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Chapter 5 Structure Theorems and Canonical Forms†
CHAPTER
5
Structure Theorems and Canonical Forms
5.1 INTRODUCTION
One of the basic tenets of science, in general, and mathematical physics, in particular, is that fundamental system properties should be independent ofthe coordinate system used to describe them. In other words, the properties of a system or process which we are justified inxalling “basic” should be invariant under application of an appropriate group of transformations. For example,the invariance in form of Maxwell’s equations under the Lorentz group of transformations is a central aspect of relativity theory. Similarly, the invariance of the frequency of an harmonic oscillator when viewed in either a rectangular or polar coordinate system illustrates the fundamental nature of this system variable. In mathematical system theory, one of the primary objectives is to discover properties about systems which are, in some sense, fundamental or t In previous chapters it has been our policy to motivate and illuminate the basic theoretical results with numerous applications from diverse areas of human endeavor. The current chapter, however, is intended primarily for “connoisseurs” of system theory and, as a result, is almost exclusively theoretical in character with the exception of some numerical examples. The reader whose tastes d o not run toward the theoretical can safely proceed to Chapter 6 at this point without loss of continuity; however, for the sake of broadening one’s scientific and mathematical culture, we d o recommend that this chapter at least be skimmed before proceeding to the following material. 88
5.1
INTRODUCTION
89
basic to all systems of a given class. Of course, the decision as to what constitutes a “basic” property is to a certain degree a subjective one determined by the tastes and motivations of the analyst. However, coordinatefree properties certainly have a strong claim to being regarded as basic system properties and to a large extent our discussions in this chapter will be devoted to an examination of such system features. In order to isolate coordinate-free system properties, it is necessary to specify a particular group of transformations Y for study. Our problem then reduces to a study of the structural features of the system C when it is subjected to transformations from Y. For each transformation g E Y, C will assume a different form dependent on the particular g. Hopefully, some of these forms will be particularly simple in the sense that the structural features of X that are invariant under Y will be evident by inspection. Roughly speaking, such a simple form of X will be called canonical. Thus our general plan of action is to: (i) isolate certain system-theoretic features B as candidates for “basic” quantities; (ii) specify a particular transformation group Y that is to act on C; (iii) investigatewhether the features B of (i)are invariant under transformations from 9 and, if so, determine appropriate canonical forms to exhibit the features B explicitly. At this point, it is natural to ask about the utility of canonical forms. Aside from their aesthetic aspects, do such forms have any practical utility? The answer to this question is that the canonical forms have considerable practical value, particularly in so-called system identification problems. The point to be noted is that, in general, the canonical forms represent C in such a way that a minimal or nearly minimal number of parameters are used. Since the canonical form possesses the same “essential” structure as the original problem, we see that for system identification problems it is highly desirable to make use of canonical forms in order to reduce to a minimum the number of unknown parameters characterizing the system. Naturally, there are several different canonical forms and “basic” properties for a given system X,depending on the particular set 9and transformation group Y selected by the system analyst. In this chapter, we shall confine our attention for the most part to cases for which 9 = {controllability/ reachability}or B = {observability/constructibility},Y = {lineartransformations in X} (state variable transformations), Y = {linear coordinate transformations in X and R}, or Y = {linear transformations in X and R plus state variable feedback}.
5
90
STRUCTURE THEOREMS AND CANONICAL FORMS
5.2 STATE VARIABLE TRANSFORMATIONS
We begin our investigation into the invariant properties of a linear system by considering the time-varying system C = (F(t),G(t),H(t))under the action of the general linear group Y = GL(n)(which is the group of real, nonsingular n x n matrices). If we consider a fixed member T E GL(n),its action on C is given by x
+
Tx,
F ( . ) - , TF(.)T-',
G ( . ) - r TG(.),
H ( . ) + H(-)T-'. (5.1)
Our interest is in determining properties of C that remain invariant under GL(n).Our first result in this direction is as follows. Theorem 5.1 Let C = ( F ( . ) , G ( . ) , -) be completely reachable and/or completely controllable. Then the new system obtained from C by the state space change of basis (5.1) is also completely reachableicontrollable. PROOF We prove the result in the case of reachability. By Theorem 3.5,C is completely reachable if and only if the matrix @(7, o)G(o)G'(o)o'(~, o) da
is positive definite for some s < 7. Under the transformation (5.1), it is easily verified that m ( s , 7)+ T@(s, z)f',
i.e., W(s,T) is congruent to the transformed t%? Standard results from matrix theory show, however, that the property of positive-definitenessof a matrix is preserved under congruence. Thus if @(s, z) is positive definite for some s < T, then so is the F? associated with the transformed system. The proof for controllability is completely analogous, utilizing the matrix W of Theorem 3.4. Corollary If C = (F( -, H(.)) is completely constructiblelobservable, then so is any system obtainedfrom Z under the action of the state space transformations (5.1). a),
PROOF
The proof is the same as for Theorem 5.1 using the matrices M,
M of Theorem 4.1.
Theorem 5.1 shows that two representations of the same system are either both completely controllable/reachable or they both fail to possess these
5.3
91
CONTROL CANONICAL FORMS
properties. An interesting question that arises is whether or not the transformation T :X+ X is uniquely determined by the two system representations C = (F, G, H),2 = (E, G,fi). The answer is given by Theorem 5.2.
e,
Theorern5.2 Given two constant systems I: = (F, G, H),2 = (8, @, the linear transformation T:X 4 X is uniquely determined by C, I:if and only ifC, 2 are completely controllable (or completely observable). PROOF Assume Z, 2 are completely observable. Consider first the case m = 1, i.e., G is a single column vector g. By complete controllability, we know that the matrix
is nonsingular. Also, by the properties of transformation (5.1), it is easily verified that
@ = [TgITFg)..-ITF"-'g] = TV, so @ is also nonsingular. Thus form = 1 we find that T is uniquely determined as
T
= @V-'.
In the case m > 1, it makes no sense to speak about the inverse of V. Thus letting r = rank G, we form the matrix
C = [GlFG 1
* * *
1 F"-'G]
and let D = CC'. Then D is n x n and is nonsingular since Z is completely controllable (see Exercise 4 of Chapter 3). Hence, TC = implies TCC' = TD = eC' and we find that
e
T
=
CC'D-'.
5.3 CONTROL CANONICAL FORMS
Since the properties of controllability/reachability-observability/constructibility satisfy our requirements for being "basic" system attributes in that they remain invariant under a change of coordinate system in X,we now seek to simplify the controllability/observability criteria for constant systems with a special choice of basis in X. The system theory literature has seen the introduction of several different "canonical " representations of completely controllable/observable systems,
5
92
STRUCTURE THEOREMS AND CANONICAL FORMS
each representation being championed for a different purpose. Clearly, an uncountable number of such representations are possible. Our treatment will be confined to an account of the two most prevalent (and seemingly useful) such representations. The first is the so-called control canonicalform, which is a generalization of the companion matrix form of classical linear algebra. The second form we study is based on the well-known Jordan canonical form of a single matrix. This is the Lur'e-Lefschetz-Letov canonical form, introduced to extend the Jordan form to the case of a pair of matrices (F, G). Consider first the case of a single-input constant system X = ( F , g , -). Let us write the characteristic polynomial of F as i,hp(z) = zn + ulzn-l
+ ... + a,.
Introduce the set of vectors el = F"-'g = Fg en = g .
+ ulFR-2g+ ... + u , - ~ L J ,
+ alg,
Then the set {e,} forms a basis for X if and only if (F, g ) is completely controllable. The proof of this observation is immediate since { e l ,e 2 ,. . . ,e,} is a triangular linear combination of the vectors { g , Fg, . . ., F " - ' g } . Thus {el,.. .,en}form a basis for X because, by complete controllability, the same is true for { g , F g , . . .,F " - ' g } . Utilizing the vectors { e l , .. . ,en},we have the so-called control canonical form. Theorem 5.3 I n the basis { e l , .. . ,e n } ,the matrices F and g have the representations -
F=
1
0 0 0
0
0
0
.
- -u,
0
-u,-1
0 1 0
0 -u,-2
1
0 0 1 0
-un-3
...
...
... ... ...
0 0 0
- -
-
.
1 -U l -
0
0 ,
0 g = : . 0 - 1-
PROOF Compute Fe,, F e z , . . . ,Fen and note that Fel = - u,e,, by virtue of the Cayley-Hamilton theorem ($F(F) = 0).
5.3
93
CONTROL CANONICAL FORMS
The origin of the control canonical form is closely related to the standard trick of converting the nth order linear differential equation
d"y + a 1 d"-'y + . . . + a n y = u ( t ) dt"
dtn-
to a system of n first-order equations
+
(5.3)
dxn/dt = - a n x l - a n - l x 2 - ... - a ] x n u(t). The passage from (5.2) to (5.3) is accomplished by introducing the state variables
xi = di-'y/dti-',
i = 1, 2,. .., n.
(5.4)
There are three points that Theorem 5.1 improves upon regarding the transformation (5.2) -,(5.3) and the classical discussions surrounding it : (i) It is not immediately apparent that (5.2) represents a completely controllable system. (ii) It is sometimes believed (and even taught) that (5.4) is the only way to convert (5.2) into the state variable form. This is false. The system C,which represents the minimal realization of the input/output function u(t) -,y ( t ) defined by the equation n
rn
i=O
i=O
1ai diy/dti = 1pi d'uldt',
m < n, a0 = 1,
(5.5)
is completely controllable and, as a.result, also admits the control canonical form. The difference between (5.5) and (5.2) is expressed through H . In the first case
y=X], while in the second case
H=[1
O...O],
5
94
STRUCTURE THEOREMS AND CANONICAL FORMS
(iii) Equation (5.4), which expresses the state variables in terms of derivatives, works well for (5.2) but it is not valid, in general. For example, it does not work for (5.5). Thus what is important here is the control canonical form and not the special formula (5.4). Now we turn to the system-theoretic extension of the Jordan canonical form, the Lur'e-Lefschetz-Letov canonical form. Recall from linear algebra that any n x n matrix A may, in an appropriate coordinate system, be expressed as
[,"
A1
A =
..* 0
0
01,
A2
0
0
Ak
where each Jordan block A ihas the form A i l 0 0
..
0
..
..
,
i = l , 2 ,..., k,
O O l i
the libeing the characteristic roots of A. The generalization is stated next. Theorem 5.4 If the single-input system ( F , g, -) is completely controllable, then there exists a unique nonsingular matrix T such that the matrix F = T F T - ' and the vector 3 = Tg have the Lur'e-Lefschetz-Letov form
where the F j are Jordan blocks with nj rows and columns of the form
5.3
95
CONTROL CANONICAL FORMS
and the vectors g j ,j = 1,2,. . . ,k have n j components and are given in theform
The numbers Ljand nj are determinedfrom the characteristic polynomial of F by
n k
xF(z) =
.i= 1
(2
- ij)”’.
PROOF We sketch the necessary steps, leaving the reader to fill in the details as exercises. The general plan is to use the chain of implications:
(i) The controllability matrix V associated with the control canonical form is nonsingular. (ii) V nonsingular implies that the equations T F - FT = 0,
Tg = d ,
admit a unique solution T for every vector d. (iii) For any other completely controllable pair
(F, S) such that
xdz)= x F @ ) , there exists a unique nonsingular matrix T such that
F
=
TFT-’,
8 = Tg.
(iv) Step (ii) implies the Lur’e-Lefschetz-Letov form. Note that Step (iii) is necessary to ensure uniqueness of the transforming matrix T. The multi-input versions of these canonical forms are similar. For example, the control canonical form is given by the next theorem.
Theorem 5 3 Let X = ( F , G, - ) b e a completely controllable constant linear system. Then there exists a nonsingular matrix T such that the transformed system Z = (F, G, -) is given by
F
=
TFT-‘,
G = TG,
5
STRUCTURE THEOREMS AND CANONICAL FORMS
\
k 2 rows
m‘ columns
m - m‘ columns
k l rows
k 2 rows
+ +
+
The numbers {ki} are positive integers such that k l k 2 ... k, = n, while m’ = rank G < m. The elements marked “ x ” in the forms F, represent invariants of the action and are determined by F , G and the transformation T .
We shall not prove this result here since we will discuss it in some detail in a later section. It should be noted, however, that the form for simplifies
5.4
97
OBSERVER CANONICAL FORMS
in the generic situation in which G is of full rank (m = m’). The numbers k i are identical to the well-known Kronecker indices for a pencil of matrices as we will elaborate in Section 5.6. EXERCISES
1. Prove Theorem 5.2 according to the scheme of implications given in its statement. 2. Define a relation W on completely controllable pairs ( F , g ) by the rule
(Fl, S l ) 9 ( F 2 3 g 2 )
if and only if there exists a nonsingular matrix T that F 2 g 2 = T g l . Show that
=
TF,T-’,
(a) W is an equivalence relation, i.e., it is reflexive, symmetric, and transitive. (b) If ( F , g ) is completely controllable, then so is any pair in the same equivalence class as ( F , 9). (c) If ( F , g) and (F, g) are two completely controllable pairs, then there exists a nonsingular matrix T such that ( F , g ) 9 (F, 3). (We shall greatly extend this result in a later section to include a much richer set of transformations than just basis changes in the state space X.) 3. Determine the Lur’e-Lefschetz-Letov form for the multi-input case.
5.4 OBSERVER CANONICAL FORMS
According to the duality principle described in the last chapter, we may associate a “dual” canonical form to each of the controller forms presented in the previous sections. It is clear that the same procedures employed in deriving the control canonical form and the Lur’e-Lefschetz-Letov form may be employed for 0btaining.a corresponding observer form by making the transformations F -P F‘, G + H’.Rather than boring the reader with this repetitious exercise, we content ourselves solely with a presentation of the result. The obseroer canonical form, corresponding to the form of Theorem 5.3, is given by the next theorem. Theorem 5.5 Let the constant system C = ( F , -, H ) be completely observable. Then there exists a coordinate transformation T in X such that the
5
98
system form
= (F,-,
R) = (TFT-', -, H T - ' ) assumes the observer canonical
p, columns
0 0
. * *
... .. .. .. . . . 0 0 ... 1 0
STRUCTURE THEOREMS AND CANONICAL FORMS
0
x
0 x
...
1
.
; ;
.
I I
.
I
x
;
...
p 2 columns
... ... . . .. , 0 0 ... 0 0 0 0 .
.
,
.
x : x i I . .
I I
.
I
...
p1 rows
x :
p 2 rows
I
I
.
..
(Here, p' = rank H _< p, while the positive integers {pi}satisfy the same constraints as the {Ai}, i.e., pi = n.) The Lur'e-Lefschetz-Letov observer form is obtained in a similar manner by an appeal to duality and the controller form. EXERCISES
1. Explicitly write the Lur'e-Lefschetz-Letov observer canonical form. 2. In either the observer canonical form or in the Lur'e-Lefschetz-Letov
observer form, what happens to the input matrix G under the change of coordinates T in X ? Is it possible to simplify the canonical form further by use of additional transformations, e.g., in R?
5.5
99
INVARIANCE OF TRANSFER FUNCTIONS
5.5 JNVARIANCE OF TRANSFER FUNCTIONS
Earlier we saw that the input/output characteristics of a constant linear system having m inputs and p outputs could be described by a p x m matrix W(t),the impulse-response matrix, or its Laplace transform Z(A), the socalled transfer function matrix. A basic structural question is to determine what aspects of the transfer matrix remain unchanged when the system C is transformed in one way or another. In this section we confine our attention to the group of state space coordinate transformations T. Since the transfer function relates the output to the input spaces without any direct connection through X,we would suspect that a change of basis in X would leave Z(A) invariant. In fact, ifthis were not the case, it would imply that the input/output behavior of Z was dependent on the coordinate system chosen in X,implying that the input/output map is not an intrinsic feature of the system. Since in many cases, the mapfis our only experimental information about Z, such a situation would completely destroy any attempt to construct an internal model of the observed process. Thus our objective in this section is to validate the feasibility of model building by proving the invariance of Z(A) under coordinate changes in X. Recall the definitions of W and Z. We have the system Z such that 2 = FX
+ Gu,
y = Hx.
An elementary calculation gives the connection between y and u as y(t) = H feF(r-s)Gu(s) ds, 0
when x(0) = 0. If every component of u equals zero except u,{t), i.e.,
(l/&)ej, = (00,
on an interval 0 s t 5 elsewhere,
E
5
100
STRUCTURE THEOREMS AND CANONICAL FORMS
then y(t) = HeF'(l/E)/ie-F"Gejds,
Passing to the limit as E
.--, 0, we
t2
E
> 0.
obtain the relation t 2 0.
y ( t ) = HeF'Gej,
Generally speaking, the ( i , j)th element of the matrix W(t)= HeF'G
gives the ith component of the output yi(t)corresponding to the unit impulse input u(t) = 6(t)ej,where 6 is the Dirac measure of weight 1 concentrated at t = 0. The matrix W ( t )completely defines all connections between the inputs and outputs of a controllable and observable system. Thus, for arbitrary input u(t),we have y(t) = /:W(t - S ) U ( S ) ds,
t 2 0,
when x(0) = 0.
Since the correspondence between u(t) and y(t) has the form of a convolution, it is more convenient to apply the Laplace transform 9to the functions u and y. Denoting these transforms by V(A)and Y(A),we obtain the matrix transfer function Z(A):
Z(A) = 9 ( W ( t ) )=
/om
W(t)e-"' dt.
The connection between the input and output then assumes the form Y(A)= Z(A)V(A). More explicitly, the transfer matrix is given by
Z(A) = H(AZ - F)-'G. By Cramer's rule, we see that the elements of Z are rational functions of A, with the degree of the denominator equal to the dimension of C (before any possible cancellation of terms in the numerators and denominators takes place). In fact, the denominator of each component in Z(A)equals xF(A), the characteristic polynomial of F . Our basic invariance result is stated next.
Theorem 5.6 The matrix impulse response function W ( t )(and consequently, its Laplace transform Z(A)) is invariant with respect to linear changes of coordinates in the state space X .
5.6
CANONICAL FORMS AND THE BEZOUTIANT MATRIX
101
PROOF Let T E GL(n)represent a change of coordinates in X.Under such a transformation, the system matrices F, G, H transform as
F + TFT-',
G + TG,
H + HT-'.
Substituting the transformed elements into the definition of W(t),we have W ( t )= (HT-')eTFT-"(TG).
The theorem follows upon application of the well-known identity eA' = I
t2 + At + A* 2! - + .-.,
convergent for all t . The result for Z(A)follows from the 1-1 correspondence between W(s)and Z(4. EXERCISES
1. Compute the transfer matrix when the system C = (F, G, H) is:
(a) in control canonical form, (b) in the Lur'e-Lefschetz-Letov canonical form.
Do the same for the observer canonical forms. What is the significance of a component in the transfer function having a numerator with a factor in common with xF(l)?(Hint: Consider the single input/single output case first.) 3. Is there any system-theoretic interpretation attached to the vanishing of one or more components in Z(A)? 4. Construct a completely controllable and completely observable system X whose transfer function is 2.
Z(4 =.
5.6 CANONICAL FORMS AND THE BEZOUTIANT MATRIX
Both the observer and controller canonical form studied thus far came about as the result of a change of coordinate systems in the state space X. A natural question to pose is whether there is some interesting and useful connection between the two forms. More precisely, if T,and T, represent
5
102
STRUCTURE THEOREMS AND CANONICAL FORMS
the basis changes for the controller and observer canonical forms, respectively, we have F, = T,FT,', Fc = T,FT,', and our question becomes: What are the properties of a matrix B such that
BFcB-' = F,? It is clear that such a matrix B exists (it equals 7',,T ; '); however, it is not immediately apparent that B possesses any interesting properties. Our purpose in this section is to show that not only does B have interesting featuresjustifyingour attention, but that it is intimately related to the classical Bezoutiant matrix of polynomial algebras. This fact enables us to obtain some useful results concerning transfer functions. Consider two polynomials a(s) and p(s) such that
a(s) = a. P(s) = bo
+ a,s + u2s2 + . - .+ a,sn, + b,s + b2s2 + + b,Sm * - .
with real coefficients. Without loss of generality, assume that Q, = 1 ( a is monic) and m < n. Then the Bezoutiant matrix of a and B is then x n matrix
BCa
PI
=I
(a0b,)*
where
(a0b2)*
(a0b2)*
(aOb3)*
bOb3)*
bOb4)*
+ (alb2)*
+ (alb3)*
(a0b3)*
+ (alb3)*
bob,),
: I-
+ (alb4), + b4b3)* ...
i
(aibj)* = aibj - ajbi.
More explicitly, if k = min(i,j) - 1, then the element [BIij is given by (aobi+j-l)*
+ ( a l b i + j - z ) * + ... + ( a k b i + j - k - 1 ) * ,
where we take aj = 0, b, = 0 f o r j > n'and p > m. It is well known that the polynomials a(s) and p(s) are relatively prime if and only if Bra, P] is of full rank and that the rank deficiency of Bra, fl] equals the degree of the greatest common factor between a and p. To relate the Bezoutiant to linear systems, we consider a single-input/ single-output system Z = (F, g , h) having transfer function p(s)/a(s). The controller canonical form of this system is given by = Fcxc + gcu, y = hC&>
i c
5.6
' 0 0 F,
103
CANONICAL FORMS AND THE BEZOUTIANT MATRIX
i
=
L
0 -ao
1
... ...
0 0
0
0
...
1
-a,
-az
1
0
0
-4-
(Note: here the coefficients ai are labeled to agree with the labeling in the polynomial m
)
7
1-
The dual of the controller form, the observer canonical form is obtained from C, = ( F , , g , , hc) by means of the relations
F,
=
F,',
go = h,',
h,
= 9,'.
It is a simple task to verify that the transfer function for both systems equals Z with transfer function b(s)/a(s).Any one of these systems is referred to as a realization of the transfer function p(s)/a(s). Let us introduce the controllability and observability matrices % and 0 as
P(s)/a(s). In fact, there will be an uncountable number of systems
W(F,9 ) = [ g I F g I
-
*
. IF"- IS],
8 ( h , F ) = W(F', h').
A basic role in relating the Bezoutiant to the observer and controller canonical forms is played by the following result.
Theorem 5.7
The Bezoutiant matrix B satisfies the identity
- B(a, P) = TP(F,), where
P(Fc) = b o l r
T =
-a1
+ b l F c + ... + bmFc"' -a2
...
- a,-l
-1
-
-a2 1
-a,,-,
--1
-1
0
-
5
104
STRUCTURE THEOREMS AND CANONICAL FORMS
The basic result relating V, (= V ( F , , gJ), 0,(=V'(F,', h,')), go(= V(Fo, go)) and 8,( =W(F,', h,')) follows. Theorem 5.8 The Bezoutiant matrix B(a, P) satisjes the following identities (a) (b) (c) (d)
B(a, B) = -%:
'O,, B(a,P) = --%,02,
B(a, P) = -0,'o,, B(a,p) = -VOW;'. PROOF Part (a) follows immediately from the relations W;' P(FJ = 0,.
=
T,
Part (b) follows from (a) by using the symmetry of B(a, P) since B(a, /I) = B'(a, p) = -0,'(V,')-'. This fact, together with the duality relations 0,' = Vo,V,' = 0, establishes (b). The duality relations plus the invertibility of T = Vc- imply (c), which in turn implies (d).
'
Conclusion (c) of the theorem leads to a very simple proof of the fact that any observable realization of X = (F, g, h) can be transformed to the observer canonical form by a change of state variables as x&) = Mx(t), where M is the constant nonsingular matrix
M = O,'O(F, h). Thus, if the controller canonical form is observable, from conclusion (c) we see that B(a, P), the Bezoutiant, is precisely'the transformation matrix needed to form the observer canonical form. EXERCISE
1. Establish the corresponding results for multi-input/multi-output systems. 5.7 THE FEEDBACK GROUP .AND INVARIANT THEORY
We have seen that the group of linear coordinate transformations in X, the state space, enables us to reduce the apparent complexity of a given system C significantlyby reducing it to a canonical form in which the inherent structure of X is more apparent. The obvious question at this stage is whether a further simplification is possible if we augment our transformation group by allowing not only basis changes in X, but also other coordinate and structural changes in C. For a variety of reasons, some of which we shall see below, the most interesting new transformations are changes of basis in the input space R,
5.7
105
THE FEEDBACK GROUP AND INVARIANT THEORY
and application of special inputs of the form u(t) = -Lx(t), where L is an arbitrary n x m constant matrix. Such inputs are termed "feedback" since their effect is to generate the input not as an explicit function of time, but as a function of the state x. Thus our new group of transformations consists of: (I) coordinate changes in X ;(11) coordinate changes in R; (111) arbitrary constant feedback laws. This set of transformations forms what is generally called the feedback group 8.Given a system C = (F, G, H), the action of 8on C is
(I) (coordinate
F
change in X) (11) (coordinate F change in R) (111) (feedback F law)
-+
TFT-', G
-+
F,
G
-+
F
GL,
-
-+
-+
TG, H H
GV-',
G
-+
G,
HT-',
H,
-+
H
-+
-+
H,
I VI
IT1 # 0, # 0,
L arbitrary.
A particular element of 8 is determined by specifying the matrices T, V, and L. Clearly, the choice V = I, L = 0, T nonsingular reduces to the case of state coordinate changes considered above. Thus the state space basis changes form a subgroup of LF. We have seen that under coordinate changes in X,the only invariants are the coefficients {ai}of the characteristic polynomial of F (or equivalently, the characteristic values of F) and the elements of H T - ' (assuming rank G = m).Since the feedback group allows more flexibility in modifying the structure of C,the number of invariants is certainly no greater than under the subgroup of state space basis changes. Our purpose is to determine the precise invariants. We begin by considering the ordered set of vectors
91,...,gm;Fgl,Fgz,...,Fgm,...; F"-lgl,.-.,F"-lgm, (5.6)
where g i = ith column of G. Under the assumption that the system C
=
(F,G, -) is completely reachable, the list (5.6)contains precisely n linearly independent vectors. Consider the '' Young's diagram ''
106
5
STRUCTURE THEOREMS AND CANONICAL FORMS
+
where li is the number of crosses in column ( i l), and k j is the number of crosses in row j. Here the rule for placing crosses in the diagram is as follows: begin with row 1 and place a cross if element ( i , j ) is linearly independent of all vectors previously considered. Then go to row 2 and repeat the process, etc. By the complete reachability of C, this procedure will result in exactly n crosses being placed in the diagram. The integers { k j } , {li} must satisfy s- 1
S
The vectors picked by the above procedure, namely, { g j , . .., F k j - ’ g j : j
EM},
where M is a subset of { 1,2, . . .,m} containing exactly s elements, constitute a basis for X . Let
M(i)={tEM:k,>i},
,..., n - 1 .
i=O,1,2
Then M(i) has li elements and M ( 0 ) = M . By linear dependence, we have ki-1
gi
=
1ajilgj,
i = 1, 2, ..., m,
i$M.
(ii)
j s M
We now make the transformation
di = g i - 1a j i ( k , + l ) g j ,
E i
M*
(iii)
jsM(ki)
It then follows from (iii) and the definition of M(i) that gi
= &i j
2
iE
aji(kj+l)&j,
M.
EM(kd
Upon substituting (iv) into (i) and (ii), we obtain
c
ki-1 Fk‘$i
=
1
aji(r+l)Frdj, r=O jsM(r)
ajildj,
gi = jEM
i = 1,2,
EM,
..., m,
i$ M.
(iv)
5.7 THE FEEDBACK
107
GROUP AND INVARIANT THEORY
A new basis in X is then defined by ki-1
eit = Fkt-'bi -
1 1 aji(r+l)Fj-c~j,
r = t j , M(r)
i~ M,
With respect to this basis, (F, G) take the form where
i = l , 2 ,..., rn.
iEM,
j # i,
i e M,
j # i,
t
=
1,2, ..., k i .
5
108
STRUCTURE THEOREMS AND CANONICAL FORMS
If we renumber the columns of G so that ki 2 k i + then F , and G, take the forms 0 1 0 ... 0 0 0 1 ... 0 ... k , rows 0 x
F,
x
...
x
0 0
=
0 x
x
.
x
o
.
0
0
0
X
G,
=
X
-
-
0
0
0
0
0
1
X
- -
0 0 ,l
X
X
.
x
0
... 0
.
...
0
... 0
x
...
X
... 0 ... 0 ...
1
... 0 .... .. ... ... 0 ... X
0
... ...
0 X
...
x ( x x
a
.
X
k, rows
-
k , rows
k , rows
X
m - 1, columns
Theorem 5.9 Relative to the feedback group 8, the invariants of the system Z = ( F , G, H)consist of the set of integers k , , k,, . . . ,k , and the elements hij of I?. Conversely, given any set of m nonnegative integers kisuch that ki = n, and a set of np real numbers I$,, a system C is determined by transformations
cf=,
from 9, i.e., {ki}and for Z.
{h,} constitute a complete, independent set of invariants
5.7 THE FEEDBACK GROUP AND INVARIANT THEORY EXAMPLE
.=[a 4 g]
109
We apply Theorem 5.9 to the system
G=[A
4).
H=(O
1 0).
First we verify that C is completely controllable. Computing the controllability matrix %?, we have $$=
1 0 1 2 1 4 0 0 2 0 8 4 0 1 0 1 0 1
I
which has rank 3. Following the prescriptions of the theorem, we write the sequence of vectors
Clearly, the first three form a linearly independent set in R 3 . The Young’s diagram is
k,
=
2, bz = 1.
Thus we see that the Kronecker invariants for this case are k l = 2, k z The set M = {1,2). We next compute the basis (5.7). We obtain ell
= Fg,
+ alllgl.
el2 =
91,
e21
=92.
=
1.
110
5 STRUCTURE THEOREMS AND CANONICAL FORMS
The only missing quantity is the constant a1 dependency relations
c 2
F2g, = -39,
+ 4Fgl
This implies that a1
1.
This is obtained from the
1
aIjkFkgj,
j = l k=O
= -{al,ogl
+ aIl1Fg1+ a120g21.
= -4. Thus the new basis is
e l l = Fgl - 49, =
ii)
e 1 2= g 1
=(), =(8). e21 = g2
If we assume that the matrices F, G, H are originally given in terms of the standard basis in R 3 , i.e., the basis el
=k),
e2
=(:), =[). e3
and if we express the standard basis in terms of the new basis ell,e12,ezl, it is easily verified that the matrix of this transformation is given by T=
[I d :i 1
t o .
Using the basis change T, we finally obtain
A = H T - ' = ( 2 0 0). The feedback law 3 L=[o
-2 0 -11
-4
eliminates the second row of E which enables us to see that a complete set of invariants for the system are the integers k , = 2, k 2 = 1, together with the elements of the matrix fi = (2 0 0). Note that in this example it was not necessary to use any Type (11) transformation because G was of full rank.
111
MISCELLANEOUS EXERCISES
REMARKS (1) The importance of the control invariants k,, . . . ,k, is that they give the sizes of the smallest cyclic blocks into which F may be decomposed by linear “feedback ” transformations. This is the only structural obstruction to arbitrarily altering F by utilization of feedback. (2) The invariants k l , . . . ,k, are identical to the classical Kronecker indices associated with the matrix “pencil” [ z l - F I GI. In fact, Kronecker’s definition for the equivalence of two pencils is
-
Pb)
Q(z)* P(z) = AQ(z)C,
for nonsingular constant A and C. It is then easy to see that [zZ - FIG] = A[zZ - E l Q C
if and only if C has the form C
=
[
LAA- -1
i],
L arbitrary, det B # 0.
Thus Kronecker’s equivalence relation applied to the pencil [zZ - F I G ] is identical to the equivalence relation induced by the feedback group 9 on pairs (F, G). EXERCISES
1. Assume that m divides n and that each ki = n/m, i = i, . . . ,s. Determine the subgroup of 9which leaves the Kronecker canonical forms for F and G invariant. What happens if all ki are not equal? 2. Complete the proof of Theorem 5.9 by showing that the parameter set S = {ki, hij> constitutes an independent and complete set of invariants for the system C = (F, G, H), i.e., for any given set S, there is some system C whose parameter set equals S (independence) and if two systems Cl and C 2 have the same parameter set S, there exists a transformation Y E9 such that C , = F E Z(completeness). MISCELLANEOUS EXERCISES
1. Suppose that the pair ( F , G) is not completely controllable and that G # 0. Show that there exists a basis change in X such that F and G are transformed into
with the pair ( F l l ,G , ) being completely controllable. Show that the rank of F 1 equals the rank of W(F, G).
5
112
STRUCTURE THEOREMS AND CANONICAL FORMS
2. Define the single-input/single-output system C = ( F , g , h ) to be nondegenerate if it is both completely controllable and completely observable. Show that the following statements are equivalent: (a) C is nondegenerate. (b) The transfer function of C(Z(A)= h(Al - F ) - ' g ) is irreducible and the degree of the denominator equals n, the dimension of C. (c) If any other system 2: of the same dimension has the same transfer function, then there exists a nonsingular T such that the two systems are related as
3. Let the system C be in Lur'e-Lefschetz-Letov canonical form, ie., Fl
F2
G=
F2 2
G.=
Gij=
Prove that C is controllable if and only if the set of s(i) rn-dimensional row vectors gfil' g/i2
9
* * * 9
gJiscij
form a linearly independent set, i = 1,2, . . . ,q. (Note that this set is composed of all the first rows of the matrices G i jassociated with a given submatrix Fi.) 4. Show that a single-input system C = (F, g , -) is completely controllable if and only if the transfer matrix Z(A) = (AZ - F ) - ' g is irreducible, i.e., the numerator and denominator have no common factors.
113
MISCELLANEOUS EXERCISES
Using this result, check the system
for complete controllability. 5. (a) Let F , G be fixed and denote the range of G by '9, i.e., Y = {x E R" : x = Gy for some y E R"}. Further, let (FJ'3) = Y + FY ... F"-'Ydenote thecontrollablesubspaceofXfor thepair(F, G). Ifdim '3 = m, show that there exist subspaces { c X such that
+
+
v]
(i) dim('3 n q) = 1, i = 1,. . . ,m ; (ii) (FIY) = V, 0 V, 0 . - .0 V , . (b) If (F, G) is a controllable pair, what is the relation between the list of integers {dim V,, dim V,, . . , ,dim V,) and the Kronecker indices k , , k2, ..., k,? 6. In the theory of representations of the general linear group GL(R"),it can be shown that each set of m nonnegative integers k i 2 0 satisfying zy= k i= n corresponds to a certain representation of GL(Rm).What is the connection between this fact and the control canonical form for a multi-input system? 7. Prove the following structure theorem. Let ( F , G) be completely reachable, m = rank G, and k , 2 k , 2 . . . 2 k , the ordered control invariants of ( F , G). Let $,, . . . , +q be arbitrary monic polynomials such that
,
(a) + i l + i - l , i = 1 ,..., 4 - 1 , q s m ; (b) deg (1/, 2 k , , deg deg$, 2 k ,
+,+
+ k,, etc.
Prove that there exists a feedback control law L such that F - G L has the invariant factors . . . ,)(lq. Conversely, show that the invariant factors of F - G L always satisfy (a) and (b). 8. Let k be an algebraically closed field and assume that the system C = ( F , G, - ) and an arbitrary control law L are defined over k. Define a control law J to be purelyfeedforward if and only if xF = x F - G J , while a control law K is purelyfeedback if and only if deg +,(F) = deg @i(F - G K ) , i = 1,2,. . . ,I'. A control law M is neutral if and only if it is purely feedforward and purely feedback (i.e., a purely feedforward law leaves the characteristic values of F unchanged, while a purely feedback law leaves the degree of the invariant factors of F unchanged).
+,,
(a) Prove that L may be written as L=J+K,
where J and K are unique modulo a neutral law M.
114
5
STRUCTURE THEOREMS AND CANONICAL FORMS
(b) Show, by a counterexample, that this result fails if k is not algebraically closed. (Recall: A field k is algebraically closed if any polynomial with coefficientsin k also has all of its roots in k. Thus the field of complex numbers is algebraically closed, but not the field of reals.) NOTES AND REFERENCES
Section 5.1 A systematic exposition of the algebraic point of view in modern physics is given by Zaitsev, G.. “Algebraic Problems of Mathematical and Theoretical Physics.” Nauka, Moscow, 1974.
A similar point of view (in English) is presented by Hermann, R., “Interdisciplinary Mathematics.” Vols. I-IX. Math. Sci. Press, Brookline. Massachusetts. 1972-1977.
The interested reader is especially urged to consult Volumes 111, VIII, and IX for material specifically devoted to algebraic and geometric aspects of system theory. For a more classical point of view on system problems, but with deep algebraic overtones, see Popov, V.. “Hyperstability of Control Systems,” Springer-Verlag, Berlin and New York, 1973.
The material of this chapter employs only the elementary tools of linear algebra and matrix theory to develop the theory of canonical structures for linear systems. For a much deeper and more thorough treatment utilizing the full machinery of abstract algebra, the reader should consult Chapter Nine and the references contained therein. The Lur’e-Lefschetz-Letov canonical form is extensively used in the book by Popov cited above. For the single-input case, see also
Section 5.3
Lefschetz. S.,“Stability of Nonlinear Control Systems.” Academic Press. New York. 1965.
The control canonical form for single-input systems is treated in detail by Kalman, R.,Falb, P., and Arbib, M., “Topics in Mathematical System Theory.” McGraw-Hill, New York. 1969.
The multi-input case may be found in Kalman, R.,Kronecker invariants and feedback, in “Ordinary Differential Equations” (L. Weiss, ed.), pp. 459-471. Academic Press, New York, 1972.
NOTES AND REFERENCES
115
The explicit use of invariant-theoretic arguments to calculate the control invariants under state variable coordinate changes is treated by Popov, V., Invariant description of linear, time-invariant controllable systems, SIAM J. Control 10, 252-264 (1972).
See also the work by Weinert, H., Complete sets of invariants for multivariable linear systems, 7th Princeton Conf. Informat. Theory and Systems. Princeton. New Jersey. 1973.
A good treatment of the relationship between canonical forms, invariants and input/output models is given in Guidorzi, R.,Complete sets of independent invariants and canonical forms for linear systems identification. Faculty of Engineering Report, U. of Bologna, January 1978.
Section 5.6
The results of this section follow
Sidhu, G. S., “A Note on the Bezoutiant Matrix and the Controllability and Observability of Linear Systems” (unpublished manuscript, 1973).
A classical treatment of the Bezoutiant matrix in the context of polynomial algebras is given by Bijcher, M., “Introduction to Higher Algebra.” Dover, New York, 1964.
An efficient constructive algorithm for transforming one observer canonical form to another is detailed in Caroli, M. and Guidorzi, R., Algebraical links between block-companion structures: A new efficient algorithm, Ricerche di Auto, 4, 1-17 (1973).
A fascinating sociological account of the rise and fall of invariant theory as a fashionable mathematical activity is found in
Section 5.7
Fisher, C. S.,The death of a mathematical theory, Arch. History Exact Sci. 3, 137-159 (1966). Fisher, C. S., The last invariant theorists, Arch. European Sociology 8, 216-244 (1967).
The feedback group and its invariant-theoretic implications for system problems is extensively pursued by Wonham, W., and Morse, A., Feedback invariants of linear multivariable systems, Proc. IFAC Symp. Multivariable Systems, Dusseldorf, October 1971. Wang, S . , and Davison, E., Canonical forms of linear multivariable systems. SIAM J . Control 14,236250 (1976).
Hazewinkel, M., and Kalman, R., “Moduli and Canonical Forms for Linear Dynamical Systems,” Rep. 7504/M, Erasmus Univ., Rotterdam, April 1974. Hazewinkel M., O n the (internal) symmetry groups of linear dynamical systems, Vieweg Tracts in Pure and Applied Physics, 4, 362-404 (1980).
116
5
STRUCTURE THEOREMS AND CANONICAL FORMS
For a general treatment of the overall concept of genericity and its use for linear system theory, see Tchon, K., On generic properties of linear systems: An overview, Kybernetikn, 19, 467-474 (1983).
For results concerning the determination of the Kronecker invariantsdirectly from input/output data, rather than from a system given, as in the text, in internal form, the paper by Guidorzi, R., Canonical structure in the identification of multivariable systems, AuromuticuJ . IFAC 11, 361-374 (1975).
is recommended.
5
Structure Theorems and Canonical Forms
5.1 INTRODUCTION
One of the basic tenets of science, in general, and mathematical physics, in particular, is that fundamental system properties should be independent ofthe coordinate system used to describe them. In other words, the properties of a system or process which we are justified inxalling “basic” should be invariant under application of an appropriate group of transformations. For example,the invariance in form of Maxwell’s equations under the Lorentz group of transformations is a central aspect of relativity theory. Similarly, the invariance of the frequency of an harmonic oscillator when viewed in either a rectangular or polar coordinate system illustrates the fundamental nature of this system variable. In mathematical system theory, one of the primary objectives is to discover properties about systems which are, in some sense, fundamental or t In previous chapters it has been our policy to motivate and illuminate the basic theoretical results with numerous applications from diverse areas of human endeavor. The current chapter, however, is intended primarily for “connoisseurs” of system theory and, as a result, is almost exclusively theoretical in character with the exception of some numerical examples. The reader whose tastes d o not run toward the theoretical can safely proceed to Chapter 6 at this point without loss of continuity; however, for the sake of broadening one’s scientific and mathematical culture, we d o recommend that this chapter at least be skimmed before proceeding to the following material. 88
5.1
INTRODUCTION
89
basic to all systems of a given class. Of course, the decision as to what constitutes a “basic” property is to a certain degree a subjective one determined by the tastes and motivations of the analyst. However, coordinatefree properties certainly have a strong claim to being regarded as basic system properties and to a large extent our discussions in this chapter will be devoted to an examination of such system features. In order to isolate coordinate-free system properties, it is necessary to specify a particular group of transformations Y for study. Our problem then reduces to a study of the structural features of the system C when it is subjected to transformations from Y. For each transformation g E Y, C will assume a different form dependent on the particular g. Hopefully, some of these forms will be particularly simple in the sense that the structural features of X that are invariant under Y will be evident by inspection. Roughly speaking, such a simple form of X will be called canonical. Thus our general plan of action is to: (i) isolate certain system-theoretic features B as candidates for “basic” quantities; (ii) specify a particular transformation group Y that is to act on C; (iii) investigatewhether the features B of (i)are invariant under transformations from 9 and, if so, determine appropriate canonical forms to exhibit the features B explicitly. At this point, it is natural to ask about the utility of canonical forms. Aside from their aesthetic aspects, do such forms have any practical utility? The answer to this question is that the canonical forms have considerable practical value, particularly in so-called system identification problems. The point to be noted is that, in general, the canonical forms represent C in such a way that a minimal or nearly minimal number of parameters are used. Since the canonical form possesses the same “essential” structure as the original problem, we see that for system identification problems it is highly desirable to make use of canonical forms in order to reduce to a minimum the number of unknown parameters characterizing the system. Naturally, there are several different canonical forms and “basic” properties for a given system X,depending on the particular set 9and transformation group Y selected by the system analyst. In this chapter, we shall confine our attention for the most part to cases for which 9 = {controllability/ reachability}or B = {observability/constructibility},Y = {lineartransformations in X} (state variable transformations), Y = {linear coordinate transformations in X and R}, or Y = {linear transformations in X and R plus state variable feedback}.
5
90
STRUCTURE THEOREMS AND CANONICAL FORMS
5.2 STATE VARIABLE TRANSFORMATIONS
We begin our investigation into the invariant properties of a linear system by considering the time-varying system C = (F(t),G(t),H(t))under the action of the general linear group Y = GL(n)(which is the group of real, nonsingular n x n matrices). If we consider a fixed member T E GL(n),its action on C is given by x
+
Tx,
F ( . ) - , TF(.)T-',
G ( . ) - r TG(.),
H ( . ) + H(-)T-'. (5.1)
Our interest is in determining properties of C that remain invariant under GL(n).Our first result in this direction is as follows. Theorem 5.1 Let C = ( F ( . ) , G ( . ) , -) be completely reachable and/or completely controllable. Then the new system obtained from C by the state space change of basis (5.1) is also completely reachableicontrollable. PROOF We prove the result in the case of reachability. By Theorem 3.5,C is completely reachable if and only if the matrix @(7, o)G(o)G'(o)o'(~, o) da
is positive definite for some s < 7. Under the transformation (5.1), it is easily verified that m ( s , 7)+ T@(s, z)f',
i.e., W(s,T) is congruent to the transformed t%? Standard results from matrix theory show, however, that the property of positive-definitenessof a matrix is preserved under congruence. Thus if @(s, z) is positive definite for some s < T, then so is the F? associated with the transformed system. The proof for controllability is completely analogous, utilizing the matrix W of Theorem 3.4. Corollary If C = (F( -, H(.)) is completely constructiblelobservable, then so is any system obtainedfrom Z under the action of the state space transformations (5.1). a),
PROOF
The proof is the same as for Theorem 5.1 using the matrices M,
M of Theorem 4.1.
Theorem 5.1 shows that two representations of the same system are either both completely controllable/reachable or they both fail to possess these
5.3
91
CONTROL CANONICAL FORMS
properties. An interesting question that arises is whether or not the transformation T :X+ X is uniquely determined by the two system representations C = (F, G, H),2 = (E, G,fi). The answer is given by Theorem 5.2.
e,
Theorern5.2 Given two constant systems I: = (F, G, H),2 = (8, @, the linear transformation T:X 4 X is uniquely determined by C, I:if and only ifC, 2 are completely controllable (or completely observable). PROOF Assume Z, 2 are completely observable. Consider first the case m = 1, i.e., G is a single column vector g. By complete controllability, we know that the matrix
is nonsingular. Also, by the properties of transformation (5.1), it is easily verified that
@ = [TgITFg)..-ITF"-'g] = TV, so @ is also nonsingular. Thus form = 1 we find that T is uniquely determined as
T
= @V-'.
In the case m > 1, it makes no sense to speak about the inverse of V. Thus letting r = rank G, we form the matrix
C = [GlFG 1
* * *
1 F"-'G]
and let D = CC'. Then D is n x n and is nonsingular since Z is completely controllable (see Exercise 4 of Chapter 3). Hence, TC = implies TCC' = TD = eC' and we find that
e
T
=
CC'D-'.
5.3 CONTROL CANONICAL FORMS
Since the properties of controllability/reachability-observability/constructibility satisfy our requirements for being "basic" system attributes in that they remain invariant under a change of coordinate system in X,we now seek to simplify the controllability/observability criteria for constant systems with a special choice of basis in X. The system theory literature has seen the introduction of several different "canonical " representations of completely controllable/observable systems,
5
92
STRUCTURE THEOREMS AND CANONICAL FORMS
each representation being championed for a different purpose. Clearly, an uncountable number of such representations are possible. Our treatment will be confined to an account of the two most prevalent (and seemingly useful) such representations. The first is the so-called control canonicalform, which is a generalization of the companion matrix form of classical linear algebra. The second form we study is based on the well-known Jordan canonical form of a single matrix. This is the Lur'e-Lefschetz-Letov canonical form, introduced to extend the Jordan form to the case of a pair of matrices (F, G). Consider first the case of a single-input constant system X = ( F , g , -). Let us write the characteristic polynomial of F as i,hp(z) = zn + ulzn-l
+ ... + a,.
Introduce the set of vectors el = F"-'g = Fg en = g .
+ ulFR-2g+ ... + u , - ~ L J ,
+ alg,
Then the set {e,} forms a basis for X if and only if (F, g ) is completely controllable. The proof of this observation is immediate since { e l ,e 2 ,. . . ,e,} is a triangular linear combination of the vectors { g , Fg, . . ., F " - ' g } . Thus {el,.. .,en}form a basis for X because, by complete controllability, the same is true for { g , F g , . . .,F " - ' g } . Utilizing the vectors { e l , .. . ,en},we have the so-called control canonical form. Theorem 5.3 I n the basis { e l , .. . ,e n } ,the matrices F and g have the representations -
F=
1
0 0 0
0
0
0
.
- -u,
0
-u,-1
0 1 0
0 -u,-2
1
0 0 1 0
-un-3
...
...
... ... ...
0 0 0
- -
-
.
1 -U l -
0
0 ,
0 g = : . 0 - 1-
PROOF Compute Fe,, F e z , . . . ,Fen and note that Fel = - u,e,, by virtue of the Cayley-Hamilton theorem ($F(F) = 0).
5.3
93
CONTROL CANONICAL FORMS
The origin of the control canonical form is closely related to the standard trick of converting the nth order linear differential equation
d"y + a 1 d"-'y + . . . + a n y = u ( t ) dt"
dtn-
to a system of n first-order equations
+
(5.3)
dxn/dt = - a n x l - a n - l x 2 - ... - a ] x n u(t). The passage from (5.2) to (5.3) is accomplished by introducing the state variables
xi = di-'y/dti-',
i = 1, 2,. .., n.
(5.4)
There are three points that Theorem 5.1 improves upon regarding the transformation (5.2) -,(5.3) and the classical discussions surrounding it : (i) It is not immediately apparent that (5.2) represents a completely controllable system. (ii) It is sometimes believed (and even taught) that (5.4) is the only way to convert (5.2) into the state variable form. This is false. The system C,which represents the minimal realization of the input/output function u(t) -,y ( t ) defined by the equation n
rn
i=O
i=O
1ai diy/dti = 1pi d'uldt',
m < n, a0 = 1,
(5.5)
is completely controllable and, as a.result, also admits the control canonical form. The difference between (5.5) and (5.2) is expressed through H . In the first case
y=X], while in the second case
H=[1
O...O],
5
94
STRUCTURE THEOREMS AND CANONICAL FORMS
(iii) Equation (5.4), which expresses the state variables in terms of derivatives, works well for (5.2) but it is not valid, in general. For example, it does not work for (5.5). Thus what is important here is the control canonical form and not the special formula (5.4). Now we turn to the system-theoretic extension of the Jordan canonical form, the Lur'e-Lefschetz-Letov canonical form. Recall from linear algebra that any n x n matrix A may, in an appropriate coordinate system, be expressed as
[,"
A1
A =
..* 0
0
01,
A2
0
0
Ak
where each Jordan block A ihas the form A i l 0 0
..
0
..
..
,
i = l , 2 ,..., k,
O O l i
the libeing the characteristic roots of A. The generalization is stated next. Theorem 5.4 If the single-input system ( F , g, -) is completely controllable, then there exists a unique nonsingular matrix T such that the matrix F = T F T - ' and the vector 3 = Tg have the Lur'e-Lefschetz-Letov form
where the F j are Jordan blocks with nj rows and columns of the form
5.3
95
CONTROL CANONICAL FORMS
and the vectors g j ,j = 1,2,. . . ,k have n j components and are given in theform
The numbers Ljand nj are determinedfrom the characteristic polynomial of F by
n k
xF(z) =
.i= 1
(2
- ij)”’.
PROOF We sketch the necessary steps, leaving the reader to fill in the details as exercises. The general plan is to use the chain of implications:
(i) The controllability matrix V associated with the control canonical form is nonsingular. (ii) V nonsingular implies that the equations T F - FT = 0,
Tg = d ,
admit a unique solution T for every vector d. (iii) For any other completely controllable pair
(F, S) such that
xdz)= x F @ ) , there exists a unique nonsingular matrix T such that
F
=
TFT-’,
8 = Tg.
(iv) Step (ii) implies the Lur’e-Lefschetz-Letov form. Note that Step (iii) is necessary to ensure uniqueness of the transforming matrix T. The multi-input versions of these canonical forms are similar. For example, the control canonical form is given by the next theorem.
Theorem 5 3 Let X = ( F , G, - ) b e a completely controllable constant linear system. Then there exists a nonsingular matrix T such that the transformed system Z = (F, G, -) is given by
F
=
TFT-‘,
G = TG,
5
STRUCTURE THEOREMS AND CANONICAL FORMS
\
k 2 rows
m‘ columns
m - m‘ columns
k l rows
k 2 rows
+ +
+
The numbers {ki} are positive integers such that k l k 2 ... k, = n, while m’ = rank G < m. The elements marked “ x ” in the forms F, represent invariants of the action and are determined by F , G and the transformation T .
We shall not prove this result here since we will discuss it in some detail in a later section. It should be noted, however, that the form for simplifies
5.4
97
OBSERVER CANONICAL FORMS
in the generic situation in which G is of full rank (m = m’). The numbers k i are identical to the well-known Kronecker indices for a pencil of matrices as we will elaborate in Section 5.6. EXERCISES
1. Prove Theorem 5.2 according to the scheme of implications given in its statement. 2. Define a relation W on completely controllable pairs ( F , g ) by the rule
(Fl, S l ) 9 ( F 2 3 g 2 )
if and only if there exists a nonsingular matrix T that F 2 g 2 = T g l . Show that
=
TF,T-’,
(a) W is an equivalence relation, i.e., it is reflexive, symmetric, and transitive. (b) If ( F , g ) is completely controllable, then so is any pair in the same equivalence class as ( F , 9). (c) If ( F , g) and (F, g) are two completely controllable pairs, then there exists a nonsingular matrix T such that ( F , g ) 9 (F, 3). (We shall greatly extend this result in a later section to include a much richer set of transformations than just basis changes in the state space X.) 3. Determine the Lur’e-Lefschetz-Letov form for the multi-input case.
5.4 OBSERVER CANONICAL FORMS
According to the duality principle described in the last chapter, we may associate a “dual” canonical form to each of the controller forms presented in the previous sections. It is clear that the same procedures employed in deriving the control canonical form and the Lur’e-Lefschetz-Letov form may be employed for 0btaining.a corresponding observer form by making the transformations F -P F‘, G + H’.Rather than boring the reader with this repetitious exercise, we content ourselves solely with a presentation of the result. The obseroer canonical form, corresponding to the form of Theorem 5.3, is given by the next theorem. Theorem 5.5 Let the constant system C = ( F , -, H ) be completely observable. Then there exists a coordinate transformation T in X such that the
5
98
system form
= (F,-,
R) = (TFT-', -, H T - ' ) assumes the observer canonical
p, columns
0 0
. * *
... .. .. .. . . . 0 0 ... 1 0
STRUCTURE THEOREMS AND CANONICAL FORMS
0
x
0 x
...
1
.
; ;
.
I I
.
I
x
;
...
p 2 columns
... ... . . .. , 0 0 ... 0 0 0 0 .
.
,
.
x : x i I . .
I I
.
I
...
p1 rows
x :
p 2 rows
I
I
.
..
(Here, p' = rank H _< p, while the positive integers {pi}satisfy the same constraints as the {Ai}, i.e., pi = n.) The Lur'e-Lefschetz-Letov observer form is obtained in a similar manner by an appeal to duality and the controller form. EXERCISES
1. Explicitly write the Lur'e-Lefschetz-Letov observer canonical form. 2. In either the observer canonical form or in the Lur'e-Lefschetz-Letov
observer form, what happens to the input matrix G under the change of coordinates T in X ? Is it possible to simplify the canonical form further by use of additional transformations, e.g., in R?
5.5
99
INVARIANCE OF TRANSFER FUNCTIONS
5.5 JNVARIANCE OF TRANSFER FUNCTIONS
Earlier we saw that the input/output characteristics of a constant linear system having m inputs and p outputs could be described by a p x m matrix W(t),the impulse-response matrix, or its Laplace transform Z(A), the socalled transfer function matrix. A basic structural question is to determine what aspects of the transfer matrix remain unchanged when the system C is transformed in one way or another. In this section we confine our attention to the group of state space coordinate transformations T. Since the transfer function relates the output to the input spaces without any direct connection through X,we would suspect that a change of basis in X would leave Z(A) invariant. In fact, ifthis were not the case, it would imply that the input/output behavior of Z was dependent on the coordinate system chosen in X,implying that the input/output map is not an intrinsic feature of the system. Since in many cases, the mapfis our only experimental information about Z, such a situation would completely destroy any attempt to construct an internal model of the observed process. Thus our objective in this section is to validate the feasibility of model building by proving the invariance of Z(A) under coordinate changes in X. Recall the definitions of W and Z. We have the system Z such that 2 = FX
+ Gu,
y = Hx.
An elementary calculation gives the connection between y and u as y(t) = H feF(r-s)Gu(s) ds, 0
when x(0) = 0. If every component of u equals zero except u,{t), i.e.,
(l/&)ej, = (00,
on an interval 0 s t 5 elsewhere,
E
5
100
STRUCTURE THEOREMS AND CANONICAL FORMS
then y(t) = HeF'(l/E)/ie-F"Gejds,
Passing to the limit as E
.--, 0, we
t2
E
> 0.
obtain the relation t 2 0.
y ( t ) = HeF'Gej,
Generally speaking, the ( i , j)th element of the matrix W(t)= HeF'G
gives the ith component of the output yi(t)corresponding to the unit impulse input u(t) = 6(t)ej,where 6 is the Dirac measure of weight 1 concentrated at t = 0. The matrix W ( t )completely defines all connections between the inputs and outputs of a controllable and observable system. Thus, for arbitrary input u(t),we have y(t) = /:W(t - S ) U ( S ) ds,
t 2 0,
when x(0) = 0.
Since the correspondence between u(t) and y(t) has the form of a convolution, it is more convenient to apply the Laplace transform 9to the functions u and y. Denoting these transforms by V(A)and Y(A),we obtain the matrix transfer function Z(A):
Z(A) = 9 ( W ( t ) )=
/om
W(t)e-"' dt.
The connection between the input and output then assumes the form Y(A)= Z(A)V(A). More explicitly, the transfer matrix is given by
Z(A) = H(AZ - F)-'G. By Cramer's rule, we see that the elements of Z are rational functions of A, with the degree of the denominator equal to the dimension of C (before any possible cancellation of terms in the numerators and denominators takes place). In fact, the denominator of each component in Z(A)equals xF(A), the characteristic polynomial of F . Our basic invariance result is stated next.
Theorem 5.6 The matrix impulse response function W ( t )(and consequently, its Laplace transform Z(A)) is invariant with respect to linear changes of coordinates in the state space X .
5.6
CANONICAL FORMS AND THE BEZOUTIANT MATRIX
101
PROOF Let T E GL(n)represent a change of coordinates in X.Under such a transformation, the system matrices F, G, H transform as
F + TFT-',
G + TG,
H + HT-'.
Substituting the transformed elements into the definition of W(t),we have W ( t )= (HT-')eTFT-"(TG).
The theorem follows upon application of the well-known identity eA' = I
t2 + At + A* 2! - + .-.,
convergent for all t . The result for Z(A)follows from the 1-1 correspondence between W(s)and Z(4. EXERCISES
1. Compute the transfer matrix when the system C = (F, G, H) is:
(a) in control canonical form, (b) in the Lur'e-Lefschetz-Letov canonical form.
Do the same for the observer canonical forms. What is the significance of a component in the transfer function having a numerator with a factor in common with xF(l)?(Hint: Consider the single input/single output case first.) 3. Is there any system-theoretic interpretation attached to the vanishing of one or more components in Z(A)? 4. Construct a completely controllable and completely observable system X whose transfer function is 2.
Z(4 =.
5.6 CANONICAL FORMS AND THE BEZOUTIANT MATRIX
Both the observer and controller canonical form studied thus far came about as the result of a change of coordinate systems in the state space X. A natural question to pose is whether there is some interesting and useful connection between the two forms. More precisely, if T,and T, represent
5
102
STRUCTURE THEOREMS AND CANONICAL FORMS
the basis changes for the controller and observer canonical forms, respectively, we have F, = T,FT,', Fc = T,FT,', and our question becomes: What are the properties of a matrix B such that
BFcB-' = F,? It is clear that such a matrix B exists (it equals 7',,T ; '); however, it is not immediately apparent that B possesses any interesting properties. Our purpose in this section is to show that not only does B have interesting featuresjustifyingour attention, but that it is intimately related to the classical Bezoutiant matrix of polynomial algebras. This fact enables us to obtain some useful results concerning transfer functions. Consider two polynomials a(s) and p(s) such that
a(s) = a. P(s) = bo
+ a,s + u2s2 + . - .+ a,sn, + b,s + b2s2 + + b,Sm * - .
with real coefficients. Without loss of generality, assume that Q, = 1 ( a is monic) and m < n. Then the Bezoutiant matrix of a and B is then x n matrix
BCa
PI
=I
(a0b,)*
where
(a0b2)*
(a0b2)*
(aOb3)*
bOb3)*
bOb4)*
+ (alb2)*
+ (alb3)*
(a0b3)*
+ (alb3)*
bob,),
: I-
+ (alb4), + b4b3)* ...
i
(aibj)* = aibj - ajbi.
More explicitly, if k = min(i,j) - 1, then the element [BIij is given by (aobi+j-l)*
+ ( a l b i + j - z ) * + ... + ( a k b i + j - k - 1 ) * ,
where we take aj = 0, b, = 0 f o r j > n'and p > m. It is well known that the polynomials a(s) and p(s) are relatively prime if and only if Bra, P] is of full rank and that the rank deficiency of Bra, fl] equals the degree of the greatest common factor between a and p. To relate the Bezoutiant to linear systems, we consider a single-input/ single-output system Z = (F, g , h) having transfer function p(s)/a(s). The controller canonical form of this system is given by = Fcxc + gcu, y = hC&>
i c
5.6
' 0 0 F,
103
CANONICAL FORMS AND THE BEZOUTIANT MATRIX
i
=
L
0 -ao
1
... ...
0 0
0
0
...
1
-a,
-az
1
0
0
-4-
(Note: here the coefficients ai are labeled to agree with the labeling in the polynomial m
)
7
1-
The dual of the controller form, the observer canonical form is obtained from C, = ( F , , g , , hc) by means of the relations
F,
=
F,',
go = h,',
h,
= 9,'.
It is a simple task to verify that the transfer function for both systems equals Z with transfer function b(s)/a(s).Any one of these systems is referred to as a realization of the transfer function p(s)/a(s). Let us introduce the controllability and observability matrices % and 0 as
P(s)/a(s). In fact, there will be an uncountable number of systems
W(F,9 ) = [ g I F g I
-
*
. IF"- IS],
8 ( h , F ) = W(F', h').
A basic role in relating the Bezoutiant to the observer and controller canonical forms is played by the following result.
Theorem 5.7
The Bezoutiant matrix B satisfies the identity
- B(a, P) = TP(F,), where
P(Fc) = b o l r
T =
-a1
+ b l F c + ... + bmFc"' -a2
...
- a,-l
-1
-
-a2 1
-a,,-,
--1
-1
0
-
5
104
STRUCTURE THEOREMS AND CANONICAL FORMS
The basic result relating V, (= V ( F , , gJ), 0,(=V'(F,', h,')), go(= V(Fo, go)) and 8,( =W(F,', h,')) follows. Theorem 5.8 The Bezoutiant matrix B(a, P) satisjes the following identities (a) (b) (c) (d)
B(a, B) = -%:
'O,, B(a,P) = --%,02,
B(a, P) = -0,'o,, B(a,p) = -VOW;'. PROOF Part (a) follows immediately from the relations W;' P(FJ = 0,.
=
T,
Part (b) follows from (a) by using the symmetry of B(a, P) since B(a, /I) = B'(a, p) = -0,'(V,')-'. This fact, together with the duality relations 0,' = Vo,V,' = 0, establishes (b). The duality relations plus the invertibility of T = Vc- imply (c), which in turn implies (d).
'
Conclusion (c) of the theorem leads to a very simple proof of the fact that any observable realization of X = (F, g, h) can be transformed to the observer canonical form by a change of state variables as x&) = Mx(t), where M is the constant nonsingular matrix
M = O,'O(F, h). Thus, if the controller canonical form is observable, from conclusion (c) we see that B(a, P), the Bezoutiant, is precisely'the transformation matrix needed to form the observer canonical form. EXERCISE
1. Establish the corresponding results for multi-input/multi-output systems. 5.7 THE FEEDBACK GROUP .AND INVARIANT THEORY
We have seen that the group of linear coordinate transformations in X, the state space, enables us to reduce the apparent complexity of a given system C significantlyby reducing it to a canonical form in which the inherent structure of X is more apparent. The obvious question at this stage is whether a further simplification is possible if we augment our transformation group by allowing not only basis changes in X, but also other coordinate and structural changes in C. For a variety of reasons, some of which we shall see below, the most interesting new transformations are changes of basis in the input space R,
5.7
105
THE FEEDBACK GROUP AND INVARIANT THEORY
and application of special inputs of the form u(t) = -Lx(t), where L is an arbitrary n x m constant matrix. Such inputs are termed "feedback" since their effect is to generate the input not as an explicit function of time, but as a function of the state x. Thus our new group of transformations consists of: (I) coordinate changes in X ;(11) coordinate changes in R; (111) arbitrary constant feedback laws. This set of transformations forms what is generally called the feedback group 8.Given a system C = (F, G, H), the action of 8on C is
(I) (coordinate
F
change in X) (11) (coordinate F change in R) (111) (feedback F law)
-+
TFT-', G
-+
F,
G
-+
F
GL,
-
-+
-+
TG, H H
GV-',
G
-+
G,
HT-',
H,
-+
H
-+
-+
H,
I VI
IT1 # 0, # 0,
L arbitrary.
A particular element of 8 is determined by specifying the matrices T, V, and L. Clearly, the choice V = I, L = 0, T nonsingular reduces to the case of state coordinate changes considered above. Thus the state space basis changes form a subgroup of LF. We have seen that under coordinate changes in X,the only invariants are the coefficients {ai}of the characteristic polynomial of F (or equivalently, the characteristic values of F) and the elements of H T - ' (assuming rank G = m).Since the feedback group allows more flexibility in modifying the structure of C,the number of invariants is certainly no greater than under the subgroup of state space basis changes. Our purpose is to determine the precise invariants. We begin by considering the ordered set of vectors
91,...,gm;Fgl,Fgz,...,Fgm,...; F"-lgl,.-.,F"-lgm, (5.6)
where g i = ith column of G. Under the assumption that the system C
=
(F,G, -) is completely reachable, the list (5.6)contains precisely n linearly independent vectors. Consider the '' Young's diagram ''
106
5
STRUCTURE THEOREMS AND CANONICAL FORMS
+
where li is the number of crosses in column ( i l), and k j is the number of crosses in row j. Here the rule for placing crosses in the diagram is as follows: begin with row 1 and place a cross if element ( i , j ) is linearly independent of all vectors previously considered. Then go to row 2 and repeat the process, etc. By the complete reachability of C, this procedure will result in exactly n crosses being placed in the diagram. The integers { k j } , {li} must satisfy s- 1
S
The vectors picked by the above procedure, namely, { g j , . .., F k j - ’ g j : j
EM},
where M is a subset of { 1,2, . . .,m} containing exactly s elements, constitute a basis for X . Let
M(i)={tEM:k,>i},
,..., n - 1 .
i=O,1,2
Then M(i) has li elements and M ( 0 ) = M . By linear dependence, we have ki-1
gi
=
1ajilgj,
i = 1, 2, ..., m,
i$M.
(ii)
j s M
We now make the transformation
di = g i - 1a j i ( k , + l ) g j ,
E i
M*
(iii)
jsM(ki)
It then follows from (iii) and the definition of M(i) that gi
= &i j
2
iE
aji(kj+l)&j,
M.
EM(kd
Upon substituting (iv) into (i) and (ii), we obtain
c
ki-1 Fk‘$i
=
1
aji(r+l)Frdj, r=O jsM(r)
ajildj,
gi = jEM
i = 1,2,
EM,
..., m,
i$ M.
(iv)
5.7 THE FEEDBACK
107
GROUP AND INVARIANT THEORY
A new basis in X is then defined by ki-1
eit = Fkt-'bi -
1 1 aji(r+l)Fj-c~j,
r = t j , M(r)
i~ M,
With respect to this basis, (F, G) take the form where
i = l , 2 ,..., rn.
iEM,
j # i,
i e M,
j # i,
t
=
1,2, ..., k i .
5
108
STRUCTURE THEOREMS AND CANONICAL FORMS
If we renumber the columns of G so that ki 2 k i + then F , and G, take the forms 0 1 0 ... 0 0 0 1 ... 0 ... k , rows 0 x
F,
x
...
x
0 0
=
0 x
x
.
x
o
.
0
0
0
X
G,
=
X
-
-
0
0
0
0
0
1
X
- -
0 0 ,l
X
X
.
x
0
... 0
.
...
0
... 0
x
...
X
... 0 ... 0 ...
1
... 0 .... .. ... ... 0 ... X
0
... ...
0 X
...
x ( x x
a
.
X
k, rows
-
k , rows
k , rows
X
m - 1, columns
Theorem 5.9 Relative to the feedback group 8, the invariants of the system Z = ( F , G, H)consist of the set of integers k , , k,, . . . ,k , and the elements hij of I?. Conversely, given any set of m nonnegative integers kisuch that ki = n, and a set of np real numbers I$,, a system C is determined by transformations
cf=,
from 9, i.e., {ki}and for Z.
{h,} constitute a complete, independent set of invariants
5.7 THE FEEDBACK GROUP AND INVARIANT THEORY EXAMPLE
.=[a 4 g]
109
We apply Theorem 5.9 to the system
G=[A
4).
H=(O
1 0).
First we verify that C is completely controllable. Computing the controllability matrix %?, we have $$=
1 0 1 2 1 4 0 0 2 0 8 4 0 1 0 1 0 1
I
which has rank 3. Following the prescriptions of the theorem, we write the sequence of vectors
Clearly, the first three form a linearly independent set in R 3 . The Young’s diagram is
k,
=
2, bz = 1.
Thus we see that the Kronecker invariants for this case are k l = 2, k z The set M = {1,2). We next compute the basis (5.7). We obtain ell
= Fg,
+ alllgl.
el2 =
91,
e21
=92.
=
1.
110
5 STRUCTURE THEOREMS AND CANONICAL FORMS
The only missing quantity is the constant a1 dependency relations
c 2
F2g, = -39,
+ 4Fgl
This implies that a1
1.
This is obtained from the
1
aIjkFkgj,
j = l k=O
= -{al,ogl
+ aIl1Fg1+ a120g21.
= -4. Thus the new basis is
e l l = Fgl - 49, =
ii)
e 1 2= g 1
=(), =(8). e21 = g2
If we assume that the matrices F, G, H are originally given in terms of the standard basis in R 3 , i.e., the basis el
=k),
e2
=(:), =[). e3
and if we express the standard basis in terms of the new basis ell,e12,ezl, it is easily verified that the matrix of this transformation is given by T=
[I d :i 1
t o .
Using the basis change T, we finally obtain
A = H T - ' = ( 2 0 0). The feedback law 3 L=[o
-2 0 -11
-4
eliminates the second row of E which enables us to see that a complete set of invariants for the system are the integers k , = 2, k 2 = 1, together with the elements of the matrix fi = (2 0 0). Note that in this example it was not necessary to use any Type (11) transformation because G was of full rank.
111
MISCELLANEOUS EXERCISES
REMARKS (1) The importance of the control invariants k,, . . . ,k, is that they give the sizes of the smallest cyclic blocks into which F may be decomposed by linear “feedback ” transformations. This is the only structural obstruction to arbitrarily altering F by utilization of feedback. (2) The invariants k l , . . . ,k, are identical to the classical Kronecker indices associated with the matrix “pencil” [ z l - F I GI. In fact, Kronecker’s definition for the equivalence of two pencils is
-
Pb)
Q(z)* P(z) = AQ(z)C,
for nonsingular constant A and C. It is then easy to see that [zZ - FIG] = A[zZ - E l Q C
if and only if C has the form C
=
[
LAA- -1
i],
L arbitrary, det B # 0.
Thus Kronecker’s equivalence relation applied to the pencil [zZ - F I G ] is identical to the equivalence relation induced by the feedback group 9 on pairs (F, G). EXERCISES
1. Assume that m divides n and that each ki = n/m, i = i, . . . ,s. Determine the subgroup of 9which leaves the Kronecker canonical forms for F and G invariant. What happens if all ki are not equal? 2. Complete the proof of Theorem 5.9 by showing that the parameter set S = {ki, hij> constitutes an independent and complete set of invariants for the system C = (F, G, H), i.e., for any given set S, there is some system C whose parameter set equals S (independence) and if two systems Cl and C 2 have the same parameter set S, there exists a transformation Y E9 such that C , = F E Z(completeness). MISCELLANEOUS EXERCISES
1. Suppose that the pair ( F , G) is not completely controllable and that G # 0. Show that there exists a basis change in X such that F and G are transformed into
with the pair ( F l l ,G , ) being completely controllable. Show that the rank of F 1 equals the rank of W(F, G).
5
112
STRUCTURE THEOREMS AND CANONICAL FORMS
2. Define the single-input/single-output system C = ( F , g , h ) to be nondegenerate if it is both completely controllable and completely observable. Show that the following statements are equivalent: (a) C is nondegenerate. (b) The transfer function of C(Z(A)= h(Al - F ) - ' g ) is irreducible and the degree of the denominator equals n, the dimension of C. (c) If any other system 2: of the same dimension has the same transfer function, then there exists a nonsingular T such that the two systems are related as
3. Let the system C be in Lur'e-Lefschetz-Letov canonical form, ie., Fl
F2
G=
F2 2
G.=
Gij=
Prove that C is controllable if and only if the set of s(i) rn-dimensional row vectors gfil' g/i2
9
* * * 9
gJiscij
form a linearly independent set, i = 1,2, . . . ,q. (Note that this set is composed of all the first rows of the matrices G i jassociated with a given submatrix Fi.) 4. Show that a single-input system C = (F, g , -) is completely controllable if and only if the transfer matrix Z(A) = (AZ - F ) - ' g is irreducible, i.e., the numerator and denominator have no common factors.
113
MISCELLANEOUS EXERCISES
Using this result, check the system
for complete controllability. 5. (a) Let F , G be fixed and denote the range of G by '9, i.e., Y = {x E R" : x = Gy for some y E R"}. Further, let (FJ'3) = Y + FY ... F"-'Ydenote thecontrollablesubspaceofXfor thepair(F, G). Ifdim '3 = m, show that there exist subspaces { c X such that
+
+
v]
(i) dim('3 n q) = 1, i = 1,. . . ,m ; (ii) (FIY) = V, 0 V, 0 . - .0 V , . (b) If (F, G) is a controllable pair, what is the relation between the list of integers {dim V,, dim V,, . . , ,dim V,) and the Kronecker indices k , , k2, ..., k,? 6. In the theory of representations of the general linear group GL(R"),it can be shown that each set of m nonnegative integers k i 2 0 satisfying zy= k i= n corresponds to a certain representation of GL(Rm).What is the connection between this fact and the control canonical form for a multi-input system? 7. Prove the following structure theorem. Let ( F , G) be completely reachable, m = rank G, and k , 2 k , 2 . . . 2 k , the ordered control invariants of ( F , G). Let $,, . . . , +q be arbitrary monic polynomials such that
,
(a) + i l + i - l , i = 1 ,..., 4 - 1 , q s m ; (b) deg (1/, 2 k , , deg deg$, 2 k ,
+,+
+ k,, etc.
Prove that there exists a feedback control law L such that F - G L has the invariant factors . . . ,)(lq. Conversely, show that the invariant factors of F - G L always satisfy (a) and (b). 8. Let k be an algebraically closed field and assume that the system C = ( F , G, - ) and an arbitrary control law L are defined over k. Define a control law J to be purelyfeedforward if and only if xF = x F - G J , while a control law K is purelyfeedback if and only if deg +,(F) = deg @i(F - G K ) , i = 1,2,. . . ,I'. A control law M is neutral if and only if it is purely feedforward and purely feedback (i.e., a purely feedforward law leaves the characteristic values of F unchanged, while a purely feedback law leaves the degree of the invariant factors of F unchanged).
+,,
(a) Prove that L may be written as L=J+K,
where J and K are unique modulo a neutral law M.
114
5
STRUCTURE THEOREMS AND CANONICAL FORMS
(b) Show, by a counterexample, that this result fails if k is not algebraically closed. (Recall: A field k is algebraically closed if any polynomial with coefficientsin k also has all of its roots in k. Thus the field of complex numbers is algebraically closed, but not the field of reals.) NOTES AND REFERENCES
Section 5.1 A systematic exposition of the algebraic point of view in modern physics is given by Zaitsev, G.. “Algebraic Problems of Mathematical and Theoretical Physics.” Nauka, Moscow, 1974.
A similar point of view (in English) is presented by Hermann, R., “Interdisciplinary Mathematics.” Vols. I-IX. Math. Sci. Press, Brookline. Massachusetts. 1972-1977.
The interested reader is especially urged to consult Volumes 111, VIII, and IX for material specifically devoted to algebraic and geometric aspects of system theory. For a more classical point of view on system problems, but with deep algebraic overtones, see Popov, V.. “Hyperstability of Control Systems,” Springer-Verlag, Berlin and New York, 1973.
The material of this chapter employs only the elementary tools of linear algebra and matrix theory to develop the theory of canonical structures for linear systems. For a much deeper and more thorough treatment utilizing the full machinery of abstract algebra, the reader should consult Chapter Nine and the references contained therein. The Lur’e-Lefschetz-Letov canonical form is extensively used in the book by Popov cited above. For the single-input case, see also
Section 5.3
Lefschetz. S.,“Stability of Nonlinear Control Systems.” Academic Press. New York. 1965.
The control canonical form for single-input systems is treated in detail by Kalman, R.,Falb, P., and Arbib, M., “Topics in Mathematical System Theory.” McGraw-Hill, New York. 1969.
The multi-input case may be found in Kalman, R.,Kronecker invariants and feedback, in “Ordinary Differential Equations” (L. Weiss, ed.), pp. 459-471. Academic Press, New York, 1972.
NOTES AND REFERENCES
115
The explicit use of invariant-theoretic arguments to calculate the control invariants under state variable coordinate changes is treated by Popov, V., Invariant description of linear, time-invariant controllable systems, SIAM J. Control 10, 252-264 (1972).
See also the work by Weinert, H., Complete sets of invariants for multivariable linear systems, 7th Princeton Conf. Informat. Theory and Systems. Princeton. New Jersey. 1973.
A good treatment of the relationship between canonical forms, invariants and input/output models is given in Guidorzi, R.,Complete sets of independent invariants and canonical forms for linear systems identification. Faculty of Engineering Report, U. of Bologna, January 1978.
Section 5.6
The results of this section follow
Sidhu, G. S., “A Note on the Bezoutiant Matrix and the Controllability and Observability of Linear Systems” (unpublished manuscript, 1973).
A classical treatment of the Bezoutiant matrix in the context of polynomial algebras is given by Bijcher, M., “Introduction to Higher Algebra.” Dover, New York, 1964.
An efficient constructive algorithm for transforming one observer canonical form to another is detailed in Caroli, M. and Guidorzi, R., Algebraical links between block-companion structures: A new efficient algorithm, Ricerche di Auto, 4, 1-17 (1973).
A fascinating sociological account of the rise and fall of invariant theory as a fashionable mathematical activity is found in
Section 5.7
Fisher, C. S.,The death of a mathematical theory, Arch. History Exact Sci. 3, 137-159 (1966). Fisher, C. S., The last invariant theorists, Arch. European Sociology 8, 216-244 (1967).
The feedback group and its invariant-theoretic implications for system problems is extensively pursued by Wonham, W., and Morse, A., Feedback invariants of linear multivariable systems, Proc. IFAC Symp. Multivariable Systems, Dusseldorf, October 1971. Wang, S . , and Davison, E., Canonical forms of linear multivariable systems. SIAM J . Control 14,236250 (1976).
Hazewinkel, M., and Kalman, R., “Moduli and Canonical Forms for Linear Dynamical Systems,” Rep. 7504/M, Erasmus Univ., Rotterdam, April 1974. Hazewinkel M., O n the (internal) symmetry groups of linear dynamical systems, Vieweg Tracts in Pure and Applied Physics, 4, 362-404 (1980).
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5
STRUCTURE THEOREMS AND CANONICAL FORMS
For a general treatment of the overall concept of genericity and its use for linear system theory, see Tchon, K., On generic properties of linear systems: An overview, Kybernetikn, 19, 467-474 (1983).
For results concerning the determination of the Kronecker invariantsdirectly from input/output data, rather than from a system given, as in the text, in internal form, the paper by Guidorzi, R., Canonical structure in the identification of multivariable systems, AuromuticuJ . IFAC 11, 361-374 (1975).
is recommended.