
 9781315214658, 9781351830546, 9781351821858, 9781482276640, 9781420030396, 9780824754709

Table of contents :

Introduction Linear Time-Invariant SISO Systems Linear Time-Invariant MIMO Systems Time-Varying Linear Systems Discrete-Time Linear Systems Infinite Dimensional Linear Systems SISO Nonlinear Systems Multivariable Nonlinear Systems Mobile Robots Flatness and Optimal Trajectories Optimal Planning with Constraints Non-Differentially Flat Systems

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Differentially Flat Systems

CONTROL ENGINEERING A Series o f Reference Books and Textbooks Editor

FRANK L. LEWIS, Ph.D. Moncrief-O’ Donnell Endowed Chair and Associate Director o f Research Automation & Robotics Research Institute University o f Texas, Arlington

1. Nonlinear Control of Electric Machinery, Darren M. Dawson, Jun Hu, and Timothy C. Burg 2. Computational Intelligence in Control Engineering, Robert E. King 3. Quantitative Feedback Theory: Fundamentals and Applications, Con­ stantine H. Houpis and Steven J. Rasmussen 4. Self-Learning Control of Finite Markov Chains, A. S. Poznyak, K. Najim, and E. Gdmez-Ramirez 5. Robust Control and Filtering for Time-Delay Systems, Magdi S. Mahmoud 6. Classical Feedback Control: With MATLAB, Boris J. Lurie and Paul J. Enright 7. Optimal Control of Singularly Perturbed Linear Systems and Applications: High-Accuracy Techniques, Zoran Gajic and Myo-Taeg Lim 8. Engineering System Dynamics: A Unified Graph-Centered Approach, Forbes T. Brown 9. Advanced Process Identification and Control, Enso Ikonen and Kaddour Najim 10. Modern Control Engineering, P. N. Paraskevopoulos 11. Sliding Mode Control in Engineering, edited by Wilfrid Perruquetti and Jean Pierre Barbot 12. Actuator Saturation Control, edited by Vikram Kapila and Karolos M. Grigoriadis 13. Nonlinear Control Systems, Zoran Vukic, Ljubomir Kuljaca, Dali Donlagic, Sejid TeSnjak 14. Linear Control System Analysis and Design with MATLAB: Fifth Edition, Revised and Expanded, John J. D ’Azzo, Constantine H. Houpis, and Stuart N. Sheldon 15. Robot Manipulator Control: Theory and Practice, Second Edition, Re­ vised and Expanded, Frank L. Lewis, Darren M. Dawson, and Chaouki T. Abdallah

16. Robust Control System Design: Advanced State Space Techniques, Second Edition, Revised and Expanded, Chia-Chi Tsui 17. Differentially Flat Systems, Hebertt Sira-Ramirez and Sunil K. Agrawal Additional Volumes in Preparation

Differentially Flat Systems Hebertt Sira-Ranrrirez Center fo r Research and Advanced Studies o f the National Polytechnic Institute Mexico City, Mexico

Sunil K. Agrawal University o f Delaware Newark , D elaware , U.S.A.

CRC Press Taylor & Francis Group S

Boca Raton London New York

C R C Press is an im print of the Taylor & Francis Group, an inform a business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway N W , Suite 300 B oca Raton, FL 33487-2742 © 2004 by Taylor & Francis Group, LLC CRC Press is an im print o f Taylor & Francis Group, an Inform a business N o claim to original U.S. G overnm ent works T his b o o k contains inform ation obtained from authentic and highly regarded sources. Reason­ able efforts have been made to publish reliable data and inform ation, but the author and publisher cannot assume responsibility for the validity o f all materials or the consequences o f their use. The authors and publishers have attem pted to trace the copyright holders o f all material reproduced in this publication and apologize to copyright holders if perm ission to publish in this form has not been obtained. If any copyright m aterial has not been acknow ledged please write and let us know so we may r ectify in any future reprint. Except as perm itted under U.S. C opyright Law, no part o f this b o o k may be reprinted, reproduced, transm itted, or utilized in any form by any electronic, m echanical, or other means, now know n or hereafter invented, including p h otocop yin g, m icrofilm ing, and recording, or in any inform ation storage or retrieval system, without w ritten perm ission from the publishers. For perm ission to p h o to co p y or use material electronically from this work, please access www. copyright.com (h ttp://w w w .cop y rig h t.com /) or contact the C opyright Clearance Center, Inc. (CCC), 222 R osew ood Drive, Danvers, M A 01923, 978-750-8400. C C C is a not-for-profit organiza­ tion that provides licenses and registration for a variety o f users. For organizations that have been granted a p h o to co p y license by the C CC, a separate system o f payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation w ithout intent to infringe. Visit the Taylor & Francis W eb site at http://www.taylorandfrancis.com and the CRC Press W eb site at http://www.crcpress.com

Series Introduction Many textbooks have been written on control engineering, describing new techniques for controlling systems, or new and better ways of mathemati­ cally formulating existing methods to solve the ever-increasing complex problems faced by practicing engineers. However, few of these books fully address the applications aspects of control engineering. It is the intention of this new series to redress this situation. The series will stress applications issues, and not just the mathemat­ ics of control engineering. It will provide texts that present not only both new and well-established techniques, but also detailed examples of the ap­ plication of these methods to the solution of real-world problems. The au­ thors will be drawn from both the academic world and the relevant applica­ tions sectors. There are already many exciting examples of the application of control techniques in the established fields of electrical, mechanical (including aero­ space), and chemical engineering. We have only to look around in today’s highly automated society to see the use of advanced robotics techniques in the manufacturing industries; the use of automated control and navigation systems in air and surface transport systems; the increasing use of intelli­ gent control systems in the many artifacts available to the domestic con­ sumer market; and the reliable supply of water, gas, and electrical power to the domestic consumer and to industry. However, there are currently many challenging problems that could benefit from wider exposure to the applica­ bility of control methodologies, and the systematic systems-oriented basis inherent in the application of control techniques. This series presents books that draw on expertise from both the aca­ demic world and the applications domains, and will be useful not only as academically recommended course texts but also as handbooks for practitio­ ners in many applications domains. Differentially Flat Systems is another outstanding entry in Dekker’s Control Engineering series.

Preface Differential flatness (or, flatness, in short) is a property of some controlled dy­ namic systems which allows to trivialize the trajectory planning tasks, without solving differential equations, while optionally simplifying the feedback con­ troller design problem to that of a set of decoupled linear time invariant sys­ tems. Roughly speaking, flatness is equivalent to controllablity and, hence, most systems of interest will exhibit this property. The flatness property allows for a complete parametrization of all system variables (states, inputs, outputs) in terms of a finite set of independent variables, called the flat outputs, and a finite number of their time derivatives. Flat outputs, and their derivatives, determine the states possibly with the help of the inputs and their derivatives. Thus, in a flat system a special observability property is enjoyed by the state with respect to this set of artificial outputs. The number of flat outputs is equal to the num­ ber of control inputs. Generally speaking, flat outputs are internal variables to the system and, hence, they are a function of the states and of a finite num­ ber of derivatives of the input components. The unique feature of allowing a parametrization of all system variables, makes of flatness a tool for analysis re­ vealing the nature of each system variable in its isolated relation with a centrally important set of variables from the viewpoint of controllability and observabil­ ity. The invertible parametrization, involved in the flat outputs definition, thus creates a local bijection between system state solutions and arbitrary trajecto­ ries in the flat output space. By specifying the desired flat outputs trajectories, the nominal state and input trajectories are completely defined without solv­ ing differential equations. Thus, flatness allows,in an off-line manner, to check and adjust the free features of the desired behavior to state, input, or output, space restrictions with relative ease. The importance of the flatness property would be easily underestimated if it were not for the fact that, in most studied physical systems, the flat outputs enjoy a distinctive, clear cut and appealing meaning on which to base the desired behavior of the system. Establishing the flat outputs is, generally speaking, hard\ since no systematic method exists for their determination, except in the linear systems case and affine nonlinear single input case. However, in most instances, one gets by with the aid of inspection, educated guessing and physical intuition (something similar can also be said about Lyapunov functions for stability assessment). In this book, we attempt an examination and exploitation of the flatness property in a variety of controlled dynamical systems. Our road map takes




us from the most elementary class of systems progressing towards the more complex class. The idea is to underline, through many examples, most of them describing physical systems, the prevailing usefulness of the flatness property and the simplicity involved in its exploitation. To make the material accessible, the emphasis will not be placed on the theoretical aspects of flatness, which can be quite demanding, but, rather, on how to uncover it, how to use it, and how to enjoy its power and elegance in planning and design tasks. The book is therefore intended not primarily for the researcher, nor for the highly mathematically oriented designer or engineer, but, mainly, for the student, for the professor who would like to grasp and find examples, as an aid in his teaching of some of the material, and, hopefully, for the practical engineer facing non-standard control projects. Chapter 1 introduces the idea of flatness and presents a historic survey of some of the most relevant contributions to the area, and its theoretical ancestry, in the available literature. Our view of the history of flatness is, of course, far from complete but within the collection of cited references the reader will find deeper roots of the idea and their influence on recent and ongoing developments. Chapter 2 is devoted to examine flatness in the context of linear time in­ variant single-input single-output systems (SISO). The two most popular rep­ resentation of SISO linear systems are: transfer function (input-output) form and state space form. In this elementary class of systems, the relation between flatness and controllability is the simplest one; they are equivalent. This fact is easy to show in any of the two representations being studied. The flat output is intimately connected with the Bezout equation in the transfer function rep­ resentation while, in state space form, it enjoys a direct relationship with the controllability matrix. The flat output is obtained by the linear combination of the states provided by the last row of the inverse of the controllability matrix. We also find out that the flat output is an observable output of the highest relative degree. This characterization is perhaps the simpler one. The problem of controlling non-minimum phase outputs is greatly simplified by the indirect use of the flatness concept. This is specially so in rest-to-rest trajectory tracking maneuvers. In arbitrary trajectory tracking problems for a non-minimum phase output the problem may be solved by resorting to backwards time integration (i.e. stable integration) of the differential equation arising from the differential parametrization of the non-minimum phase output in terms of the flat output. The chapter provides a number of physically oriented examples. A high-gain hierarchical control method is explained in the context of the stabilization prob­ lem for an inverted pendulum on a cart and for the double inverted pendulum on a moving cart. An important problem in designing controllers is that of deal­ ing with perturbation inputs, specially when they are not of the matched type. We demonstrate through a couple of simple physically motivated examples how flatness can still be assessed and used in the context of randomly perturbed systems. Chapter 3 undertakes the issue of flatness in multiple-input, multiple-output (MIMO) time invariant (square) systems. We begin by characterizing flatness in systems in polynomial matrix representation. In this representation, unimod­



ularity, coprimeness Bezout’s identitiy and controllabiliy are intimately related to flatness. We also explore the characterization of flatness in the context of rational transfer matrices. We refresh the concepts of system poles and zeros in the context of MIMO systems. The input-output representation is also allowed to be non-square and this case immediately ties in with polynomial systems representation naturally leading to quasi state representation. The chapter con­ siders also systems in state space form and the relation of flatness with the “square” controllability matrix is clearly exhibited. Again the flat outputs are obtained by the linear combination of the state variables with selected row vec­ tors of the inverse of the controllability matrix. Precisely those determined by the Kronecker controllability indices of the system. Chapter 4 centers around time-varying linear systems of single or multiple inputs nature. The only studied representation is the state space description. The case of uniform controllability is first studied and the relation with Sil­ verman’s controllability criterion is established in the same manner as in the time-invariant case. Some physical system examples ranging from power con­ verters to linearization of the non-holonomic car are presented as instances of controller design problems. The non-uniformly controllable case is also stud­ ied following interesting known results established in the recent control systems literature. Chapter 5 is devoted to study flatness in the context of discrete time li­ near systems (SISO and MIMO cases). Any simple definition of flatness proves to be intimately related to the need for ensuring causality. It turns out that flatness can be defined in a causal way and in an anti-causal manner. Surpris­ ingly enough, we find some interesting physical examples where the anti-causal flatness option, related to a difference forward parametrization of the system variables in terms of the flat output and a finite number of its advances, yields a feasible (causal) controller. In this chapter we, again, undertake the issue of flatness within the two most popular representations of linear time invari­ ant discrete time systems; the input-output representation and the state space representation. Chapter 6 closes the broad class of linear systems by briefly studying flatness in some infinite dimensional linear systems. We heavily rely on results and examples found in the literature devoted to this fascinating issue. We are very superficial in this chapter as many of the developments presented are the subject of recent publications by other authors and colleagues. No general theory still exists of flatness for the important class of infinite dimensional systems. We only study some elementary facts of flatness in linear delay differential systems and present a collection of examples related by systems described by partial differential equations. None of the results appearing in this chapter are credited to any of the authors of this book and only a glimpse is provided at the power, simplicity, elegance and generality of the concept of flatness. A comment on the implications of flatness in some physically significant systems, described by linear (and some nonlinear) partial differential equations with boundary control, is in order. Flatness, when it is verified, provides a very precise and easy to understand tool for the digital computer simulation of distributed systems with



boundary control. The simulations though do not entitle the integration of differential equations, nor complicated numerical schemes dealing with finite difference methods at all. This is, in itself, an achievement of great importance in assessing nominal boundary control inputs from desired behavior. Chapter 7 initiates the study of flatness in nonlinear systems. It under­ takes the simplest case; that of SISO nonlinear systems. This is the last class of systems where the concept of flatness is still clearly related to the system controllability. A system which enjoys the strong accessibility property is flat. The converse is not true as there are many nonlinear systems (specially those included in the class of underactuated systems) which enjoy the strong acces­ sibility property and are proven not to be flat. We point out that flatness is not necessarily circumscribed to systems which are linear (affine) in the control input. The concept equally handles systems which are nonlinear in the control. For affine in the control systems the geometric conditions of exact feedback linearizability are obtained in terms of the involutivity of a certain distribution of vector fields and the linear independence of a set of vector fields intimately related to the (nonlinear) “controllability” matrix of the system. The gradient of the flat output is still found as the last row vector of the inverse of such a controllability matrix. Perhaps, the simplest class of nonlinear systems enjoying the strong accessibility property, which are also not flat, are constituted by the, so called, Liouvillian systems. These are systems in which the defect variables are integrable in terms of differential functions of the “largest” linearizing out­ put (the output playing the role of the flat output in the largest linearizable subsystem). This class of systems are studied in Chapter 12. Chapter 8 continues the study of flatness now for the case of multivariable nonlinear systems. We first study the flatness property for the subclass of sys­ tems linearizable by means of static state feedback and control input coordinates re-definition. Flatness, when it is verified, clearly solves the problem of deter­ mining the control input extension which is required as a necessary condition for static feedback linearizablity of the suitably extended system. In this chap­ ter we present a series of physically oriented examples which have constituted challenging examples for application of developed dynamic feedback controller design techniques. Namely, the PVTOL example, the rocket example, the uni­ cycle example and many others. We illustrate, with some simple application examples, an interesting criterion, known as the ruled manifold criterion, which constitutes a necessary condition for flatness. Chapter 9 concentrates on a quite studied class of nonlinear multivariable systems. Namely, that of mobile robots. We model and establish the flatness property of a number of well known mobile robot examples, such as the nonholonomic car, with one or several trailers, the sleigh, the cycab. We also study some other mobile robots and illustrate flatness based controllers. In particular we undertake the prismatic-prismatic-revolute (PPR) robot with a flexible pointing arm. We present, in full detail, the assessment of flatness and controller designs for the hovercraft system model and an interesting well-known walking toy model, or biped. In Chapter 10, Calculus of Variation techniques are used to find extrema



of functionals under different boundary conditions involving fixed and free end times and end states. These necessary conditions for extrema are extended to functionals with higher derivatives, relevant to optimization problems for differentially flat systems. The problems of auxiliary constraints are treated using calculus of variation techniques. Chapter 11 extends the results of calculus of variations to address optimal planning problems in the presence of inequality constraints. This chapter uses slack variables to change inequality constraints into equality constraints. The necessary conditions for optimality are obtained through the use of Lagrange multipliers. This approach is applied to dynamic systems described in both first-order and higher-order forms. Numerical algorithms are provided for solv­ ing the dynamic optimization problems using collocation. Solution of feasible planning problem for differentially flat systems are presented motivated from near real-time numerical solution with the planning problem mapped into the differentially flat space. Poly topic approximations are made to the inequality constraints to decrease the computational complexity of the problem in real-time implementation. Laboratory implementations are presented. Chapter 12 studies several examples belonging to the class of Liouvillian sys­ tems. In this class of systems, the “linearizing output” of the largest linearizable subsystem still plays a central role in trajectory planning issues. We provide a collection of system examples which are not differentially flat. An effective feedback controller design, for arbitrary trajectory tracking, for such systems is quite challenging and continues to inspire interesting new approaches to such problems. The book contains a number of straightforward exercises and some homework problems of varying complexity in some of the chapters. Some of the topics treated are just not suitable for homework exercises from which to quickly learn of our own progress and abilities. The basic idea is to let the reader try by (her)himself some designs, simulations and additional analysis studies in the most traditional usages of flatness. Only a very few problems require knowledge beyond that exposed in the book. The material in this book has been used for one-semester courses at a grad­ uate level. People attending those courses usually had wide varying interests within the engineering discipline: mechanical, electrical, chemical and industrial engineers. Most of them were able to quickly relate the material to their own field of interest. The principal aim in writing this book is to explain, in the simplest terms, the concept of flatness and to be able to use it, either in design or analysis tasks, on system models. We hope to have fulfilled such a quest if only partially. Writing this book has been a sustained effort of which the authors have also derived great pleasure. We are aware of the possibilities of many mistakes. For all these, we sincerely apologize and do seek forgiveness by asking for the kind help of the readers in pointing out those blunders to us.

Hebertt Sira -Ramirez SunilK Agrawal

Acknowledgments The writing of this book would have been much harder without the support, encouragement and cooperation of many people and Institutions, both in America and Europe. The first author, HSR, would like to pay sincere tribute, and recognition, to Professor Michel Fliess, whose friendship, advice and support have been a treasured asset over many years. Professor Fliess' influence has been determinant in undertaking this writing job. I sincerely continue to enjoy his many original and far’ reaching contributions to the Automatic Control discipline. HSR would also like to express his gratitude to all his colleagues, and former students, of his Alma M ater; the Universidad de Los Andes (ULA), in Merida, Venezuela, where the idea of this book started to take shape after his retirement, from that beloved University, in 1995. Special thanks are due to Richard Marquez, a former undergraduate and Master's student of the ULA, who latter became a PhD student of Prof. Michel Fliess. Richard has been a continuous source of inspiration and insight. Thanks are due to Mario Spinetti’Rivera for his enormous help, during HSR's long absence from Venezuela, in many diverse matters. The encouragement of Orestes Llanes*Santiago is also gratefully acknowledged. The Centro de Investiga­ tion y Estudios Avanzados del Instituto Politecnico Nacional (CinvestavIPN), in Mexico City, has been the academic home o f HSR during the last five years. The kind support and generosity of this institution are gladly acknowledged. Among his colleagues at Cinvestav, special thanks are due to Dr. Gerardo Silva-Navarro, a dearest friend, whose continuous support and encouragement have been particularly helpful at the lowest energy points of this writing project. He has been particularly kind and brave enough to go ahead and try, in the company of able students, some of the author's preaching about flat systems in experimental setups. Dr. Rafael Castro-Linares and Dr. Vicente Parra-Vega have shared, at different points, the enthusiasm of the author for some of the ideas and techniques appearing in this book. Thanks are also due to the many graduate students in Cinvestav who have endured the teaching and reading of this material in the form of notes and handouts. A sincere acknowledgment to the Consejo Nacional de Ciencia y Tecnologia, (CONACYT-Mexico) for the generous funding of Research Projects- 32681'A and 42231-Y, which have made possible, in one way or another, some of the developments and work printed out in this book.




The author HSR has benefited from the support of L'Ecole Centrale de Lille, through several summers, where he has had the opportunity to discuss, through seminars and informal chats, some of the topics regarding flatness with his colleagues Professors W. Perruquetti and J.P. Richard. He has also enjoyed long visits to the Gage Laboratoire of L’Ecole Polytechnique, in Palaisseau, France. The generosity of its director, Professor Marc Giusti, and the kindness of Professor Francois Ollivier are gratefully acknowledged. The author acknowledges fruitful discussions, the opportunity of delivering and attending seminars, and visits, with Professors P. Rouchon, J. Levine, J. Rudolph, H. Mounier and E. Delaleau. Thanks are due to the Mechanical Engineering Department, of the University of Delaware at Newark, for inviting HSR to teach portions of the material in this book during the Fall Semester 2001. The graduate students of Prof. Sunil Agrawal at the UoD have kindly provided precious feedback on the contents of preliminary notes. HSR wants to express his warmest recognition to his wife, Maria Elena Gozaine-Mendoza, for her immense generosity, support and tender affection. She has stoically endured, with compassion, the complete distraction of the author, from earthly matters, required to conceive, give shape, and finish this book. SKA would like to deeply acknowledge his graduate students Dr. Tawiwat Veeraklaew, Dr. Nadeem Faiz, Dr. Madhu Annapragada, Dr. Xiaochun Xu, Dr. Armando Ferreira, Dr. Stephen Pledgie, Dr. Yongxing Hao, Pana Claewplodtook and his current students Kaustubh Pathak, Kalyan Mankala, Jin Yan, So-ryek Oh whose dissertations and theses have explored the wonderful world of trajectory optimization for higherorder and differentially flat systems. He would like to thank his post­ doctoral visitors Dr. Maxmilian Schlemmer and Dr. Jaume Franch who have contributed significantly to the development of this work. He is very thankful to Prof. Richard Murray for the opportunity to spend a sabbatical in his group at Caltech in 1988 that led to the development of polytopic approximation of nonlinear constraints for differentially flat systems. He would like to acknowledge his research colleagues Prof. Brian C. Fabien, Prof. Joel Burdick, Prof. Vijay Kumar, Prof. Jorge Angeles, Prof. Peter Hagedorn, Dr. Robert Murphey, Elena Messina, who have been a source of inspiration and generous supporters of his work. The financial support of National Science Foundation through various research grants, including Presidential Faculty Fellow Grant in 1994, the support of Alexander von Humboldt Foundation through a Friedrich Willheim Bessel Prize in 2002, and other research grants from AFRL, NIST are gratefully acknowledged. Last, he would like to very sincerely thank Prof. Hebertt SiraRamirez for his friendship, his initiation into this book project, and keeping up with the ups and downs of completing this manuscript. He would like to thank his wife Seema for her companionship and Neelima and Naman for their patience and support.

Contents Series Introduction Preface Acknowledgments

iii v xi




Linear Time"Invariant SISO Systems 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9


Introduction Systems in Transfer Function Form A Relation with the Behavioral Approach Systems in State Space Form The Non-Minimum Phase Issue Uses of the Differential Parametrization The Issue of Additive External Perturbations Historical Notes and References Problems

Linear Time-Invariant MTMO Systems 3.1 3.2 3.3 3.4 3.5 3.6 3.7



Introduction Systems in Polynomial Matrix Representation Systems in Rational Transfer Matrix Representation Non-square Systems in Polynomial Matrix Description Systems in State Space Representation Historical Notes and References Problems

Time-Varying Linear Systems 4.1 4.2 4.3 4.4 4.5 4.6 4.7

11 11 12 16 17 28 33 49 57 57

69 69 70 74 83 84 95 95


Introduction Flatness of SISO Time-Varying Linear Systems Trajectory Tracking in Nonlinear Systems Flatness of MIMO Time-Varying Linear Systems The Non-Uniformly Controllable Case Historical Notes and References Problems


101 102 108 121 126 130 130

Discrete-Time Linear Systems 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Introduction Systems in Transfer Function Form The Non-Minimum Phase Case Linear SISO Controllable Systems in State Form A Shape Control Problem in a Rolling Mill Historical Notes and References Problems

Infinite Dimensional Linear Systems 6.1 6.2 6.3 6.4

Introduction Linear Delay Differential Systems Systems Described by Partial Differential Equations Historical Notes and References

SISO Nonlinear Systems 7.1 7.2 7.3 7.4

137 137 137 144 146 161 163 164 169 169 170 175 187 191 191 192 194

Introduction Definitions Feedback Linearizable SISO Systems Some General Results on the Flatness of SISO Nonlinear Systems 7.5 The Flatness Property in the Analysis of Nonlinear Systems 7.6 Tracking Arbitrary Trajectories 7.7 Finding the Flat Output


Multivariable Nonlinear Systems


8.1 8.2 8.3 8.4 8.5 8.6 8.7

Introduction Systems Linearizable by Static State Feedback Systems Linearizable by Dynamic Feedback Some General Results for MIMO Systems An Alternative View of Flatness Historical Notes and References Problems

Mobile Robots 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

Introduction Modelling Nonholonomic Cars Flatness Based Control of a Non-Holonomic Car The PPR Planar Robot The Hovercraft System The Walking Toy Historical Notes and References Problems

202 206 214


221 229 251 255 257 258 269 269 270 284 287 296 307 314 314


Contents 10

Flatness and Optimal Trajectories 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9


Optimal Planning with Constraints 11.1 11.2 11.3 11.4 11.5


Introduction Functionals of a Single Function Functionals of n Functions Functionals with Higher Derivatives Functionals with Constraints Trajectory Optimization Problems Higher*Order or Differentially Flat Forms Historical Notes Problems

First-Order Systems with Inequalities Higher*Order Systems with Inequalities Numerical Solution via Nonlinear Programming Feasible Planning in Near Real-Time Historical Notes

Non-Differentially Flat Systems 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12


Introduction Liouvillian Systems Fuel Consumption in a Car The Monge Equation The Variable Length Pendulum A Simplified Model of a Helicopter The Underactuated Ship The Soft Landing Problem A High Frequency Control Approach Some Other Non-Differentially Flat Systems Historical Notes and References Problems

323 323 325 329 333 338 342 356 370 372

377 377 383 388 396 408

411 411 411 413 416 417 424 430 438 445 452 455 455 461

Chapter 1

Introduction Differential flatness is a natural idea associated with under-determined sys­ tems of differential equations. It represents the possibility of completely pa­ rameterizing every system variable in terms of a privileged finite set of “free” variables in the system. In order to understand the basic features of flat­ ness, consider a system of n linear algebraic equations in n + m unknowns £ = ( x i , . . . , x n,x n+ i , . . . , x n+m) = (x ,x n+ i , . . . , x n+m) = O r,/), written as Ax + B f = 0, B ^ 0, rank[A, B] = n Assume that A is invertible and that B is full rank m. Clearly, all the solutions for x may be written in terms of the unknown vector / as x = - A ~ 1B f i.e., all solutions are parameterizable in terms of / . On the other hand, since the n x m matrix B is full rank, the variable / is expressible in terms of the components of x, as in the formula / = ~ ( B t B ) ~1B t A x The linear system [A, Z?]£ = 0 exhibits then a privileged set of variables, namely / = (xn+ i , •■•, £n+m), which is a set of internal variables to the system of equa­ tions (we will later prefer to use the adjective endogenous) capable of making x take on any arbitrary desired (finite) value. Suppose now that the matrix A is not invertible, say it is rank n - m . Then, in general, / is no longer free to take any value as it will become zero for any desired x which lies in the null space of A. As a consequence, it is no longer true that the components of x can be completely parameterized in terms of / . The property of being able to make x take on any desired value, by proper choice of / , is lost. Note that, in the above discussion, the role of the vector / may have been played, in principle, by another subset of m variables in £ for which the corresponding A matrix is invertible and the corresponding B matrix is full rank. 1



Control systems are frequently represented in terms of linear, or nonlinear, sets of differential state equations with a certain number, m, of control inputs. Typically, the number of states n is larger than the number m of controls and the number of differential equations is equal to the number of state variables. Thus, the control inputs, which are unknown functions that usually require to be determined, constitute a set of additional variables that render the system of equations under-determined. Flat systems, or differentially flat systems, exhibit a property which is highly reminiscent of the property exhibited by the previous elementary under-deter­ mined algebraic example. It is not surprising that such a property may exist in controlled systems as they constitute, generally speaking, under-determined systems of equations themselves. The number of control inputs being respon­ sible for the under-determination. It is clear that the invertibility of A and the full rank of the matrix B in the linear algebraic example will find much more stringent, but still natural, conditions in the case of controlled differential equations. In essence, these conditions will be summarized by the controllability property of the given system. Due to the fact that controllability is a fundamental desirable property of controlled dynamic systems (whether, continuous, discrete, linear, nonlinear, finite dimensional or not) then, the possibilities of finding the flatness property in a given dynamic system, will be tantamount of having the controlled system satisfy some form of the ubiquitous controllability property. Since, on the other hand, controllability is strongly related to being able to have the system state trajectory reasonably do whatever we want, within a finite interval of time, then, flatness will be strongly related to being able to off-line plan feasible state trajectories and to devising corresponding feedback controllers that make the system state precisely follows those desired trajectories. A striking advantage in the recognition of flatness is that, both, the trajectory planning and the con­ troller specification tasks becomes surprisingly simple. Flatness is then related to the fact that the entire set of trajectories (solutions) of the system are in a smooth, one-to-one, correspondence with free trajectories lying in an m di­ mensional space, the space of the flat outputs. As a consequence, the specified desired trajectories for the flat outputs uniquely determine the state trajecto­ ries and the nominal control inputs behavior. Clearly, the solving of differential equations is sidestepped, at the planning stages, since differential expressions, involving the flat outputs, need to be evaluated for obtaining the state trajecto­ ries and the input variables behavior. These facts would have limited relevance if it were not true that, in the context of realistic physical examples, the flat outputs invariably have a specific and concrete meaning. Generally speaking, this set of fictitious outputs represents a key set of physical variables whose measurability is either granted or desirable. E. Cartan and D. Hilbert are the forefathers of flatness from the context of under-determined sets of differential equations. In their work, they either searched for nonlinear space, and time, coordinate transformations which ren­ dered the studied system easily integrable, or for a particular set of variables which completely parameterized the system solutions without solving differen­

3 tial equations ([1], [2] [19]). The precise formulation of differential flatness in the control systems context is due to the work of Professor Michel Fliess and his colleagues: Jean Levine, Philippe Martin and Pierre Rouchon. The first fundamental articles, and devel­ opments, appeared a decade ago in 1993. The first journal article [8] written by the team, in French, is devoted to flatness of nonlinear systems and the asso­ ciated idea of defect (i.e., the lack of flatness). The setting of the contribution is that of differential algebra, a topic not easily found in engineering curricula throughout the world. In this article the idea of flatness appears as a natural outcome of the equivalence problem formulated in a differential algebraic con­ text. The article clearly shows, through a non-trivial multivariable non linear system example describing an overhead crane, that flatness is suitable for deal­ ing with physical systems even if they are not described in traditional differential equations form but as a set of differential equations subject to a set of algebraic restrictions. The notion of defect, or the lack of flatness, is explored in [9] along with a set of interesting physical examples such as the Kapitsa pendulum, the ball and beam and many others. A high frequency control approach is proposed to uncover the flatness of the average system in a variety of these examples. A complete exposition of all the developments concerning flatness, defect and a collection of some of the many challenging physical examples, that the new theory was capable of handling, appeared in an article by Fliess, Levine, Martin and Rouchon (FLMR) in 1995 [10]. It soon became clear that the differential algebraic setting could be recast in a purely differential geometric setting in­ volving infinite jet spaces, diffieties (an abbreviation for “differential varieties” ) and Cartan fields. This generalization naturally englobed the idea of space and time coordinate transformations and brought to the attention the relevance of Lie-Backlund transformations in dynamic systems equivalence problems and in feedback linearization. The complete recasting of flatness in this new context appeared in an article by FLMR in [11]. An earlier version of this approach had already appeared in 1993 (See Fliess et al. [12]). Several conference, or workshop, papers also appeared around that time exploiting different aspects of the theory with numerous examples. An independent contribution, in a sim­ ilar line of thought, can be found in Pomet [25]. A closely related approach is represented by the work of Rathinam [28] which studies flatness from an abso­ lute equivalence setup due to E. Cartan. In this contribution by Rathinam the flatness associated to Lagranian systems with one control input less than the degrees of freedom of the system is characterized via the use of configuration variables. The author addresses such flatness as configuration flatness (See also Rathinam and Murray [29]). Contributions to feedback linearization, from the viewpoint of absolute equivalence, were also furnished by Sluis [32]. The ideas of absolute equivalence are also used, in a Pfaffian setup, for the characterization of flatness in van Nieuwstadt et al. in [35]. It is fair to say that the insight into the algebraic and geometric formu­ lations of flatness was greatly motivated from the algebraic theory of linear systems based on modules. This trend was initiated by Fliess in [5]. The idea of relating controllability to the freeness of the system module is intimately tied



to the notion of a module basis. The basis of a free module is central to the flatness property, and to the characterization of flat outputs, in any linear time invariant, or time varying, system. The relation is particularly clear from the developments found in the article by Fliess [6] where it is shown that Willems’ concept of controllability, which, roughly speaking, consists in being able to smoothly tie past state trajectories to future state trajectories *, is equivalent to the freeness of the corresponding system module. In turn, the module theo­ retic formulation of linear systems arose as a particular application area of the Differential Algebraic approach in the study of continuous and discrete (non­ linear) dynamic systems (See Fliess [3]). Complete accounts of this algebraic line of thought can be found in some subsequent works by Fliess of which we mention [4], and the tutorial effort represented by the work of Fliess and Glad [7]A key contribution in the history of flatness is the establishment of a general necessary condition for the assessment of flatness. This characterization is due to Rouchon [31] and has become popularly known as the ruled manifold condition. This condition, of geometric flavor, and quite easy to check, says that at each point of the state space, the set of state velocity vectors obtained for all possible control inputs must be a ruled submanifold of the tangent space at that point for the system to be flat. So, if the condition is not satisfied the system cannot be flat, otherwise nothing can be concluded. Flatness is undoubtedly related to the general problem of systems equiva­ lence. As a consequence, flatness is intimately related to feedback linearization. A crucial problem in the control of nonlinear systems, even if they are exactly feedback linearizable, is the one related to the trajectory planning problem. A view of flatness, different to that of feedback linearizability, requires to view this key property as an useful and easy way to generate nominal solution trajectories of the controlled system in light of the desired control objectives. The works of van Nieuwstadt [33] and van Nieuwstadt and Murray [34] represent a con­ tribution centered around this deeper idea of flatness and go forth to consider important computational issues involved in the use of flatness for the generation of solution trajectories in nonlinear control systems. Some recent interesting contributions entitle the extension of the concept of flatness to delay systems and to systems described by classical linear, and some nonlinear, partial differential equations with boundary control. This exciting lines of developments cast a renewed importance to flatness as a crucial system property with various practical implications. In the case of linear delay systems, several important contributions must be mentioned: The work of H. Mounier [22], Fliess and Mounier [15], and Mounier and Rudolph [23] encompass the theoretical foundations of the natural extension of flatness to delay systems, as well as several successful case studies where the feasibility of the flatness based control idea is clearly presented. In the area of systems described by Partial Differential Equations, controlled from the boundary conditions, a number of 1For an extensive treatment o f this and many other important topics, within a theoretical viewpoint that has becom e to be known as the Behavioral approach, the reader may look into the book by Polderman and W illem s [24] and the references therein.

5 classical physical systems cases have been studied. In these cases the concept of flatness may be clearly portrayed and assessed in terms of an infinite differen­ tial parametrization of all system variables in terms of a physically meaningful output function. A classical example, dealing with the flexible beam, was first examined in Fliess et al. [16] along with interesting simulations. The flatness of a distributed parameter model of a tubular reactor is studied and used for nom­ inal boundary control assessment and simulation in Fliess et al. [17]. Recent outstanding contributions in the area of nonlinear infinite dimensional systems are those by N. Petit which include a, real life, industrial plant application [26] (See also Petit et al. [27] for interesting developments). Flatness continues to be a challenging area with many open windows for contributions. Flatness invariably has something new to offer in many sub­ disciplines within the Automatic Control field with an incidence in industrial environments (See Rothfuss et al. [30]). As examples, the area of Predictive Control, known to be one of the most applicable methodologies in industrial applications, has recently being re-interpreted and assessed from the flatness viewpoint, with substantial profits, in a most interesting work by M. Fliess and R. Marquez [13], [14]. These developments have also laid the foundations for Generalized Proportional Integral control, or GPI control for short, which evades the need for asymptotic observers and synthesizes controllers from iterated in­ tegrals of inputs and outputs in a rather practical manner. A quite powerful development in the use of flatness, which we will unfortunately not have the space and time to report in this book, is constituted by the contribution of Hagenmayer and Delaleau [18] where the notion of exact feedforward linearization based on flatness is exploited to obtain a feedback trajectory tracking controllers with great ease. In fact, the developments of GPI control naturally complement the idea of exact feedforward linearization and evade the need for asymptotic observers. Several experimental implementations of the idea have been reported in the work of Hagenmayer [20]. Flatness simplifies the trajectory tracking problem and trivializes the feed­ back controller design task. Nevertheless, flatness can be advantageously com­ bined with many other areas and methodologies of the Control Systems arsenal; Flatness and Passivity, Flatness and Sliding Modes, Flatness and Optimal Con­ trol, are some of the possibilities that have been explored by the authors of this book with satisfying results not achievable by the original techniques alone. The relevance of flatness in the popular backstepping based controller design has been reported in Martin et al. [21]. However, flatness extends beyond the off­ line trajectory planning and feedback tracking, or stabilizing, controller design capabilities. Since, much as controllability, it is a property intimately tied to the system structure, flatness also allows one to inspect some other important properties of the system from the underlying parametrization of system vari­ ables in terms of the privileged set of flat outputs. Thus, important properties, such as constant input detectability, equilibrium parameterizability, minimum or non-minimum phase properties of particular outputs, etc., can all be directly assessed from the perspective of flatness. This means that flatness is also an interesting, and non traditional, analysis tool. Interestingly enough, flatness



also provides a comparison tool to assess feasibility of nominal control inputs and nominal state trajectories even in systems which happen to be non flat, but which enjoy a flat approximation. Many interesting and important contributions, or foundational works in flat­ ness or related areas, have not been surveyed in this introduction. The literature on flatness continues to grow and it is practically impossible to include all the fine contributions that pop up in journals, conferences, workshops and tutorial courses around the world. In this, and many other respects, we sincerely ask for the kind understanding of readers and authors alike.

Bibliography [1] E. Cartan, “Sur l’equivalence absolue de certain systemes d’equations differentielles et sur certain famillies de curbes” in (Evres Completes, pp. 1163-1168, Gauthier-Villars, 1953. [2] E. Cartan, Les Systemes Differentielles Exterieurs et leur Applications Geometriques, Hernan 1945. [3] M. Fliess, “Automatique et corps differentieles” Forum Mathematique, Vol. 1, pp. 227-238, 1989. [4] M. Fliess, “Generalized Controller Canonical Forms for Linear and Nonlinear Dynamics” IEEE Transactions on Automatic Control Vol. AC-35, pp. 9941001, 1990. [5] M. Fliess, “Some basic structural properties of generalized linear systems” , Systems and Control Letters, Vol. 15, pp. 391-396, 1990. [6] M. Fliess, “A remark on Willems’ trajectory characterization of linear con­ trollability” , Systems and Control Letters, Vol. 19, pp. 43-45, 1992. [7] M. Fliess and T. Glad, “An algebraic approach to linear and nonlinear con­ trol” in H. J. Trenteleman and J. C. Willems (Eds.) Essays in Control: Per­ spectives in the Theory and Applications, pp. 223-267. Birkhauser, Boston, 1993. [8] M. Fliess, J. Levine, Ph. Martin and P. Rouchon, “Sur les systemes non lineaires differentiallement plats” Comptes Rendus Hebdomadaires des Sceances de VAcademie de Sciences , Serie I, Vol. 316, pp. 619-624, 1993. [9] M. Fliess, J. Levine, Ph. Martin and P. Rouchon, “Defaut des systemes non lineaires et commande haut frequence” Comptes Rendus Hebdomadaires des Sceances de VAcademie de Sciences , Serie I, Vol. 316, pp. 513-518, 1993. [10] M. Fliess, J. Levine, Ph. Martin and P. Rouchon, “Flatness and defect of non-linear systems: introductory theory and examples” International J. of Control, Vol. 61, No. 6, pp. 1327-1361, 1995. 7



[11] M. Fliess, J. Levine, Ph. Martin and P. Rouchon, “A Lie Backlund ap­ proach to equivalence and flatness of nonlinear systems” IEEE Transactions on Automatic Control, Vol. 44, pp. 922-937, 1999. [12] M. Fliess, J. Levine, Ph. Martin and P. Rouchon, “Linearisation par bouclage dynamique et transformations de Lie-Backlund” Comptes Rendus Hebdomadaires des Sceances de TAcademie de Sciences , Serie I, Vol. 317, pp. 981-986, 1993. [13] M. Fliess and R. Marquez, “Towards a module theoretic approach to discrete-time linear predictive control” International J. of Control, Vol. 73, pp. 606-623, 2000. [14] M. Fliess and R. Marquez, “Un approche intrinsieque de la commande predictive lineaire discrete” Journal Europeen des Systemes Automatises, Vol. 35, pp. 127-147, 2001. [15] M. Fliess and H. Mounier, “Quelques Proprietes Structurelles de Sistemes Lineaires a Retard Constants” , C.R. de TAcademie des Sciences de Paris, Serie I, Vol. 1-319, pp. 289-294, 1994. [16] M. Fliess, H. Mounier, P. Rouchon and J. Rudolph, “Systemes Lineaires sur les operateurs de Mikusinsky et commande d’une poutre flexible” . In ESAIM Proc. “Elasticity, viscoelaticite et controle optimal 8-eme entretien du Centre Jacques Cartier, Lyon, pp. 157-168, 1996. [17] M. Fliess, H. Mounier, P. Rouchon and J. Rudolph, “A distributed param­ eter approach to the control of a tubular reactor: A multivariable case” in Proceeding of the 37th IEEE Conference on Decision and Control, Tampa, Florida, pp. 439-442, 1998. [18] V. Hagenmeyer and E. Delaleau, “Exact feedforward linearization based on differential flatness ” [19] D. Hilbert, “Uber den begriff der klasse von differentialgleichungen” , Mathematishe Annalen, Vol. 73, pp. 95-108, 1912. [20] V. Hagenmeyer, Porsuite de Trajectoire par commande non lineaire robuste fondee sur la platitude differentielle, PhD Thesis, L’Universite Paris XI, Orsay, 2002. [21] P. Martin, R. M. Murray and P. Rouchon, “Flat Systems” , Plenary Lec­ tures and Minicourses, in Proc. European Control Conference 1997, G. Bastin, M. Gevers (Eds.), CIACO, Ottingnies-Louvain-la-Neuve, 1997. [22] H. Mounier, Proprietes structurelles de systemes Lineaires a retards: As­ pects theoriques et practiques, PhD Thesis, Universite Paris-Siid, Orsay, 1995. [23] H. Mounier, J. Rudolph, “Flatness based control of Nonlinear Delay sys­ tems on an example of a class of chemical reactors” , International Journal of Control, Vol. Vol. 71, pp. 871-890, 1998.



[24] J. W. Polderman and J. C. Willems, Introduction to Mathematical Systems Theory: A Behavioral Approach, Texts in Applied Mathematics, Vol. 26, Springer-Verlag, New York, 1998. [25] J. B. Pomet, “A Differential Geometric Setting for Dynamic Equivalence and Dynamic Linearization” In Geometry in Nonlinear Control and Differ­ ential Inclusions. B. Jakubczys, W. Respondek and T. Rzezuchowski (Eds.), Banach Center Publications, Warsow, 1995. [26] N. Petit, “Systemes a Retards, Platitude en Genie des Procedes et Controle de certaines Equations des Ondes” PhD Thesis, Ecole des Mines de Paris, May 2000. [27] N. Petit, Y. Creff and P. Rouchon, “Motion Planning for two classes of nonlinear systems with delays depending on the control” in Proceeding of the 37th IEEE Conference on Decision and Control, Tampa, Florida, pp. 10071011, 1998. [28] M. Rathinam, Differentially Flat Nonlinear Control Systems, PhD Thesis, California Institute of Technology, Pasadena, California, 1997. [29] M. Rathinam, R. M. Murray, “Configuratoin Flatness of Lagrangian Sys­ tems Underactuated by One Control” , SIAM Journal on Control and Opti­ mization, Vol. 36, No. 1, pp. 164-179, 1998. [30] R. Rothfuss, H. M. Heinkel, M. Lasa and P. Tirgari, “Ingenierie des Systemes - Le Role du Controle dans le Developement des Systemes Mechatroniques” Premiere Conference Internationale Francophone d’Automatique (CIFA-2000), pp. 14-25, Lile, France July 2000. [31] P. Rouchon, “Necessary Condition and Genericity of Dynamic Feedback Linearization” , Mathematics of Control, Signals and Systems, Vol 5, pp. 345358, 1995. [32] W. M. Sluis, Absolute Equivalence and its Application to Control Theory, PhD Thesis, University of Waterloo, Ontario, 1992. [33] M. van Nieuwstadt, Trajectory Generation for Nonlinear Control Systems, PhD Thesis, California Institute of Technology, Pasadena, California, 1996. [34] M. van Nieuwstadt and R. M. Murray, “Approximate trajectory generation for for differentially flat systems with zero dynamics” In Proc. 34th IEEE Conference on Decision and Control, New Orleans, pp. 4224-4230, 1995. [35] M. van Nieuwstadt, M. Rathinam and R. M. Murray, “Differential Flatness and Absolute Equivalence” in Proc. of the 33th IEEE Conference on Decision and Control, Vol. Lake Buena Vista, FI. pp. 326-332, 1994.

Chapter 2

Linear Time-Invariant SISO Systems 2.1


In this chapter, we explore the concept of flatness in the context of linear time invariant dynamic systems provided with a single input and a single output. These systems are commonly addressed as SISO systems (from Single Input, Single Output). Usually, it is desired to stabilize the output of the system or to have it track a desired reference trajectory. This is greatly facilitated if the system is flat, regardless of the nature of the internal dynamics associated with the output variable (zero dynamics, residual dynamics). In the very particular context of linear systems, the connection between flatness and the concept of controllability is, perhaps, the clearest one: A linear time invariant system is flat if and only if the system is controllable. Linear SISO systems may be rep­ resented in terms of rational proper transfer functions or in matrix state space form. We show that for either type of representation, the concept of flatness is equivalent to that of controllability. The identification of the flat output in a linear system becomes of particular importance since the corresponding dif­ ferential parametrization associated with the flat output allows one to reduce any stabilization or tracking problem to a corresponding problem defined on the flat output. A challenging problem is that of having a non-minimum phase output track a desired reference. For rest-to-rest maneuvers flatness is particu­ larly helpful and provides an elegant indirect solution to the problem. For other more complicated problems, such as tracking arbitrary output trajectories in a non-minimum phase flat system, the problem may prove to be difficult. An interesting feature of flatness is that all system properties, pertaining input to state, state to output and input-output relations, may be inferred from the dif­ ferential parametrization of the system variables. In some instances this may be easier than handling the original system equations. Systems are always af­ fected by external (exogenous) perturbations. The flatness property may still




be suitably exploited in the regulation of such prevailing class of systems. For this, one relies on the fact that for sufficiently benign perturbations, the flatness property always results in the possibilities of having a “matched” system. Section 2.2 deals with flatness of linear time-invariant SISO systems pre­ sented in transfer function form. Section 2.4 develops a simple way to identify the flat output in a given state space representation. The issue of regulating a non-minimum phase output is treated in Section 2.5. The uses of the differen­ tial parametrization provided by flatness to assess the minimum or nonminimum phase behavior of particular outputs and also to assess other properties of the system are treated in Section 2.6. The handling of unknown external perturba­ tions affecting the system and the exploitation of the underlying flatness prop­ erty for controller design are explored in Section 2.7. The chapter closes with some comments on historical notes and references and some proposed problems.


Systems in Transfer Function Form

Consider a linear SISO system represented in transfer function form, y(s) = G (s)uis)1. In this context, the system variables directly related to the external world are constituted only by the input u and the output y. We say, in this particular case, that a variable of the system is endogenous if it can be expressed as a linear combination of the input, the output and a finite number of their time derivatives. Otherwise, the variable is said to be exogenous. An endogenous variable is then said to be a differential function of the input and the output variables. A SISO system is said to be flat (we also sometimes refer to such systems as differentially flat), if there exists an endogenous variable, called the flat output, here denoted by / , such that the input u and the output y can in turn be expressed as a linear combination of the flat output and a finite number of its time derivatives, i.e., that y and u are differential functions of the flat output / . Naturally, any other endogenous variable of the system automatically enjoys the same property with respect to the variable / . We customarily say that the flat output / differentially parameterizes all system variables. In the case of linear systems, flatness is directly related to the controllabil­ ity of the system, as demonstrated from a general viewpoint by Fliess in [7]. The existence of this relationship has been proven using Module Theory. The interested reader can find a terse exposition of this topic in Fliess and Glad [9]. We can give, following Fliess et al [14], a more direct and intuitive approach to show the important connection between controllability and flatness in the case of linear (input-ouput) systems represented in classical transfer function form. Consider a linear SISO system, given by its transfer function representation v(s) = ^ y u(s)


A well known result (see the book by Kailath [17]) establishes that the system is controllable if and only if the polynomials n(s) and d(s) are coprime, i.e., 1Strictly speaking the variable s is to represent the differential operator ^



they have no non-trivial common factors. In such a case, by Bezout’s theorem, there exist polynomials a(s) and b(s), called the Bezout polynomials, such that the following identity is satisfied for all s € C a(s)n(s) + b(s)d(s) = 1


Implicitly define f(s ) as a new variable, /( « ) = ^ y u(s)


It is clear that the system input, u(s) and the system output y(s) may be written in terms of f(s ) as, y{s) = n (s)f{s),

u{s) = d (s)f(s)


Multiplying both sides of the Bezout identity, (2.2.2), by the variable, /(s ), one obtains, f(s )


a (s)n {s)f(s) + b(s)d{s)f(s)


a(s)y(s) + b(s)u(s)


The last equation says that we have found an endogenous variable, / , which depends only on the system’s input and output, and a finite number of their time derivatives, i.e. a variable which is a differential function of the system variables, such that, in turn, all system original variables (namely, the input and the output) are expressible as differential functions of the found variable, / . The variable / , evidently, qualifies as a flat output. Thus, given a linear controllable system, a flat output / can always be found in terms of an endogenous relation of the system variables u,y. On the other hand, given a flat linear system, with flat output / , then, by the very definition of a flat output, the input and the output can be expressed as differential func­ tions (which, obviously, must be linear) of the flat output, i.e. y(s) = m (s)f(s), u(s) = p (s )f(s) and moreover the flat output f(s ) is endogenously generated, i.e. there exist polynomials a(s) and b(s) such that f (s ) = a(s)y(s) 4- b(s)u(s). It follows that f(s ) = [a(s)m(s) + b(s)p(s)]f(s). Hence a(s)m(s) + b(s)p(s) = 1 which means that m(s) and p(s) are coprime. The transfer function repre­ sentation of the system y(s)/u(s) = m(s)/p(s) is therefore controllable. We summarize this result in the following proposition, P rop osition 2.2.1 A linear, time invariant, input-output system, given in transfer function representation ( 2 .2 . 1), is flat if and only if the numerator and denominator polynomials in the transfer function are coprime. In other words, a linear system of the form ( 2 .2 . 1) is flat if and only if the system is controllable. Suppose that a linear input-output scalar system of the form, (2.2.1), has a numerator polynomial n(s) which is constant, while d(s) is any polynomial of



degree > 1. Then, clearly, the system is controllable and, hence, flat. Thus, for all linear SISO systems whose transfer function representation exhibits a constant numerator, a flat output is given by the system output y , or, by any constant multiple of such output. R em ark 2.2.2 In fact, given any controllable linear system in transfer function form, y(s) = [n(s)/ d(s)]u(s), the flat output can be chosen as any constant multiple of the variable: f (s ) = [l/d(s)]u(s). A flat output is then given by: F (s) = [ai/d(s)]u(s), for any k, ^ 0. E xam ple 2.2.3 Consider the following linear, unstable, minimum phase, sys­ tem y(s) = ^ —ru(s) s —1


The system is evidently controllable as the numerator and denominator polyno­ mials are coprime. The Bezout polynomials, corresponding to n(s) = s + 1 and d(s) = s —l are given by the constant polynomials, a(s) = 1/2 and b(s) = —1/ 2. The flat output, in this case, is then given by f = \ y -\ u


It followsthat f allows a differential parametrization of allsystem variables: u = f — / and y = f 4- / . The state realization of the system interms of the flat output f is thus f = / 4- u, y = 2 / 4- u (2.2.8) E xam ple 2.2.4 Consider the linear system:

y{s) = ^ y u(s) = ^ T ( f + 6) , + 6M(s)


with a / 5,a / 1. Since the system is controllable, it is flat and the flat output, f , is given by

= , , + * : \ )> + b

We have that u(s)


[s2 + (1 + b)s 4- b] f (s )



(s + a )f(s)

The flat output f(s ) is given, in terms of u and y by, xf \ 5 — a -f^ + l 1 /(S ) " ~ (a - b)(a - 1) V + (a - b){a - 1) W



{ 2 .2 . 11 )

This lastexpression, which is of the form: f = p(s)y 4- q(s)u, isobtained by finding thecoefficients, m, n and p, for a(s) = ms 4- n, b(s) = p,which satisfy



Bezout's identity: a(s)n(s) 4- b(s)d(s) = 1. I n other words, (ms + n)(s + a) + p(s 2 4- (1 4- b)s 4- 6) = 1 and, hence, (m + p)s 2 + (am + n + p(l + b))s + an + pb = 1 V s


Equating the coefficients of equal powers of s in both sides of the identity, yields: m 4- p



am 4- n + p (l 4- b)







which is a linear system of equations in m, n and p, with a and b being known.


Flatness and controllable realizations

Consider a linear time-invariant SISO system described by y(s) =

bmsm + 6m_ is m_1 + •■• + &() -u(s), m < n sn + ansn~l -t h o0


with coprime polynomials in the numerator and denominator. output, / , may be given by, K f(s)

sn T an_ is n 1 4- ••■4- ao

Then, a flat



for any, arbitrary, nonzero constant k . The representation of the given inputoutput system, in terms of a differential equation and a scalar output equation is given by dn f

dn - i f

- do/ =

i ^ +an- 1d ^ + dm f


dm - l f

bm“ TTTT + bm-1 _>^yn_ 1~ + dtm~ eft”


h b0f

The choice of the flat output / as in (2.2.15), naturally leads to the state space controllable canonical representation. Indeed, letting x\ = / , £2 = / >•••> Xn = f ( n~l\ we obtain the following state representation of the system, ( x1 \ d_ eft

( Xi + bu,

= A


y —c

\ Xn ) with ( A




= 0 V “ flo 1 ( b0

1 —an—1 /

0 -eii 0

••• 0 )




0 v 1 y (2.2.18)



It is not difficult to see that for this representation, the Kalman controllability matrix, [6, .46,..., .4n-16], is full rank ra, independently of the system parameters. Clearly, then, a flat linear time-invariant system always admits a representation in controllable canonical form. From this development, it follows that a redefinition of the system input, by means of a state-dependent input coordinate transformation, the fundamental integration structure of the system emerges in a clear manner. Indeed, set a new control input v to be defined as : Kit

OqX \

0\X< 2

. ..

Oiji —\ X n


The new system, with the redefined input, is then reduced to a system in Brunovsky’s canonical form (0


0 \

0 \ 0

0 0


0 \ X +


0 )


v i /

A Relation with the Behavioral Approach

In recent times, a new mathematical formalism has been developed for describ­ ing dynamic systems, known as the Behavioral Approach. The developments in this area were initiated by Professor J. C. Willems and comprised in a re­ cent book by Polderman and Willems [22]. Roughly speaking, in this approach manifest (i.e., external) and latent (i.e. internal) variables are used to describe dynamic systems along with the behavioral equations. The behavior is the solu­ tion set of the behavioral equations. Two kinds of system representations are advocated: Image representation, which is of the form: u = rn(-^)l and Ker­ nel representation, of the general form: r(-jfj)u = 0. (see [22] for precise and interesting details). Consider a linear SISO system with manifest variables y, u in the input ouput behavior representation:

where p and q are differential polynomials with no common factors. A flat output is any latent variable, /, which allows for the following image representation

v = q(J t) l '

u = p {7 t) l

We can then say, given the relations between controllability and flatness, that a behavior in kernel representation, r ( ^ ) u = 0, admits an image representation, lj = m ( A ) f if and only if the system is flat.




Systems in State Space Form

Most frequently, since the advent of Kalman’s state space representation of linear systems, linear dynamic systems are given in the following, first order, vector differential equation form x = Ax + bu,

x 6 Rn, u € R


with A and b being, respectively, an n x n constant matrix and b being an n-vector of constant entries. Suppose that the characteristic polynomial of the constant matrix A, written in the complex variable s, is given by, Sn 4- CVn_ i S n 1 + ••• -f oq S + CVo

A state space coordinate transformation of the form, z = Tx, with T defined as the inverse of the Kalman controllability matrix, T = [M & ,...,A n" 16]“ 1 can be proposed. The system, in new coordinates, z, reads, z = Az + 7 U, A = T A T ~1, 7 = Tb


where, 0 0 0 1 0 0 0 1 0


•' •• •' • ■■

0 0 0

-OL0 -a 1


-a 2


-O L n - 2 -O Ln - l


0 1



• ■ •

• ■ 0 0



_0 0 0




1 0

The state coordinate / = zn completely parameterizes the transformed state variables, and hence, the original variables x, as well as the input u. Indeed, it is easy to verify that the transformed variables and the input u can be written in terms of / and a finite number of its time derivatives, Zn-l


f + O n -lf

Z n-2


/ + Q n - l / + (X n -if

Zl = u =

/( « - D +Q n_ 1/ ( » - 2) + . . . + a i / f M + a n _ 1fln -i) + ... + aof

(2 4 3 )

As a consequence, all the original states x can also be parameterized in terms of the output / . The transformed state / = zn is then the flat output. We then have,



P rop osition 2.4.1 The flat output of the linear controllable system in state space form: x = Ax -f bu is given, modulo a constant factor, by the linear combination of the states ob­ tained from the last row of the inverse of the Kalman controllability matrix [b,Ab,...,An- l b] i.e. }= { 0



1 ] [b ,A b ,---,A n~ 1b] 1X

It is clear then that in a SISO linear system, the flat output can always be made to depend only on the state variables of the system. This has an important consequence regarding the observability of the flat output. Suppose for the moment that the flat output / is a function only of the state vector x. Moreover, since the system is linear let us assume that / is a linear function of the state vector x, f = Xx for a certain row vector lambda of dimension 1 x n. The problem of finding the flat output is now how to find this row vector A. Let us write down the following vector of derivatives of / f

= Xx


= \x = XAx + Abu


= XA2x + XAbu + Abu

/ ( " - 1)


A.4n- 1x + A .4"-26u + --- + A6u(n- 2>


In matrix notation, this set of relations read as / f f




\ XAn~l )

\ f ^ (


0 Ab XAb

0 0 Ab

n -3* \ XAn~2b XAn~sb

0 \ / 0 0

u u

\ (2.4.6)

( n - 2)

Xb )

\ tl'

Since from here we should be able to obtain x in terms of / and of its time derivatives alone, all the entries in the matrix relating / , and the derivatives of / , to u and the derivatives of u, should vanish.



We then have that A must satisfy Ab = 0, XAb = 0,

••• \An~2b = 0


i.e., A is orthogonal to every vector column in the full rank controllability matrix, except the last one, to which it must be aligned (XAn~l b ^ 0). Otherwise, A would be orthogonal to a set of n linearly independent vectors in Rn and this would mean that A would be zero, which is impossible. Also, note that




\ (2.4.8)

V AAn_i y must be invertible. This means that the flat output is an observable output, which is also of relative degree n. If the matrix above is not invertible, / satisfies a differential equation with no relation to x or u, then / is not an endogenous variable. Indeed, from the relation / f i


A A.4

\ x


Indeed, if the observability matrix is not invertible, then there exists a non-zero row vector 7 such that / /



\ x = 0


Then / is seen to evolve on its own and it is not related to any variable in the sys­ tem. This means that / is an exogenous variable. It is therefore uncontrollable. Clearly a contradiction with the controllability of the system.


A DC motor

The differential equations characterizing a classical DC motor system are given by (2.4.11) (2.4.12)



where I is the armature circuit current, v is the voltage applied to the armature circuit, acting as a system input, and oj is the angular velocity of the motor shaft. The constants L, R, ke are, respectively, the inductance of the armature circuit, R is the resistance and ke is the counter-electromotive force constant of the motor. The constants J, B and km are, respectively, the inertia of the rotor, the viscous friction coefficient, assumed to be constant, and the DC motor torque constant. A set of state variables for the system may be chosen to be: x\ = / , x 2 = uj. Denoting by u the input voltage v, we have,

I U M ? :t)UMi)"


The controllability matrix of the system, and its inverse, are readily obtained C = ( I



~£ V j L //


k Km \\

& 0 v

f ) -r L .


The flat output F is given by any scalar multiple of the linear combination F = ( 0

l ) C ~ 1x = ^ u i K>m


The flat output can then be taken to be the motor shaft angular velocity, cu. This flat output provides the following differential parametrization of the system variables and the control input Xi



JL ~ ( LB + RJ\ ■ ( RB v= J ~ + ( — k— ) F + \\ Th,m~ + ej) F Km \ Km J



The differential parametrization clearly degenerates into a static parametriza­ tion of the system equilibria in terms of the flat output equilibria. Letting F — F — constant we obtain B xi = — F, Km

_ x 2 = F,

_ (R B . u = I — + kt \ Km

The differential parametrization derived from the flatness property, also characterizes many system properties. For instance, the zero dynamics asso­ ciated with, xi, is JF -I- B F = 0. We may say, then, that x x is a minimum phase variable when regarded as a system output. The variable x2, on the other hand, is seen to have no zero dynamics whatsoever. The system zero state observability or zero state detectability properties can also be established from the differential parametrization provided by flatness (see Byrnes et al. [4] for the fundamental definitions). Suppose we regard x 2 as the system output and let the control input u be identically zero, along with the condition x 2 — F = 0. We readily obtain that, F = F — 0. It follows,



then, that the state x\ satisfies: x\ = 0, i.e. the whole system state vector is zero. The system is seen to be zero state observable. On the other hand, if we take x\ as the system output and let u = 0 along with the condition x\ = 0, we obtain, from the last condition, an exponentially asymptotically stable dynamics governed by F + (B/J)F — -f (B /J))F = 0. However, the condition u = 0 is equivalent to -


R\ •

( RB


F+(7+l)F+U +^f)F'° ( d =

R\ ( d


+ i )


kekm ^

\dt + 7 ) F + - 7 r F ’ 0

Both conditions: u = 0 and x\ = 0 are seen to be, again, equivalent to: F = F = 0, it follows that the system is zero state observable from x\ regarded as an output. Taking as the system output the angular velocity y = x , X c2k‘2 ki(ki + k 2) + —

ckik 2 X2 ~ - m T * 3


From these relations, one readily obtains the state variables as differential func­ tions of the flat output. k2f


+ cf

c2k\ + k \ M i k2f + c f c2k\ + k\M\






(ki + k2) f + c f + M j c2 k\ + k\Mx (ki + k2) f + c f + M xjW c 2k\ + k%Mi "


The control input is also obtained as a function of the states, and therefore as a function of the various time derivatives of the flat output. The control input variable can be then computed from the following equality f(4 )




( k2M \ M 2{k\ + k2)2 + c2k i M 2(ki + k2) + c2M i k i k 2 H- k%M2 \

~ \ +

M fw 2

( cM jM 2(ki - k2)(k\ + k2) - c3Aq (Mi + M2) - ck\M\ \ J X2 m 2m 2

; 21




A cart with two inverted penduli

Consider the mechanical system shown in Figure 2.2. We use the Euler-Lagrange formalism to derive the mathematical model of this system. m2

Figure 2.2: A cart with two penduli.

The kinetic energies associated with the components of the system are given by, ICm -

= ^ "*i(*gi + J/gi). ^2 = ^rn2 {x 2G2 + y2 G2)


where x Gi = x + Li sin 0ij y a = Lt cos#*, i = 1,2 Using theselast relations we obtain the following expression for energy, /C =

(2.4.23) the kinetic

\ {M + mi + m 2)x 2 4- m iLix#i cos#j 4- \-m\L\6 \ Z


4-77121/2X^2 COS ^2 + ^ 2 ^ 2 ^ 2

The potential energy is given by the sum of the potential energies of each in­ verted pendulum resting on the cart P = TTiigLi (cos 0\ - 1) 4- Tn2gL 2(cos02 — 1)


Using the Euler-Lagrange formalism we obtain the followingmodel for the system (M + rai 4- m2)x 4- raiLi ^ cos#i - (#i)2 sin#ij



+ m 2L 2 ^2 cos 02 - (#2)2 sin02j = / m iL ix cos0i 4 m\L\0\

—migLi sin0i =


m 2L2x cos#2 A rn2L\62 —m 2gL 2 sm0 2 =



To normalize the system representation, we define e= y -, Li


, £ = j~ , Li

d r = \ff~ d t, y Li


= — , mi

u= 4 m\g


We obtain, abusively using the “dot” notation for derivatives with respect to r, (1 4 n 4 ac)£ 4 0i cos 0i - (0i )2 sin 0i 4 /xe [02 cos 02 - (02)2 sin02J



£ cos 0i 4- 0i - sin 0i



£ cos 02 4 602- sin 02


0 (2.4.27)

An equilibrium point of the system is given by f = X,

01 = 0, 02 = 0, ix = 0


Defining the following incremental variables

£(5 =


£ — Xy


01 , 02(5 = 02 ,

Us =



We obtain the following tangent linearization of the system dynamics around the equilibrium point in implicit form: £ 0 |h2s + A ^ | 4" €

(2 .7 .1 5 )

with e being a strictly positive small constant, asymptotically stabilizes a to a small vicinity of zero, where the asymptotically stable dynamics, xi = —Axi, is approximately valid. This guarantees asymptotic stability of xi = ^A(s) and B(s) are right coprime then, there exist square non-singlular matrices A1(s) and jV( s) such that the Bezout identity (3.2.4) is satisfied. Multiplying out this Bezout identity by the vector £(s) on the right, we have: ,M(s).4(s)£(s) +A /'(s)£(s)£(s) = £(s) which, by virtue of the relations in (3.2.5), yields the following expression for £ in terms of y and w, £(s) = M( s )y( s) +Af(s)u(s) This shows that the vector £ is endogenously generated.


System outputs and flat outputs

The general result, concerning the natural identification of the flat output vector with the system’s output vector, y, is represented by the following statement found in the work of Fliess and Marquez [9]. A square system of the form: Wi(s) u2 {s)

2/1 ( s )

2/2 (s ) _ 2/m ( s )

= V(s) um(s)

admits y as a flat output vector if and only if the following two conditions are satisfied: 1. The system is controllable, i.e. V(s) and Q(s) are left coprime 2. The matrix V(s) is unimodular.



P roof Assume the output zy qualifies as a flat output and suppose, contrary to what we want to prove, that the two conditions in the theorem are not satis­ fied. Then, the matrices Q(s) and V(s) have a non-unimodular greatest com­ mon left divisor matrix, that we denote by C(s), i.e. Q(s) = C(s)Qo(s) and V(s) = C(s)Vo(s) with Q q ( s ) and V q ( s ) being coprime. Moreover, the follow­ ing relations are valid for certain set of system variables, £, Qo{s)y(s) - Vo(s)u(s) = f(s),

£ (s)f(s) = 0

Since £ is not unimodular L(s)£(s) = 0 has nontrivial solutions. Given that zy is the flat output, there exists a differential parametrization for u in terms of y, i.e. u(s) — M(s)y(s). This means that £(s) = [Qo(s) - P0(s).M (s)]y(s) It follows, upon left multiplication by £(s) of the previous expression, that [S(s) - V ( s ) M («)] y(s) - 0 and, therefore, y cannot qualify as a flat output due to the fact that its com­ ponents satisfy differential equations which are independent of the rest of the system variables. The system is therefore controllable and the matrix V is unimodular. Suppose now that the system is controllable and the matrix V(s) is unimod­ ular. It follows that pre-multiplying the system equations by the left inverse of V(s), denoted by IZ(s), we obtain 1l(s)Q(s)y{s) = u(s)

The coprimeness of Q(s) and V{s) implies that the components of y cannot be solutions of differential equations which are independent of u. The vector z y, which is of the same dimension than u qualifies then as a flat output. E xam ple 3.2.4 The system

is controllable since the matrices appearing in the square system are left coprime and the matrix on the right hand side is unimodular. The vector y, thus, qualifies as a vector of flat outputs. E xam ple 3.2.5 In the previously discussed linearized model of an orbiting satel­ lite, the square polynomial matrix representation (3.2.6) is such that the matrices are coprime, the matrix in the right hand side is uni­ modular. Hence, the variables zi$ and Z25 are the flat outputs.




Systems in Rational Transfer Matrix Rep­ resentation

If a square matrix G(s) can be written as the product Q~ 1(s)P(s) with Q{s), V(s) being left coprime, then G(s) is said to have a left coprime factorization in terms of Q(s) and V(s). If a square matrix G(s) can be written as the product A(s)B ~ 1(s) with A(s), B(s) being right coprime, then G(s) is said to have a right coprime factorization in terms of *4(s) and B(s). A square system in rational transfer matrix representation is given by an expression of the form