142,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The theory of dynamic systems is addressed in this book in accordance with the "modern" approach, heir to algebraic analysis, which has been implemented since the last decade of the 20th century. After a reminder of the evolution of the representation of systems based on transfer functions or matrices, the duality of controllability and observability is revisited, and new results are produced concerning time-varying discrete-time systems. To complete and improve the existing analyses, the poles and zeros of linear systems and their interconnections are presented in a new way, as well as the problem of systems governed by functional differential equations (of retarded or neutral type) and their stabilization. This book also proposes known and original mathematical complements.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 476
Veröffentlichungsjahr: 2024
Ähnliche
Table of Contents
Cover
Table of Contents
Title Page
Copyright Page
Preface
List of Notations
1 Representation of Systems: A Historical Overview
1.1. Transfer functions and matrices
1.2. State-space representation
1.3. “Geometric” approach
1.4. Polynomial matrix description
1.5. The behavioral approach
1.6. Module of a system
1.7. The formalism of algebraic analysis
2 Linear Systems: Concepts and General Results
2.1. Control systems
2.2. Strict equivalence of Rosenbrock systems
2.3. Controllability, observability and their duality: the algebraic point of view
2.4. Reachability, observability and their duality: Kalmanian point of view
3 Poles and Zeros of Linear Systems, Interconnectedness and Stabilization
3.1. Poles and zeros of continuous or discrete invariant linear systems
3.2. Poles and zeros of interconnected systems
4 Systems with Differential Equations and Difference Equations
4.1. Systems governed by functional differential equations
4.2. Time-invariant linear systems with lumped delays
4.3. Time-invariant linear systems with distributed delays
Appendix The Mathematics of the Theory of Systems
A.1. Laplace transform
A.2. -semi-groups of operators
A.3. Variations on the theme of injective cogenerators
A.4. Complements of linear algebra
References
Index
Other titles from ISTE in Mathematics and Statistics
End User License Agreement
List of Illustrations
Chapter 3
Figure 3.1. Diagram of a control system
Figure 3.2. Series interconnection of Σ
1
, Σ
2
Figure 3.3. Parallel interconnection of Σ
1
and Σ
2
Figure 3.4. Elementary feedback interconnection
Figure 3.5. Example with diagram
Figure 3.6. General closed-loop system
Guide
Cover Page
Table of Contents
Title Page
Copyright Page
Preface
List of Notations
Begin Reading
Index
Other titles from ISTE in Mathematics and Statistics
WILEY END USER LICENSE AGREEMENT
Pages
iii
iv
ix
x
xi
xii
xiii
xiv
xv
xvi
xvii
xviii
xix
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
285
286
287
288
289
290
291
292
293
New Mathematical Methods,Systems and Applications Set
coordinated byHenri Bourlès
Volume 1
System Theory – A Modern Approach 1
Linear Ordinary and Functional Differential Equations
Henri Bourlès
First published 2024 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2024The rights of Henri Bourlès to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2023951603
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-985-3
Preface
This book studies dynamical systems, with or without delays, governed by differential equations, difference equations or differential-difference equations. The approach adopted is a “modern” approach, which, as the author sees it, has slowly imposed itself from the last decade of the 20th century: in this approach, a system can be represented either by a set of equations, or, even better, by the abstract object that summarizes these equations – in the linear case, this is a module – or it can be represented by the set of solutions to these equations, which make up the “behavior” of the system. We move from equations to solutions using a “functor” – a concept that has become commonplace since the advent of the theory of categories in the second half of the 20th century. In favorable cases, where the module has “good properties” (an algebraic condition), as well as the function space in which the solutions are found (an analytic condition), this functor is a “duality”. The knowledge of solutions is then essentially equivalent to that of solutions. This allows automation engineers (theoreticians as well as practitioners) to make use of all resources, whether algebraic or analytic, in accordance with the spirit of “algebraic analysis”.
No dynamical system is truly linear: an RLC circuit, for example, is nonlinear because a resistance has a nonlinear characteristic when it functions over a large enough range. Further, this kind of circuit is no longer governed by a differential equation if we take into account the propagation of the electromagnetic field (obeying the partial differential wave equation). However, in the majority of cases, all these phenomena can be ignored, at least in the first analysis. If we are working with an intrinsically nonlinear system, it is often effective to design an appropriate control law, to linearize it around a nominal working point or around a nominal trajectory; the model obtained is a linear system that is time invariant in the first case, and time varying in the second case. Thus it is reasonable, at least in the first step (and this is what we will do in this book) to consider the case of linear systems. Nonetheless, there is an abundance of literature on nonlinear systems (e.g. see [ISI 95, KHA 95]). The robust control of these systems is studied in [FRC 06].
In the rest of this text, “system” means a “dynamical system” (which clearly excludes, among others, philosophical, logical and computer systems). Despite this restriction, the concept of a system covers highly diverse realities that require very different methods of analysis. To take a well-known example, from the time of Poincaré the solar system has posed the highly complex problem of N bodies and the resulting chaotic behaviors [ARN 06]. But in automation engineering, we are chiefly interested in systems designed by and for humans (cars, planes, robots1, etc.). They are generally instrumented, equipped with actuators to control them and sensors to measure their responses to prompts, and we will therefore call them control systems as in [BLS 10]. It is possible, however, to have situations with an installation where the most relevant control and measurement variables have not yet been studied. Thus, one of the first tasks of the automation engineer is to carry out this study for a system, which is therefore not yet a control system. The methodology to follow, for this question [BLS 96], can only be explained after a detailed study of the “structure at infinity” of systems.
A system, as envisaged here, is thus a part of the material world. Engineers and scientists represent such a system mathematically, using equations (differential equations, recurrence equations, etc.) or other mathematical objects. This is why it is appropriate to say that a system is represented (or characterized, governed, or, where applicable, defined) by these equations (or by a set of matrices, a module, etc.). However, the system itself is, of course, distinct from its representation, and, as philosophers say, transcends its representation2.
Although this book is not a history of automation (on which subject the reader can consult, notably, [BEL 64, BEN 79, BEN 93, BIS 09, MAY 70]), for pedagogic reasons it seemed necessary to provide an overview of the evolution of this discipline in Chapter 1. Over the 20th century, the formalism used to represent a system has evolved from the conventional framework of transfer functions (an approach where Bode was a key player) to that, as seen earlier, of algebraic analysis (behaviors, modules, functors, etc., introduced into automation by Willems, Fliess, Oberst and their collaborators or followers) via the Kalman state-space representation. This was done, first of all, to account for a system in its totality (which cannot be done with transfer functions) and also to avoid trapping the automation engineer within a particular form of equations and variables. This procedure is analogous to the adoption of “covariant” formulations – as physicists refer to it – in differential geometry: see [BLS 19].
Consequently, in Chapter 2, it is possible to formulate the concept of a system within the appropriate framework (neither too general, to avoid pointless complications, nor too specific), and to then use the full amplitude of the concepts of controllability and observability, which have been fundamental to automation from Kalman onwards. The latter demonstrated that for time-invariant linear systems, these notions are “duals” of one another and this “duality principle” is revisited. Its validity is confirmed, both from the algebraic point of view (“A-dual”) as well as the “Kalmanian” (“K-dual”) point of view for linear, continuous-time systems (which could be time-varying), but “discrete-time” situations lead to more nuanced conclusions.
Chapter 3 begins by explaining, from the intrinsic viewpoint based on modules, the theory of infinite zeros and poles of time-invariant linear systems governed by differential or difference equations. This discussion improves upon (and corrects) the discussion presented by Bourlès and Fliess [BLS 97, FLI 96]. A study of the interconnection of systems leads to the parameterization of the Youla-Kuc̆era stabilizing compensators. Only proper compensators can be made, and this chapter concludes by studying proper stabilizing compensators using different approaches, including the approach recommended by Vidyasagar [VID 85], based on algebra. Nonetheless, the reader must remember that the stabilization of systems is only a relatively minor goal for automation engineers, who aim to design robust regulators [ZHO 95, MCF 90] that will guarantee, in addition to stability in perturbed situations, rejection of disturbances and reference tracking, and so on (see for example [BLS 10]).
Systems governed by differential-difference systems (of retarded or neutral type) are studied in Chapter 4. The essential concepts are first introduced in the general case (nonlinear), but it is the linear case that is given special attention. Time-invariant linear systems of retarded-neutral type can be defined over a ring , which was introduced and studied by Gluesing-Luerssen [GLU 97, GLU 01a]; it has properties that are quite similar to the ring over which time-invariant linear systems governed by ordinary differential equations are defined. Thus, it is possible to develop, for “-systems” (without rewriting [GLU 01a]), a theory that is nearly identical to that of “-systems”, as described in Chapter 3, including the parameterization of stabilizing compensators. However, a significant difference is that in order to be feasible, “-compensators” must not only be proper, but also non-anticipative (or even “strongly non-anticipative”, using the terminology from [GLU 01a]), and unfortunately, this double condition cannot be obtained by methods that are as simple as those described at the end of Chapter 3: see section 4.3.5(III). This question is still largely an open one [DIL 09].
Most of the concepts and mathematical results required to understand this book have been discussed in [BLS 17, BLS 18, BLS 19], and we make many references to these. However, to avoid over-burdening the text, it seemed useful to create an appendix with additional mathematical explanations. This appendix, independent of the rest of the book, but constantly referenced throughout, contains reviews and additional information on the Laplace transform (notably, the introduction of a space that is canonically isomorphic in the space of ultradistributions, which is the natural space of Laplace transforms of all distributions), a section on -semi-groups of operators (which play a key role in the theory of systems governed by differential-difference equations), and finally two additional sections on algebra: the first on homological algebra, more specifically on injective cogenerators of finitely presented modules; the second on general algebra. This chapter contains a few original results, or at least results that the author could not find elsewhere in the literature.
June 2023
Notes
1
To paraphrase the Gospel, “robots were made for humans, humans were not made for robots”, which is something our society tends to forget.
2
However, the abuse of language, “without which any mathematical text runs the risk of pedantry, not to say unreadability” (Bourbaki) cannot always be avoided and we will therefore speak of a “state-space system” or a “Rosenbrock system” and so on, despite the fact that these are only the
mathematical representations
of the system being discussed.
List of Notations
:=: equal by definition
≅, : bijection or isomorphism
: set of integers 0, 1, 2, …
G× (G: Abelian group or ): set G – {0}
∂: operator d/dt, p. 2
s: indeterminate or Laplace variable, p. 2
: Laplace transform of y, p. 2
Q(A): field of fractions of A, p. 2
δ0 : relative degree, p. 2
Σ: normal form (or Smith normal form), p. 3
rk: rank, p. 3
: sequence of roots of the polynomial a, p. 6
+: disjoint union or concatenation, p. 5
−: inverse operation of +, p. 5
σ: shift forward operator, p. 6
z: indeterminate or -transform variable, p. 6
A, B, C, D: state-space system matrices, p. 7
{C, A, B, D} or {C, A, B}: state-space system, p. 7
φ (t;t0, x0;ω), p. 8
, p. 8
Γ(A, B): controllability matrix, p. 10
Ω(C, A): observability matrix, p. 10
im (M) , ker (M):image, kernel of a matrix with entries in , p. 11
, : stable part of the complex plane, its closure, p. 14
, : space of linear maps, p. 14
: linear maps of a state-space system, p. 14
: state-, input-, output-space, p. 14
σ: spectrum, p. 16, 212
, : unstable subspace, stable subspace, of , p. 18
Σ≡ [P, Q, T, U]:Rosenbrock representation, p. 23
: system matrix or (Rosenbrock matrix), p. 23
, p. 27
: space of distributions on the real line, p. 27
: space of analytic functions in , p. 27
: behavior, p. 27
M = cokerA (•R): module of a system, p. 33
[w]A, [w], p. 33
hW: HomC (−, W) functor, p. 34, 228
AT = T−1A, MT = M ⊗A AT, , p. 40
ΣT: T-system, p. 40
, p. 41
, p. 43
: space of inputs, space of outputs, p. 43
(K, δ) , (K, σ): differential ring, difference ring, p. 46
K[s; δ] , K[∂; δ] , K[z; σ] , K[s; δ, σ]: skew polynomial rings, p. 46
K [z, z−1; σ]: skew Laurent polynomial ring, p. 46
X: fundamental matrix, p. 58, 64
Γκ: controllability matrix, p. 58, 64
, p. 59, 65
ρ: σ−1 delay operator, p. 63
, , p. 66
ad: adjoint map, p. 68
, p. 70
M⋆,, p. 71
Σ⋆: A-dual, p. 72
Ωκ: observability matrix, p. 76
Φ(t2, t1):transition matrix, p. 78, 84
, Wc(t1, t2): reachability, controllability Gramians, p. 80, 85
: ring of analytic functions in , p. 81
[s;δ], p. 81
ΓN (t), p. 84
, Wo (t1,t2): observability, constructibility Gramians, p. 87, 92
Σ⊛: K-dual, p. 94
{t.p.}T, {t.z.}T: sequence of poles, sequence of transmission zeros, p. 99
{i.d.z.}T: sequence of input-decoupling zeros, p. 101
{s.p.}T: sequence of system poles, p. 104
{o.d.z.}T: sequence of output-decoupling zeros, p. 104
{i.o.d.z.}T: sequence of input–output-decoupling zeros, p. 109
{h.m.}T: sequence of hidden modes, p. 111
{i.z.}T, {s.z.}T: sequence of invariant zeros, sequence of system zeros, p. 113, 115
, p. 139
, , p. 138
ǁfǁ∞, p. 138
θ(f), p. 139
, p. 139
, , p. 142
: ring of proper rational fractions, p. 142
: space of continuous functions from [a, b] into , p.145
, supt∈]−∞,0] |ϕ(t)|, p. 146, 150
: space of uniformly continuous bounded functions from ]−∞, 0] into , p.150
det (Δ (λ)): characteristic function, p. 156
: x (t) ↦ x (t − rj), delay operator p. 159
, n, p. 159
p* (s) = det (A(s, e−s)) , p. 164
, p. 165
: algebra of entire functions, p. 172, 203
, p. 172
C, , , , p. 174
D, p. 177
R○, ○R, p. 173
: Laplace transform, p. 188, 193, 198, 206
ea: function s ↦ e−as, p. 188
α0 (f): abscissa of convergence, p. 188
, : Laplace transform of f, of T, p. 188, 198, 206
m: monomial function t ↦ t, p. 191
, : convergence interval, p. 193, 196
: convolution algebras, p. 198
τa: shift operator f (t) ↦ f (t − a), p. 200
PW: Paley–Wiener algebra, p. 203
χω : t ↦ eiωt, p. 207
: algebra of bounded operators in , p. 211
: ideal of consisting of compact operators in , p. 212
: domain of u, p. 212
: operator that is not necessarily bounded, p. 212
: set of closed operators in , p. 212
ρ (A): resolvent set of A, p. 212
Pσ (A), Cσ (u) , Rσ (u): point, continuous, residual spectrum of A, p. 212
r (u), s (u): spectral radius, spectral bound, p. 212
: resolvent, p. 213
a (λ): ascent or index, p. 213
N (λ; u): generalized eigenspace of λ, p. 213
u|N , u|N, p. 214
A0: infinitesimal generator of the -semi-group (T (t))t≥0, p. 217
ω0 (T ): type of the -semi-group (T (t))t≥0, p. 219
: space of continuously differentiable functions from [−r, 0] into , p.223
JΛ, , : Riesz projector, its image, its kernel, p. 227
AMod: category of left A-modules, p. 229
Ab: category of Abelian groups, p. 229
: first Weyl algebra, p. 230
W0: canonical cogenerator, p. 231, 233
U⊥, , p. 232
A, p. 233
ACoh: category of coherent left A-modules, p. 235
AModfp: category of finitely presented left A-modules, p. 236
AModfg: category of finitely generated left A-modules, p. 238
, : set of matrices, set of n × n matrices with entries in A, p. 254
GLn (A): subset of consisting of invertible matrices, p. 254
: sequence of Smith zeros of the polynomial matrix A relative to AT, p. 257
: sequence Smith zeros of the AT-module MT, p. 258
char {T}: characteristic variety of the torsion A-module T, p. 259
∂[A] ,∂ri [A] ,∂kj[A]: degree, row degree, column degree of A, p. 261
1Representation of Systems: A Historical Overview
1.1. Transfer functions and matrices
1.1.1. Transfer functions
The first step in developing the theory of systems was taken in 1945, with the publication of [BOD 45], a reference book in which Bode summarized his own work – the notorious asymptotic diagram, the concepts of gain and phase margins, amplitude-phase relations – and the work of others, including his colleagues at Bell Telephone Laboratories. Among these was Black – credited with inventing the concept of “feedback”, in 1927 [BLA 34] – and Nyquist – who in 1932 developed the famous stability criterion that is now known by his name [NYQ 32]; Bode [BOD 45] also presented original results (such as the theorem on the integral of the sensitivity function, the full impact of which was only understood in the early 1980s). This book was complemented in 1947 by [JAM 47], whose formalism seems clearer to today’s readers and who introduced the Black–Nichols abacus in the form it is still used today.
The underlying formalism in these works is that of transfer functions. Let Σ be a linear time-invariant, continuous-time control system1 governed by the differential equation:
where u and y are, respectively, the input and the output of the system, and Dl, Nl are polynomials with indeterminate s (therefore elements of the ring ):
where b0 ≠ 0. A system Σ with this structure is sometimes called an input–output system (as we will see further on, there are also other types of representation). Since it has only one input and one output, this system is said to be single-input–single-output (SISO).
In equation [1.1], the indeterminate s is substituted by the differentiation operator ∂ = d/dt; additionally, the symbols2y and u can be substituted by sufficiently regular functions y = t ↦ y (t) and u : t ↦ u (t), for example of class C∞. The transfer function of this system is the rational fraction such that, if and denote the Laplace transforms (assuming these exist: see section A.1.1) of such functions, u and y, with the only additional hypothesis being zero initial conditions: y(i) (0) = 0 (i = 0, …, n − 1), u(j) (0) = 0 (j = 0, …, n′ − 1), we have3:
hence:
This rational fraction is assumed to be irreducible over A (i.e. the only common factors of Dl and Nl are units of A, in this case, non-zero constants). The transmission poles and transmission zeros of Σ (or, through an abuse of language, of G) are the roots in of Dl and Nl, respectively (by taking into account multiplicities, that is, by counting an m-order root m times).
The rational fraction (where Q(A) denotes the field of fractions of A) is said to be proper if n′ ≤ n, strictly proper if n′ < n and biproper if n = n′. The difference δ0 = n – n′ is called the relative degree of G (in algebra, −δ0 is called the degree of G, extending the common concept of polynomial degrees to rational fractions).
1.1.2. Transfer matrices
(I) Transmission poles and zeros
The formalism seen earlier can be generalized for the case where the input u and the output y of the system Σ are columns with m and p entries, respectively: Nl and Dl are then polynomial matrices whose dimensions are, respectively, p × m and p × p, Dl assumed to be regular, that is, invertible over the field of rational fractions , and is a rational p × m matrix (“transfer matrix”). Such a system is said to be MIMO (multi-input–multi-output). Transmission poles and zeros were defined in this general context by McMillan in 1952 [MCM 52]: let d be a least common denominator of the elements of G, so that:
where is a polynomial matrix: . It is known ([BLS 17], section 3.4.2) that is equivalent to its normal (or Smith normal) form:
diag(e1, …, er, 0, …, 0) denoting, by convention, with E = diag(e1, …, er) ∈ Ar×r where 4, and e1, …, er ∈ A× are the non-zero invariant factors, such that e1 | e2 | ... | er; consequently, there exist unimodular square matrices (i.e. whose determinant is a unit of A) U, V such that , and it is thus obtained that:
where the rational fractions ni/di are irreducible over A, n1 | n2 | … | nr, dr | dr−1 | … | d1.
DEFINITION 1.–
The matrix
(of dimension
p
×
m) is called the
Smith–McMillan form
of G over
A
.
The
transmission poles
(respectively,
transmission zeros
) of system
Σ
(or of G) are the roots in
of the polynomials d
i
(respectively, n
i
), i
= 1, …,
r, taking multiplicities into account.
REMARK 2.–
All abbreviations used in this book are usual in the literature on systems theory.
DEFINITION 3.–
The systemΣis said to be proper (respectively, strictly proper) if its transfer matrix G = (gij) is proper (respectively, strictly proper), that is, if the entries gij are all proper (respectively, strictly proper).
When G is proper (or even strictly proper), it is generally not true that the entries ni/di of the Smith–McMillan form of G are proper.
EXAMPLE 4.–
The Smith–McMillan form of:
is:
REMARK 5.–
It is an experimental fact that only proper systems are physically realizable. The systems to be controlled are even generally strictly proper; this is true for mechanical systems, given their inertia, as well as electric systems, electromechanical systems or thermal systems, for the same reason (see [BLS 10], Chapter 1). The exception seems to be an elementary electric circuit made up of a resistance R at the ends of which a potential difference U is applied (the system input), with the output variable being the intensity i of the current going across the resistance. However, the conventional equation U = Ri (according to which the system is proper but not strictly proper) is not valid unless the propagation of the electromagnetic wave is neglected (the “true” system equation bringing into play the wave propagation equation; the non-instantaneous nature of the propagation can again be interpreted as “electric inertia”).
However, it must be noted that with inappropriate modeling, it may be found that a system has an improper transfer matrix [WIL 97].
REMARK 6.–
It can be seen that the system
Σ
having the transfer matrix
[1.7]
has transmission poles
−1
and
−2,
both with multiplicity of
2,
and has transmission zero
−2
with multiplicity
1
. Following Rosenbrock
[ROS 70]
, it is agreed to write:
and {p.t.} and {z.t.} are said to be, respectively, the sequence of transmission poles and the sequence of transmission zeros ofΣ(strictly speaking, these are not sets: see infra, notation 7).
In general, if a is a non-zero polynomial that factorizes into prime factors over
as per:
where λ ≠ 0, r ≥ 0, pi ≠ pj if i ≠ j and, then it is possible to write that the sequence of zeros or roots of a is:
and according to Oberst et al. ([OBE 20], p. 185) that its support is:
Let us proceed more axiomatically:
NOTATION 7.–
Given two sets A, B, A + B is their disjoint union (denoted asorin [BLS 17], section 1.2.6(II). If a ∈ A ∩ B, this element a is duplicated in A + B; for example, if A = B = {a}, considering a copy B′ = {a′} of B, then A + B := A ∪ B′ = {a, a′}. Since in practice, there is no need to distinguish between a and a′, it is agreed to write A + B = {a, a}; this is no longer a set but a sequence of elements (of which the order does not matter); generally speaking, A + B is the concatenation of the sequences A, B. This notation, used notably by Rosenbrock [ROS 70], is convenient since conversely, A − B (where B is a subsequence of A) denotes the sequence obtained by deleting from A those terms that are elements of B, hence (A + B) − B = A; for example, ({a} + {a}) – {a} = {a, a′ } – {a} = {a′} = {a}, or by abusing the writing ({a} + {a}) – {a} = {a, a} – {a} = {a}. The set of elements of the sequence {a, a} is the support of this sequence, namely, {a}.
DEFINITION 8.–
Letbe a commutative field. Thenand its supportare determined by the following three conditions: (1) ifis a constant polynomial (which may be zero), then; (2) if a ≠ 0 is an irreducible polynomial, thenis the set of roots of a in an algebraic closure of; (3) if a, b, then (+ denoting the concatenation: see notation 7):
(II) Blocking zeros
Blocking zeros (b.z.) were defined in the following way, by Ferreira and Bhattacharyya [FER 77] in 1977, based on the Smith–McMillan form of G:
DEFINITION 9.–
The sequence of blocking zeros {b.z.} overAis.
Evidently we have:
with equality in the single-input–single-output case. The following result is clear:
LEMMA 10.–
We have:
1.1.3. The discrete-time case
This formalism can also be applied to discrete-time systems: the differential equation [1.1] then becomes a recurrence equation, where the differentiation operator ∂ is replaced by the shift-forward operator σ : x (t) ↦ x (t + 1) (this operator is often denoted by q in the literature); the Laplace variable s is replaced by the variable z of the -transform (which dates back to the middle of the 18th century, but whose application to sampled systems can largely be traced back to the 1950s, with the works of Hurewizc, Tsypkin, Lawden, Barker, Ragassini, Zadeh, etc.: see the summary by Jury and Tsypkin [JUR 71]). A discrete-time system whose transfer matrix (in the indeterminate z) is proper (respectively, strictly proper) is said to be causal (respectively, strictly causal). A causal system is also said to be non-anticipative.
1.2. State-space representation
1.2.1. Continuous-time state-space systems
The second step in the development of the theory of systems began in 1960, with Kalman’s work [KAL 60b, KAL 60c, KAL 63]. Control systems5 were envisaged in a state-space representation (sometimes called a state-space system by abuse of language):
written {C, A, B, D} (and {C, A, B} if D = 0); x is the state (n components) and u and y have the same significance as in sections 1.1.1 and 1.1.2(I); A, B, C are matrices of real numbers, whose dimensions are n × n, n × m, and p × n, respectively, and (often, ). This formalism encompasses the preceding one: indeed, in moving from the Laplace domain, having substituted6u, x, y in [1.10] with sufficiently regular functions u : t ↦ u (t), x : t ↦ x (t), y : t ↦ y (t), under the hypothesis of zero initial conditions, (namely, x (0) = 0 if ), we obtain:
yielding [1.4] with:
Reciprocally, given a control system Σ having a rational transfer matrix , there exists a non-unique state-space representation {C, A, B, D}, for which we have relation [1.11] (see for example [BLS 10], section 7.4). The transfer matrix G(s) is proper (respectively, strictly proper) if and only if (respectively, D = 0). D is called the direct term, and by replacing G by G − D, it is easy to return to the case where D = 0.
1.2.2. Discrete-time state-space systems
A discrete-time state-space system [KAL 60b]Σ is governed by equations of the form:
(see section 1.1.3 for the notations), and it is thus easy to transpose what has been said above regarding continuous-time systems to discrete-time systems. This system is said to be reversible if is invertible, because in this case x = A−1 (σx − Bu). Again, this state-space system is denoted as {C, A, B, D} (and {C, A, B} if D = 0).
1.2.3. Controllability and observability
Using state-space representations, Kalman [KAL 60b, KAL 69, Chapter 2] could introduce the fundamental concepts of reachability, controllability, observability and constructibility, which cannot be defined using a transfer function formalism. To simplify these statements, it is assumed that D = 0, which changes nothing in the results.
In what follows, we place ourselves in the case of a state-space system {C, A, B}, which may be time-varying (i.e. with matrices A, B, C possibly depending on time) and it is convenient to introduce the following notations: let there be any two points x1, x2 in and two times t1, t2 with t2 > t1 ( in the continuous-time case; in the discrete-time case). The semi-closed interval with origin t1 and extremity t2 is denoted by [t1, t2[; in the discrete-time case this is, of course, an interval of ([BLS 17], section 2.1.2(I)). Let . Then, φ (t; t0, x0; ω) denotes the solution to the state equation (continuous-time) or x (t +1) = Ax (t) + Bu (t) (discrete time) at time t ∈ [t1, t2] when the system input u is equal to ω and x (t0) = x0. Further, η (t; t0, x0; ω) denotes the output y at time t ∈ [t1, t2].
It is assumed that ω is such that φ (t;t0, x0;ω) and η (t; t0, x0; ω) are well-defined for (almost)7 all t ∈ [t1, t2] and all ; in the continuous-time case, this signifies that ω is a sufficiently regular function, for example belonging to ([BLS 19], section 1.5.2(I)).
We write if there exists ω (as above) for which φ(t2; t1, x1; ω) = x2.
DEFINITION 11.–
The state
(or the event
(
t
2
, x
2
)
) is
reachable
on
[
t
1
, t
2
[
for
Σ
if
. The state
(or event
(
t
1
, x
1
)
) is
controllable
on
[
t
1
, t
2
[
for
Σ
if
.
The state(or the event (t1, x1)) is non-observable on [t1, t2[ forΣif η (t; t1, x1; 0) = 0, for (almost) all t ∈ [t1, t2[. The state(or event (t2, x2)) is non-constructible8 on [t1, t2[ forΣif η (t; t2, x2; 0) = 0, for (almost) all t ∈ [t1,t2[.
The state-space system
Σ
is
completely reachable
(respectively,
completely controllable
) on
[
t
1
, t
2
[
if any point in
is reachable (respectively, controllable) for
Σ
over this interval. This system
Σ
is
completely observable
(respectively,
completely constructible
) on
[
t
1
, t
2
[
if only the origin of
is non-observable (respectively, non-constructible) for
Σ
on this interval.
REMARK 12.–
Intuitively, a system is completely observable on
[
t
1
, t
2
[
if, with control u
= 0
, the knowledge of the output y on
[
t
1
, t
2
[
makes it possible to determine x
(
t
1
)
(or, in an equivalent manner, without assuming u
= 0
, if the knowledge of
(
y, u
)
on
[
t
1
, t
2
[
makes it possible to determine x
(
t
1
)
). In this case, x
(
t
2
)
can be derived by integration of the state equation (an integration that is a simple recurrence law in discrete-time), therefore the system is completely constructible on
[
t
1
, t
2
[
. In the continuous-time case, the integration can also be carried out backward from x
(
t
2
)
to x
(
t
1
)
, and consequently,
complete observability
and
complete constructibility
are synonymous concepts. This is the same for complete controllability and complete reachability. On the other hand, in the discrete-time case, x
(
t
1
)
can be derived from x
(
t
2
)
only in the case of a reversible system.
In general, in discrete-time, complete observability is thus a stronger concept than complete constructibility
, and similarly,
complete reachability is a stronger concept than complete controllability
.
It is the
semi-closed
interval
[
t
1
, t
2
[
(and not the closed interval
[
t
1
, t
2
]
) that should be considered in
definition 11
, and this is essential in the discrete-time case (this is unimportant in the continuous-time case, since
{
t
2
}
has then zero measure): for reachability, the controls that can transfer
(
t
1
, 0)
to
(
t
2
, x
2
)
are u
(
t
1
), …,
u
(
t
2
−
1)
. The dual concept of observability (see
infra
,
section 2.4.4
) brings into play the outputs y
(
t
1
), …,
y
(
t
2
−
1)
.
In
[BLS 10]
(sections 10.4.2 and 10.4.4), a state-space system
Σ
is said to be
controllable, 0-controllable, observable and 0-observable
, respectively, if there exists t
2
>
t
1
such that
Σ
is
completely reachable, completely controllable, completely observable, completely constructible
on
[
t
1
, t
2
[
. This terminology will be retained in this book, except when it is essential to revert to the Kalmanian terminology, in
section 2.4
.
Definition 11 is valid for a time-varying system. Kalman showed that in the time-invariant case, this definition is equivalent to very simple algebraic conditions:
THEOREM 13.– (Kalman criterion)
The systemΣis controllable (in the specific sense of remark 12(3)) if and only if rk (Γ (A, B)) = n; it is observable if and only if rk (Ω (A, B)) = n, where Γ and Ω are the controllability and observability matrices given by:
DEMONSTRATION.–
The demonstration is given in [KAL 60b], (see also [WOL 74], theorem 3.5.3) for controllability of a continuous-time system. It is also given in
section 2.3.2
in a more general context.
Let us study the controllability
9
of a discrete-time system
[1.12]
, as this case is the simplest, while also being very instructive. First of all, consider the case where
t
0
= 0 through a shift of the origin of time. Let {
x
(
k
)}
k
≥0
and {
u
(
k
)}
k
≥0
denote the sequence of states starting from any initial state and the sequence of control variables. Thus:
and by induction, it is obtained:
According to the Cayley–Hamilton theorem ([BLS 17], section 2.3.11(VII)), An is a linear combination (with coefficients in ) of In, A, …, An−1, therefore rk (Γk) = rk (Γn) for all k ≥ n. Consequently, for all , there is a control sequence {u (k)}k≥0 such that we have x (k) = x1, if and only if rk (Γn) = n and k ≥ n. The Kalman criterion is thus obtained, with Γ(A, B) = Γn.
Let us now study the observability of the discrete-time system
[1.12]
. As the control is zero, we have
x
(
j
+ 1) =
Ax
(
j
), hence:
and consequently:
The initial state x0 is therefore determined by the sequence {y (j)}0≤j≤k−1 if and only if is right invertible, that is, of rank n. Using the same reasoning as in (2) above, this condition is satisfied if and only if rk and k ≥ n.
The result below can be derived from this demonstration (where the terminology given in remark 12(3) is used).
THEOREM 14.–
The discrete-time state system [1.12] is 0-controllable if and only if im (An) ⊂ im (Γ (A, B))10, and it is 0-observable if and only if ker (Ω (A, B)) ⊂ ker (An).
For short, the pair (A, B) is said to be controllable if and only if rk (Γ (A, B)) = n, and the pair (C, A) is said to be observable if and only if rk (Ω (A, B)) = n; hence the Kalman duality theorem:
THEOREM 15.–
(C, A) is observable if and only if (AT ,CT)is controllable.
1.2.4. Poles of a state-space system
Consider again a state-space system Σ given by {C, A, B}.
DEFINITION 16.–
The system poles (s.p.) are the eigenvalues of A (taking into account multiplicities). The sequence of system poles is the sequence of zeros of a = det (sI − A) (definition 8).
Matrix A is similar to a diagonal table diag of Jordan blocks . The πi (i = 1, …, q) are the elementary divisors of A, and by embedding in to simplify the notation, they are of the form:
where is an eigenvalue of A (therefore, a system pole) and μ (πi) is its multiplicity, as a root of πi (see section A.4.2(II)).
(I)
In the continuous-time case, the state equation is written, through a change in basis, in the form of a decoupled system of equations . Thus, up to change of basis, we have:
To simplify the notation, the index i is omitted, and is written Jπ, μ (πi) is written μ and xi0 is written x0. This thus leads to making explicit the solution:
According to [BLS 17], section 3.4.4:
(II)
In the discrete-time case, through a change of basis, the state equation x (k + 1) = Ax (k) takes the form of a decoupled system of equations (i = 1, …, q), hence, up to change of basis:
Using the same conventions as above, the solution:
is made explicit, based on the fact that λIμ commutes with the nilpotent matrix Jπ − λIμ, because of the Newton binomial, and of the equality:
REMARK 17.–
It can thus be seen that the system poles, whether for continuous-time or discrete-time, make it possible to make explicit the “free response” of the system (taking into account the multiplicities μ (πi), i = 1, …, q), namely, the behavior of its state as a function of time, starting from any initial condition x0, when the control is identically zero.
(III)
The set of transmission poles is included in the set of system poles, with equality if and only if the system (or, for brevity {C, A, B}) is controllable and observable.
1.2.5. Stability of linear time-invariant systems
Consider a linear time-invariant state-space system {C, A, B, D}. Let be the value of the state of this system at time t, when its control u is zero and x0 is its state at the initial time. This initial time can be assumed, without loss of generality, to be equal to 0 through a shift of the origin of the time axis.
DEFINITION 18.–
The system,Σ, considered here, is stable in the sense of Lyapunov if, for all δ > 0, there exists ε > 0 such that ∥x (t)∥ ≤ δ for all t ≥ 0 if ∥x0∥ ≤ ε. This system is exponentially stable (or asymptotically stable)11 if for all, limt→+∞ ∥x (t)∥ = 0.
In the continuous-time case, one has x (t) = eAt x0, and in the discrete-time case x (t) = At x0.
In the continuous-time case, we put and . In the discrete-time case, we put and 12.
The expressions for eAt and At, reviewed in section 1.2.4, make it possible to state the following theorem:
THEOREM 19.–
The system
Σ
is stable in the sense of Lyapunov if and only if all its poles belong to
, those belonging to the frontier
(if any) all having multiplicity 1 (as roots of the minimal polynomial of A).
This system is exponentially stable if and only if all its poles belong to
.
A system is said to be stable in the input–output sense if it is proper and all its transmission poles belong to . By abuse of language, it is then said that its transfer matrix is stable.
1.3. “Geometric” approach
1.3.1. Formalism of the geometric approach
In what follows, if , are two real finite-dimensional vector spaces, denotes the space of linear maps from into and .
In [1.10], where D is assumed to belong to , the matrices A, B, C, D represent linear maps , , , in bases of the state-space , of the input-space and the output-space . During the 1970s, under the impetus of Wonham and his collaborators, Kalmanian formalism was reformulated into the more intrinsic language of vector spaces and linear maps [WON 85]. This reformulation would have remained a simple algebraic exercise, but it led to new results, including, notably, the “internal model principle” (see [FRA 76] and [FRA 77]) – though an equivalent for this can also be found in the usual matrix formalism (see [DAV 76]).
The state-space system [1.10] can be written more intrinsically in the geometric approach:
There is a shift to discrete-time, by replacing with x (t + 1), the other terms being unchanged.
The concept of pole can be defined in this formalism. The eigenvalues λi of the endomorphism coincide with those of the matrix A, which represents it in any base of , and the πi (i = 1, …, q), invariant under change of basis, only depend on the endomorphism of which they are therefore the elementary divisors. The λi are the system poles. Let where μi = μ (πi); then ([BLS 10], section 13.3.4, theorem 527):
If denotes the restriction of to , has πi as its only elementary divisor. It is convenient to write the endomorphism in the form of a “diagonal sum” (ibid, p. 480):
Let be a vector such that e1 ≠ 0. Then:
is a basis, said to be -cyclic13, of ; in this basis, the endomorphism is represented by the Jordan block (ibid, p. 487).
1.3.2. Reachable and non-observable subspaces
The reachable subspace of is the set of reachable states (definition 11(i)); it is given by:
The non-observable subspace is the set of non-observable states, and it is given by:
The Kalman criterion can be reformulated as follows: the system is controllable if and only if , and it is observable if and only if .
1.3.3. State-feedback controls, observers
(I) Pole placement by state-feedback
Consider a linear time-invariant state-space system and, to clarify ideas, assume that it is a continuous-time system given by [1.14]. With a state-feedback control:
, where v plays the role of a command signal, the equation for the closed-loop state-space system becomes:
In bases of and , is represented by a “gain matrix” , the “state-feedback gain matrix”.
A subset Λ = {λ1, …, λn} of the complex plane is symmetric with respect to the real axis if it is invariant by conjugation . Given an endomorphism of , its spectrum , that is, the set of its eigenvalues in ([BLS 18], section 3.4.1 (II)) has this property.
The following result was obtained by Wonham [WON 67] in 1967 (the system may be a continuous-time system or discrete-time system, with the conventions given in remark 12(3)).
THEOREM 20.–
The following conditions are equivalent:
for any subset
Λ = {
λ
1
, …
, λ
n
}
of the complex plane, symmetric with respect to the real axis, there exists
such that
(in other words,
“places the set of poles of the closed-loop system at
Λ
”);
is controllable.
If is controllable, then Λ only determines uniquely for m = 1. If m > 1, there exist many resulting in the same pole placement.
However, it must not be believed that these different control laws have the same properties, especially with respect to the robustness they confer upon the closed-loop system (see [BLS 10], section 8.1.4).
(II) Observers
For a state-space system [1.14] of which only the output y is measured (and whose control variable, u, is also known), it is useful to compute an approximation of the state x (t), whether this is to control the system or for dependability, fault-diagnosis, etc. The concept of state observer first emerged, in a stochastic context, in the form of the discrete-time Kalman filter in 1960 [KAL 60a], and then for a continuous-time filter in 1961 [KAL 61] (“Kalman-Bucy filter”). We will study deterministic observers here and their theory is chiefly covered in Luenberger’s work, in his articles between 1964–1966 [LUE 64, LUE 66] (complete state observer, reduced-order observer; see also [LUE 71]). In the following paragraphs, we will restrict ourselves to cases with complete state observers.
Consider again the continuous-time state-space system [1.14], with the discretetime case being similar. A complete state observer provides an estimated state governed by a differential equation of the form:
where . In bases of and , is represented by a “gain matrix” , the “observer gain matrix”. Let be the observation error, defined by . By subtracting [1.16] from the state equation, we obtain:
and the observer poles are thus the eigenvalues of . The eigenvalues of this endomorphism are identical to those of its transpose, consequently the observer poles are the eigenvalues of . The result that follows, valid for a continuous- or discrete-time system (using the conventions in remark 12(3)), is therefore a consequence of the Kalman duality theorem (theorem 15) and theorem 20:
THEOREM 21.–
The following conditions are equivalent:
for any set
Λ = {
λ
1
, …
, λ
n
}
in the complex plane, symmetric with respect to the real axis, there exists
such that
(in other words,
“places the set of poles of the observer at
Λ
”);
is observable.
1.3.4. Canonical Kalman decomposition, stabilizability and detectability
The formalism of the geometric approach is useful in characterizing the canonical Kalman decomposition, initially obtained by Kalman in [KAL 63].
(I) Decomposition along controllability–stabilizability
Let be the state-space, be the reachability subspace and be a complementary subspace of in . Let be a basis of and be a basis of . In the ordered basis of and a basis of , the linear maps and are represented by matrices A and B of the form:
where (Ac, Bc) is controllable. Since , induces an endomorphism of which is a representative matrix ([BLS 17], section 2.2.3(I)).
Consider the state-feedback control [1.15] and let be the matrix that represents in the basis and a basis of . We have:
and . The pair (A, B) is controllable if and only if the matrix is empty. According to theorem 20, it is possible to choose K such that σ (Ac − BcKc) is included in . On the other hand, if (A, B) is not controllable, since the state-feedback control has no action on , it is possible to exponentially stabilize the system with this control variable if and only if is included in . This observation leads to the concept of stabilizability defined in lemma-definition 22, equivalent to that given by Wonham in [WON 67] in 1967 and reiterated by Hautus [HAU 70] in 1970:
Let be the minimal polynomial of . It is possible to factorize α as follows:
where the roots of α− all belong to and where the roots of α+ all belong to the complement of this set in . The endomorphism of obtained by substituting for the indeterminate in polynomial α+ (respectively, α−) is written (respectively, ). Writing and , we have:
The subspaces and are stable through , and the restriction (respectively, ) is an endomorphism of (respectively, ) for which the minimal polynomial is α+ (respectively, α–); and are called the unstable subspace of and the stable subspace of , respectively.
LEMMA-DEFINITION 22.–
The following conditions are equivalent:
;
.
The system (or pair)
) is
stabilizable
if one of the above equivalent conditions is satisfied.
DEMONSTRATION.– 1) Point (i) clearly signifies that the unstable modes are controllable ([WON 85], theorem 2.3), and (ii) the non-controllable modes are stable. These propositions are equivalent, each one being the contraposition of the other.
(II) Decomposition along observability–detectability
Let be the non-observable subspace. The orthogonal of is the subspace:
of the dual of ([BLS 18], section 3.10.2(II))14. Since , the endomorphism of induces an endomorphism .
Let be a complement of in , be a basis of and be a basis of . In the ordered basis of and a basis of , the linear maps and are represented by matrices of the form:
where the pair is controllable and where is a representative matrix of . Thus, we have in the bases and , duals of and :
where the pair (Co, Ao) is observable.
Consider the complete state observer [1.16] and the error equation [1.17]. In the bases , the linear map is represented by a matrix and:
Consequently, there exists an exponentially stable observer if and only if is included in , which leads to the concept of detectability in lemma-definition 23, equivalent to that given by Wonham [WON 67] in 1967 (nonetheless, the term “detectability” was only introduced by Hautus [HAU 70] in 1970); it is also valid for a discrete-time system.
LEMMA-DEFINITION 23.–
The following conditions are equivalent:
;
.
The system (or the pair)
) is
detectable
if one of the above equivalent conditions is satisfied.
DEMONSTRATION.– 1) Point (i) can also be written as ; equivalently, , therefore (i) is equivalent to (ii) according to lemma-definition 22. ▄
By construction, is detectable if and only if is stabilizable.
(III) General decomposition
According to Wonham ([WON 85], Chapter 3, exercise 3.6), let , , , be subspaces of such that:
We have:
and consequently (the signs c, , o, , placed as indices, signify, respectively: controllable, non-controllable, observable, non-observable). The subspaces , , their sum and their intersection are unique, while the other subspaces are only unique up to isomorphism.
By choosing an ordered basis , in , where are bases of , we obtain the canonical Kalman decomposition ([KAL 63], theorem 5): the linear maps , , are represented by matrices (each * denoting any submatrix of the appropriate dimension):
where:
is controllable, where:
is observable and where:
is controllable and observable. We thus have the equalities:
and the transmission poles are the eigenvalues of Aco.
1.4. Polynomial matrix description
1.4.1. PBH test (Hautus criterion)
Necessary and sufficient conditions for controllability and observability were independently obtained by Popov, Belevitch and Hautus in the late 1960s [BEE 68, HAU 69, POP 73] (“PBH test” or “Hautus criterion”):
THEOREM 24.–
The pair
(
A, B
)
is controllable if and only if:
The pair
(
C, A
)
is observable if and only if:
DEMONSTRATION.– It is sufficient to demonstrate (i). It must first be noted that rk (sIn − A) < n if and only if s is an eigenvalue of A.
Assume that there exists a left eigenvector , η ≠ 0, associated with the eigenvalue λ, such that ηT B = 0. We then have ηT AkB = λk AB = 0 (k = 0, 1, …, n − 1), therefore rk (Γ (A, B)) < n.
Conversely, assume rk (Γ (A, B)) < n. There thus exists a non-zero row vector such that:
Let be a non-zero polynomial of minimal degree such that ξTψ (A) = 0. Such a polynomial exists, with 1 ≤ d° (ψ) ≤ n, according to the Cayley–Hamilton theorem ([BLS 17], section 2.3.11(VII)). If λ is a root of ψ, it is possible to write ψ (s) = (s − λ) φ (s) with 0 ≤ d° (φ) ≤ n − 1. By writing ηT = ξT φ (A), we have η ≠ 0 and ηT (A – λIn) = ξT φ (A) (A – λIn) = ξT ψ (A) = 0. According to [1.23], we get ηT B = 0, and, consequently, rk ([λIn − A B]) < n.
If rk ([sIn − A B]) < n or rk ([sIn − AT CT]) < n, s is necessarily a system pole and is called a non-controllable pole in the first case, and a non-observable pole in the second.
These poles, which are never transmission poles (sometimes, by abuse of language of language, it is said that “they are cancelled” when calculating the transfer function or matrix), had already been discovered by Kalman (under a different name) using his canonical decomposition ([KAL 63], theorem 5).
1.4.2. Rosenbrock representation
(I) The Rosenbrock formalism
Conditions [1.21] and [1.22] bring in polynomial matrices. This formalism was generalized in 1970 by Rosenbrock [ROS 70], who envisaged a representation of a system Σ in the form:
where ξ is a column with l entries, called the pseudo-state (or partial state), and P, Q, T, U are matrices with dimensions l × l, l × m, p × l and p × m, respectively, with det (P ) ≠ 0 (the Rosenbrock representation[1.24] will sometimes be referred to as a Rosenbrock system, through an abuse of language).
The entries of these matrices belong to (this is why [1.24] is still called a “polynomial matrix description of system Σ”15); in [1.24], the indeterminate s is substituted by the differentiation operator ∂ for continuous-time systems, and by the shift-forward operator σ for discrete-time systems. We write:
