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English Pages 1096 [1095] Year 1972
MODERN CONTROL THEORIES NONLINEAR, OPTIMAL AND ADAPTIVE SYSTEMS by
Prof. Dr. Corresponding
Member
FRIGYES CSAKI of the Hungarian
Academy
of Sciences
i\
COLLEGE
AKADUMIAI KIADO, BUDAPEST
1972
a y
The
original
“Korszeru szabalyozaselmelet
Nemlinearis, optimalis es
adaptiv rendszerek” published by Akademiai Kiado, Budapest 1970
Translated by P. Szoke Translation revised by B.
® Akademiai
Balkay
Kiad6, Budapest 1972
Printed in Hungary
CONTENTS
Symbols
13
Preface
17
27
References
PART 1.
1
31
Introduction 1.1 Characteristic features of
nonlinear systems
1.1.1
Importance of nonlinear systems
31
1.1.2
Fundamental equations State and phase equations
31
1.1.3
33
special features of nonlinear systems
Some
1.1.5
Classification of nonlinearities
38
1.1.6
Analysis of nonlinear systems
40 40
PART
Methods of 2.1
2
linearization
Linearization about an operating point; tangential approximation Determination of the linearized coefficients by expansion 2.1.1
An
2.1.3
method of linearization The most usual ways of algebraic linearization
2.1.4
Linearization of characteristic curves
2.1.2
2.2
36
1.1.4
References
2.
31
alternate
45 46 46 52
54
2.1.5
54 Determination of linearized coefficients by leastsquare approximation 57
2.1.6
The
2.1.7
Summary
first test
method of Lyapunov
61
66
References
67
Harmonic linearization 2.2.1 Fundamental assumptions of the describing function method Fundamental relations 2.2.2
68
77
2.2.5
Generalized describing functions Describing functions of some simple nonlinearities Approximate determination of the describing function
2.2.6
An alternate approximate method for the determination of describing
2.2.3
2.2.4
68 69 78 86
functions
90
2.2.7
Stability test
93
2.2.8
Examples for the uses of describing functions The root locus of a nonlinear system Drawbacks of the describingfunction method Compensation in nonlinear systems The harmonicbalance method
2.2.9
2.2.10 2.2.11
2.2.12
103
110 114 119
120
CONTENTS
6
2.2.13
Harmonic
linearization of state equations
123
References
132
2.3 Statistical linearization
136
2.3.2
Statistical linearization of nonlinear characteristics
140
2.3.3
142
2.3.5
Expressions of linearized gains Examples of statistical linearization A variant of linearizedgain calculation
2.3.6
Statistical linearization of
2.3.7
Statistical
relations
136
some
typical nonlinearities
systems analysis
2.4.2 2.4.3
2.4.4 2.4.5 2.4.6
147 156
157 162 169
Combined describing functions 2.4.1
171
Dualinput describing functions Incremental locus Approximate dualinput describing function
171
Combined harmonic and random linearization A review of combined linearization formulae Approximate combined linearization
183
References
174 178
185 187 189
PART
3
Transient processes 3.1 Graphical methods
193
194
3.1.1
Firstorder linear system
194
3.1.2
Linear integrator Single energystorage nonlinear system Feedback control systems
197
Supplementary comments The secant method Supplementary comments to the secant method The tangent method
201
3.1.3
3.1.4 3.1.5 3.1.6 3.1.7 3.1.8
References 3.2
128
Fundamental
References
3.
125
2.3.1
2.3.4
2.4
121
Comparison of the three methods 2.2.15 Other describing functions 2.2.16 The inverse problem 2.2.14
Numerical methods 3.2.1
3.2.2 3.2.3
3.2.4 3.2.5 3.2.6 3.2.7
3.2.8
198 199
202
205 209 211
212
Taylor series expansion The Euler method The modified Euler method
212
The Adams method The Milne method The RungeKutta method Predictorcorrector methods Checking the numerical methods
217
215 216 219
220 222 226
3.2.9
Leastsquare fitting
227
3.2.10
The use of 2 forms in the evaluation of transient processes Naumov’s graphoanalytical method
232
3.2.11
References
229
235
CONTENTS 3.3 Analytical 3.3.1
3.3.2 3.3.3
3.3.4 3.3.5 3.3.6
3.3.7 3.3.8
3.3.9
3.3.10 3.3.11
3.3.12 3.3.13
3.3.14 3.3.15 3.3.16
237
methods
Variation of parameters Expansion according to a small parameter special case of expansion according to a small parameter Finding of periodic solutions by the perturbation method The method of reversion The method of Lighthill and Temple The method of collocation The Galerkin method The RitzGalerkin method
A
The Lie series method The asymptotic series method The TaylorCauchy transformation The method of recurrence relations The method of complex convolution The method of successive integrations Method of Lalesco’s nonlinear integral equations
References 3.4 Closedform solutions for transient processes Directly integrable differential equations 3.4.1 3.4.2 Linear differential equations of the first order Separable differential equations 3.4.3 3.4.4 3.4.5
Introduction of a homogeneous variable Solution of equations derived from a total differential
Introduction of an integrating factor Introduction of a new variable 3.4.7 Incomplete secondorder differential equations 3.4.8 The Bernoulli equation 4.1 3.4.9 3.4.10 The Riccati equation 3.4.11 The EulerCauchy equation 3.4.6
238 241
244 246 249
250 253 256 258 261
263 263
265 268
270 271
275 276 277
278 279 279
280 281 281
282 283
283
284
3.4.13 Elliptic functions
284 285
3.4.14 Hyperelliptic functions
289
References
290
3.4.12 Solution of variablecoefficient linear differential equations
PART 4.
7
4
The stateplane and phaseplane method The stateplane and phaseplane method Writing up the phase equations 4.1.1 4.1.2
4.1.3
4.1.4 4.1.5 4.1.6 4.1.7
4.1.8 4.1.9
Determination of the phase portrait by computation Graphical methods for constructing phase trajectories evolute methods
The Time calibration of phase trajectories The Poincare analysis of singular points Some comments on the Poincare method Energy relationships and the phase portrait Phase trajectory construction by energy considerations
293 294 296 301 312 322 326 333
348 351
353
CONTENTS
8
4.1.10 Limitcycle examination
356
References
362
4.2 Piecewise linear systems
4.2.2
Saturation or limitation Dead band or threshold
4.2.3
Variable gain
4.2.4
Backlash or hysteresis Adhesion and Coulomb
4.2.1
4.2.5
4.2.6 4.2.7
364 364 367 368 371
374
friction
Variable damping Examination of piecewise
377 linear systems
380 380
References 4
4.3 Onoff control systems
382
4.3.1
The block diagram of
4.3.2
Analytical determination of phase trajectories
385
4.3.3
Examples of onoff control The method of characteristic curves The method of point transformation
398
4.3.4 4.3.5 4.3.6
4.3.7 4.3.8
onoff control
Limit cycle calculation in relay systems Optimal relay control system Minimumtime systems
References
PART 5.
Lyapunov’s second or direct method 5.1.1 The fundamental ideas of stability 5.1.2 The notion of signdefiniteness 5.1.3
5.1.4 5.1.5
5.1.6
5.1.7
5.2
Lyapunov functions The Lyapunov theorems Examples for the application of the Lyapunov method to autonomous systems Proof of the Lyapunov method Application of the Lyapunov method to nonautonomous systems
5.1.8
Practical stability
5.1.9
Eventual stability
400 409 416 425 433
439 441
442 445 448
450 456 466
470 472 473
References
475
Determination of the Lyapunov functions Determination of Lyapunov functions for autonomous linear systems 5.2.2 Determination of Lyapunov functions for autonomous nonlinear systems 5.2.3 The Krasovskii method
477
5.2.1
5.2.4 5.2.5 5.2.6 5.2.7
5.3
389
5
Stability of nonlinear systems 5.1
384
The Aizerman method The variablegradient method for generating Lyapunov functions The Zubov method Determination of Lyapunov functions for nonautonomous systems
477
480 483
489 491
494 498
References
500
Canonical forms and transformations 5.3.1 The basic equations of direct control
504
501
CONTENTS
0
5.3.2
The fundamental equations of
5.3.3
Closed formulae of canonical transformation for direct control
518
Closed formulae of canonical transformation for indirect control for constructing the Lyapunov function. Introduction
5.3.4
indirect control
512 523
5.3.5
Methods
528
5.3.6
Construction of the Lyapunov function for indirect control Construction of the Lyapunov function for direct control Lur’e’s polynomial transformation Special cases of Lur’e’s polynomial transformation
529
5.3.7 5.3.8
5.3.9
and zero
539
547
550
5.3.10 Simplified stability criteria 5.3.11 Pole shifting
537
shifting
551
560 References 561 5.4 Synthesis by the Lyapunov method 561 5.4.1 Synthesis on the basis of an integral criterion 5.4.2 Syn thesis of linear excited systems on the basis of the integral 563 criterion 5.4.3
Synthesis of closedloop control systems
565
5.4.4
Synthesis of an excited nonlinear system
566
5.4.5
The parameteridentification method
569
5.4.6
Synthesis of nonlinear adaptive control systems Synthesis of asymptotically stable optimal nonlinear systems
571
5.4.7
Estimation of the damping rate of a transient process References 5.5 Sampleddata systems 5.4.8
579 586 587
5.5.1
Stability definitions
587
5.5.2
Stability theorems
688
5.5.3
The relation between the Routh Hurwitz criterion and the Lyapunov 593 method 595 systems The Krasovskii method for discretedata
5.5.4
Synthesis of discretedata systems Estimating the transient process References 5.6 Absolute stability 5.5.5
5.5.6
599
600 601
602
5.6.1
Definition of absolute stability
603
5.6.2
The Pojdov
605
5.6.3
5.6.4 5.6.5 5.6.6
5.6.7 5.6.8
5.6.9
5.7
573
criterion and its proof Geometric interpretation of Popov’s criterion Extension of the Popov criterion Application examples of Popov’s criterion Absolute stability of the control process in nonlinear systems The stability degree of nonlinear systems The integral criterion for nonlinear systems Relationship between the Popov and Lyapunov methods
References Absolute stability of nonlinear discretedata systems 5.7.1 Absolute stability in discretedata systems
Popov
5.7.2
Interpretation of the
5.7.3
Generalization of the stability criterion
5.7.4
The necessary and
5.7.5
Estimation of the degree of stability
criterion
sufficient conditions of absolute stability
609
614 617 621
624 626 628
629 631
631
635 636 637
637
10
CONTENTS 5.7.6 5.7.7
Quadratic estimation Relationship between the
638
Popov
criterion
and the Lyapunov method
in discretedata control systems
639
References
641
5.8 Generalization of the
5.8.1
5.8.2 5.8.3
frequency method 642 Absolute stability of systems including a nonlinearity of limited slope 642 Modified stability criteria in the frequency domain 646 Absolute stability of multivariable systems 650
Generalization for time variable nonlinearities References 5.8.4
PART 6.
654 655
6
Optimal systems
663
6.1 Application of the calculus of variations to the solution of
optimal control
problems 6.1.1
665
The principal theorems of the
classical calculus of variations
665
Variants of the optimal control problems 6.1.3 Optimal control of timeinvariant systems 6.1.4 Linear optimization problems References 6.2 Pontryagin’s principle 6.1.2
6.2.1
Pontryagin’s
maximum
682 693 696
707 709
principle
6.2.2
Some examples
6.2.3
The Pontryagin minimum
involving the
maximum
709 principle
727
principle
735
Optimal control of linear timeinvariant controlled plants 6.2.5 Some features of optimal systems 6.2.6 Synthesis of minimumtime systems 6.2.7 Design of fueloptimal systems 6.2.8 Design of energyoptimal control 6.2.9 Optimal control with a hyperspheretype constraint 6.2.10 Optimal control of systems with transportation lags 6.2.4
751
763
820 830 837 ~"
References 6.3
745
846 N
855
Dynamic programming 6.3.1
6.3.2
(maximum) 6.3.3
principle
878
Connection between dynamic programming and the calculus of variations
6.3.4
867
Fundamentals 867 Connection between dynamic programming and the minimum
881
Connection between the Lyapunov functions and dynamic program
ming References 6.4 Functional analysis in the solution of optimal control problems 6.4.1 Optimal control of single variable plants 6.4.2 Optimal control of multivariable controlled plants 6.4.3 Optimal control of timevariable multivariable controlled plants
884 888 891 891
895 899
6.4.4
Complementary comments
900
6.4.5
Some numerical examples
901
References
904
CONTENTS
PART 7.
11
7
Adaptive control systems 7.1
909 910 912 912 914 914 915 916
Variants of adaptive control systems 7.1.1 Passive adaptation 7.1.2 Input variable adaptation 7.1.3 Extremal or optimizing systems 7.1.4 System variable adaptation 7.1.5 Systemcharacteristic adaptation 7.1.6
Supplementary comments
References
917
Some examples
918
of adaptive systems Highgain adaptive systems 7.2.2 Adaptive systems with a prescribed damping factor 7.2.3 Adaptive missile acceleration control 7.2.4 Input signal selfadaptation of a tracking servo 7.2.5 Modelreference adaptive systems References 7.3 Optimizing methods 7.3.1 Fundamental concepts of optimizing systems 7.3.2 Some types of optimizing systems 7.3.3 Analysis of quasistationary processes 7.3.4 Methods of searching in complicated optimizing systems References 7.4 The theoretical bases of adaptation, learning, and optimizing 7.4.1 The criteria of optimality 7.4.2 The adaptation process and its algorithm 7.4.3 Adaptation under constraints 7.4.4 Pattern recognition 7.2
7.2.1
924 926 929
930 931
932 945
950 954 956 956 957
958
959
Identification
962
7.4.6
Adaptive filters Adaptive (dual) control
963
7.4.7
964 966
PART
8
Appendix 8.1
922
7.4.5
References
8.
918 919
971
Some fundamental principles of matrix calculus and vector Some fundamental theorems of matrix algebra
analysis
8.1.1
and quadratic forms
972 972 975
8.1.2
Bilinear
8.1.3
Norms
976
8.1.4
Fundamentals of vector analysis
981
8.1.5 Some rules of differentiation References 8.2 State variables, state equations 8.2.1
8.2.3
988 989
Deduction of the transfer matrix of a controlled plant from the state equations
8.2.2
985
990
Deduction of the state equations from the transfer function or 991 transfer matrix 996 State equations of feedback systems
CONTENTS
12 8.2.4
Normal plants
997
8.2.5
Canonical form
998
8.2.6
Determination of the phase variable form
1002
1004
References 8.3 Solution of the state differential equations 8.3.1
1007
Solution of a timeinvariant linear homogeneous vector differential
equation 8.3.2 Determination of the fundamental matrix 8.3.3 Determination of the fundamental matrix in cases with multiple eigenvalues 8.3.4 Solution of the timeinvariant inhomogeneous state equations References
1007 1009
1014 1015
8.4.2
1016 1017 Solution of time variable homogeneous state equations 1017 Solution of the timevariable inhomogeneous differential equation 1019
8.4.3
The adjoint system
*
8.4 Variablecoefficient differential equations 8.4.1
1021
Determination of the transition matrix References 8.5 Reachable states, controllability, observability 8.5.1 Reachable states
1022
8.4.4
1026 1027 1027
and observability
8.5.2
Definition of controllability
8.5.3
Controllability of linear timeinvariant systems
1028
8.5.4
Observability of linear timeinvariant systems
1031
8.5.5
Normal plants
1033
1027
References 8.6 State 8.6.1
8.6.2
1034
and phase equations of sampleddata systems 1035 Determination of homogeneous phase equations 1035 Determination of the phasevariable form from the pulseddata transfer function
8.6.3
General state equations of sampleddata systems
1040
8.6.4
Solution of linear state equations
1043
8.6.5
^transformation
1045
8.6.6
Determination of the transition matrix Determination of the transition matrix of a timevariable plant
1047
8.6.7 8.6.8
Controllability
and observability
References 8.7
1038
H
Some connections with 8.7.1
8.7.2
>
1048 1050
1052
theoretical mechanics
1053
Fundamental concepts and connections The Lagrange equation
1053
References
1056 1061
List of references
1063
Index
1087
SYMBOLS
=
>
is equivalent to does not equal equals by definition; denotes equals identically is not identically equal to is greater than
than
greater or equal to is less or equal to symbol of factorial for all (universal quantifier) there exists (existential quantifier) is
0 x< 0 x
1,
=
[sgn xv
f
1,
.
.
x,
sgn
,
.
x
a: n
T ]
1
Xn)
>
•
•>
xn)
,
•>
•
Xn)
(2.1.114)
•
This nonlinear system of equations can be written as a single vector fune tion with a vector argument
y
= g(x)
Let us assume that the arbitrary nonlinear functions glt ,gq can be exdetermined series the operating Taylor around point by panded into Xq]. Then, with the expansion completed and the higherX [X1; 2 order terms neglected, .
.
=
X
M
,
.
.
.
.
= gM~g(X) +
y
— %i)
(*/ ,i=
1
OXi \
M
(2.1.115)
Here d l±\
9g dXj is
M
a vector of q of components, that If treatment ,
is
dx i
dXj is,
a
_
qx 1 column
matrix.
around the operating point
restricted to small variations
then
y
 g(X) = y
Axi
(2.1.116)
M
fT, dXj
The above expression establishes a linear relationship between the variations Ayj and Ax,, resp. Ay and Ax. The proportionality factors (that is, the linearized coefficients) are furnished by the partial derivatives at the point of operation. With the variations Ax,, considered as input variables, and the variations of Ayj as output variables, one can plot the block diagram illustrated
by
Fig. 2. 1.12,
where Xj*
= 1, 2, = 1, 2, j
=
i
dx.
M
.
.
.
n
.,
.
(2.1.117)
.,
q
represents the gains, readily determined from the partial derivatives. For small variations around the operating point M, the equation system of the linear model of the multivariable transfer element is
Ay
K^2Ax2
Ky^Ax^

Ay
^ X2\Ax^
j
K
Ayq
X\
f
J^q^A x 2
j
22
.
.
.
(
K^n Axn
,

X2n Axn
,
\
Ax2
~
(2.1.118) J
.
.
.
j
K.qn A xn
,
DETERMINATION OF LINEARIZED COEFFICIENTS or,
5J
written in a vectorequation form,
Ay where Ay
^ KzJx = A Ax,
(2.1.1—19)
qx 1 and Ax an nx 1 column
vector (column matrix), and n), with the gains is a
y —
g(pci>
*
(2.1.51)
LINEARIZATION ABOUT AN OPERATING POINT
58
The dependent
variable at the point of operation
Y = g(X X2 x,
This value
is
for
m
g(Xxk
X
known
.
.
,
.,
Xn
M
is
(2. 1.52)
) .
arbitrary sampling points around the operating
M:
point
Xk — The
,
2k ,
Xnk
•
•
—
m
)
(1c
,
1,2,..
m
)
(2.1.5—3)
.
M and the
between the operating point
differences
.,
sampling points
are expressed as
X
X = Ax x
lfc
lk
(1c
=
1, 2,
(1c
=
1,2,..
Xn — Axnk Y  Y = Ay
.
.
.,m)
(2. 1.54)
Xnk
and
k
The
linear
model
is
k
.,
m).
(2.1.55)
again assumed to have the form
Ay
m % Ki Axi = R
1
Ax1 +...
j=i
+ Kn Axn
(2.1.56)
.
Now let
us form the sum of the squared deviations for the sampling points, using the coefficients of the linearized model and the outputs of the model and the nonlinear element:
m
E = 2 (K fc
A
=
1
K
Ax lk +
first
dE
__
BKX
2 %(K
BKj
sum
E
is
that
all
dE
_
_
BK 2
K ^ Ax x
X
Axlk
k=\
+X
mm
divided
fc= 1
d
by
lk
2
Ax2k
2
=
(i
(2. 1.58)
Kn
Axik
+ X 2 ^ x k dx 2
2
1. 2,
j
Kn Axnk — Ayk
.
.
.
.
.
.,n)
ik
H
.
.
f"
Kn
fe=l
The hnearized coefficients the system of n equations is, if
+
and rearranged:
(i=
(that
of the quadratic
(2.1.57)
is,
BE
or,
+ ...+ K„ Ax„ k  Ayk f
partial derivatives be zero:
BE that
Axkk
minimum
necessary condition for the
the
k
K
1,2, ...,»)
=
)
Axik
mm k=
Axnk Axik
1
—0 (2. 1.59)
— £ Ayk Ax k=
ik
(2.1.510)
.
can be determined from sampling has been correct the determinant of the equation system does not equal zero). f
(i
1, 2,
.
.
.
(2.1.510), if the
,ri)
DETERMINATION BY LEASTSQUARE APPROXIMATION
= g(x),
For a singlevariable function y will be
59
the equation of the linear model
Ay^KAx, where the linearized
(2.1.511)
coefficient or equivalent gain is
m
2 AVk A*k
K = ^
m>
,
1
.
(
2
.
1 . 5  12 )
2
fc=
For a function of two independent model equation is
variables, y
^ K Ax + K Ax
Ay
1
2
l
2
= y(x
lt
x2 ), the linearized
,
where (
m
'
l*»
m
2
\2 Ay k Ax lk
K =
m
j
(^«)
(*>
m
x
2
2 M*«)
/
— 2 k=
2
2
k=
/c=
m
Axlk Ax2k j
m
m
1
r
 2 Ay k Ax 2k 2 k=l
Axlk Ax 2k (2.1.513)
'
{2 Ayk Ax2k
\
m
2
k=1
\
(Ax u ) 2
\
m
m (
 \2
(
Ay k Axlk
(fc=i
/
j
K.,=
m
m
2
2
k=
k= 1
2 Ax
m
2
{^x2k ) 2
lk
Ax 2k
l*»
Ax ik dx2 k
/
12
fc=
I
(2.1.514) If the nonlinear relation can be expressed analytically, the sums may be replaced by integrals [T20], e.g. in the case of the single variable nonlin
earity y
= y(x
)
AX
K=
f
Ag(Ax)Axd (Ax) .
f
AX
(Ax)
2
d (Ax)
The same procedure can be followed for two or more variables. The method of least squares is much more complicated and seldom
(2.1.515)
therefore
resorted to in practice.
Example
2.1.5.
Let us determine the transfer function of a flyweight tachometer [T20] by the leastsquare method. As has been pointed out before, the reduced force is fr
= McEa>
2 .
LINEARIZATION ABOUT AN OPERATING POINT
60
For small variations about the point of operation,
Fr where
FR =
MCRQ\.
= MCR(Q +
Aco)\
0
Simphfying by this expression
Afr
We
Afr
= MCR ( 2 Q
0
Ao
+
(Aco)*)
yields,
.
wish the linearized model equation to assume the form
KAco
Afr
The gain can be determined by equation
K=
J
M R(2D Aco +
(Aco ) 2
0
c
(2.1.515):
)
Acod{Aco)
J {Aa>yd{Aco) Integrated, this becomes
mcrq (aQo 0
3 )
— 2 Mc R O 0
K= {AQ0 f o
Thus the method of
least squares yields in this case a result identical with that obtained by the tangential approximation, since force is a quadratic function of angular velocity.
Example
2. 1.5.2
Let us linearize the relation
V at an operating point
X
= 9{x) = xn
=?a 0.
Tangential approximation gives
—
= nxn ~
n~ = nX nX"' l
x
x=X
x=*X
and hence
Ay
Now
x
Ax
.
the method of least squares yields
Y that
= nXn ~
f
Ay
==
(X
f
Ax) n
is,
Ay
= {X + Ax) n  X n
Hence, AX
K=
((X
— J
.
+ Ax) n  X n ) Axd{Ax) .
AX J
AX
(Ax) 2 d(Ax)
FIRST TEST
now
Integration
METHOD OF LYAPUNOV
61
yields
my h=i
2
n
2h+l
2h
—
Xn2h+l (/\X) 2h +
X
l, J
where the symbol If n = 1, then
]
[
K=
refers here to the integer part of the 1; if,
on the other hand, n
K=
= 2X

—
(
A X)
= 2,
mixed
fraction.
then
.
3
3
Thus the methods of least squares and of tangential approximation give the same result. If n = 3, then the gain 2
K=
X*(AX) 3
(AX) +4 *
—3 (AX)
5
= 3X + 4(zlX) 5 2
3
X, also on the variations of AX and by the tangential approximation. if n = 4,
d.epends, besides
2 ,
therefore differs from
that furnished Similarly,
4x 3 (^x) 3 K=
+45 x zj
4.
THE FIRST TEST METHOD OF LYAPUNOV by the
introduction of state or phase variables, any nth order nonlinear differential equation It is well
(see Section 1.1.3) that,
Cn
X
may
— f(x,
(2. 1.61)
(« 1)
#
X, ...
,
X
)
be reduced to the simultaneous set of nonlinear
of first order
differential equations
LINEARIZATION ABOUT AN OPERATING POINT
f>2
or to the nonlinear vector differential equation
x
= f(x)
(
2
1
.
.
6 3 )
Let us assume that the nonlinearity can be expressed by a singlevalued analytic function: then, at a given equilibrium operating point M,
X = [X v x
.
2,
X„]
.
T
and thereabout, the first partial derivatives of the nonlinear functions are finite, continuous, and unique. The nonlinear functions f\
fn or,
—
•
= fn(xV a
'2>
'> **7i)
•
•>
*
•
(
2 1 6 4 )
(
2
.
.
Xn)
written in vector form, the nonlinear vector function f
= f(x)
.
1 . 6 —5 )
can therefore be hnearized by making use of the first two terms of the Taylor series, the socalled first approximation. With the higherorder terms neglected, the approximate linearmodel equation of the nonlinear system becomes
Ax
M that
is,
Af
^ Azlx
(
2
.
1
.
(
2
.
1
.
6 6 ) 6 7 )
where the elements of the matrix A are: Aj = dfj/dXi. According to Lyapunov’s first stability theorem [L2, L5, L8], if the nonlinear system is substituted by its linear model of first approximation, and the characteristic equation of the differential equation t
(n)
Ax
= Af(Ax
(n— 1)
#
,
Ax
,
.
.
.
Ax)
thus obtained, or of the vector differential equation Azlx Ax
=
(
2 1 68 ) .
.
(2.1.6—9)
has roots whose real parts are nonzero, then the stability of the nonlinear system can always be decided on the basis of the linear approximation (in other words, the higherorder terms do not influence stability). Thus, if the roots of the characteristic equation of the scalar (differential equation system) or vector differential equation furnished by the linear approximation are negative, or have negative real parts, then the nonlinear system is stable, that is, if deflected from its equilibrium operating point, it will return thereto. On the other hand, if some of the roots are positive or have positive real parts, the nonlinear system will be unstable and, if deflected from its point of operation, will never return to equilibrium.
FIRST TEST
METHOD OF LYAPUNOV
63
If the real part of any one of the roots is zero, this criterion cannot be applied. In such cases, stability may depend, for example, on the direction of displacement. In control engineering, roots with a real part close to zero are undesirable (since then the control system is liable to exhibit excessive hunting) and, therefore, the exclusion of roots with zero for their real part will not be uncomfortable for practice. It should be emphasized that the Lyapunov stability theorem applies only to small deviations about the point of equilibrium. The characteristic roots of the linear differential equation system
Axx
=A
11
A ln Axn
Ax1
(
Axn
= A nlAx + 1
.
.
2
.
1 . 6  10 )
+ A nn Axn
.
that is, the linear differential equation (2. 1.69) obtained by the linearization of the nonlinear vector differential equation (2.1. 63) and the nonlinear differential equation system (2. 1.62) can be determined from the characteristic equation:
= A — «I =
D(s)


0
=
K{s)
or
si
= 0.
—[A
(2.1.611)
\
\
The
characteristic roots are, therefore, the eigenvalues of the matrix A. Writing out the determinant in detail gives
=
D(s)
=0 A ni
Now
this determinant
D(s)
Here the
=
(s) n
An
(
2
.
1 . 6  12 )
A„„—S
2
expands to
+ anits)" +
+ %(«) + a =
1
.
.
0
.
0.
(2.1.613)
coefficients dfi—\
=
^11
= ^11
“t
^22 “f*
•
+
...
+
=
^21 ^22
An\A n 2 sums of the
determinant

A
principal minors of
itself). 

*
“t
1,
^nn n—l
i
f
n
(2.1.614)
^A n n—l ,
A^iA^ aQ
•
An—
^12
^21 ^21
are the
.
•
•
A ^
77,77
A in
•
•
•
•
•
•
•
^ 2 n
first,
second, etc. order (a 0
is
the
LINEARIZATION ABOUT AN OPERATING POINT
64
Example
2. 1.6.1
Let us examine whether the nonlinear system described by the differential equation d2 « ax \C sin x U +D
=
d£ 2
d£
stable (such a differential equation describes, for example, a synchronous motor or generator, or an oscillating system consisting of a nonlinear spring,
is
=
x. mass). By the way, the output is y In equilibrium state, the time derivatives are zero, and the equation giving the coordinates of the operating point is
damper and
C
X=
sin
Let us assume that the system
U.
slightly displaced
from
X sin Ax ^ sin X +
cos
is
its
operating
point:
x
=X=
AX.
Since
d 2X
=0
dt 2 and sin
x
= sin X cos Ax +
cos
the linearized differential equation of the variations 6? Ax
dAx
D
f
dl2
The
—0
•
Ax,
be
.
dt
characteristic equation
is
then,
—
s2
+ Ds + C cos X = 0
K(s)
By
„ „ G cos X Ax .
,
will
X
the RouthHurwitz criterion, the condition of stability of this linear equation is
differential
D> and
this
Example
is
C cos X
>0
the stability criterion also for the original nonlinear equation.
2. 1.6.2
Let us examine the nonlinear by means of phase variables. Introducing x±
= x,
x2
= x, #2
x2
and
0;
differential
equation of the above example
the phase equations become >
= —O sin x — Dx + U l
2
therefore, fi(xi>
x2
)
—
X2>
f2 ( xi,
X2 )
= —C
sin
xx
— Dx2 +
U.
'
FIRST TEST
METHOD OF LYAPUNOV
65
The equilibrium state of the system can obviously be determined from the equations a?!
x2
= 0, =0 .
Hence, the coordinates of the point of operation are
v
•
U
i
G
X = 2
0.
Now let us examine the stability of the system for slight displacements about the point of equilibrium. The Jacobian matrix
_
J(f,x)
dxT
M and the
1
— G cos X
M
x
—D
linearized system of differential equations can be written as
Axx Ax2 The
0
df
'
is
= Ax — —C cos X 2
,
1
Ax1
characteristic equation determined
K{s)= D(s)
—
=
— DAx
2
from equation 1
C cos Xj
.
—s
Ds
2 f
(2.1.612)
is:
Ds + G cos^! =
0
of course, identical with the one in the previous example, just as are the stability conditions.
it is,
Example
2. 1.6.3
Let us examine the stability of the simple control system illustrated in Fig. 2. 1.61, for slight deviations about the state of equilibrium r(t) 0, if
—
ue
= eg(e)
and
u
r(t)
^
eftJ
°, K > 0, K < a a
> °,
x
2
.
Problem 2. 1.6.1 Determine the conditions of stability of a nonlinear system characterzed by the Van Der Pol differential equation
— dl
2
2£
• •
;
;
'
*
HARMONIC LINEARIZATION
;
*\
’ \
Oiving to the difficulties of their treatment, nonlinear systems are usually reduced to linear ones, if possible. One solution (linearization in the time domain) was discussed in the foregoing section. Another possibility is the socalled harmonic linearization in the frequency domain. Here the nonlinear transfer element is substituted by a linear one, equivalent as far as the fundamental component is concerned. Linearization in the time domain and the frequency domain are not competitive but complementary procedures. The linearization about a point of operation involves continuous characteristic curves easy to differentiate, and small deviations. Harmonic linearization, on the other hand, can be successfully employed also in cases of discontinuous curves and substantial deviations but presupposes the possibility of quasistationary oscillations. In the present chapter, the various methods of harmonic linearization will be discussed. In addition to a detailed treatment of the socalled describing function method, most frequently used in control engineering, the socalled harmonicbalance method and the harmonic linearization of the stateequation coefficients will also be considered [G6, G7, G16, G28, 03,
P14, P27, T8, W10].
FUNDAMENTAL ASSUMPTIONS OF THE DESCRIBING FUNCTION METHOD 2.2.1
If a nonlinear transfer element is excited by a sinewave input signal, then the output signal will contain, in addition to the fundamental, also some harmonics. The describing function expresses the amplitude ratio and phase shift of the output fundamental as compared to the sinewave input. For a multivalued nonlinearity, the describing function is a complex quantity composed of the amplitude ratio (the absolute value), and the phase shift (an angle, or argument) that is, it will describe not only the magnitude ratio but also the phaseshift conditions. For a singlevalued nonlinearity, on the other hand, the describing function furnishes only the magnitude ratio of the output fundamental to the harmonic input. Owing to the presence of a nonlinearity, the describing function depends on the magnitude and, sometimes, on the frequency of the pure sinewave input [ 1 —7 j Thus the nonlinear element is linearized by the describing function and the subsequent analysis or synthesis is based on the magnitude or, possibly, frequencydependent linearized element. Thus, by making use of the describing function, the nonlinear system can be readily studied in the frequency domain with the frequencyresponse methods developed for linear systems, and both analysis and synthesis are fairly easy to perform. The
DESCRIBING FUNCTION METHOD
69
most important advantage of the method is the theoretically unlimited number of time constants in the linear part. From the viewpoint of accuracy, the greater the number of time constants, the better. The method of describing functions is based on the following assumptions [17]:
The output
and the frequency of its fundamental is idensinewave input. This means that subharmonic generation is excluded. It is generally assumed, furthermore, that the nonlinearity is symmetrical and, consequently, the output wave has no constant component of zero frequency (the method can be generalized, however, to cover asymmetrical nonlinearities as well). 2° Only the fundamental of the output must be taken into account as the higher harmonics will be attenuated by the linear transfer elements of the system to such a degree as to become negligible (the linear part of the system is, in the frequency domain considered, of a lowpass character). 3° The nonlinear element does not vary with time. Timevariable (nonautonomous) elements have no describing function, as they may have no quasistationary periodic output, either. A frequencydependent element, on the other hand, does have a describing function which is, both amplitudeand frequencydependent. 4° Only one nonlinear element is permissible in the control system: the other elements must all be linear. If the system has two or more nonlinearities, it is most reasonable to combine them into a single one, and determine the describing function for this combined element (although the describing function method can be extended to cover several nonlinearities too). The method of describing functions has also its limitations [17]: l°No adequate technique has as yet been devised for checking the accuracy of the result. 2° The method will yield quantitative and qualitative estimates only. Thus the stability or instability of the system can be assessed, and so can the existence, the magnitude and the frequency of the limit cycle. An analysis performed in the frequency domain, will not, however, furnish even an approximate indication of the dynamic behaviour of the system in the time domain, that is, on overshoot, transient processes, settling time, etc. At the same time, it should be emphasized that the describing function method is the only simple practical procedure for the treatment of nonlinear systems higher than secondorder. The method of describing functions is particularly useful, if the describing function itself is not oversensitive to curve shape variations. 1°
is
periodic,
tical to that of the
2.2.2
FUNDAMENTAL RELATIONS
Figure 2. 2. 21 is the block diagram of two simple singleloop control systems. The nonlinearities have been combined into a single nonlinear element. Let the sinewave input of the nonlinear element be [C16, L41] u(t )
= B sin
cot
.
(
2 2 2 1 ) .
.
HARMONIC LINEARIZATION
70
b)
Fig. 2.2.21. Simple singleloop nonlinear control
systems
periodical output variable of the nonlinear element can usually be expanded into a Fourier series:
The
y(t)
= A0
f
2 Mn cos ncot B
n sin
n cot)
(2. 2.22)
n=
Assuming a symmetrical nonlinearity, the from the following equations: A.q
=0
coefficients
can be determined
,
n y(t)

1
cos n
cot
dcot
(2. 2. 23)
,
J 0 71
y(t) sin

n cot d cot
(2. 2. 24)
J' 0
Generally,
A n = A n (B,jco)
and
Bn (B,jco),
both amplitude and frequencydependent.
that
is,
the coefficients are
FUNDAMENTAL RELATIONS
When
71
using the describingfunction method, only the fundamental of
the output signal
is
taken into consideration: y(t)
=B
sin
x
cot
+A
x
cos
cot
(2. 2. 25)
.
Introducing the expressions
oi sin cpx
= ysrMf, Ax
— —^
)
B
=
(2.2.216)
If the symmetrical nonlinearity is singlevalued, then therefore 0, and hence Gx x
q'
=
=B
A t =0,
cp x
=
0,
;
N(B,jco)
B
= q(B,jco) =
(2.2.217)
.
The above expressions show that the describing function N(B,jco) of the nonlinear element can be treated hereafter in the same way as the frequency function
G(jco) of a linear transfer element, except that the depen
dence on the amplitude
B
of the input variable must also be taken into
account. It will be repeatedly emphasized that harmonics are not reckoned with because a strong damping by the linear transfer elements is assumed. Consequently, the harmonic content of the feedback variable may be con
sidered as negligible.
Example
2.2.2.
Let us determine the describing function of a nonlinearity resemb lin g a simple limiter or a sudden saturation (Fig. 2. 2.22). [11, 12]. The characteristics of electronic amplifiers are approximately of this type or, with a considerable simplification, so are the magnetization curves of iron cores. Let the linear section have a slope N or, in other words, let it have a gain N The sinewave input of amplitude I? 6 gives rise to a truncated sinewave output
K
K
>
.
= Knu = KnB sin y = KN b; y = Knu = Kn B sin y
etc.,
a
where
= sin ~ The
1
b
is
cot
a b/B), into the describing function is
.
K =
>
K
N(B)
4
k
71
B
.
K
of the ideal relay, characterized by
V
Example
— 9(u) = k sgn u.
2.2.2.2
Let us determine the describing function of a backlash or hysteresis type nonlinearity [8, 17, 23, 35, 37]. The nonlinear characteristic and the input and output variables are presented in Fig. 2.2.23. When changing the sign
Fig
.
2. 2. 23.
Establishing the describing function of a hysteresis type nonlinearity
FUNDAMENTAL RELATIONS
75
of the input variable, the output will remain unchanged for a period whose h length is governed by the backlash. Thus the output is, for B
= Kn {B sin — h); y = Kn (B — h); + h); y = Kn (B sin
0
.r,
[jr [7~ sm {h £
!h
h*!
B^B'
’
2
\
i
Three
y

yj!L.
k
position
Relay
d
IT, d
e
e
with
d b
0
n
Pit
*
*
!L» n rt±)\
it
B
1
.i
\
\2ir
Deficiency
Z
if
d~B
F l
oi*s\n'd/B
m V yd:
Negative
t
el »
j3=rs\if e/B
k
Hysteresis
Compound
'
,
V v>
n
1
0
\
Ft
0
ut
Kn
B^b
if
,
i
~
P
.
r 'f +s
sxnFd
i
'
dsiri'b/B
+ Wsm
*
*l
an
4s in*cT
r Hma*?
HARMONIC LINEARIZATION
80
we have seen before. Figure 2. 2.41 represents the describing function divided by the slope, that is, the ratio N(B)/Kn vs. the reciprocal relative amplitude b/B. Incidentally, the expression N(B)/Kn is often is
real as
called the normalized describing function.
b B Fig.
If
tude
2. 2. 41.
Describing function of saturation (limitation)
B 0 u)
(2.2.523)
relations can be written
sin cot
.
up
for
HARMONIC LINEARIZATION
90
Example 2. 2. 5.1 The equation V of a nonlinear element (2.2.512), (2.2.513),
?
= g(n)
u,
u>
0
u
,
0
0)
2 . 2 6— 1 ) .
and the
AN ALTERNATE APPROXIMATE METHOD
minus sign to the quadrants II and III (when d (u/B)
mind that
= 
dVl(w/5) a r
1
(
^L
Yl
91
0, u > 0; by 2 and by g t if u < 0, u > 0 (Fig.
Let, furthermore, function g(u) be denoted if
u>
0,
w
< 0;
by g z
,
if
< 0, u < 0;
u
then gx g x B is
=—
A,
=—
,
= j
 g2
f ( 9l tiB J
)
(2.2.610)
d«.
0
Naturally,
if g(u) is single valued
Example 2.2.6. In Example
2. 2. 2. 2,
there
the value of
A
is
x
no hysteresis
loop,
and A x
=
0.
was 
A
x
= KnB
71
On
the other hand,
by equation
'A]
_u
(2.2.610)
and
— Kn (B —
A which
'(h) 2
is
h)
Fig. 2. 2. 23,
2h
the same result. 2.2.7
STABILITY TEST
of the describing function method, the approximate stability examination of nonlinear systems can readily be performed, the approximate conditions of limit cycle generation can be stated, and the amplitude and frequency of the oscillation at the input of the nonlinear element can also be defined [G6, G7, G16, G28, P14]. Figure 2. 2. 71 represents the block diagram of two versions of a simple singleloop control system. The nonlinear element is substituted by the describing function N(B, jco). The frequency responses of the two closed
By means
loop systems will be, approximately,
W
a {jco)
N(B,jco)G x (ja>)
= 1
+
N(B,
jco)
(2.2. 71)
Gx (jco) G2 (jco)
and
W
b (jo>)
=
[jw)
(2. 2. 72)
l+N(B,jco)Gx (j(o)G.2 (jco) undamped, that
respectively. The closedloop control system will be at the limit of stability if 1
+
N(B,jco) Gx {jco)
G2 (jco)
=0
.
is,
(2.2.73)
HARMONIC LINEARIZATION
94
a)
b)
Fig. 2.2.7—1. Block diagrams of singleloop control systems
If this condition is satisfied, the system will have a permanent hunting, that is, a limit cycle will be produced. Oscillation amplitude and frequency at the input of the nonlinear element can be determined on the basis of equation (2 2.73). Multiloop control systems are to be reduced to a simple singleloop type [29]. The simplification of a block diagram is shown in Fig. 2.2.7—2. Now let us return to Fig. 2.2. 71. At the input of the nonlinear element, the expression u(t) sin cot Im U(B,jco)eJ at (2.2.74)
=B
=
=
was assumed to
=
arise (in part (a) of the figure: e u, in part (b): y u). After the opening of the control loop before the nonlinear element, the fundamental component of the return signal will assume the form Vi(t)
=C
x
sin(co£
+
)=H(B,ja>) where
=G
,
(2. 2. 76)
G(jco) has the same role 1 (ja>)G2 (jco). The function H(B,ja>) in nonlinear systems as the frequency response G(jco) in linear ones, but
STABILITY TEST
Fig.
2. 2.72.
Simplification of the block diagram of a multiloop control
system including a single nonlinear element
95
HABMONIC LINEABIZATION
96
H
a twovariable function of frequency co and input amplitude B, wherefore a set of curves will be obtained instead of a single Nyquist diagram [P27]. Figure 2.2.73 shows three curves of the set. ^"0 is passed on the right when running Bs the point 1 1 If B through the respective Nyquist plot in the direction of increasing frequencies, Bj, which indicates the stability of the system. At an amplitude of (
B,
jco) is
in this case,
=
— =— +
,
B=
Fig. 2. 2.73. Frequencyresponse diagram of a control system containing a nonlinear element
—
passed on the left, thus indicating the instability of the sysnonB L) the system is in a neutral state, that is, the linear system will generate a limit cycle, with an amplitude L and a frequency co co L at the input of the nonlinear element. The foregoing considerations apply to minimumphase and stable in itself H{B, jco) functions. In other words, the function H(B,jco) is supposed to have zeros and poles in the upper half of the complex co plane only (which means that the function H(B, s) has only lefthandplane zeros and poles). Similarly to the common Nyquist criterion, the method outlined above can readily be extended to cases where the limitations referred to are absent. Hence, on the basis of the generalized amplitude and frequencydependent functions H(B,jco) or characteristics, the following statements may be made: If the equation
the point 1 tem. Finally,
is
if
=B
B=B
=
l+H{BL ,ja>L = )
0
H(BL ,jco L
or
BL
)
= 1
(2.2.77)
then one (or more ) limit cycle( s may be generated in the nonlinear system. This limit cycle (or limit cycles) will be either convergent (permanent) or divergent (dissolving). These are often called stable and unstable limit cycles, respectively. If the condition of limit cycle generation is satisfied and, for the adjacent 0 of the diagram G(BL f bB, jco) produced by a slight increase bB amplitude BL the system appears as stable, then the limit cycle will prove to is satisfied for
a given pair (or pairs)
,
co L ,
>
,
STABILITY TEST
97
B
be convergent as, in such cases, the amplitude will decrease to L ConverseSB, joo) indicates an unstable system, the limit cycle is ly, if G{B l .
+
divergent. system, that Finally, if for values min H{B, jco) indicates a stable 1 is to the left of the plot H{B, jco) in the case of a minimumis, the point phase stable loop, then any finite initial conditions will give rise to bounded output variables in the system. On the other hand, if stability sets in for amplitudes m ax then the system will exhibit a stable be0. This, however, does not invariably mean a stable haviour about point of equilibrium since the system may oscillate at very high frequencies
B>B
—
B=
B), and indeed passes 1 at any amplitudes does not pass through the point this point on the right, then the system is stable under any finite initial conditions. This means a case of global stability. Instead of the socalled singlelocus procedure outlined above, the socalled doublelocus technique is often resorted to (rather exceptional in the case of linear systems, this technique is quite usual for the nonlinear cases) [3033]. There is no need to plot the entire set of curves, but one amplitudedependent diagram and one frequencyresponse curve will generally suffice. The stability test may be performed by using either the Nyquist, the inverse Nyquist, or the Nichols diagram.
—
B
of the Nyquist diagram transfer function of the linear part resultant the If in equation (2. 2. 73), of the open loop is Gi(joo)G2 (jco) —G(joo), then the limit condition of stability may be written as 2. 2.7.1 Stability
check by
means
G{'jco)
=
JL
(2. 2. 78)
~N{B,ja>)
stability test, the polar sponse locus, is first plotted in the
For the
diagram G(jco), called the frequency recomplex plane by the usual method; the
locus of the negative reciprocal of the describing function
is
plotted after
wards. In order to simplify the procedure, frequencyindependent describing lfN(B) is functions N(B) are at first assumed (Fig. 2.2.7—4). Generally, 1/N(B) will a complex quantity. For a real describing function N(B), the curves on arrows The plane. complex the follow the negative real axis of respectively. amplitudes, point in the direction of increasing frequencies and For minimum phase stable transfer functions G(s), the criterion of
—
—
closedloop system stability
is
as follows:
—
1/N(B) does not intersect the The system is stable if and only if the locus polar diagram G(jco) and running through the main branch of the latter in the direction of increasing frequencies the former will in its full extent be passed ,
on
the right.
7
HARMONIC LINEARIZATION
98
The example illustrated by Fig. 2. 2. 74 refers to a stable system since the locus 1/N{B) is situated on the stable side (8) of the locus G(jco). I indicates the unstable domain. The Nyquist criterion is clearly a special case relating to linear systems of the criterion described above. The describing function of the proportional linear element is a constant independent of amplitude 1/N(B) is reduced to a certain point of the negative and hence, real axis.
—
—
B
Fig.
2. 2. 7 4.
by the Nyquist diagram
means
of the inverse Nyquist diagram the initial equation (2. 2. 73) in the form
2.2.7 .2 Stability test by
Let us write up
Stability testing
_L
— N(B)
(2. 2. 79)
G(jco)
and
plot, in the
complex plane, the inverse frequency response locus
as well as the negative describingfunction locus
Fig. 2.2.7 5. Stability testing
by the
—N(B)
l/G(jco),
(Fig. 2. 2. 75).
inverse Nyquist diagram
STABILITY TEST
99
Now
the stability criterion can be defined as follows: is stable if and only if the negative describingjunction locus does not intersect the inverse Nyquist locus for various amplitudes,
The system
l/G(jco) but is situated completely within the S domain at the left of the latter one, when the HG(jco) locus is traversed in the direction of increasing frequencies. Of course, this criterion applies again to such systems only where the transfer function G(s) of the open loop has no righthalfplane poles or zeros.
apparent that the condition defined above is the generalization of the common inverse Nyquist stability criterion. Figure 2. 2. 75 illustrates a stable example. It
is
2.2.7 .3 Stability test by the Nichols diagram Stability can be checked also on the basis of the Nichols (log gain vs. phase angle or phase margin) diagram. Plotting the frequencyresponse
and the negative function describing 1/N(B) on the basis of absolute values expressed in decibels, and phase angles (or phase margins) (Fig. 2.2. 76), we get the following criterion of stability: When running through the frequencyresponse plot G(jco) in the direction of increasing frequencies, we must pass the am1/N(B) plitudedependent locus on its left (i.e. the latter has to lie on the righthand side S of the former). As an explanation, let us add diagram
G(j(o)
reciprocal
—
—
=
G that the origin 20 log 0, (corresponddiagram the of 0 q> of the 1 ing to the point complex plane) is similarly to the right of the Nichols plot of a Fig. 2.2.7 6. Stability testing by the Nichols diagram stable system. The reference direction is reversed here because and the Nichols InCr ]n 6r cp 7i on the horizontal diagram has 20 log G on the vertical and
0 and
1 whereas 1 yield co 2 sec c 1 The frequencies thus obtained are in 1 gives oo 4.9 sec 25 sec H l good agreement with the results farther above.
For example,
K =
TM —
Problem 2.2.9. Figure 2. 2. 96
1
sec
=
and
Kh =
=
5 sec
,
.
the block diagram of a simple control system. In addition to the nonlinear device (ideal relay), there is also a dead time element, besides the integrator with delay. Investigate the stability of the closedloop system by both the rootlocus method and other methods. is
Fig. 2.2. 96. Block diagram of a control system
2.2.10
(cf.
Problem
2. 2. 9.1)
DRAWBACKS OF THE DESCRIBINGFUNCTION METHOD
of describing functions is approximate in that it neglects the higher harmonics arising at the output of the nonlinear element, that is,
The method
assumes a pure sine wave output variable. This approximate method provides good enough results if the higher harmonics neglected do not in effect deteriorate the control process [G9, G28, T20]. As an example, let us consider the describing function of saturation, whose formula is given by equation (2. 2.41). Let us express the amplitude ratio of the third harmonic and the input sine wave; that is, let us plot a describ
it
DRAWBACKS OF THE DESCRIBINGFUNCTION METHOD
115
ing function for the third harmonic:
N (B) = S
4 _
_
3n
Kn 
2 " 3/2
B
N
The normaHzed describingfunction diagrams for N(B)/Kn and 3 (B)/Kn respectively, are shown in Fig. 2.2.10—1. It is seen that, for weak input ,
the third harmonic may be neglected whereas stronger input signals, for which the output variable approximates a squarewave shape, result in a third harmonic of an order of magnitude identical with that of the fundamental. (Low pass linear elements will of course reduce the third harmonic much more, than the fundamental.) The same applies to the signals,
fifth
harmonic.
Fig. 2.2.101. Describing functions of saturationtype nonlinearities
Figure 2.2.102 shows the diagrams of the fundamental and of the third and fifth harmonic for the characteristic of an ideal relay (or dry friction) Fig. 2.2.103 shows the same for a dead band. These figures warrant conclusions similar to those above. The method is of an uncertain accuracy also if the two curves are shaped as in Fig. 2.2.104, illustrating the describing functions of backlash and its third harmonic. If the control system has a convergent limit cycle of small amplitude (one of the points of the range of weak inputs in figure), then the system may or may not be acceptable depending on the permissible control deviation. Figure 2.2.104 reveals, however, that the neglected third harmonic is of a significant value just in this signal amplitude range, and if the filtering provided by the linear part is insufficient the result 8*
DRAWBACKS furnished by the
OB’
THE DESCRIBINGFUNCTION METHOD
117
method will not approximate the real system closely enough.
As an example, the describing function may indicate stability for a certain nonlinear control system which is in effect unstable. The method may yield an inaccurate result even if the frequency response locus of the linear
Fig. 2.2.104. Fundamental and third harmonic development for backlash vs. the input variable
plant and the curve of the negative reciprocal describing function intersect at a small angle. As an illustration, Fig. 2.2.105 presents four Nichols diagrams: (a) the point of intersection determining the limit cycle is clearly defined, (b) the point of intersection is acceptable, (c) the intersection is uncertain at a small angle, and (d) the two curves run parallel without intersection. The two latter examples show with dashed fines where the curves corresponding to an actual nonlinearity wouldrun (it must be noted here, however, that the dashed curve is only of a theoretical interest, as it would necessitate the taking into consideration of all the higher harmonics and also of the linear system as a filter). In case (c), where the intersection of the approximate curve 1/N(B) with the locus G(jco) indicates the existence of a convergent limit cycle, there is no limit cycle in the actual system and, in case (d), where the two
—
HARMONIC LINEARIZATION
118
cj
Fig. 2.2.105. Limit cycle
d)
examination by the Nichols diagram
intersection or limit plots run parallel, the approximation indicates no oscillations. permanent cycle whereas, in reality the system will exhibit To summarize, it may be stated that in higherorder systems, where the zeros, that is, linear part of the openloop system has several poles and functions describing filtering effect is satisfactory, the method of
where the
secondacceptable, and the results obtained will be correct. In first, not alwill part linear the by and possibly thirdorder systems filtration of demethod the ways be satisfactory and, therefore, results provided scribing functions should be checked by other methods.
is
COMPENSATION IN NONLINEAR SYSTEMS 2.2.11
119
COMPENSATION IN NONLINEAR SYSTEMS
In nonlinear systems, there are two theoretically possible ways of compensation. First, we may deliberately introduce into the linear system a nonlinear compensator unit to improve dynamic behaviour and, on the other hand, the instability of a nonlinear system may be eliminated by the introduction of linear compensators [3437].
Fig 2.2.111 Block diagram of nonlinear feedback compensation .
Compensation in nonlinear systems is best checked by means of an analog computer as the entire dynamic behaviour has to be studied. For preliminary investigations, aimed at the elimination of instability or of the limit cycle, the method of describing functions will be found useful. By applying linear signal compensators, the frequency response of the linear part may be modified and intersection with the amplitudedependent locus can be avoided while the loop gain is held constant or indeed increased.
In nonlinear control systems, nonlinear cascade or feedback compensation
is
also often used.
Figure 2.2.111 illustrates the internal feedback compensation of a nonlinear system. Its characteristic equation is 1
if
+ G HN h {B) + GxG H N(B) = 2
x
the nonlinear signal compensator
N(B) then the
critical condition
N(B) the other hand,
if
0.
(2.2.111)
chosen so that
= N h {B),
(2.2.112)
of system stability v 1
On
is
0
ill
be
= Gx(G H, + H). 2
the linear compensator
H=GH 2
0
is
(2.2.113)
chosen so that (2.2.114)
HARMONIC LINEARIZATION
120
then the
critical
condition of stability
may
1
be written as
= G G Ho. x
N(B)Tn^)
2
(2.2.115)
In the third case, when both conditions (2.2.112) and (2.2.114) are fied, the critical condition is given by
= GjGnHn 2
2N(B) The most
interesting case is
when
N„(B) is satisfied.
.
(
satis
2 2 11 — 6 ') .
.
the condition
= 1  N(B)
(2.2.117)
Equation (2.2.115) then takes the form
—1 — GG H X
2
0
,
that is, the nonlinear system can be linearized by means of an appropriate nonlinear compensation satisfying (2.2.117). The type of nonlinear compensator realizing this aim may be pinpointed by means of the inverse describing function method. By means of equations (2.2.113) or (2.2.115), and (2.2.116) or (2.2.117), respectively the effect of the signal compensator on system stability is relatively easy to check.
2.2.12
Of
THE HARMONICBALANCE METHOD
the harmonic linearization methods, that of the describing functions is of greatest importance in control engineering. Sometimes, however, an old method, the method of harmonic balance can also be used [1, 64], [Kl, K3, P27]. This method is suitable for the demonstration of the possible existence of a limit cycle only and will not indicate the convergence or divergence of the limit cycle. The method consists, essentially, in substituting the expressions 2 co sincotf, etc., into the homogeneous coB cos cot, u u sin cot, u nonlinear differential equation and, neglecting higher harmonics, making the coefficients of the sine and cosine terms separately equal to zero. It is these conditions that determine the amplitude L and frequency co L of the limit cycle. Divergence or convergence of the limit cycle has to be decided separately. all
=B
—
=— B
B
Example 2.2.12.1 In Example 2. 2. 2. 3, the assumption x linearized equation
B
2
=
0 leads to the harmonically
HARMONIC LINEARIZATION OF STATE EQUATIONS
The condition of harmonic balance

(1
co
is
B=
2 )
BT 4
Hence, the limitcycle data are
=B
121
0
= 0.
coB
,
BL =
=
2, co L
1.
sin cot, the harmonically linearized differential Incidentally, since u equation may be written also in the form
ii
B2
—s
'
u
4
=0
u
4
2.2.13
HARMONIC LINEARIZATION OF STATE EQUATIONS
linearization method described in the above section can be generalized to the linearization of state equations as well. Such equations are often encountered in control engineering and, consequently, their har
The harmonic
monic linearization is an important problem [P27] Let us start from the homogeneous nonlinear vector x
=
Ax
g(x)
j
differential equation
(2.2.131)
.
assumed that g(x) is an odd nonlinear function, components is represented by an odd function:
It
is
= 0;
g(0) g(
— x) = — g(x);
g{ (
g
t(
— xv — x = 1, 2 2,
0,
{i
,
.
.
.
.
,
gA x\> x2 >
)
.,
that each of
= 0;
0)
.,
— xn =
.,
.
.
0
i.e.
its
(2.2.132) •
•
•>
xn)
(2.2.133)
n).
Let us find a linearized vector equation x
= Hx+Kx,
(2.2.134)
B
are functions of the amplitudes it T and of the frequency vector b n] that is, of the amplitude [2?i, 2 substituted sequentially co, respectively The components of vector x are
whose components
Hij( b, co); A’ ; ; (b, 
co)
B
—
by the values g
(
=
Bj ( sin (i
=
cot
+
l, 2,
•
,
ipi);
.
.
.,
•
y>i
n)
,
B
=
0
(2.2.135)
.
Thus, by substituting into the nonlinear function g(x), neglecting higher harmonics, and determining the vector resultants of the individual components,
we
get
g
^C where
B — Cn ix
i
ix
cos
(B1
sin
sin
{cot
jj)
sin
(cot
+
ipj)
dcot,
(2.2.1314)
0 2n
K = ij
~^
B
J
9ij
(Bj sin (cot
J
+ Vj
))
cos
(cot
+ Vj
)
d cot
.
(2.2.1315)
o
The
stability of
a harmonically linearized system can be determined from
the roots of the characteristic equation
D(s)
= H + sK — si = 
In possession of the culated.
BouthHurwitz

0
criterion, the roots
(2.2.1316)
need not be
cal
In the case in hand, the Hurwitz determinants depend on the amplitudes By In linear systems, the last Hurwitz determinant but one, n_ x usually assumes zero value and hence the characteristic equation has two conjugate complex roots. In the case in hand, therefore, it is once more best to plot the surface 0 in the parameter space of the coefficients n_x and Hij} Hij, to determine whether the curve obtained by varying the
H
.
H
=
,
HARMONIC LINEARIZATION OF STATE EQUATIONS
123
amplitudes Bj in the domain 0
ex2
—
exxx2
.
=—
ex\x2 is of equation (2.2.1313), the only nonlinearity g 21 (xx x 2 the multivariable type. Thus, by the multivariable version of (2.2.1314),
By
,
)
2n
^J>
H = *1
2 sin 8i
cot
X B.2 sin (cot
+
ip
2 ))
sin
cot
d cot
or
B—0 cos 3eBx— y> 2
1
and from that of
(2.2.1315), 2n
K =2X
nooB
r
J
f
2 cot X B 81 5? sin
2
sin
(
cot
+
*p 2 ))
cos
cot
do#
or
B2 eB x—
K =
y 4co
21
Since
x2
=x
x,
K = — eB /4; 2
2X
.
 sin
ip
2
.
=
B =
H ——1,
and ti/2, coBx and equation phase and the harmonically linearized
we
have
2
—x x2 = — x
xx
2.2.14
2
2X
is
,
x
j
e
x2
•
COMPARISON OF THE THREE METHODS
three methods of harmonic linearization are equivalent but their application involves more or less computation. This will be shown by an example: that of the control system with backlash discussed in Example 2.2.8. 1.
The
HARMONIC LINEARIZATION
124
Example 2.2.8. 1 illustrates the use of the describing function method. The same problem was studied in Examples 2. 2. 9. 2 and 2. 2. 9. 3 by the rootlocus method based on a describing function. Now two further alternatives will be presented: the harmonic balance method, and the harmonic linearization of state equations.
Example
2.2.14.1
Let us solve Example
2.2.8. 1
command
If the reference
by the harmonic balance method.
r(t)
zero, the differential equation of the
is
nonlinear control system becomes
B
KL g(u) = 0
+ = B sin
it f
KL = KEKM If u cot, and the nonlinearity is substituted by the describing function, then, taking coefficients A x and Bx from Example 2. 2. 2. 2 and assuming B >• h.
where
.
no
kl kn 7t
71
_
2
+
B
[B
sini
2
2
—j
}
1
—
1
h
h2
B
B
u
=0
2
The condition of existence of a limit cycle is (since the sine and cosine, or real and imaginary terms of the equation obtained by substituting u = B sin cot or BeJwt have to become equal to zero):
—
co 2
Tm
j
—7i
(
Sin 1
1
[2
B
[
and 4
.
K L KN 71(0
With a time constant of
K = K =
TM =
h2
(5
B
\
n
2 ,
the convergent limit cycle data for 0.18 Ti and co H 4.69, whereas for they are B c 0.50 h and coc 1.59, in good earlier results. (The divergent limit cycle data are also
25 sec 1 are 1 5 sec
a gain of h a gain of L agreement with our
lh
1 sec,
BH =
—
=
obtained.) Limitcycle convergence itself of other considerations.
=
must be decided on the
basis
Example
2.2.14.2 Let us solve Example 2.2.8. 1 equations.
On
substitution of xx
=u
become x,
=
by harmonic
and x2
=
u, the
linearization of the state
phase variable equations
will
COMPARISON OP THE THREE METHODS
now
125
the coefficients of the harmonically linearized equations ~
1
*®2
^2
>
— #21
1 *®1 ~f"
#21
•a:.
*®1
M are to be found. Assuming give
B0; u
ff
u)
= J 9iu )p(u — oo 00
=J 9oi( m u>^u)
Kso(
ff(u)
)
H (u) p^u) du
J
KS0 (m
cr
u,
,
t
2 I g (u) Pxi U )
m u> T do).
(2. 3. 74)
(? 0
J
j 00 4
=
time (r 0), the correlation function provides the meansquare value of the variable or if u 0, its variance:
For zero
s hifting
m =
= al
) dco "J
ol
.
(2.3.7 6
2n J
Between the powerdensity spectra of the output and input of the system
illustrated in Fig. 2. 3. 71, there
is
linear
the relationship (2.3.7— 7)
Fig.
2. 3. 71.
Powerdensity spectra
of the linear system
This is the socalled indexchanging rule [C26]. The outputsignal variance will be 00
o 2y
= y*(t) =y {t)= — 2
2rt 1
11 *
!
J — 00
&yy(ja>) dco.
(2.3.7S)
164
STATISTICAL LINEARIZATION
As seen above, one of the
statistical linearization
methods creates a connection between the standard deviations at the input and output of the nonlinear static element:
= K*s ol
(2.3.7—9)
,
statistical analysis may be extended to cover nonlinear systems too. If the transfer functions and characteristics of the openloop system are known, the aim of the analysis is to determine statistical characteristics such as standard deviations at zero mean value of the variables arising at different points of the closed loop.
whence
Example 2.3.7. The nonlinearity in the control system a limiter with KN = 1, k = b = 2.
illustrated
by
2. 3. 72
Fig.
is
a)
Fig. 2.3.72a. Block diagram of a control system
(cf.
Example
2. 3. 7.1)
b)
Fig. 2.3.72b. Block diagram of the same system, with the statistical parameters of the variables and the describing functions
The complete input
signal
r(t)
of the closedloop system
falls
into
two
parts: r(t)
=m
nr (t)
r f
m
r is the expected value of the complete input variable, and n r (t) the random noise component of zero expectation. Since the process is stationary, the expectation is independent of time. The distribution of the complete input is a normal distribution of mean value m, (Fig. 2.3.7— 3). Let the powerdensity spectrum of the noise component of the complete input signal be
where is
N
®nnU™) 1
2
+0) 2
N 1
+jco
N 1
—joi
STATISTICAL SYSTEMS ANALYSIS
165
Let us study first the effect of the expected value of the complete input signal on the closedloop control system. The transfer function defining the input of the limiter is KeS_ s f
for the useful signal
K e Km
=W
a {8)
component, and
Ke* s
for the noise ic

Ke K s
component since the nonlinearity may be considered as a statKM and Ks for the useful signal and noise component, re
element of gain
spectively. The expected value of the input signal u(t) of the nonlinear element is zero. This is easy to see since the control system is of Type 1 and therefore the steadystate value of the actuating error is zero. Hence the nonlinearity transfers no useful signal in the steady state. let us turn to the effect of the standard deviation of the complete input signal. Taking into consideration the powerdensity spectrum and the transfer function of the noise component, and since s ja>, the powerdensity spectrum at the input of the nonlinearity will be
Now
=
= W (— 8) W {s) 0nn {s) =  ke* N N Ell s Ke Kg 1 s — + Ke Kg 1 — s KeNs_ KeNs 8* + {\+Kb K8 8 Ke Ks s* (1 + Kb Kg) s + Ke Ks ®uu{s)
b
b

f
f
)