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12 Difference input-output relations. Generators
12 Difference input-output relations. Getmators In this chapter we shall briefly present fundamental properties of thc difference input-output relations introduced in chapter 10. Because tht theory is rather similar to the theory of differential input-output relations. we can pass over many details, only referring to chapter 6. However, there are some important dissimilarities which we shall study more thoroughly. The most important is the concept of causality, which was not explicitly needed in connection with differential systems. However, causality is closely related to the concept of properness of differential systems. However, we shall also consider noncausal difference input-output relations, as we considered nonproper differential input-output relations. Here we are interested only in regular input-output relations and regular generators which determine unique transfer matrices. As with differential input-output relations, we shall define input-output equivalence and transfer equivalence on the set of regular generators and the latter also on the set of regular input-output relations. The canonical forms for these equivalences differ slightly from the corresponding canonical forms in the continuous-time case. Choice of signal space X In this chapter the signal space 2 is assumed to be regular and sufficiently rich and we consider only the case T = Z. Without loss of generality X can be taken as Cz. The case T = No is almost completely analogous to the theory of differential input-output relations presented in chapter 6. Only the properness and controllability concepts have a slightly different meaning from the systems theory point of view. 28 1
Algebraic Theory for Multivariable Linear Systems
12.1 Regular difference input-output relations and regular generators In section 10.2 we defined the difference input-output relation generated by a pair ( B ( z ) , A ( r ) )of polynomial matrix operators as a set S of pairs ( u , y ) E %?x X9 satisfying the matrix difference equation A(.z)y = B(’1)u.
1
This equation can be regarded as a C[ *]-module equation but also as a C(;j-module equation. Because the latter structure has some advantages, we shall use it here. In particular, this implies that we can cancel powers of ’1 if they exists on both sides of equation (1).This property is considered more precisely in section 12.2. Now, the partitioned rational matrix operator [ A ( t ) : -B(’1)], A ( # )E C(alSx9, B(.z)E C(t]lsxr is called a generator for the difference input-output relation S. Note that in what follows, we usually do not make any notational distinction between matrices over C[z] and C(z], neither their entries. Regular generators for regular difference input-output relations. Order
A difference input-output relation S is said to be regular if it has a generator [A(#) i -B(1)] with A ( t ) square and detA(r) # 0. Consequently, such a generator is called regular. If detA(’1) E C(&]is written in the form a,C
+ a,+,**~1+
. . . + a,P
= (a,
+ a,+1.2+. . . + a,rn-m)zm
with a, # 0, a, # 0, m ,n E Z, the number n - m is called the order of the generator and of the corresponding regular difference input-output relation S. A regular difference input-output relation has the property that it determines unique systems (cf. section 10.2), which, however, are not necessarily causal. Another important property is that a regular difference input-output relation S C X r X 2 9 has the whole %‘ as its domain. 2
Note. (i) A regular difference input-output relation S C X r x %?generated by the regular generator [ A ( * )i -B(*)] is a (linear) mapping Xr+ 2 9 if and only if A ( * ) is C(r]-unimodular (cf. appendix A l ) , i.e. if and only if detA(r) = c,znfor some c f 0, n E Z. (ii) Let S1 and S2 be two difference input-output relations generated by [Al(*) i -Bl(r)] and [ A * ( * )i - B ? ( z ) ] , respectively. If [ A l ( r ) i -B1(4)] 282
Difference input-output relations. Generators 12.2
is regular, Az(s) is square and SzC S1, then [Az(s) i -Bz(k)] is also regular. 0
12.2 Input-output equivalence. Canonical forms for inputoutput equivalence. Causality Two fundamental theorems
The relationships between difference input-output relations and their generators are in principle similar to the corresponding properties of differential systems. There are, however, some dissimilarities, which make it necessary for us to present briefly the discrete-time counterparts of theorems (6.2.1) and (6.2.2). The theorems can be proved following the lines of section 6.5. 1 Theorem. Let S1,SzC %?x %9 be the difference input-output relations generated by [ A l ( s ) i -B1(s)] and [Az(r) i - B z ( ~ ) ] respectiuely, , andsuppose that [Al(*) i -Bl(r)] = L(r)[Az(2) i -Bz(r)] forsome matrix L ( r ) ouer C(.z]. Then (i) S2 C SI, (ii) if L (s) is square and C(a]-unimodular, then S1 = SZ, (iii) ifS1 = Sz, L ( r ) is square, anddetA1(4) # 0 (or detA2(z) # 0), then 0 detAZ(t) #O(detA,(s) # 0) and L ( r ) is C(s]-unimodular. 2 Theorem. Let S1, S2 C %r x %'q be the difference input-output relations generated by [Al(s) i -B1(r)] and [Az(s) i -&(+)I, respectively, with [A1(+)i -B1(s)] regular and Az(s) square. Zf SzCS1, then [Az(r) i - &(a)] is regular and there exists a unique matrix L(4) ouer C(t], d e t L ( s ) # 0, such that [Al(a) i -B1(4)] = L(s)[A2(s) i -&(a)], and if, 0 in particular, S1 = Sz,then L (4) is C ( s]-unirnodular. Input-output equivalence
=
C(a]-row equivalence
The above theorems lead to the following conclusion (cf. section 6.2). Two regular generators M1 P [Al(#) i -Bl(t)] and M ZA [Az(s) i -B2(s')] are said to be input-output equivalent if they generate the same difference input-output relation. As a consequence of theorems (1) and (2) we have the result that the regular generators M1 and M Z are inputoutput equivalent if and only if they are C(c]-row equivalent, i.e. if and only if there exists a C(.z]-unimodular matrix P ( s ) such that [Az(s) i -B2(4)] = P(t)[Al(+) i -Bl(+)]. 283
Algebraic Theory for Multivariable Linear Systems
Canonical forms for input-output equivalence According to the previous result every set of canonical forms for C(s]-row equivalence also constitutes a set of canonical forms for input-output equivalence. Thus, for instance, the set of CUT-forms (“canonical upper triangular”-forms) of regular generators is a set of canonical forms for input-output equivalence (see appendix A2). Note $at the CUT-form of a polynomial matrix considered as a matrix over C(eJ[can ’ be different from its CUT-form when it is considered as a matrix over C [ a ] . A regular generator [A(*) i - B ( s ) ] of CUT-form over C ( * Jis such that A(*) possesses the same properties with respect to the degrees of the entries as the CUT-form for polynomial matrices explained in appendix A2. The normalization is in this case most conveniently performed so that the constant terms of the diagonal entries of A(c) are equal to 1. Then A(*) and B ( t ) can be written as A ( * ) = A0 + A1s + A $
+ . . . + A,P, B ( + ) = (Bo + Bis + . . . + B,s”)/*”,
3 4
where A ( ) , A l , . . . ,A,, Bo, B1,. . . , B, are matrices over C with A(,,A l , . . . ,A , upper triangular and A(, invertible with diagonal entries equal to 1.
Causality The canonical form introduced above is closely related to causality (see Hirvonen et a f . , 1975). If the generator [A( ) i - B ( t ) ] for a difference input-output relation S is of CUT-form, and we write A ( & )and B ( ’ L )in the forms (3) and (4), respectively, then S is generated also by a recursion equation Aoy(k) + A # ( k
5
=
BOu(k + p )
-
1) + . . . +A,y(k - m )
+ Blu(k + p - 1) + . . . + B,u(k
+ p -n),
k E Z,
which can be uniquely solved for y ( k ) because A0 is invertible. Using this it is easily seen that S is causal if and only if p 0, i.e. if and only if [A(*) i - B ( * ) ] is a polynomial matrix over C [ t ] .According to the foregoing we shall call a regular generator causal if its CUT-form is a polynomial matrix over C [ * ] .
12.3 The transfer matrix. Properness and causality Let [ A ( * )i - B ( a ) ] be a regular generator for a difference input-output relation S. Considering A(*) and B ( s ) as matrices over the field C ( 4 ) of 284
Difference input-output relations. Generators 12.4
quotients of C ( t ] and using the fact that A ( t ) is invertible, we can define the transfer matrix 1
% ( a ) = A(t)-'B(*)
determined by [ A ( r ) i - B ( a ) ] . %(t) is also uniquely determined by S, which seems to be a straight consequence of theorem (12.2.2). However, the proof of theorem (12.2.2) is based on this property, so that it must be proven in a different way (cf. section 6.5). Properness
The properness of a transfer matrix %(*) = A ( * ) - ' B ( * )determined by a regular generator [ A ( * )i - B ( a ) ] has no great importance from the systems theory point of view, in contrast with the continuous-time case. Instead, if we consider % (t) as a matrix g( 1/z) over the field C ( 1 / a ) of quotients of C [ @ ] (see appendix A2), then the properness of $(l/z) is closely related to the causality of [ A ( r ) i -B(t)]. In fact, it is quite easy to show that [ A ( t ) i - B ( a ) ] is causal if and only if %(1/t) = A ( t ) - ' B ( a ) is a proper matrix over C ( l / h ) . As a consequence we shall say that [ A ( * )i - B ( * ) ] is strictly causal, if $( l / a ) is strictly proper, and anticausal if %(l/t) is a polynomial matrix over C ( I/i). 2
3
Remark. In the case T = No we can also define a transfer matrix
%(a)= A ( q ) - ' B ( d determined by a regular generator [ A ( ? ) i - B ( q ) ] . Then % ( q ) is proper if and only if the input-output relation S generated by [ A( q ) i - B ( q ) ] is causal (cf. section 10.2). This differs slightly from the continuous-time case, because there every regular input-output relation is also causal, irrespective of its properness. 0
12.4 Transfer equivalence. Canonical forms for transfer equivalence. Controllability Two regular generators [ A l ( & )i - B 1 ( t ) ] and [Az(t) i - B 2 ( 4 ) ] are called transfer equivalent if they determine the same transfer matrix, i.e. 1
A l ( t ) - ' B l ( t ) = Az( t ) - l B z ( * ) .
This defines an equivalence relation on the set of all regular generators. The set of regular generators [ A ( * )i - B ( a ) ] of CUT-form such that A ( * ) and B(t) are left coprime matrices over C ( * ] constitutes a set of 285
Algebraic Theory for Multioariable Linear System
canonical forms for the transfer equivalence. Of course there are many other sets of canonical forms for the transfer equivalence, but coprimeness is a common property of them all. Transfer equivalence on difference input-output relations
Because our signal space % is sufficiently rich it contains all signals k w czk, k E Z, c, z E C . Therefore the input-output relation S C 2' x %q generated by a regular generator [A(a) i - B ( r ) ] contains all pairs ( u , y ) E 2' x %q such that
u = c z ( , ) , y=A(z-')-'B(z-')cz(.)
2
= %(z-')cz(.),
where c E C' and z E C - (0)satisfies detA(z-') # 0. Using this we can easily show that the transfer matrix % ( r ) is uniquely determined by the input-output relation S. More exactly, we have the following theorem.
3
Theorem. Let S1 and S2 C %' x %q be the regular difference input-output relations generated by the regular generators [Al(i) i - B l ( p ) ] and [A2( r ) i -&($)I, respectively. Let the corresponding transfer matrices be . the foldenoted by %I(*) A Al(r)-'Bl(r) and % 2 ( r ) B A ~ ( r ) - ~ B 2 ( 1 )Then lowing statement is true:
4
If S2 C S1, or S1 C S2, then %I(*) = % 2 ( r ) . 0 As a consequence we can define the transfer equivalence on the set of
regular difference input-output relations, too. The set of input-output relations S, generated by the regular generators [A(#) i - B ( r ) ] with A ( t ) and B ( r ) left coprime constitute a set of canonical forms for this equivalence. Furthermore, every canonical form is the first (minimal) element of a transfer equivalence class, if this is partially ordered by set inclusion (cf. theorem (6.4.14)). Controllability
Let S be a regular difference input-output relation generated by the regular generator [A(r) i - B ( a ) ] . We shall say that S is controllable if A($) and B ( * ) are left coprime matrices over C ( * ] . 5
Remark. In the case T = Nothe transfer equivalence concepts are analogous to the case T = Z. However, the controllability is a little more complicated. Let us say that the regular difference input-output relation S generated by [A(?) i - B ( p ) ] is reachable if A(?) and B ( p ) are left coprime matrices over C [ a ] ,and S is controllable if A(?) and B( a) are left coprime when 286
Difference input-output relations. Generators 12.4
they are considered as matrices over C(p]. Thus S is controllable if all common left divisors L ( a ) of A ( p ) and B( y) have determinants of the form det L ( a ) = can, c E C - {0},n E No. These definitions coincide with the ordinary definitions of controllability and reachability associated with 0 the state-space representation.
287
Algebraic Theory for Multivariable Linear Systems
12.1 Regular difference input-output relations and regular generators In section 10.2 we defined the difference input-output relation generated by a pair ( B ( z ) , A ( r ) )of polynomial matrix operators as a set S of pairs ( u , y ) E %?x X9 satisfying the matrix difference equation A(.z)y = B(’1)u.
1
This equation can be regarded as a C[ *]-module equation but also as a C(;j-module equation. Because the latter structure has some advantages, we shall use it here. In particular, this implies that we can cancel powers of ’1 if they exists on both sides of equation (1).This property is considered more precisely in section 12.2. Now, the partitioned rational matrix operator [ A ( t ) : -B(’1)], A ( # )E C(alSx9, B(.z)E C(t]lsxr is called a generator for the difference input-output relation S. Note that in what follows, we usually do not make any notational distinction between matrices over C[z] and C(z], neither their entries. Regular generators for regular difference input-output relations. Order
A difference input-output relation S is said to be regular if it has a generator [A(#) i -B(1)] with A ( t ) square and detA(r) # 0. Consequently, such a generator is called regular. If detA(’1) E C(&]is written in the form a,C
+ a,+,**~1+
. . . + a,P
= (a,
+ a,+1.2+. . . + a,rn-m)zm
with a, # 0, a, # 0, m ,n E Z, the number n - m is called the order of the generator and of the corresponding regular difference input-output relation S. A regular difference input-output relation has the property that it determines unique systems (cf. section 10.2), which, however, are not necessarily causal. Another important property is that a regular difference input-output relation S C X r X 2 9 has the whole %‘ as its domain. 2
Note. (i) A regular difference input-output relation S C X r x %?generated by the regular generator [ A ( * )i -B(*)] is a (linear) mapping Xr+ 2 9 if and only if A ( * ) is C(r]-unimodular (cf. appendix A l ) , i.e. if and only if detA(r) = c,znfor some c f 0, n E Z. (ii) Let S1 and S2 be two difference input-output relations generated by [Al(*) i -Bl(r)] and [ A * ( * )i - B ? ( z ) ] , respectively. If [ A l ( r ) i -B1(4)] 282
Difference input-output relations. Generators 12.2
is regular, Az(s) is square and SzC S1, then [Az(s) i -Bz(k)] is also regular. 0
12.2 Input-output equivalence. Canonical forms for inputoutput equivalence. Causality Two fundamental theorems
The relationships between difference input-output relations and their generators are in principle similar to the corresponding properties of differential systems. There are, however, some dissimilarities, which make it necessary for us to present briefly the discrete-time counterparts of theorems (6.2.1) and (6.2.2). The theorems can be proved following the lines of section 6.5. 1 Theorem. Let S1,SzC %?x %9 be the difference input-output relations generated by [ A l ( s ) i -B1(s)] and [Az(r) i - B z ( ~ ) ] respectiuely, , andsuppose that [Al(*) i -Bl(r)] = L(r)[Az(2) i -Bz(r)] forsome matrix L ( r ) ouer C(.z]. Then (i) S2 C SI, (ii) if L (s) is square and C(a]-unimodular, then S1 = SZ, (iii) ifS1 = Sz, L ( r ) is square, anddetA1(4) # 0 (or detA2(z) # 0), then 0 detAZ(t) #O(detA,(s) # 0) and L ( r ) is C(s]-unimodular. 2 Theorem. Let S1, S2 C %r x %'q be the difference input-output relations generated by [Al(s) i -B1(r)] and [Az(s) i -&(+)I, respectively, with [A1(+)i -B1(s)] regular and Az(s) square. Zf SzCS1, then [Az(r) i - &(a)] is regular and there exists a unique matrix L(4) ouer C(t], d e t L ( s ) # 0, such that [Al(a) i -B1(4)] = L(s)[A2(s) i -&(a)], and if, 0 in particular, S1 = Sz,then L (4) is C ( s]-unirnodular. Input-output equivalence
=
C(a]-row equivalence
The above theorems lead to the following conclusion (cf. section 6.2). Two regular generators M1 P [Al(#) i -Bl(t)] and M ZA [Az(s) i -B2(s')] are said to be input-output equivalent if they generate the same difference input-output relation. As a consequence of theorems (1) and (2) we have the result that the regular generators M1 and M Z are inputoutput equivalent if and only if they are C(c]-row equivalent, i.e. if and only if there exists a C(.z]-unimodular matrix P ( s ) such that [Az(s) i -B2(4)] = P(t)[Al(+) i -Bl(+)]. 283
Algebraic Theory for Multivariable Linear Systems
Canonical forms for input-output equivalence According to the previous result every set of canonical forms for C(s]-row equivalence also constitutes a set of canonical forms for input-output equivalence. Thus, for instance, the set of CUT-forms (“canonical upper triangular”-forms) of regular generators is a set of canonical forms for input-output equivalence (see appendix A2). Note $at the CUT-form of a polynomial matrix considered as a matrix over C(eJ[can ’ be different from its CUT-form when it is considered as a matrix over C [ a ] . A regular generator [A(*) i - B ( s ) ] of CUT-form over C ( * Jis such that A(*) possesses the same properties with respect to the degrees of the entries as the CUT-form for polynomial matrices explained in appendix A2. The normalization is in this case most conveniently performed so that the constant terms of the diagonal entries of A(c) are equal to 1. Then A(*) and B ( t ) can be written as A ( * ) = A0 + A1s + A $
+ . . . + A,P, B ( + ) = (Bo + Bis + . . . + B,s”)/*”,
3 4
where A ( ) , A l , . . . ,A,, Bo, B1,. . . , B, are matrices over C with A(,,A l , . . . ,A , upper triangular and A(, invertible with diagonal entries equal to 1.
Causality The canonical form introduced above is closely related to causality (see Hirvonen et a f . , 1975). If the generator [A( ) i - B ( t ) ] for a difference input-output relation S is of CUT-form, and we write A ( & )and B ( ’ L )in the forms (3) and (4), respectively, then S is generated also by a recursion equation Aoy(k) + A # ( k
5
=
BOu(k + p )
-
1) + . . . +A,y(k - m )
+ Blu(k + p - 1) + . . . + B,u(k
+ p -n),
k E Z,
which can be uniquely solved for y ( k ) because A0 is invertible. Using this it is easily seen that S is causal if and only if p 0, i.e. if and only if [A(*) i - B ( * ) ] is a polynomial matrix over C [ t ] .According to the foregoing we shall call a regular generator causal if its CUT-form is a polynomial matrix over C [ * ] .
12.3 The transfer matrix. Properness and causality Let [ A ( * )i - B ( a ) ] be a regular generator for a difference input-output relation S. Considering A(*) and B ( s ) as matrices over the field C ( 4 ) of 284
Difference input-output relations. Generators 12.4
quotients of C ( t ] and using the fact that A ( t ) is invertible, we can define the transfer matrix 1
% ( a ) = A(t)-'B(*)
determined by [ A ( r ) i - B ( a ) ] . %(t) is also uniquely determined by S, which seems to be a straight consequence of theorem (12.2.2). However, the proof of theorem (12.2.2) is based on this property, so that it must be proven in a different way (cf. section 6.5). Properness
The properness of a transfer matrix %(*) = A ( * ) - ' B ( * )determined by a regular generator [ A ( * )i - B ( a ) ] has no great importance from the systems theory point of view, in contrast with the continuous-time case. Instead, if we consider % (t) as a matrix g( 1/z) over the field C ( 1 / a ) of quotients of C [ @ ] (see appendix A2), then the properness of $(l/z) is closely related to the causality of [ A ( r ) i -B(t)]. In fact, it is quite easy to show that [ A ( t ) i - B ( a ) ] is causal if and only if %(1/t) = A ( t ) - ' B ( a ) is a proper matrix over C ( l / h ) . As a consequence we shall say that [ A ( * )i - B ( * ) ] is strictly causal, if $( l / a ) is strictly proper, and anticausal if %(l/t) is a polynomial matrix over C ( I/i). 2
3
Remark. In the case T = No we can also define a transfer matrix
%(a)= A ( q ) - ' B ( d determined by a regular generator [ A ( ? ) i - B ( q ) ] . Then % ( q ) is proper if and only if the input-output relation S generated by [ A( q ) i - B ( q ) ] is causal (cf. section 10.2). This differs slightly from the continuous-time case, because there every regular input-output relation is also causal, irrespective of its properness. 0
12.4 Transfer equivalence. Canonical forms for transfer equivalence. Controllability Two regular generators [ A l ( & )i - B 1 ( t ) ] and [Az(t) i - B 2 ( 4 ) ] are called transfer equivalent if they determine the same transfer matrix, i.e. 1
A l ( t ) - ' B l ( t ) = Az( t ) - l B z ( * ) .
This defines an equivalence relation on the set of all regular generators. The set of regular generators [ A ( * )i - B ( a ) ] of CUT-form such that A ( * ) and B(t) are left coprime matrices over C ( * ] constitutes a set of 285
Algebraic Theory for Multioariable Linear System
canonical forms for the transfer equivalence. Of course there are many other sets of canonical forms for the transfer equivalence, but coprimeness is a common property of them all. Transfer equivalence on difference input-output relations
Because our signal space % is sufficiently rich it contains all signals k w czk, k E Z, c, z E C . Therefore the input-output relation S C 2' x %q generated by a regular generator [A(a) i - B ( r ) ] contains all pairs ( u , y ) E 2' x %q such that
u = c z ( , ) , y=A(z-')-'B(z-')cz(.)
2
= %(z-')cz(.),
where c E C' and z E C - (0)satisfies detA(z-') # 0. Using this we can easily show that the transfer matrix % ( r ) is uniquely determined by the input-output relation S. More exactly, we have the following theorem.
3
Theorem. Let S1 and S2 C %' x %q be the regular difference input-output relations generated by the regular generators [Al(i) i - B l ( p ) ] and [A2( r ) i -&($)I, respectively. Let the corresponding transfer matrices be . the foldenoted by %I(*) A Al(r)-'Bl(r) and % 2 ( r ) B A ~ ( r ) - ~ B 2 ( 1 )Then lowing statement is true:
4
If S2 C S1, or S1 C S2, then %I(*) = % 2 ( r ) . 0 As a consequence we can define the transfer equivalence on the set of
regular difference input-output relations, too. The set of input-output relations S, generated by the regular generators [A(#) i - B ( r ) ] with A ( t ) and B ( r ) left coprime constitute a set of canonical forms for this equivalence. Furthermore, every canonical form is the first (minimal) element of a transfer equivalence class, if this is partially ordered by set inclusion (cf. theorem (6.4.14)). Controllability
Let S be a regular difference input-output relation generated by the regular generator [A(r) i - B ( a ) ] . We shall say that S is controllable if A($) and B ( * ) are left coprime matrices over C ( * ] . 5
Remark. In the case T = Nothe transfer equivalence concepts are analogous to the case T = Z. However, the controllability is a little more complicated. Let us say that the regular difference input-output relation S generated by [A(?) i - B ( p ) ] is reachable if A(?) and B ( p ) are left coprime matrices over C [ a ] ,and S is controllable if A(?) and B( a) are left coprime when 286
Difference input-output relations. Generators 12.4
they are considered as matrices over C(p]. Thus S is controllable if all common left divisors L ( a ) of A ( p ) and B( y) have determinants of the form det L ( a ) = can, c E C - {0},n E No. These definitions coincide with the ordinary definitions of controllability and reachability associated with 0 the state-space representation.
287