6 Differential input-output relations. Generators

Differential input-out put relations. Generators In this chapter we shall study some fundamental properties of differential input-output relations ( c f . section 4.2). We shall, in particular, be interested in regular input-output relations and regular generators for such relations. Now, given a regular generator, this generator determines a unique corresponding regular differential input-output relation and also a unique corresponding transfer matrix. On the other hand, given a regular differential input-output relation, it is found that there is a whole class of regular generators generating this relation, but only one corresponding transfer matrix. Finally, given a transfer matrix, there is a whole class of regular generators, as well as a whole class of regular differential input-output relations corresponding to this transfer matrix. These relationships then lead us in a natural way to certain equivalence relations-to input-output equivalence on the set of all regular generators, and to transfer equivalence on the set of all regular generators as well as on the set of all regular differential input-output relations. Canonical forms for these equivalences are also considered. It turns out that the canonical forms devised for the transfer equivalence on the set of all regular differential input-output relations possess an interesting minimality property. The material presented in this chapter is to a large extent based on results presented by Hirvonen, Blomberg, and Ylinen, 1975, and Ylinen, 1975. The transfer equivalence on the set of all regular generators and the corresponding canonical forms are also discussed in a paper by Aracil and Montes, 1976, in connection with the problem of factoring a transfer matrix in a unique way. Various forms of system equivalence introduced and discussed by Rosenbrock, 1970, Morf, 1975, and others are in a certain sense related to our input-output equivalence. The relationship between 88

Differential input-output relations. Generators 6.I

these different equivalences will be discussed in a subsequent chapter. Choice of signal space 2t

The signal space 2t is, throughout this chapter, assumed to be a regular signal space according to definition (5.1.3) possessing the richness property (5.1.5). % may, in particular, be equal to C" or 53.

6.1 Introduction. Regular differential input-output relations and regular generators In section 4.2 we introduced the concept of a differential input-output relation S C 2tr x 2 t q generated by a pair (A(p), B ( p ) ) of polynomial matrix operators with A ( p ) square and detA(p) # 0. In the sequel we shall also encounter input-output relations generated by more general pairs of polynomial matrix operators. So, consider the matrix differential equation 1

A(P)Y = B ( p ) u ,

where u E 8', y E 2 t q , and where A ( p ) and B ( p ) are polynomial matrix operators (cf. (4.1.3)) of sizes s by q and s by r respectively. Interpreting the signal space 2t as a C[p]-module (cf. section 5 . 2 ) , ( 1 ) can be regarded as representing a set of module equations. The differential input-output relation generated by ( l ) , or alternatively by the pair ( A ( p ) ,B ( p ) ) , is now defined as the set S of all input-output pairs ( u ,y ) E Z r x 2 t q satisfying equation ( l ) , i.e. as the relation S C X r X Xq given by (cf. (4.2.2)) 2

S 4 { ( u , y)l(u, y) E %r x S is of course a subspace of %?' X of a partitioned matrix equality

3

and A(p)y

B(p)u}. 2'7. Writing A(p)y = B ( p ) u in the form %q

=

I:[

[A@) i - B ( p ) J

.--..= O ,

and interpreting [A@) i - B ( p ) ] as a partitioned polynomial matrix operator V + ' -F ,i.e. as a linear mapping Xq X %?+ %', it is seen that (2) can be written simply as 4

S-'= ker[A(p) i - B ( p ) ] , where S-' denotes the converse of S, i.e. S-'4 { ( y , u ) l ( u , y)

E S}. We

89

Algebraic Theory for Multivariable Linear Systems

shall also say that S is generated by [A(p) i - B ( p ) ] , and that [ A ( p ) i - B ( p ) ] is a generator for S . S is clearly uniquely determined by its generator, but-as was already pointed out in note (4.2.6), (i)-the same S can generally be generated by infinitely many generators. 5

Remark. The concept of a generator for a differential input-output relation S as introduced above is closely related to the concept of the “system matrix” as used by Rosenbrock, 1970. We shall return to this question in chapter 7. 0 Regular generators for regular differential input-output relations. Order

A differential input-output relation S is said to be regular, if there is a generator [A(p) i - B ( p ) ] for S with A ( p ) square and d e t A ( p ) # 0. Such a generator is likewise called regular, and d (det A ( p )) is called the order of the generator and of the corresponding regular differential input-output relation S (cf. section 6.2 below). It has previously been pointed out (see note (4.2.6), (ii)) that a regular differential input-output relation possesses the remarkable property of determining a natural corresponding dynamic differential system (with respect to a fixed time point t o ) .Regular input-output relations also possess other attractive properties which make them particularly suitable as mathematical models of real systems. Thus a regular differential input-output relation S C %” x 23 possesses the property of having the whole of 2’ as its domain (cf. note (5.4.1), ( i ) ) . This property is very convenient when compositions of input-output relations are studied (cf. chapter 7). A regular S also allows a finite dimensional state-space representation (cf. chapter 7). We shall, in what follows, mainly be interested in regular differential input-output relations and regular generators. 6

Note. (i) A regular differential input-output relation S C T X X q generated by the regular generator [A(p) i - B ( p ) ] is clearly a linear mapping E-Xq if and only i f A ( p ) : P+Xqhas an inverseA(p)-’: X q + X q , i.e. if and only if the matrix A ( p ) is unimodular (cf. note (5.4.1), (iv)). (ii) Let S1 C TI x X q l and S2 C X r 2 x %qz be two differential input-output relations generated by [ A l ( p ) i - B l ( p ) ] and [Az(p) i - B 2 ( p ) ] respectively. If S1 n S2 is nonempty, then rl = r2 and q1 = q2. If, in addition, A l ( p ) and A2(p) are square, then both the generators are of the same size. Let S1 now be regular and A l ( p ) square with d e t A l ( p ) # 0. If S2 C SI and A2(p) is square, then detAz(p) # 0. This is seen as follows. First note 90

Differential input-output relations. Generators 6.2 that S2 C S1 also implies that &(O) C S1(0), i.e. that kerAz(p) C kerAl(p). K e r A l ( p ) is finite dimensional because d e t A l ( p ) # 0 (cf. note (5.4.1),(iii)). It is concluded that kerA2(p) is also finite dimensional and consequently detA2(p) # 0. The above properties mean that if S1is a regular differential input-output relation, then every other differential input-output relation S2, which is a subset of S1 and which is generated by some [A2(p) i -B2(p)] with A2(p) square, is also regular. Further, if [ A l ( p ) i -B1(p)] with Al(p) square is a generator for a regular S1, then this generator is necessarily regular. (iii) Let S be a differential input-output relation, and let (u, y) E S. Then every other ( u , yl) E S is such that y - yl E S(O), and conversely, every (u, y + z) with z E S(0) belongs to S. (iv) In a generator [A(p) i -B(p)] as given above we may even allow B ( p ) to be equal to the empty matrix, i.e. the matrix having no rows and no columns. This case corresponds to a situation where there is no input 0 present.

6.2 Input-output equivalence. Complete invariants and canonical forms for input-output equivalence Two fundamental theorems

There are a number of important relationships between differential input-output relations and their generators. The most significant of them are contained in the following fundamental theorems. The proofs of the theorems are given in section 6.5. 1

Theorem. Let Sl C %' x 2P be the differential input-output relation generated by [Al(p) i - B I ( ~ ) ] ,A l ( p ) E C [ p ] S ' x q ,Bl(p) E C[p]'lXr, and let S2 C %' x 99 be the differential input-output relation generated by [A2(p) i -Bz(p)], A2(p) E B2(p) E C [ p ] s 2 x r .Suppose further that [Ai(p) i -Bi(p)] = L(p)[Az(p) i -&(p)I for some L ( P ) EC[PIS1Xs2. Then the following statements are true: (i) S2 c S1, (ii) if s1 = s2 and L ( p ) is unimodular, then S1 = SZ, (iii) if S1 = S2, s1 = s2 = q , and detAI(p) # 0 (or detA2(p) # 0), then 0 detA2(p) # 0 (detAl(p) # 0) and L ( p ) is unimodular.

2

Theorem. Let S1 C Z r X X q be the regular differential input-output relation generated by [AI(P) i -Bl(p)], Al(p) E C[p]qxq, detAI(p) f 0, Bl(p) E C[p]qXr,and let S2 C X r X 23 be the differential input-output rela91

Algebraic Theory for Multivariable Linear Systems

, z(p)E C[plqXr. tion generated by [A2(p) i - B z ( p ) ] , A2(p) E C [ p I q x 4 B Then the following statement is true: If S2 C S1, then detA2(p) # 0 , moreover, there exists a unique L ( p ) E C [ p ] q x 4 , det L ( p ) # 0 , such that [Al(p) i - B , ( p ) J = L ( p ) [ A 2 ( p ) i - B 2 ( p ) ] ,and if, in particular, S1 = SZ, 0 then L ( p ) is unimodular. Note that statement (iii) of theorem (1) is contained in theorem (2). Generators generating the same regular differential input-output relation The theorems presented above lead to the following important conclusion. Let S be the regular differential input-output relation generated by the regular generator [ A l ( p )i - B l ( p ) ] , i.e. with A l ( p ) square and d e t A I ( p ) # 0. Let [A2(p) i - B z ( p > ] be another partitioned polynomial matrix operator of the same size as [ A l ( p ) i - B l ( p ) ] . Then [Az(p) i - B 2 ( p ) ] is also a regular generator for S if and only if [Al(p) i - B l ( p ) ] and [A2(p) i - B 2 ( p ) ] are row equivalent (cf. appendix A2 for details concerning this equivalence), i.e. if and only if there exists a unimodular polynomial matrix operator P( p ) of proper size such that [A2(p) i - B z ( p ) ] = P ( p ) [ A I ( p ) i - B , ( p ) ] . Row equivalent regular generators are thus of the same order = the order of the regular differential input-output relation S generated by them. Input-output equivalence

=

row equivalence

The above considerations suggest the following formalization (cf. Fig. 6.1). Let 3 denote the set of all regular differential input-output relations, and A4 the set of all regular generators for the elements of 3, i.e. define

3

.A.A{[A(p) i - B ( p ) J l A ( p ) E C [ p I q x q , detA(p) # O , B ( p ) E C [ p l q X r forsomeq,rE{l,2,3 , . . .

3 A {SlS-'

4

=

ker M

-

for some M E M } .

Further let the assignment

M

5

}},I

S , S-I

=

ker M .

define the mapping ( a surjection with respect to 3)

f:A4+3.

6

The generators M I , M2 E A4 are now said to be input-output equivalent if they generate the same input-output relation S E 3 , i.e. if f ( M l ) = f ( M z ) . The result quoted above then implies that input-output equivalence in the $ The case r = 0 would require special treatment.

92

Differential input-output relations. Generators 6.2

present case means the same thing as row equivalence. These terms are therefore used as synonyms in the sequel as far as regular generators are concerned. Complete invariant for input-output equivalence on M

Input-output equivalence is of course an equivalence relation on M , it is in fact the natural equivalence determined by the mappingfabove. According to the terminology of MacLane and Birkhoff, 1967, chapter VIII, fcan thus be called a complete invariant for input-output equivalence. Canonical forms for input-output equivalence on A

From the fact that input-output equivalence in this case coincides with row equivalence, it also follows that the set of CUT-forms (“canonical upper triangular”-forms) of the elements of M constitute a set of canonica[forms, denoted by A*, for input-output equivalence (see appendix A2 for a detailed discussion of the CUT-form of polynomial matrices). This means that every equivalence class on M for input-output equivalence contains a unique element of A*. Let, for M E M , the corresponding CUT-form f

Fig. 6.1. Commutative diagram. 3 is the set of all regular differential input-output relations, A4 is the set of all regular generators for the elements of 3 , A* is the set of all CUT-forms contained in A.

93

Algebraic Theory for Multivariable Linear Systems

be denoted by M*, and let g:A-+A*

7

be the mapping (a surjection with respect to A * )defined by the assignment 8

MI-+ M * . g is then also qualified to be called a complete invariant for input-output equivalence on A.

Finally, we note that the restriction o f f t o A*, i.e. flA*,is the mapping assigning a unique element of 9, to every element of A*. flA* is clearly thus has an inverse surjective with respect to 9t' and injective-it (flA*)-': %+A*. The situation is illustrated by the commutative diagram shown in Fig. 6.1. 9

Note. Above we defined input-output equivalence only for regular generators. Of course, also nonregular generators can be said to be inputoutput equivalent if they generate the same differential input-output relation. Quite generally then it holds that row equivalent generators are always also input-output equivalent (cf. theorem ( l ) , (ii)). The converse 0 is, however, not generally true.

6.3 The transfer matrix. Proper and strictly proper transfer matrices, generators, and differential input-output relations The transfer function and transfer matrix are well-known concepts in the operational calculus based on the use of the Laplace transform. We shall here introduce generalized forms of these concepts. The transfer matrix determined by a regular differential input-output relation

Consider a regular differential input-output relation S C X' x 2'9, and let [A@) i - B ( p ) ] , A ( p ) E C [ p I q x qdetA(p) , # 0 , B ( p ) E CLp]qx', be a regular generator for S. Next let A @ ) and B @ ) be identified with the corresponding rational matrices over C @ ) , i.e. over the field of quotients of C [p ] (cf. note (5.2.1), (ii)). We thus have A ( p ) E C ( p ) q X q , det A @ ) # 0, and B ( p ) E C( p ) q xrBut . thenA(p) is invertible in C(p)qxq, and the rational matrix

WP) = A b ) - ' B @ )

1 94

Differential input-output relations. Generators 6.4

is well-defined as an element of C ( p ) q x r .We shall call %(p)the transfer matrix determined by [ A @ ) i -B(p)].It turns out (cf. section 6.4) that %(p)is, in fact, also uniquely determined by S. %(p)is therefore alternatively said to be the transfer matrix determined by S. In the single input-single output case the term “transfer ratio” is occasionally used instead of the term “transfer matrix”. It should be emphasized that no mapping in which signal spaces are involved is assigned to the transfer matrix at this stage-here the transfer matrix is just a matrix with entries from the field C@). Later on (cf. part I11 of the book) the transfer matrix will be interpreted, under certain circumstances, as a mapping between Cartesian products of suitable signal spaces. Proper and strictly proper transfer matrices, generators and differential input-output relations

According to appendix A2, a transfer matrix %(p)as given by (1) above can be written in a unique way as 2

%(p)

+ K(p),

where K ( p ) is a polynomial matrix and where every entry of %‘(p) is such that its numerator is of lower degree than its denominator (the zero is supposed to fulfil this condition). %(p)is now said to be proper if K ( p ) is a constant matrix and strictly proper if K ( p ) is the zero matrix. Generators and differential input-output relations are likewise said to be proper or strictly proper depending on the properties of the transfer matrix determined by the generators and input-output relations in question. % O ( p ) in (2) is called the “strictly proper part” of %(p). Conditions for a regular generator [ A ( p ) i -B(p)] to be proper are discussed in appendix A2. It is a well-known fact that a realistic description of the behaviour of real systems at high frequencies implies the use of models based on proper or strictly proper input-output relations.

6.4 Transfer equivalence. Complete invariants and canonical forms for transfer equivalence. Controllability Transfer equivalence on the set At of all regular generators Consider the set At of all regular generators as given by (6.2.3). In section 6.2 we defined the input-output equivalence = row equivalence on this set. 95

Algebraic Theory for Multivariable Linear Systems

Now there is also another important equivalence on A, called “transfer equivalence”. This transfer equivalence and related concepts are discussed in detail in appendix A2. We shall recapitulate here briefly the relevant results. Let M I A [ A l @ ) i - B 1 ( p ) ] and M 2 4 [ A z ( p )i -&@)I be two elements of A. These elements are said to be transfer equiualent if they determine the same transfer matrix %(p)(cf. (6.3.1)), i.e. if 1

AI(p)-’Bl(p)

= Az(P)-IBz(p).

Clearly, if M I and M2 are input-output equivalent (=row equivalent) then they are also transfer equivalent (the converse does not generally hold). Complete invariant for the transfer equivalence on A

Let 5 denote the set of all transfer matrices determined by the elements of A (cf. Fig. 6.2), i.e.

5 P {%(p)l%(p) = A ( p ) - ’ B ( p ) for some [ A ( p ) i - B ( p ) ] E A},

2

and let the assignment

3

[A@)i

-W)l + + m - ’ B @ )

define the mapping (a surjection with respect to 9) f,:A+ 5.

4

The transfer equivalence on A is then the natural equivalence determined by f,, and fl is thus qualified to be called a complete invariant for this equivalence. Canonical forms for transfer equivalence on A

It is further shown in appendix A2 that there is a set At C A* C A of CUT-forms which qualifies as a set of canonical forms for the transfer equivalence on At., i.e. there exists a mapping (a surjection with respect to At.:) 5

g,:AM.-,W

which is a complete invariant for the transfer equivalence on A. The image g , ( [ A ( p )i - B 0 7 ) ] ) EM,* of [ A ( p ) i - B ( p ) ] € A , denoted by [ A ( p )i - B ( p ) ] t , is obtained as follows. Write

96

Differential input-output relations. Generators 6.4

Fig. 6.2. Commutative diagram.

92 is the set of all regular differential input-output relations, A4 is the set of all regular generators for the elements of 9,

At: is the set of all CUT-forms contained in A4 of the form [ A @ ) i -B@)] with A @ ) , B(p) left coprime, T is the set of all transfer matrices %(p)= A(p)-'B(p) constructed from the elements [ A @ ) i -B@)] of A4, amis the set of all minimal input-output relations contained in

a.

where L ( p ) is a GCLD (greatest common left divisor) of A @ ) and B @ ) (a method for finding an L ( p ) is given in appendix A2). Then it holds that 6

gt([A(p) i - B ( p ) ] ) A [ A ( p ) i - B ( p ) ] ?

= [ A I@)

i

-B I@)] *,

where [ . ] * denotes the CUT-form of [ * ] (cf. section 6.2). The restrictionf,lAlu,*off, to A? is again the mapping assigning a unique element of 9 to every element of At:. ftlA? has clearly an inverse 97

Algebraic Theory for Multivariable Linear System ( ftlA:)-': 3+.ht: . Given %(p)E 3,the corresponding element M E A: can be found by factoring %(p)as

%(PI = A ( p ) - ' B ( p ) ,

7

where A (p) , B ( p ) are left coprime polynomial matrices (applicable factorization procedures are described in appendix A2), and then taking the CUT-form [ A ( p ) i - B ( p ) ] * of [A($) i - B ( p ) ] as M . The situation is illustrated by Fig. 6.2, where the broken line encircles the quantities and relationships explained so far in this section. Note that At, 3,A*, and f denote quantities already introduced in section 6.2. Transfer equivalence on the set 3 of all regular input-output relations

We shall now also introduce a transfer equivalence on the set % of all regular differential input-output relations. Let SI,S2 E 3 be generated by MI A [ A l @ ) i - B l ( p ) ] and M2 4 [ A z ( p ) i - B z ( p ) ] E A4 respectively. Then S1 and S2 are said to be transfer equivalent if M I and Mz are transfer equivalent, i.e. if Al(p)-' B l ( p ) = A*(p)-' B2(p). This new transfer equivalence is indeed an equivalence relation on 3. This follows readily from the following theorem. The theorem is proved in section 6.5, and the proof of theorem (6.2.2) given in the same section is actually based on this result. 8

Theorem. Let S1 and S2 C T x 2 P be the regular differential input-output relations generated by the regular generators [ A l @ ) i - B l ( p ) ] and [ A 2 @ )i - B 2 ( p ) ] respectively. Let the corresponding transfer matrices be A 2 ( p ) - l B2(p). Then the denoted by P Al(p)-' B l ( p ) and %2(p) following statement is true: If S2 C S1, or S1 C S2, then

% I @ ) = %~(p).

0

-

Complete invariants for transfer equivalence on 3

Using the notations of the above theorem it follows that the assignments and S2 % 2 ( p ) define a mapping (a surjection with respect to 9)

S1 c,

9

(pt:3+3,

and that, by construction, 10

ft

=

(pt

of.

It can thus be seen that the transfer equivalence on % according to the 98

Differential input-output relations. Generators 6.4

definition given above is just the natural equivalence determined by qt. qt is accordingly a complete invariant for the transfer equivalence on 3. Another complete invariant for this equivalence is the mapping (cf. Fig. 6.2) 11 (ftIAT1-l 0 ($4, which assigns a unique element of A; to every element of 3. Canonical forms for the transfer equivalence on 9t

12

It remains to find a suitable set of canonical forms for the transfer equivalence on 3,i.e. a mapping 9t+ 9t such that the transfer equivalence on 9t is the natural equivalence determined by this mapping. To begin with, let 3, C 9t denote the range off (AT, i.e. the set of all regular differential input-output relations that can be generated by regular generators [ A @ ) i - B ( p ) J of CUT-form with A @ ) , B ( p ) left coprime. Clearly (cf. (10)) ftlAT = ( q t l 3 . m ) 0 ( f IAT)? and the mapping ( a surjection with respect to 3,)

13

YtP ( q t I % ) - ' o ~ t : ~ + ' % n C ~

fulfils the requirement mentioned above. This means that yt is a complete invariant for the transfer equivalence on 9t and that 3, constitutes a set of canonical forms for the transfer equivalence on 3.Fig. 6.2 is now completely explained. Minimal differential input-output relations

The elements of 3,possess a minimality property as stated in the following theorem. 14

Theorem. Let 8,C 9 tdenote an equivalence class for the transfer equivalence and let S , E 9,denote the canonical form determined by 8,, i.e. on 9, S , E gt and S , = yt ( S ) for an arbitrary S E Et. Finally let 8,be partially ordered by set inclusion. Then S , is the first element of 8,. Moreover, $39, constitutes the set of all first elements of the equivalence classes for the 0 transfer equivalence on 3.

15 Proof. For S, to be the first element of 8, we shall have S,E8, and S, C S for every S E gt.Now the first condition is fulfilled by construction, and we have to consider only the second one. 99

Algebraic Theory for Multivariable Linear Systems

So, let % ( p ) E 5 be the transfer matrix determined by gt, i.e. % ( p ) = q ( S ) for an arbitrary S E El, and let M , [ A , ( p ) i - B , ( p ) ] EAT be the canonical form determined by % ( p ) , i.e. M m [ A m ( p ) i -Bm(p)] = (ftlAf)-' ( % ( p ) ) , with A , ( p ) , B , ( p ) left coprime. Then let S E gt be arbitrary, and let M 4 i [ A ( p ) i - B ( p ) ] E At be a regular generator for S. M and M , are then transfer equivalent by construction, and according to appendix A 2 it holds that 16

for some square polynomial matrix operator L ( p ) . From theorem (6.2.1), (i) it then follows that S, C S. The last part of the theorem follows by construction. 0 The element S, E determined by a given S E 3 or a given %(p)E 9according to S, = yl (S),or S, = (qt1%,)-' (%(p))respectively, is also called the minimal input-output relation determined by S or %(p) respectively.

a,,,

The concept of a controllable differential input-output relation

We shall use the following terminology. Let S be a regular differential input-output relation generated by the regular generator [ A @ ) i - B @ ) ] , and let S, be the minimal input-output relation determined by S. We shall then say that S is controllable if S = S,, or equivalently according to the above results, if A @ ) and B @ ) are left coprime. The controllability concept introduced above turns out to be closely related to the ordinary controllability concept used in the state-space theory. It will be shown in chapter 7 that a regular differential input-output relation is controllable in the above sense if and only if any observable state-space representation of S is controllable in the ordinary state-space sense. 17

Note. Let S be a regular differential input-output relation, and let S, be the minimal input-output relation determined by S.According to theorem (14) above it then holds that S, C S and S,(O) C S(0). In addition it can easily be shown that S = ((0) x S(0)) + S,. We shall comment on this result in chapter 7 below. 0

6.5 Proofs of theorems (6.2.1), (6.2.2), and (6.4.8) This section is devoted to the proofs of theorems (6.2.1), (6.2.2), and (6.4.8) presented above. We shall need the following simple fact. 100

Differential input-output relations. Generators 6.5

1

Note. Let U, V , W be vector spaces over C, and let f : U + V and g : V 3 W be linear mappings. Then it holds that ker g 0 f = {xlx E U and f ( x ) E ker g} 3 k e r f , and so k e r f = ker g 0 f if and only if ker g n Rf = (0). If, in particular, Rf = V , then kerf = ker g o f if and only if ker g = {O}, i.e. if and only if g is an injection. 0

2

Proof of theorem (6.2.1). Now let f, g, and g 0 f i n the above note be identified with [A2(p) i - B z ( p ) ] : Zq X Z' + Z S 2 ,L ( p ) : Z S 2 + Z s ', and L ( p ) [ A l ( p ) i - B 2 ( p ) ] = (Al(p) i - B l ( p ) ] :a"q X 2'- 2" respectively of theorem (6.2.1). Item (i) of the theorem then follows directly from k e r f C kerg of,i.e.

sz c s1.

sI = s? and L ( p ) unimodular correspond to ker g = (0) implying k e r f = kerg O f , i.e. S2 = SI.This is item (ii) of the theorem. Suppose next that SI= Sz, s1 = s 2 = q and detA,(p) f 0 (or detA2(p) # 0). This implies that S1 = S2 is regular and that both the generators [Al(p) i - B l ( p ) ] and [Az(p) i - B 2 ( p ) ] must be regular (cf. note (6.1.6), (ii)). Consequently detAz(p) # 0 (detAI(p) # 0) and thus also det L ( p ) # 0. [ A l ( p ) i - B l ( p ) ] is accordingly of full rank q , and R [Al(p) i - B l ( p ) ] is the whole of %"? (cf. note (5.4.1), (i)) corresponding to R f = V in the above note. S1 = S2, i.e. k e r f = kerg 0f, then implies that kerg = {0}, i.e. that L ( p ) is an injection. Because det L ( p ) # 0, R L ( p ) is the whole of Zq, and L ( p ) thus has an inverse P i 4 2 q . It follows that L ( p ) is unimodular as asserted in item (iii) of the 0 theorem. Theorem (6.2.2) seems to be only a slightly strengthened form of theorem (6.2.1), (iii). Therefore it is somewhat surprising to notice that the proof of theorem (6.2.2) is quite involved. Our proof of theorem (6.2.2) is based on theorem (6.4.8). We shall therefore start with this proof.

3

Proof of theorem (6.4.8). We only need to consider the case S? C Sl-the case SIC S2 is then only a matter of notation. Note to begin with that the rational matrices % ( p ) and %2(p)mentioned in the theorem are equal if and only if Yj1(s) = %(s) for almost all s E C, where Yil(s) and g2(s)are obtained from Yjl(p) and % ( p ) respectively simply by replacing p everywhere by the complex variable s. Next we recall that our signal space Z contains all complex-valued functions on T of the form fw u(t) = ce", t E T , c , s E C (cf. note (5.1.6), 101

Algebraic Theory f o r Multioariable Linear Systems

4

( i ) ) . Consider then an input-output pair ( u , y) E 2' x ZP of the form u = cles(.) y = c2es(.) c1 E C', c2 E Cq, s E C C C, where C detAl(s) # 0 and detAZ(s) # 0.

is the set of all s E C such that

It follows that

A1(p)czes(O= Al(s)c&('), 5

where Al(s) etc. are again obtained from A l ( p ) etc. by replacing p everywhere by the complex variable s. Now choose c1 and cz so that ~2 = A2(S)-1B2(S)C1 = % ~ ( S ) C , .

6

It is then easily seen that the pair ( u , y ) so obtained satisfies Az(p)y = B Z ( p ) u ,i.e. we have ( u , y) E S2. This same ( u , y) is now also an element of SIif and only if 7

W ) C 1

= %(S)Cl.

The assumption S2 C S1 thus implies that (7) must hold for all s E C and all c1 E C' implying that Yj2(s) = %(s) for all s E C , i.e. for almost all 0 s E C. Consequently % ( p ) = Yj2(p).

8

Proof of theorem (6.2.2). Note first that SZ C S1 and detA,(p) # 0 also implies that detA2(p) # 0 (cf. note (6.1.6), (ii)). Further, if there exists an L ( p ) such that [Al(p) i - B l ( p ) ] = L(p)[Az(p) i -Bz(p)J, then this L ( p ) is unique and d e t L ( p ) # 0. This follows from the equality A l ( p ) = L ( p ) A & ) and note (5.4.1), (v). Hence, let S1 and S2 be regular differential input-output relations generated by the regular generators [Al(p) i - B l ( p ) ] and [Az(p) i - B 2 ( p ) ] respectively, and let S2 C SI.According to theorem (6.4.8) it then follows that

%(PI

9

= Al(p)-'Bl(p) = %(P) = A2(p) -"p),

i.e. [ A l ( p ) i - B l ( p ) ] and [A2(p) i -B2(p)] are transfer equivalent. They thus have a common canonical form (CUT-form; cf. appendix A2) [ A b ) i -B(p)l

10 102

Differential input-output relations. Generators 6.5

withA(p), B ( p ) left coprime andA(p)-'B(p) = % l ( p ) = % ~ ( p Moreover, ). there exist square polynomial matrices M l ( p ) , M z ( p ) , detMl(p) f 0, detM2(p) # 0, such that 11

[Ai(p) i - B i b ) ] = M i @ ) [ A ( p ) i -B(p)I and

12

[AZ(p) i -B2(p>l =M2@)[A(p) i -B(p)l.

Next let M ( p ) , detM(p) # 0, denote a GCRD of M1(p) and M 2 ( p ) so that 13

M1@)

= Nl(p)M(p)

and 14

Mz(p) = N2(p)M(p) with N , ( p ) , N 2 ( p ) right coprime and detNl(p) # 0, d e t N ~ ( p )# 0. Now S2 C S1 is equivalent to S2 = S1 n SZ, and S2 = S1 is equivalent to S2 = S, f l S2, and Sl= S1 n S2. S1 n S2 is by definition the set of all pairs ( u , y ) E 2' x %q satisfying Al@)Y

= B1b)u

A2@)Y

= B2(p)u,

and i.e. a generator for S1 n S2 is given by

[ M ( p )A ( p ) i - M ( p ) B ( p ) ] .

15

According to theorem (6.2.1), (ii), other generators for SIf l S2 can be constructed from the generator ( 15) by premultiplication by a unimodular matrix. It is also known (cf. appendix A2) that

[:El]

............

with N l ( p ) and N 2 ( p ) square and right coprime, can be brought to the Smith-form (which here coincides with the CUT-form)

103

Algebraic Theory for Multivariable Linear Systems

by premultiplication by a suitable unimodular matrix. Combining these results it is seen that a generator for S1 n S2 of the form

or equivalently

[ M @ ) A ( P ) i --MbJ) B(P)l,

16

is obtained from (15). Note that (16) is a regular generator. S1 fl S2 is thus a regular differential input-output relation. Summarizing then, S1 is generated by 17

[Ai(p)

i

- B i b ) ] = N l ( p ) [ M ( p ) A ( p ) i - M ( p ) B(p)I,

S2 by 18

[Az(p) i -B2(p)l = N z @ ) [ M @ ) A ( p ) i

and S,

B07)13

n S2 by [ M ( P ) A ( p ) i --M@)B(P)l

19

Note that all these generators are regular. Theorem (6.2.1), (iii) is now applicable and leads to the following conclusions. S2 C S1, or equivalently S2 = S1 n S 2 , implies that N 2 ( p ) is unimodular and so 20

[Ai@) i -Bi(p)] =Ni(p)Nz@>-'[Az@)i -B2(p)I

with Nl(p)N2(p)-l a polynomial matrix corresponding to the L ( p ) mentioned in the theorem. S2 = S1, or equivalently S2= S1 fl S2 and S1 = S1 fl S2, implies that both N l ( p ) and N 2 ( p ) are unimodular. This corresponds to a unimodular 0 Nl(p)N2(p)-'in (20) and to a unimodular L (p).

6.6 Comments on canonical forms. Canonical row proper forms We have in the preceding paragraphs considered canonical forms for input-output equivalence and transfer equivalence based on the CUT-form of polynomial matrices. This CUT-form proves very convenient in connection with the treatment of various analysis and synthesis problems as discussed in the next chapter. However, there are certainly also other possibilities for choosing suitable 104

Differential input-output relations. Generators 6.6

canonical forms for input-output and transfer equivalence-recall that a set C C X of canonical forms for an equivalence relation E on a set X is generally a collection of representatives of the equivalence classes on X determined by E so that C contains one and only one representative of each class (MacLane and Birkhoff, 1967, chapter VIII). In the preceding paragraphs the set of CUT-forms could of course be replaced everywhere by any other suitable set of canonical forms for input-output equivalence. One important possibility in choosing the canonical forms for inputoutput and transfer equivalence would be to base them on the CRP-form (“canonical row proper” form) of polynomial matrices (cf. appendix A2). This form was discussed independently by Forney, 1975, and Guidorzi, 1975, (cf. also Beghelli and Guidorzi, 1976) and it is based on the concept of a “row proper” polynomial matrix as introduced by Wolovich, 1974. This form offers a very convenient representation of generators determining proper and strictly proper transfer matrices (cf. appendix A2). We shall discuss the application of this particular canonical form in chapter 8.

105