- Email: [email protected]
Cascade control of systems over integral domains
Systems & Control North-Holland
Letters
4 (1984)
387-390
September
1984
Cascade control of svstems over integral domains * J
Stanislaw School
of
H. iAK Electrical
Lafayerre,
IN 47907,
Received
4 April
Engineering. USA
such that Purdue
Uniuersity.
West
= degsP(s)
1984
S(0) = +oo,
Necessary and sufficient conditions for the existence of cascade compensators and dynamic equivalence of linear systems over integral domains arc derived. The considerations lead IO constructive procedures for dynamic compensation. Kqvwor& Systems over rings. Dynamic and Smith forms.
compensation,
Hermite
1. Introduction
where Z is the set of integers. A rational function t(s) E D(s) will be called proper iff S( t(s)) 2 0. The set of proper rational functions will be denoted by D,,(s). Observe that D,,(s) endowed with the degree function S( -) is a Euclidean domain. Indeed, for every t,, r, E D,,(s), both nonzero, there exist unique p and r both from D,,(s) such that
The cascade control and dynamic equivalence of linear systemswith real coefficients were extensively studied by many researchers[3-61. The definitive results in this direction were obtained by Morse [3] and Wolovich and Falb [4]. The importance of the cascade control lies in its relationship with model matching and decoupling problems. Recently there were efforts towards the generalization of some of these results to systems over rings [1,2]. In this paper we formulate necessary and sufficient conditions for the existence of cascade compensators for linear systems over integral domains. We also suggest constructive procedures for dynamic compensation if such is feasible.
I, =
2. Preliminaries
Lemma
Through the paper D will denote an integral domain, D[s] will stand for the ring of polynomials with coefficients in D, and D(s) will denote the field of rational functions
D;Xm(~),
w=Bo’
tzp
r
+
where r = 0 or 6(r) < S(t,).
Moreover for all I,, all axioms of a Euclidean domain are satisfied ([7], p. 143). The units in D,,(s) are those proper rational functions for which 6(t) = 0. Let us now consider the set of p x m matrices with elements from D,,(s). If p = m, then the set D,frXJ’(s) of p X p matrices forms a ring. The units in D&“P(s) are unimodular matrices, i.e. those matrices U(s) for which det U(s) E D,,(s). Following Vardulakis et al. [S], we can define three elementary column or row operations on a rational matrix T(S)= D,9”m(s). Having at our disposal elementary operations we can easily prove: t,
E Dp,(s),
6(r,)
Thus
1 (Column Hermite Form). Let T(s) E
rank .Js) Then
5 8(t,t,).
T(s)
= mdp, is
column
ml. (ouer D,,(s))
equiualent
to the
lower triangular form
4s)
r S-rll
where o(s), p(x) E D[s], p(s) s 0. By a(.) we will mean the degree function on D(s) defined as kD(s)-tZu{+oo) * This work was supported by the school gineering of the Purdue University. 0167-6911/84/$3.00
- degg4s),
0 1984, Elsevier
Science
of Electrical
En-
.. .
0
. ..
01
3qT]At,,(s) : s-‘= 0 -** 0:J 1.r,,(s) 0-f s-rPP i/p 5 m, or
Publishers
0
B.V. (North-Holland)
387
s-rll
t,,(s)
.X@[T] p
...
0
0
Using similar arguments as in Lemma 1 one can easily prove:
0
s-Q2
Lemma 2 (Smith Form). For any T E Dirxm(s) with rank DC,JT= r there exist D,,(s)-unimodular matrices (I E Dir”“(s) and V E DEXm(s) such that
S -r”?“,
LlW . . .
t,,(s)
$m(4
ifp>m,wheret,i(s)=Oforallj=1,2,...,k-1; or if tkj(s)#O for some j=1,2,...,k-1 then S(t,,) > 8(tkj) for all such j (k = 1, 2,. . . , fin(m, p)).
I
s-41 0 s-4,
UTV = 0 I
.
where q, 1 q2 2 . . . 2 q,.
Example 1. Let TI E Dirx2(s), where D = 2 and
3. Dynamic compensation
0
3)
s+l s+3 1 s+3
_ (s + 2)(s + 3)
1 s+3
(s+
2)(s+ 2s+7
To bring T, to its Hermite Column Form we perform the following column operations on T,:
0 T;vy,
s+3
-
= T,V,, =
s(s + l)(s + 3) 2s2+7s+3
pJf’[T,I, where
v,= [-
r
1
s+2 s 0
and
v,, =v,v*=
388
IL
s+2 S
- 2(s+
0 1 1 s+l 1
s(s + l)(s + 3)
2)
s+3
s(s + 1)
s+l
..*
0I
Let T E Dir”“(s) be a proper transfer matrix of full rank r = min( p, m). Then any m X k transfer matrix q E DGxk(s) will be called a dynamic compensator of T. Suppose we are given T, E D,4”m(~) and T, E D,4”k(s). We pose the following question: What is a necessary and sufficient condition for the existence of T, such that T,T, = T,. The answer to this question is provided by:
.
S-’
k.Y[T]
0
Proof of this fact is analogous to the one in Vardulakis et al. [5] where D = R (the set of real numbers).
s+2 2 s+3 1
1
s+l
Theorem 1. There exists T, E Dp4”k( s ) such that T,T, = T2 iff9[TI] =9’[T, 1T2]. Proof follows from Lemma 2. Remark 1. Theorem 1 suggeststhe following algorithm for dynamic compensation: Step 1. Transform T, to Smith Form: UT,V= S,. Step 2. Perform the multiplication UT,. Step 3. Find T,such that UT, = S,c = UT,VT,. Step 4. T, = VT,. In the casewhen both T, and T, are of the same dimensions, a specific kind of precompensator may be desirable. Such a cascade controller is best defined via dynamic equivalence [3,4]. We will call T, and T2 both from D,~““‘(s) equivalent under dynamic compensation (or dynamically equivalent) iff T,T,, = T, and T,T,, = T, for some T,, and T,, from DzXm(s). For the relevance of dynamic equivalence to the decoupling problem of systems over unique factorization domains, the reader is referred to Datta and Hautus [l]. See also [2].
Theorem 2. Tl, Tz E DirXrn( s) are dynamically equioalent iff.Z’[T,] =.%‘[T*]. Proof follows from Lemma 1. As a consequence of Theorem 2 we have: Corollary. Let TE DpqXm(s), p < m, rank T=p. Then T has a proper right inoerse iff.P[ T] = [I,, (01. Example 1 (continued). Given T2 E Dirx2(s), Z (Wolovich and Falb [4]),
D =
S
s+l s2+6s+7
(s + l)(s
+ 3)
3s + 10 (s + l)(s + 3) The Hermite Column Form of T, can be obtained by applying the following column operations:
Remark 2. Observe that the Hermite Column Form over D,,(s) constitutes an abstract invariant for dynamical equivalence on the set of transfer matrices. Thus based on Hermite Column Form we can construct a set of canonical forms under this relation of equivalence. The application of our results to decoupling of linear systems over a unique factorization domain (UFD) is expressed by: Theorem 3. Let lZ = (A, B, C) represent a linear, reachable system over a UFD. Then the following is equivalent: (i) Z can be decoupled by dynamic state feedback with a static precompensator G invertible ouer the UFD. (ii) .z?{ C[sZ - A]-‘B} = diag{s-‘). Proof of this theorem is a consequence of Theorem 3.8 of Datta and Hautus [l]. Example 2 (Datta
and Hautus [l]). Let
s-r+2
T+
[
--
s+r+l
(r+l)s+12+1
rs-zr’+2z+1
1
where A = s2 + (1 - z)s - z - 1, D = R[z]. Application of the right elementary operations represented by
VY2
v=fiv, i-l
brings T to the form After noninteresting multiplication, one can see that X[T,] from Example 1 is the same as .%‘[T2]. Thus T, and T, are dynamically equivalent over Z,,(s). The compensator equating T, and T, can be found as follows. Note that T,V,, = T2VR2. Hence T&V,-,’
=.W[T] Here
_ (s+z+l)s
= T,
A 1 1 v3=
The form of T,, is exactly the same as in Wolovich and Falb [4] computed with the help of the so called interactor.
=9[T].
i
0
1 ,
1
0
,
[(2r+l)s2+(5z+3)s-2z3+2r2+5r+l]s (s-z+2)A
1 0
1 .
Thus this system can be decoupled
by state feed389
back and a static precompensator Wzl.
invertible
over
References [I]
K.B. Datta and M.L.J. Hautus, Decoupling of multivariable control systems over unique factorization domains. SfAM J. Conrrol Oprim. 22 (1) (1984) 28-39. [2] M. Kono, Decoupling and arbitrary coefficient assignment in time-delay systems, Sysfems Conrrol Lerr. 3 (6) (1983) 349-354. [3] A.S. Morse, System invariants under feedback and cascade control, in: G. Marchesini and S.K. Mitter. Eds., Morhe-
390
[4] . [5]
(6)
[7]
maricoi Sysrems Theory. Lecture Notes in Economics and Mathematical Systems, No. 131 (Springer, Berlin, 1976) 61-74. W.A. Wolovich and P.L. Falb, Invariants and canonical forms under dynamic compensation, SIAM J. Conrrol Oprim. 14 (6) (1976) 996-1008. A.I.G. Vardulakis. D.N.J. Limebeer. and N. Karcanias, Structure and Smith-McMillan form of a rational matrix at infinity, Infernor. J. Conrrol35 (4) (1982) 701-725. J.M. Dion and C. Commault. Smith-McMilIan factorizations at infinity of rational matrix functions and their control interpretation. Sysrem Conrrol Lerr. 1 (5) (1982) 312-320. I.N. Herstein, Topics in Algebra (Wiley, New York, 1975).
Letters
4 (1984)
387-390
September
1984
Cascade control of svstems over integral domains * J
Stanislaw School
of
H. iAK Electrical
Lafayerre,
IN 47907,
Received
4 April
Engineering. USA
such that Purdue
Uniuersity.
West
= degsP(s)
1984
S(0) = +oo,
Necessary and sufficient conditions for the existence of cascade compensators and dynamic equivalence of linear systems over integral domains arc derived. The considerations lead IO constructive procedures for dynamic compensation. Kqvwor& Systems over rings. Dynamic and Smith forms.
compensation,
Hermite
1. Introduction
where Z is the set of integers. A rational function t(s) E D(s) will be called proper iff S( t(s)) 2 0. The set of proper rational functions will be denoted by D,,(s). Observe that D,,(s) endowed with the degree function S( -) is a Euclidean domain. Indeed, for every t,, r, E D,,(s), both nonzero, there exist unique p and r both from D,,(s) such that
The cascade control and dynamic equivalence of linear systemswith real coefficients were extensively studied by many researchers[3-61. The definitive results in this direction were obtained by Morse [3] and Wolovich and Falb [4]. The importance of the cascade control lies in its relationship with model matching and decoupling problems. Recently there were efforts towards the generalization of some of these results to systems over rings [1,2]. In this paper we formulate necessary and sufficient conditions for the existence of cascade compensators for linear systems over integral domains. We also suggest constructive procedures for dynamic compensation if such is feasible.
I, =
2. Preliminaries
Lemma
Through the paper D will denote an integral domain, D[s] will stand for the ring of polynomials with coefficients in D, and D(s) will denote the field of rational functions
D;Xm(~),
w=Bo’
tzp
r
+
where r = 0 or 6(r) < S(t,).
Moreover for all I,, all axioms of a Euclidean domain are satisfied ([7], p. 143). The units in D,,(s) are those proper rational functions for which 6(t) = 0. Let us now consider the set of p x m matrices with elements from D,,(s). If p = m, then the set D,frXJ’(s) of p X p matrices forms a ring. The units in D&“P(s) are unimodular matrices, i.e. those matrices U(s) for which det U(s) E D,,(s). Following Vardulakis et al. [S], we can define three elementary column or row operations on a rational matrix T(S)= D,9”m(s). Having at our disposal elementary operations we can easily prove: t,
E Dp,(s),
6(r,)
Thus
1 (Column Hermite Form). Let T(s) E
rank .Js) Then
5 8(t,t,).
T(s)
= mdp, is
column
ml. (ouer D,,(s))
equiualent
to the
lower triangular form
4s)
r S-rll
where o(s), p(x) E D[s], p(s) s 0. By a(.) we will mean the degree function on D(s) defined as kD(s)-tZu{+oo) * This work was supported by the school gineering of the Purdue University. 0167-6911/84/$3.00
- degg4s),
0 1984, Elsevier
Science
of Electrical
En-
.. .
0
. ..
01
3qT]At,,(s) : s-‘= 0 -** 0:J 1.r,,(s) 0-f s-rPP i/p 5 m, or
Publishers
0
B.V. (North-Holland)
387
s-rll
t,,(s)
.X@[T] p
...
0
0
Using similar arguments as in Lemma 1 one can easily prove:
0
s-Q2
Lemma 2 (Smith Form). For any T E Dirxm(s) with rank DC,JT= r there exist D,,(s)-unimodular matrices (I E Dir”“(s) and V E DEXm(s) such that
S -r”?“,
LlW . . .
t,,(s)
$m(4
ifp>m,wheret,i(s)=Oforallj=1,2,...,k-1; or if tkj(s)#O for some j=1,2,...,k-1 then S(t,,) > 8(tkj) for all such j (k = 1, 2,. . . , fin(m, p)).
I
s-41 0 s-4,
UTV = 0 I
.
where q, 1 q2 2 . . . 2 q,.
Example 1. Let TI E Dirx2(s), where D = 2 and
3. Dynamic compensation
0
3)
s+l s+3 1 s+3
_ (s + 2)(s + 3)
1 s+3
(s+
2)(s+ 2s+7
To bring T, to its Hermite Column Form we perform the following column operations on T,:
0 T;vy,
s+3
-
= T,V,, =
s(s + l)(s + 3) 2s2+7s+3
pJf’[T,I, where
v,= [-
r
1
s+2 s 0
and
v,, =v,v*=
388
IL
s+2 S
- 2(s+
0 1 1 s+l 1
s(s + l)(s + 3)
2)
s+3
s(s + 1)
s+l
..*
0I
Let T E Dir”“(s) be a proper transfer matrix of full rank r = min( p, m). Then any m X k transfer matrix q E DGxk(s) will be called a dynamic compensator of T. Suppose we are given T, E D,4”m(~) and T, E D,4”k(s). We pose the following question: What is a necessary and sufficient condition for the existence of T, such that T,T, = T,. The answer to this question is provided by:
.
S-’
k.Y[T]
0
Proof of this fact is analogous to the one in Vardulakis et al. [5] where D = R (the set of real numbers).
s+2 2 s+3 1
1
s+l
Theorem 1. There exists T, E Dp4”k( s ) such that T,T, = T2 iff9[TI] =9’[T, 1T2]. Proof follows from Lemma 2. Remark 1. Theorem 1 suggeststhe following algorithm for dynamic compensation: Step 1. Transform T, to Smith Form: UT,V= S,. Step 2. Perform the multiplication UT,. Step 3. Find T,such that UT, = S,c = UT,VT,. Step 4. T, = VT,. In the casewhen both T, and T, are of the same dimensions, a specific kind of precompensator may be desirable. Such a cascade controller is best defined via dynamic equivalence [3,4]. We will call T, and T2 both from D,~““‘(s) equivalent under dynamic compensation (or dynamically equivalent) iff T,T,, = T, and T,T,, = T, for some T,, and T,, from DzXm(s). For the relevance of dynamic equivalence to the decoupling problem of systems over unique factorization domains, the reader is referred to Datta and Hautus [l]. See also [2].
Theorem 2. Tl, Tz E DirXrn( s) are dynamically equioalent iff.Z’[T,] =.%‘[T*]. Proof follows from Lemma 1. As a consequence of Theorem 2 we have: Corollary. Let TE DpqXm(s), p < m, rank T=p. Then T has a proper right inoerse iff.P[ T] = [I,, (01. Example 1 (continued). Given T2 E Dirx2(s), Z (Wolovich and Falb [4]),
D =
S
s+l s2+6s+7
(s + l)(s
+ 3)
3s + 10 (s + l)(s + 3) The Hermite Column Form of T, can be obtained by applying the following column operations:
Remark 2. Observe that the Hermite Column Form over D,,(s) constitutes an abstract invariant for dynamical equivalence on the set of transfer matrices. Thus based on Hermite Column Form we can construct a set of canonical forms under this relation of equivalence. The application of our results to decoupling of linear systems over a unique factorization domain (UFD) is expressed by: Theorem 3. Let lZ = (A, B, C) represent a linear, reachable system over a UFD. Then the following is equivalent: (i) Z can be decoupled by dynamic state feedback with a static precompensator G invertible ouer the UFD. (ii) .z?{ C[sZ - A]-‘B} = diag{s-‘). Proof of this theorem is a consequence of Theorem 3.8 of Datta and Hautus [l]. Example 2 (Datta
and Hautus [l]). Let
s-r+2
T+
[
--
s+r+l
(r+l)s+12+1
rs-zr’+2z+1
1
where A = s2 + (1 - z)s - z - 1, D = R[z]. Application of the right elementary operations represented by
VY2
v=fiv, i-l
brings T to the form After noninteresting multiplication, one can see that X[T,] from Example 1 is the same as .%‘[T2]. Thus T, and T, are dynamically equivalent over Z,,(s). The compensator equating T, and T, can be found as follows. Note that T,V,, = T2VR2. Hence T&V,-,’
=.W[T] Here
_ (s+z+l)s
= T,
A 1 1 v3=
The form of T,, is exactly the same as in Wolovich and Falb [4] computed with the help of the so called interactor.
=9[T].
i
0
1 ,
1
0
,
[(2r+l)s2+(5z+3)s-2z3+2r2+5r+l]s (s-z+2)A
1 0
1 .
Thus this system can be decoupled
by state feed389
back and a static precompensator Wzl.
invertible
over
References [I]
K.B. Datta and M.L.J. Hautus, Decoupling of multivariable control systems over unique factorization domains. SfAM J. Conrrol Oprim. 22 (1) (1984) 28-39. [2] M. Kono, Decoupling and arbitrary coefficient assignment in time-delay systems, Sysfems Conrrol Lerr. 3 (6) (1983) 349-354. [3] A.S. Morse, System invariants under feedback and cascade control, in: G. Marchesini and S.K. Mitter. Eds., Morhe-
390
[4] . [5]
(6)
[7]
maricoi Sysrems Theory. Lecture Notes in Economics and Mathematical Systems, No. 131 (Springer, Berlin, 1976) 61-74. W.A. Wolovich and P.L. Falb, Invariants and canonical forms under dynamic compensation, SIAM J. Conrrol Oprim. 14 (6) (1976) 996-1008. A.I.G. Vardulakis. D.N.J. Limebeer. and N. Karcanias, Structure and Smith-McMillan form of a rational matrix at infinity, Infernor. J. Conrrol35 (4) (1982) 701-725. J.M. Dion and C. Commault. Smith-McMilIan factorizations at infinity of rational matrix functions and their control interpretation. Sysrem Conrrol Lerr. 1 (5) (1982) 312-320. I.N. Herstein, Topics in Algebra (Wiley, New York, 1975).