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Systems Over Rings: Geometric Theory and Applications
Copyright © IFAC Linear Time Delay Systems, Grenoble, France, 1998
SYSTEMS OVER RINGS: GEOMETRIC THEORY AND APPLICATIONS G. Conte' A. M. Perdon ••
• Dipartimento di Elettronica e Automatica, Universita' di Ancona, Via Brecce Bianche, 60131 Ancona, Italy. [email protected] Dipartimento di lvIatematica "V. Volterra ", Universita' di Ancona, Via Brecce Bianche, 60131 Ancona, Italy, [email protected]
Abstract: The aim of this paper is to present an overview of the geometric approach to the study of dynamical systems with coefficients in a ring and, in particular, to discuss the most recent results concerning noninteracting control problems it has produced. The study of systems with coefficients in a ring is motivated, in addition to an intrinsic interest on the subject, by the fact that they may be viewed as tools for investigating problems which concern parameter dependent systems or delay-differential systems. The geometric approach is shown to provide complete characterizations of the solvability conditions, together with feasible procedures for computing solutions, of noninteracting control problems, like the Disturbance Decoupling Problem and the Block Decoupling Problem. Finally, some examples which clarify the applications to problems concerning delay-differential systems are presented. Copyrighr© 1998 IFAC Resume : Ce travailpresente les resultats plus recentes obtenus par l'approche geometrique dans le cadre des systemes sur les anneaux. Les systemes sur les anneaux sont utiles comme modeles pour etudier des nombreuses classes des systemes, comme les systemes a retard et les familles des systemes lineaires qui dependent d'un parametre. L'approche geometrique permet de resoudre plusieurs problemes de commande, tel que le rejet de perturbation et le decouplage par blocks avec stabilisation. Des nouveux algorithms, qui permettent de construire pratiquement les solutions cherches, sont aussi presentes. Keywords: Systems over rings, geometric approach, noninteracting control problems
1. E,TRODUCTIO;'\
involve parameter dependent systems or delaydifferential systems. The advantages one has in using models with coefficients in a ring originate from the possibility of deriving, in a natural way, methods and techniques from the framework of systems with coefficients in the field ~ of real numbers and of using the powerful mathematical tools offered by ring and module algebra. This produced a research effort for extending to the
Dynamical systems with coefficients in a ring represent a natural generalization of linear, timeinvariant, finite dimensional, dynamical systems with coefficients in the field 1}( of real numbers. Early papers (see (I\:amen, 1976), (Sontag, 1976), (Sontag, 1981)) pointed out the potential use of systems with coefficients in a ring as an abstract tool for dealing with control problems which
175
namical system 1; with coefficients in the field of real numbers ~ is given, in continuous time, by a set of equations of the form
framework of systems with coefficients in a ring resuI ts and techniques from the framework of linear, dynamical systems with coefficients in the field ~ of real numbers. In recent years, this work turned out to be particularly successful for what concerns the development of a geometric approach and its applications to a number of control problems (see (Hautus, 1982), (Inaba et al., 1987), (Inaba and Wang, 1996), (Conte and Perdon, 1995a),(Conte and Perdon, 1995b),(Conte et al., 1996),(Conte et al., 1998b), (Conte et al., 1998b),(Conte et al., 1998a), (Conte and Perdon, 1998)). In the paper, after describing the notion of dynamical system with coefficients in a ring and illustrating some particular features due to ring and module algebra, we analyse the basic notions of the geometric approach in its extension to systems with coefficients in a ring. The main difference \vith respect to the situation encountered in the classical geometric approach (see (Wonham, 1985), (Basile and Marro, 1969)) is that dynamic properties based on the existence of feed backs with special features cannot be characterized by means of purely geometric conditions. For example, controlled invariance, which is a geometric property, does not characterize invariance with respect to a compensated dynamics. On the other hand, geometric conditions are sufficient for assuring the existence of feed backs with special features which act on a system extension. Therefore, from the point of view of the geometric approach, there is a stricter analogy between the situation concerning systems with coefficients in a ring and systems with coefficients in the field ~ of real numbers if, in the ring framework, one considers dynamic properties based on the existence of dynamic feed backs. By exploiting this fact, we show that it is possible to extend, in a suitable way, a large number of techniques and results based on the geometric approach. We illustrate this by discussing in particular a class of noninteracting control problems which includes the Disturbance Decoupling Problem and the Block Decoupling Problem with stability or coefficients assignements. The geometric approach is shown to offer satisfactory solutions, as well as feasible procedure for finding them, to those problems for systems with coefficients in quite general rings. Finally, we discuss some example which clarify how systems with coefficients in a ring and the results obtained by means of the geometric approach can be used for studying delay-differential systems and related control problems.
x(t) = Ax(t) { y(t) = Cx(t)
+ Bu(t)
(1)
or, in discrete time, by a set of equations of the form x(t {
+ 1) = y(t)
Ax(t)
= Cx(t)
+ Bu(t)
(2)
where t represents the time variable, x is a vector ~n, U is a vector in the in the state space X input space U = ~m, y is a vector in the output space Y = ~P, and A, B, C are real matrices of suitable dimensions. In both cases, forgetting the physical meaning in terms of states, inputs and outputs of equations (1) and (2), we can say that E is described abstractly by the ordered triple of real matrices (A, B, C). Generalizing this notion, it is then possible, as in (Sontag, 1976), to give the following abstract definition of a dynamical system with coefficients in a commutative ring R.
=
Definition 1. Given a commutative ring R, a linear, dynamical system E with coefficients in R, briefly a system over R, is given by an ordered triple of matrices (A, E, C) with entries in R, whose dimensions are, respectively, n x n, n x rn, p x n. The literature about systems over rings is quite large. Example of problems and applications of systems over rings are found, in particular, in (Kamen, 1976), (Sontag, 1976), (Sontag, 1981), (Conte and Perdon, 1982), (Conte and Perdon, 1995b), (Habets, 1994), (Brewer et al., 1986) and in the references therein. In order to help the intuition, it is convenient to associate to the abstract system E (A, E, C) over the ring R a set of equations of the form
=
x(t {
+ 1) y(t)
= Ax(t) + Eu(t) = Cx(t)
(3)
where t represents an integer variable, x is an element in the free R-module X = Rn, u is an element in the free R-module U = R m and y is an element in the free R-module Y = RP. For systems over rings, free modules play the role of vector spaces in the description of classical dynamical systems and, although a physical interpretation of equations (3) is generally missing, referring to them we can speak, in an obvious way, of states, inputs and outputs and of a discrete-time dynamics for E. Then, it is clear that all notions and problems that concern classical dynamical systems with coefficients in the field of real numbers
2. PRELIMINARIES In control theory, the state space representation of a linear, time-invariant, finite dimensional, dy176
1R can be stated, possibly in an abstract way, and investigated for systems over rings. Although from a technical point of view the study of systems over rings may be more difficult, they are akin to linear systems with real coeficients and, for this reason, the basic ideas of control theory find a natural extension to their framework. This makes them a versatile tool for approaching a number of physically meaningful problems and motivates a concrete interest in them.
the commutative ring R by a set of equations of the form (3), can be defined using the classical discrete-time notion of reach ability and, since the Cayley-Hamilton Theorem holds over a commutative ring, the module of reachable states < A.IImB > turns out to be the submodule of the state module X = Rn spanned by the column of the reach ability matrix
Example 1. Let us consider a parameter dependent, linear dynamical system ~, described by equations of the form
The submodule ImRr; is invariant with respect to the dynamics, but it may happen that no maximal, linearly indepents subset of columns of Rr; can be completed to a basis of X. In such case, a decomposition of I; with respect to reachability is not possible. When, in particular, RE is full rank, it is important to distinguish between the possible situations, as pointed out in the following Definition.
x(t) = A(~)x(t) { y(t) = C(Ox(t)
+ B(~)u(t)
Rr;
(4)
where the entries of A(~), B(~), C(~) are real, continuous functions of a real parameter ~. In spite of the fact that systems of this kind may frequently be necessary for modeling uncertainty or variability of a physical parameter, classical control theory does not offer specific analysis and synthesis tools for dealing with them. However, we can associate to I; the system E', defined by the triple (A(~), B(~), C(~)) over the ring C of real, continous functions of a real variable, and, in this way, we can apply to E the techniques which are avaible for systems over rings. For instance, the problem of stabilizing E for all values of ~ by means of a parameter dependent feedback can be approached by studying the possibility of stabilizing, in a suitable abstract sense, E'.
= (B
AB ... A. n - 1 B).
Definition 2. Let E be a system over the commutative ring R described by a set of equations of the form (3). Then, I; is said:
i) weakly reachable if dim(ImRr;) = dim(X); ii) reachable if ImRE = X. Weak reachability occurs if RE has full rank, while reachability requires the stronger condition that Rr; has a right inverse over R. Reachability, instead of being generic as for systems with coefficients in the field of real numbers ~, is therefore a quite strong property and only weak reachability . . IS genenc.
In dealing with systems over rings, in particular in extending the approaches developed for systems with coefficients in the field of real numbers ~, one has to pay attention to the fact that ring and module algebra is more rich and complicated than linear algebra (see (Atiyha and Macdonald, 1969), (Lang, 1984)). Since nonzero elements in a ring are not necessarily invertible, a linear dependency relation like 2:7=1 ajXj 0 between elements of a free module over a ring R does not imply that each Xi is a linear combination of the remaining ones. So, differently from what happens in the case of vector spaces, we can find sets of generators of a free module from which no basis can be extracted, as well as sets of linearly independent elements of maximal cardinality which are not sets of generators. In particular, we may have submodules of a free module which are not direct summands. From the point of view of dynamical properties, this fact has remarkable consequences, since it may prevent from decomposing a system with respect to dynamically invariant submodules. The situation concerning the submodule of reachable states deserves to be analysed in greater detail. Reachable states for a system E, described over
Example 2. Let us consider a delay-differential system E whose state dynamics is defined by the equation
x(t)
= u(t -
<5)
(5)
where x E ~, u E ~ and <5 is a fixed time delay. Introducing the delay operator b., whose action on a function oftime f(t) is described by b.f(t) = f(t - <5), we can rewrite (5) as x(t) = b.u(t) and associate to E a system E', over the ring ~[b.] of real polynomials in the indeterminate b., whose state dynamics is defined by the equation
=
X(t
+ 1) = b.u(t)
(6)
with x E 1R[~], u E ~[~]. As b. is nonzero but not invertible in ~[b.], I;' is weakly reachable and not reachable. This reflects the fact that a state of E can be reached from 0 only at a time t ~ <5. Example 3. Let us consider a parameter dependent system E defined by equations of the form (4), where the entries of (A(O, B(~), C(~)) are real polynomials in the indeterminate ~. If the 177
corresponding system E' over the ring of real polynomials :R[~l is reachable, so is E for any value of ~. If E' is only weakly reachable, i.e. the reachability matrix R!;, is full rank but not invertible, then E is reachable for almost all, i.e. all but a finite number, values of~.
that all nondecreasing sequences of submodules of a finite dimensional free module converge in a finite number of steps is satisfied by a large class of rings, called l"oetherian rings, which includes, e. g, the ring of integer numbers Z, the rings of polynomials with real coefficients :R[Xl, ... ' Xk], the Principal Ideal Domains (see (Atiyha and Macdonald, 1969), (Lang, 1984)).
In case R is an integral domain, any system over R can also be viewed as a a system over the quotient field 1\ of R. Clearly, weak reachability over R implies reachability over 1\. A concept which turns out ot be usefull in dealing with the phenomena described above is that illustrated in the next Definition.
3. BASIC GEO~IETRIC CONCEPTS
The basic notions of the geometric approach to linear dynamical systems described in (Wonham, 1985) and in (Basile and Marro, 1969) concern subspaces of the state space which have specific invariance properties. Assuming, in the rest of the Section, that R is an integral domain, we can state the Definitions and results that follow.
Definition 3. (Conte and Perdon, 1995b) Let R be an integral domain. Given an R-module X and a submodule V ~ X, the closure of V in X is the submodule V defined by V = {x E X, such that ax E V for some nonzero a ER}. If V ~ X coincides with its closure V, then V is said closed in .\.
Definition 4. (Hautus, 1982) Given a system E, defined over R by equations of the form (3), a submodule V of its state module X is said to be i) (A, B)-invariant, or controlled invariant, if and only if AV ~ V + 1mB; ii) (A, B)-invariant of feedback type if and only if there exists an R-linear map F : X -+ U such that (A + BF)V ~
The closure V of V is the smallest closed submodule of X containing V and dimR V = dimR V. If X has no torsion and V is closed, the quotient module X/V has no torsion. If R is a Principal Ideal Domain, this implies that a closed submodule V is a direct summand of X, i.e. its basis can be completed to a basis of X. The converse, namely that direct summands are closed submodules, is true in general. If R is a PID and X is free, the closure of a submodule V can be algorithmically computed. Let V denote a matrix whose columns are a basis of V and let
V = M [diag{vl
Any feedback F as in ii) above is called a friend of v. For systems with coefficients in a ring, an (A, B)invariant submodule V is not necessarily of feedback type and it cannot be made invariant with respect to a closed loop dynamics, as it happens in the case of systems with coefficients in the field of real numbers !Jr. The geometric notion of (A, B)invariance is weaker than the dynamic notion of feedback type invariance, which is the most important in applications and the most difficolt to check, and, actually, the equivalence between them holds only for direct summands of X (see (Conte and Perdon, 1998)). If R is a PID, (A, B)-invariant submodules of feedback type can be characterized in a quite simple way using the notion of closure.
o·· .,vd] N
be a Smith decomposition. Then, by the columns of M
v.
V is
generated
[~] N.
A further basic difference between ring and module algebra and linear algebra is seen in the behavior of, respectively, nondecreasing or nonincreasing sequences of submodules of a finite dimensional free module and nondecreasing or nonincreasing sequences of subspaces of a finite dimensional vector space. In particular, for a ring R, a sequence {Vdk>o of submodules of, say, Rn, with Vi ~ Vi+l (n-;ndecreasing sequence) or, respectively, Vi 2 Vi+l (non increasing sequence), does not always converge in a finite number of steps, as a sequence of subspaces of a vector space actually does. The requirement that all nonincreasing sequences of submodules of a finite dimensional free module converge in a finite number of steps is quite restrictive and practically unrealistic. On the other hand, the requirement
P1'Oposition 1. (Hautus, 1982) Let R be a PID and let E be a system over R described by equation of the form (3). An (A, B)-invariant submodule V of the state module X of E is of feedback type only if its closure V is (A, B)invariant. Given a submodule K ~ X, there exists a maximum (A, B)-invariant submodule of X contained in K, usually denoted by ).r (K), but there may not be a maximum (A: B)-invariant submodule 178
'R. =< (A + BF)
of feedback type contained in K . The computation of V· (K) is not difficult for systems with coefficients in the field of real numbers !R, since it coincides with the limit of the sequence {Vd defined by
Vo = K Vk+1 = K n A- 1(Vk
and the limit itself is reached in a number of steps lesser than or equal to the dimension of the state space. For systems with coefficients in a ring, the sequence (7), which is not increasing, may not converge in a finite number of steps and, in such case, an algorithm for computing V· (K) is not available. A result useful in the computation of V·(K), in a restrictive situation, is that given by the following Proposition.
'R. =< (A
n=
=
=
i) ii)
A further important geometric concept scribed in the following Definition.
minimum element s.(n) of the family defined by (10) for W =
A pre-controllability submodule n turns out to be a controllability submodule if and only if it is and an (A, B)-invariant of feedback type. As in the situation concerning (A, B)-invariance and invariance of feedback type, we have therefore a geometric notion, namely pre-controllability submodule, which is weaker than the corresponding dynamic notion of controllabilitysubmodule. The family of all pre-controllability submodules of X which are contained in a given submodule K is closed with respect to the sum of submodules. Therefore, it has a maximum element, which is denoted by n·(K). Letting V· denote the maximum (A, B)-invariant submodule of X contained in K, we have that n· (1{) is the minimum element of the family Sv', defined by (10) for }V V·.
(8) 1S
n is (A, B)-invariant; n coincides with the n.
Given a submodule K ~ X, there exists a smallest (C, A)-invariant submodule of X containing K, usually denoted by S·(K). If R is a Noetherian ring, nondecreasing sequences of submodules of X converges in a finite number of steps and S. (K) can be found as the limit of the sequence{ Sk h>o d~Mdby -
So = K
(10)
Definition 7. (Conte and Perdon, 1995a) Given a system E, defined over R by equation of the form (3), a submodule n of its state module X is said to be a pre-controllability submodule if
S).
Sk+1 = K + A(Sk n l{el·G).
nn > .
Since Sw is closed with respect to intersection, it has a unique minimum element, denoted by S. (W), and therefore the following definition makes sense.
Definition 5. Given a system E, defined over R by equations of the form (3), a submodule S of its state module X is said to be (G, A)-invariant, or conditionally invariant, if and only if A(S n ~
lImB
Sw = {S ~ X, such that S = wn (AS + 1mB)}.
Together with the notion of (A, B)-invariance it may be useful to consider that described in the following Definition.
A'erG)
+ BF)
A controllability submodule n is obviously an (A, B)-invariant submodule offeedback type and, in particular, equation (9) holds for every friend F of n. For this reason, the property of being a controllability submodule is quite restrictive and, due to its characterization in dynamic terms, it is very difficult to be checked. A weaker notion, that captures the geometric aspects of the previous one without implying the existence of any friend, turns therefore to be useful. In order to introduce such notion, given a system I:, defined over the Noetherian ring R by equations of the form (3), and a submodule n of its state module X, let us consider the family Sw defined by
Proposition 2. (Conte and Perdon, 1995b) Let R be a PID and let I: be a system over R described by equation of the form (3). Assume that K is closed, and denote by V = i 1k Vi the limit of the sequence {Vdk~o defined by (7). Then, if Vk Vk+l for some k, V Vk V·(J{erG) and {Vk h~o converges in a finite number of stesps to the maximum (A, B)-invariant submodule of K. Otherwise, if Vk :/; Vk+1 for all k, we have dim CV) < dim(K).
=
(9)
Controllability submodules have been considered in (Inaba et al., 1987) for systems over PID's. Following the line of (Wonham, 1985), it can be proved that n is a controllability submodule if and only if there exists a map F : .\ -+ U such that
(7)
+ 1mB)
I ImBG >
de-
Definition 6. Given a system E, defined over R by equations of the form (3), a submodule n of its state module X is said to be a controllability submodule if there exist maps F : X -+ U and G : U -+ U such that
=
179
Assuming that R is a Noetherian ring, we have the following results which allows us to check if a given submodule is a pre-controllability submodule and to compute the maximum pre-controllability submodule contained in a given one.
4.1 Disturbance Decoupling Problem
The noninteracting control problem we consider here is described by the following statement. Problem 1. Let us consider a system E, defined over the Noetherian ring R by equations of the form
Proposition 3. (Conte and Perdon, 19950) Given
a system E, defined over the Noetherian ring R by equations of the form (3), the sequence {Skh>o of submodules of X defined recursively by So = 0 Sk+l = n
n (.4.Sk + 1mB)
x(t + 1) = .4.x(t) { y(t) = Gx(t)
(11)
4. (Assan
xa(t
et al., 1998b) Given a sys-
tem E, defined over the Noetherian ring R by equations of the form (3), the sequence {nk h>o of submodules of X defined recursively by no = S· nKn.4.- l (ImB) nk = S· n K n .4.- l (nk_l + 1mB),
{
+ 1) =
u(t) = Fx(t)
+ H xa(t)
(14)
+G 2 q(t)
A necessary condition for the solution of the DDP is given (see (Wonham, 1985)) by ImD
It is very important to remark that, since the sequence {Rk h>o is nondecreasing, this last Proposition gives a practical, algorithmic way for computing n"(I{erG) if R is a Noetherian ring.
NO~INTERACTI~G
.4. l x(t) + .4. 2 x a(t) +Glq(t)
where X a E X a = Rn., .4. 1 , .4. 2 , F, H, G l and G 2 are matrices of suitable dimensions with entries R, such that the output of the compensated system EF,G does not depend on q.
(12)
where S· = S·(ImB) is the smallest (G,.4.)invariant submodule of X containing 1mB, converges in a finite number of steps to the maximum pre-controllability submodule n·u:) in K.
4.
(13)
where q E Q = Rn is a disturbance. The Disturbance Decoupling Problem (DDP) for E consists in finding an integer n a and a feedback law of the form
converges in a finite numer of steps to S. (R). Proposition
+ Bu(t) + Dq(t)
~
V"
+ 1mB
(15)
where V· denote the maximum (.4., B)-invariant submodule contained in [{erG. Condition (15) is also sufficient for the existence of a static feedback solution, namely a solution of the form (14) with n a = 0, if V· turns out to be of feedback type. Then, (15) characterizes completely the solvability of the DDP for systems with coefficients in the field of real numbers ai and, in turn, this show that the use of dynamic feedback is actually not necessary. For systems over rings, it appears difficult to characterize in geometric terms the existence of a static feedback solution, since, as already remarked, a maximum (.4., B)-invariant of feddback type in [{erG, which, in this case, would be needed, may not exist. The most complete result in this direction is obtained by imposing a restrictive condition on E, namely by requiring that
CONTROL
PROBLEMS The geometric approach is particularly suitable for dealing with a number of problems known as noninteracting control problems (Wonham, 1985), (Basile and Marro, 1969). Basically, they consist in finding a feedback which decouples specific components of the input from specific components of the output of a given system. Additional requirements include in general stability or coefficients assignement. For systems with coefficients in the field of real numbers ai, static feedback solutions exist for several noninteracting control problems. This is not the case for systems with coefficients in a rings, mainly because geometric invariance properties does not imply the existence of appropriate static feed backs. As we will see, it is therefore necessary to employ dynamic feed backs , whose action can be viewed as the action of a static feedback on a suitable dynamic extension of the original system.
V· n 1mB = {O}.
(16)
Condition (16) is verified if different input sequences for E produce different output sequences. Regarding the DDP, the importance of (16) is due to the fact that, if (15) holds, it assures the existence of a smallest element V~ in the lattice [, of the (.4., B)-invariant submodules V of l{ el'G such that ImD ~ V + 1mB. Then, we have the following result. 180
Proposition 5. (Conte and Perdon, 1995b) Given a system ~, defined over the ring R by equations of the form (13), let V· n 1mB = {O}, where V· is the maximum (A, B)-invariant submodule contained in J( erG, be satisfied. Then, the DDP for ~ is solvable with a static feedback, namely a feedback of the form (14) with n a = 0, if and only if 1mD ~ V· + 1mB and v'c, namely the smallest element in the lattice of the (A, B)-invariant submodules V of J( e7'G such that ImD ~ V + 1mB, is of feedback type.
characterization of the solvability conditions of the DDP, they cannot be easily used in practice due to the difficulty in computing V· (I{ erG) we have already mentioned. A solution to this problem comes from a recent results which is related to Proposition 4.
Proposition 8. (Assan et al., 1998a), (Assan et al., 1998b) Given a system ~, defined over the Noetherian ring R by equations of the form (13), denote by RiBD). the ma:imum pre-controlla?ility submodule 111 li. erG With respect to the Input matrix (B D). Then, the DDP for ~ is solvable if and only if ImD ~ 'R(BD)'
.c
The remark made in the previous Section about the properties of the closure of a submodule over a PID R allows us to state also the following result.
It is important to remark that, together with Proposition 4, the above Proposition provides not only a complete characterization of the solvability conditions but also an algorithmic way for computing solutions to the DDP for systems over Noetherian rings.
Proposition 6. (Conte and Perdon, 1995b) Given a system ~, defined over the PIDR by equations of the form (13), let V· n 1mB = {O}, where V· is the maximum (A, B)-invariant submodule contained in f{ erG, be satisfied. Then, the DDP for ~ is solvable with a static feedback, namely a feedback of the form (14) with n a = 0, if and only if 1mD ~ V' + 1mB and the closure V~ of Vc' namely of the smallest element in the lattice .c of the (A, B)-invariant submodules V of J( erG such that ImD ~ V + 1mB, is (A, B)-invariant.
4.2 Block Decoupling Problem The noninteracting control problem we consider here is described by the following statement.
Concerning the computation of Vc, it is interesting to note that, if (16) holds, it coincides with the module V·(J(erG) nS'(ImB + ImD). Further results along this line, obtained relaxing condition (16), are contained in (Assan et al., 1998c). Extending our interest to more general solutions than those consisting of static feed backs, we can circumvent the problems which arise if V' is not of feedback type. By allowing dynamic feedback solutions, we have in facts the following Proposition.
Problem 2. Let us consider a system ~, defined over the Noetherian ring R by equations of the form (3), and assume that the output of E is splitted into k blocks, k ;::: 2. Denoting by Yi a vector in RP', for i = 1, ... , k with L:7=1 Pi = P, the output equation of ~ becomes
Yi(t)
= GiX(t),
i
= 1, ... , k,
(17)
where Gi , for i = 1, ... , k is a Pi x n matrix with entries in R. The Block DecQupling Problem (BDP) for ~ consists in finding an integer n a and a feedback law of the form
Proposition 7. (Conte and Perdon, 1995b) Given a system E, defined over the Noetherian ring R by equations of the form (13), the DDP for ~ is solvable if and only if ImD ~ V· + 1mB, where V' is the maximum (A, B)-invariant submodule contained in J( erG.
Xa(t + 1) = Alx(t) + A:!xa(t) k
+ LGaiVi(t)
i=l
u(t) = Fx(t)
+ Hxa(t)
(18)
k
+ LGiVi(t)
i=1
The key tool in the proof of Proposition 7 consists in finding a dynamic extension ~e of ~, with state module X e = X + X a , in which the original V· gives rise to an (A, B)-invariant submodule of feedback type. In general, to do so, one has to take n a = dimV'. The static feedback which decouples the disturbance from the output of ~e, then, defines a dynamic feedback of the form (14) which decouples the disturbance from the output of ~. Although the above results provide a complete
X a E X a := Rn., Vi E Rnl',i = 1, ... ,k, AI, A:!, F, H, Gi and Gai are matrices of suitable dimensions with entries R, such that in the compensated system ~F,G each block input Vi completely controls the output Yi, but has no influence on the outputs Yj for j =/: i, i = 1, ... , k.
where
In order to state the solvability conditions for the BDP, let us define the suhmodules Ki
=
181
nj=l,j# I{ erCj and let us denote by Ri the maximum pre-controllability submodule contained in k
rankC Rr; = dim(Y)
K;. Then, a necessary condition for the solvability of the BDP is given (see (Wonham, 1985)) by R;+f{erC;=X,
i=I, ... ,k.
or, equivalently, by the fact that C R'f; 1S full row rank, and we modify accordingly the BDP (compare with (Conte et al., 1997)).
(19)
Problem 3. In the same situation as in Problem 2, the Block Decoupling Problem with Weak Output Controllability for E consists in finding an integer n a and a dynamic state feedback law of the form (18) such that in the compensated system EF,G each block input V; weakly controls the output Yi, but has no influence on the outputs Yj for j :j:. i, i = 1, ... , k.
For systems with coefficiens in the field of real numbers lR, under the restrictive hypoyhesis
n k
I{ el'C; = {O},
(20)
;=1
condition (19) implies the existence of a static feedback which solves the problem. Essentially, (19) and (20) guarantee that the various Ri have at least one common friend. Sufficiency of (19), under (20), for systems with coefficients in a ring was proved in (Conte and Perdon, 1995a) (previous partial results were given in (Inaba et a1., 1987)), exploiting the fact that (20) also says that the 'Ri are direct summands of X, and hence they are (A, B)-invariant submodules of feedback type. If (20) is removed, the situation becomes more difficult, no matter if the system has coefficients in a field or in a ring. However, in the last case, the n; cannot be assumed to be (A, B)invariant submodules of feedback type and, in principle, this complicates the problem. Actually, pre-controllability submodules play, in the BDP, a role similar to that played by (A, B)-invariant submodules in the DDP. In facts, if (19) holds, we can construct a suitable dynamic extension E e of E in which the original ni give rise to controllability submodules having a static feedback as common friend. This defines a dynamic feedback of the form (18) which solves the BDP for E. More precisely, the general result we have is the following.
This second formulation of the BDP is more natural in the abstract context of systems over rings and also when these are used for modelling delay-differential systems (see (Sename and Lafay, 1996)) or parameter depending systems. Solvability conditions for the BDP with Weak Output Controllability are given in the following Proposition. Proposition 10. (Conte et al., 1997) Given a system E, defined over the Noetherian ring R by equations of the form (3), (17), let ni, for i = 1, ... , k denote the maximum pre-controllability submodule contained in Ki= n~=l,j;I!i f{ erCj' Then, the BDP with Weak Output Controllability is solvable for E if and only if dim(n;
= dim(X),
i
= 1, ... , k.
An inportant requirement in noninteracting control problem concerns stability or coefficients assignement. For a system E, described over the ring R by equation of the form (3), an abstract notion of stability can be given in terms of the characteristic polyomial p(z) = det(zI - A) of its dynamic matrix, which belongs to the ring R[z] of polynomials in the indeterminate z with coefficients in R. Namely, chosen a multiplicatively closed subset 1£ of R[z] which contains polynomials of all degrees, we say that E is 1£-stable if p(z) belongs to 1£. In this way, 1£, which is called the set of 1£-stable polynomials, play the role of the set of Hurwitz polynomials for systems with cofficients in the field of real numbers lR. In very general terms, the system E is 1£-stabilizable if there exists a feedback that makes the compensated system 1£-stable. Analogously, the system E is coefficients assignable if there exists a feedback that arbitrarily assigns the coefficients in the characteristic polynomial of the dynamic matrix of the compensated system. Coefficients assignability implies stabilizability for any set 1£ of stable polynomials, and it plays therefore a key role in stabilization problems.
Proposition 9. (Conte et al., 1998b) Given a system E, defined over the i\oetherian ring R by equations of the form (3),( 17), let Ri, for i 1, ... , k denote the maximum pre-controllability submodule contained in K;= n~=l,j;I!i f{el·Cj . Then, the BDP for E is solvable if and only if
=
for i
+ f{ erC;)
= 1, ... , k.
It has to be remarked that, in the formulation of the Block Decoupling Problem, controllability of the output implies that the submodule of reachable states I mRr; maps onto the output module. As for reachability, this is a very restrictive condition for systems over rings, since generically, when n 2: p, one has only dim(C( < AllmB > dim(Y). In the same spirit of Definition 2 we will speak, in this last case, of l!'Eak output controllability, characterized by
)) =
182
Coefficients assignability requires reachability and it is implied by that property if dynamic feed backs are allowed. Clearly, this means to have a compensated system whose dynamic matrix is larger than that of the original system E. For this reason, coefficients assignability is better intended as the possibility of finding a suitable dynamic extension E e of E which is coefficients assignable by means of a static feedback. The basic result in this direction is given by the following Proposition.
5. APPLICATIONS TO DELAY-DIFFERENTIAL SYSTEMS System with coefficients in a ring are used for studying control problems which concern delaydifferential systems. This is done, as in Example 2, by associating to a given delay-differential system a system with coefficients in a ring in such a way that the two systems have the same signal flow graph. Then, noninteracting control problems concerning the original delay-differential system can be translated into corresponding problems for the other system and the solutions found for these can be used for deriving solutions to the original problems. In this process, different situations may be originated by different choices of the ring of coefficients. Let us illustrate this facts by means of examples.
Proposition 11. (Emre and Khargonekar, 1982) Given a system E, defined over the ring R by equations of the form (3), it is possible to find a dynamic extension E e of dimension lesser than or equal to n 2 which is coefficient assignable by a static feedback if and only if E is reachable.
The BDP with the additional requirement of coefficients assignement can be stated as follows.
Example 4. Let us consider the delay differential system E defined by the following equations
Problem 4. In the same situation as in Problem 2, the Block Decoupling Problem with Coefficients Assigllement for E consists in finding an integer n a and a dynamic state feedback law of the form (18) such that in the compensated system EF,G each block input Vi controls (or weakly controls, in the case of BDP with weak output controllability) the output Yi, but has no influence on the outputs Yj for j ::j:. i, i = 1, ... , k, and the coefficients of the characteristic polynomial of EF,G are arbitrarily assigned.
xdt) = Xl(t udt X2(t) = xdt udt ydt) = xdt Y2(t) = X2(t)
=
X2(t
+ 1) = .6.1xdt)+ .6. 2Ul(t) + U2(t) ydt) = .6. 1 xdt-) + X2(t)
(22)
with coefficients in the ring of real polynomials in two indeterminates 1R[.6. 1 , .6. 2], We want to investigate if it is possible to decouple line by line the system E and to assign arbitrarily the coefficients of the decoupled system. Since E and E' have the same signal flow graph, this can be done by solving the corresponding problem for E' over the ring 1R[.6. 1 ,.6. 2 ]. On the basis of the results shown in the previous Sections, we are therefore interested in the maximum precontrollability submodules Ri contained, respectively in K i !{ e7'Dj, j::j:. i, for i, j 1,2, with
(or if and only if i
+ 1) = .6. 1 .6. 2xdt) + X2(t)+
Y2(t) = X2(t)
fori=I, ... ,k
+ !\'erCi ) = dim(.l:'),
xdt
.6. 1udt)
Proposition 12. (Conte et al., 1998a) Given a system E, defined over the Noetherian ring R by equations of the form (3), (17), let Ri, for i 1, ... , k denote the maximum pre-controllability submodule contained in Ki= n~:;::l,#i J{ erCj. Assume that E is reachable.Then, the BDP with Coefficients Assignement for E is solvable if and only if
dim(Ri
(21)
Introducing the delay operators .6.i, defined for any function f(t) by !l;J(t) = f(t - oil, we can formally associate to I: the system E' defined by the equations
Combining the above results, it is possible to state the following Proposition.
Ri+J{erCi=X
01 - 02) + X2(t)+ od od+ 02) + U2(t) od + X2(t)
= 1, ... , k
in tha case of BDP with weak output controllability) .
=
=
It is important to remark that, together with
Proposition 4, the above Propositions provides not only a complete characterization of the soh'ability conditions but also an algorithmic way for computing solutions to the BDP for systems over i\oetherian rings.
We have that the submodules Ri, for i = 1,2, are generated, respectively, by the columns of the matrices 183
ticular, the dynamic matrix of the extended compensated system is the matrix Condition Ri + I\ erD; = X, for i = 1,2, is not satisfied, but dimR[~l ,~21 (Ri + I\ er D;) equals dimR[~l '~21X 2 for i 1,2. Therefore the Block Decoupling Problem with Weak Output Controllability and Coefficients Assignment is solvable for the system E'. Since the subm~dules Ri are not (A, B)-invariant submodules of feedback type, they are not controllability submodules. Hence, a solution of the considered BDP requires, in order to define a dynamic feedback, to consider a dynamic extension E~ of E'. Explicit computations, reported in (Conte et al., 2 1998a), show that an extension with n a suffices for finding a solution of the considered BDP. Moreover, E~ can be defined in such a way that it turns out to be coefficients assignable by means of a static state fedback, so that no further extension, as possibly indicated in Proposition 11 is needed. Then, assuming that the desired characteristic polynomial in the extended compensated system is (z + 2)4, further explicit computations (see (Conte et al., 1998a)) allow us to find a feedback matrix F~ of the form
=
=
whose characteristic polynomial is the desired one, and the transfer function matrix of the ex-
tende[de:'~~fensated sy].stem is the diagonal ma-
,
tnx
o - s+2 --
.
(24)
Introducing the delay operator ~, defined for any function f(t) by ~f(t) = f(t - (5), we can formally associate to E the system El defined by the equations
xdt+ 1) X2(t + 1) { y(t)
0 0 113 114]
[
0 e- 262 '
Example 5. Let us consider the delay differential system E defined by the following equations
=
121 fn 123 124 o 0 -2 0 o 0 0 -2
-s+ 2
'
= q(t) = xdt) + ~u(t) = X2(t)
(25)
u(t)
with coefficients in the ring lR[~] of real polynomials in one variable, We want to investigate if it is possible to decouple the disturbance q from the output of 1:. Since 1: and 1: 1 have the same signal flow graph, this can be done by solving the corresponding problem for El over the ring R[~]. Denoting the defining matrices of El by
with
A
= [~ ~] , (B
D)
= [~~ 1 ~]
,
C=[o1], and matrices
[~2]'
G
Gal
=
U],
Gal
=
[n,
G1
=
since the maximum (-4, B)-invariant submodule in I\ erG is V· = span { [ ~ ~ 1 ] }, we have
2= [_~21_ ~i]' These yield a dy-
that ImD is not contained in V· + 1mB. Hence, condition (15) is not satisfied and the DDP for 1: 1 is not solvable with a feedback having coefficients in R. On the other hand, we can see the triple of matrices (.4, (B D), G) as defining a system E 2 over the ring ~[~]r of realisable rational functions in one variable ((Sename and Lafay, 1996)), namely the ring whose elements are of the form :i~j, with p(~), q(~) E lR[~] and q(O) = qo f. O. The systems El and E 2 are different objects, although they are formally defined by the same matrices,
namic feedback law acting on E of the form
xadt) Xa2(t) U1 (t)
= -2x adt) + vdt) = -2X a2(t) + V2(t) = 113xadt) + f14Xa2(t)+ V1 (t)
U2(t)
+ V2(t)
= h1xdt) + h2J.'2(t)+
(23)
h3 Xa1(t) + h4 x a2(t)~2t'1(t) - (~2 + ~i)V2(t)
which decouples the system and assigns the coefficients of the characteristic polynomial. In par184
since the basis rings are different. However, the signal flow graph of L2 is the same as that of ~. Now, for L2 , the maximum (A, B)-invariant submodule in /{ er C is
Basile, G. and G. Marro (1969). Controlled and conditioned invariant subspace in linear system theory. Journal Optimization Theory and Applications, 3. Brewer, J. \V., J.W. Bunce and F.S. Van Vleck (1986). Linear systems over commutative rings. In: Lecture Notes in Pure and Applied Mathematics. Vo!. 104. Marcel Dekker. Conte, G., A.M. Perdon and A. Lombardo (1996). The static and dynamic decoupling problem for systems over a ring. Proceedings of IFAC IVorld Conference, San Francisco, CA. Conte, G., A.M. Perdon and A. Lombardo (1997). The decoupling problem with weak output controllability for systems over a ring. Proceedings IEEE-CDC '97, San Diego,CA. Conte, G., A.M. Perdon and A. Lombardo (19980). Block decoupling problem with coefficient assignment and stability for linear systems over a noetherian ring. Proceedings IFAC Conference on System Structure and Control, Nantes, France. Conte, G., A.M. Perdon and A. Lombardo (1998b). Dynamic feedback decoupling problem for delay-differential systems via systems over rings. IMACS Journal "Mathematics and Computers in Simulation ", special issue "Delay Systems. Conte, G. and A.M. Perdon (1982). Systems over principal ideal domains. a polynomial model approach. SIAM Journal on Control and Optimization, 20. Conte, G. and A.M. Perdon (1995a). The decoupling problem for systems over a ring. Proc. 33rd IEEE CDC '95, New Orleans,LO. Conte, G. and A.M. Perdon (1995b). The disturbance decoupling problem for systems over a ring. SIAM Journal on Control and Optimization. Conte, G. and A.M. Perdon (1998). The block decoupling problem for systems over a ring. to appear on IEEE Trans. AC. Emre, E. and P. Khargonekar (1982). Regulation of split linear systems over rings; coefficient assignment and observers. IEEE Trans. Aut. Control, A C-27. Habets, 1. (1994). Algebraic and computational aspects of time-delay systems. Ph.D. Thesis, Eindhoven University of Technology. Hautus, 11.1.J. (1982). Controlled invariance for systems over a ring. In: Lecture Notes in Control and Inf. Sci .. Vo!. 39. Springer. Inaba, H. and W. \Vaag (1996). Block decoupling for linear systems over rings. Linear Algebra and Its Applications. Inaba, H., N. Ito and T. Munaka (1987). Decoupling and pole assignment for linear systems defined over a principal ideal domain. Linear Circuits, Systems and Signal Processing: theory and applications.
~
V' = span { [ ] } . The difference with the previous sistuation is due to the fact that in ?R[~lr the element ~ - 1 is invertible. So, the solvability condition (15) for the DDP is satisfied and, in particular, u(t) = ~u(t)
is a solution to the DDP for It is now easy to see that
+ xdt)
~2'
u(t) = u(t - 6)
+ xdt)
is a solution for the DDP concerning the original delay-differential system E.
6. CONCLUSION Althoug some problems, mainly concerning the possibility of defining algorithmic procedures for constructing submodules with specific geometric properties, are still to be solved, the geometric approach to systems with coefficients in a ring appears capable of providing satisfactory solutions to many interesting control problems. In turn, this offers valid tools for dealing with delay-differential systems and with families of systems depending on parameters using algebraic methods. Further developments, besides adding new technical results, should concentrate on the construction of suitable algorithms, based on computer algebra, for implementing design procedures involving systems with coefficients in a ring.
7. REFERENCES Assan, J., J.F. Lafay and A.M. Perdon (19980). An algorithm to compute the maximum controllability submodule over a principal ideal domain. Proceedings IFAC Workshop on Linear Time Delay Systems, Grenoble, France. Assan, J., J.F. Lafay and A.~1. Perdon (1998b). Computing the maximum pre-controllability submodule over a noetherian ring. submit to Systems and Control Letters. Assan, J., J.F. Lafay and A.M. Perdon (1998c). On feedback invariance properties for systems over a principal ideal domain. in press on IEEE Transactions on Automatic Control. Atiyha, M.F. and I.G. i'.'lacdonald (1969). Introduction to commutatil'e algebra. AddisonWesley. 185
hamen, E. (1976). Lectures on algebraic system theory. linear systems over rings. NASA Contractor Report 3016. Lang, S. (1984). Algebra. Addison-Wesley. Sename, O. and J.F. Lafay (1996). Decoupling of square linear systems with delays. to appear in IEEE Automatic Control. Sontag, E. (1976). Linear systems over rings control : a survey. Ricerche di A utomatica, 7. Sontag, E. (1981). Linear systems over rings: a (partial) update survey. Proc. IFA C/81 , Eyoto. Wonham, t\L (1985). Linear multit'ariable control: a geometric approach. Springer Verlag.
186
SYSTEMS OVER RINGS: GEOMETRIC THEORY AND APPLICATIONS G. Conte' A. M. Perdon ••
• Dipartimento di Elettronica e Automatica, Universita' di Ancona, Via Brecce Bianche, 60131 Ancona, Italy. [email protected] Dipartimento di lvIatematica "V. Volterra ", Universita' di Ancona, Via Brecce Bianche, 60131 Ancona, Italy, [email protected]
Abstract: The aim of this paper is to present an overview of the geometric approach to the study of dynamical systems with coefficients in a ring and, in particular, to discuss the most recent results concerning noninteracting control problems it has produced. The study of systems with coefficients in a ring is motivated, in addition to an intrinsic interest on the subject, by the fact that they may be viewed as tools for investigating problems which concern parameter dependent systems or delay-differential systems. The geometric approach is shown to provide complete characterizations of the solvability conditions, together with feasible procedures for computing solutions, of noninteracting control problems, like the Disturbance Decoupling Problem and the Block Decoupling Problem. Finally, some examples which clarify the applications to problems concerning delay-differential systems are presented. Copyrighr© 1998 IFAC Resume : Ce travailpresente les resultats plus recentes obtenus par l'approche geometrique dans le cadre des systemes sur les anneaux. Les systemes sur les anneaux sont utiles comme modeles pour etudier des nombreuses classes des systemes, comme les systemes a retard et les familles des systemes lineaires qui dependent d'un parametre. L'approche geometrique permet de resoudre plusieurs problemes de commande, tel que le rejet de perturbation et le decouplage par blocks avec stabilisation. Des nouveux algorithms, qui permettent de construire pratiquement les solutions cherches, sont aussi presentes. Keywords: Systems over rings, geometric approach, noninteracting control problems
1. E,TRODUCTIO;'\
involve parameter dependent systems or delaydifferential systems. The advantages one has in using models with coefficients in a ring originate from the possibility of deriving, in a natural way, methods and techniques from the framework of systems with coefficients in the field ~ of real numbers and of using the powerful mathematical tools offered by ring and module algebra. This produced a research effort for extending to the
Dynamical systems with coefficients in a ring represent a natural generalization of linear, timeinvariant, finite dimensional, dynamical systems with coefficients in the field 1}( of real numbers. Early papers (see (I\:amen, 1976), (Sontag, 1976), (Sontag, 1981)) pointed out the potential use of systems with coefficients in a ring as an abstract tool for dealing with control problems which
175
namical system 1; with coefficients in the field of real numbers ~ is given, in continuous time, by a set of equations of the form
framework of systems with coefficients in a ring resuI ts and techniques from the framework of linear, dynamical systems with coefficients in the field ~ of real numbers. In recent years, this work turned out to be particularly successful for what concerns the development of a geometric approach and its applications to a number of control problems (see (Hautus, 1982), (Inaba et al., 1987), (Inaba and Wang, 1996), (Conte and Perdon, 1995a),(Conte and Perdon, 1995b),(Conte et al., 1996),(Conte et al., 1998b), (Conte et al., 1998b),(Conte et al., 1998a), (Conte and Perdon, 1998)). In the paper, after describing the notion of dynamical system with coefficients in a ring and illustrating some particular features due to ring and module algebra, we analyse the basic notions of the geometric approach in its extension to systems with coefficients in a ring. The main difference \vith respect to the situation encountered in the classical geometric approach (see (Wonham, 1985), (Basile and Marro, 1969)) is that dynamic properties based on the existence of feed backs with special features cannot be characterized by means of purely geometric conditions. For example, controlled invariance, which is a geometric property, does not characterize invariance with respect to a compensated dynamics. On the other hand, geometric conditions are sufficient for assuring the existence of feed backs with special features which act on a system extension. Therefore, from the point of view of the geometric approach, there is a stricter analogy between the situation concerning systems with coefficients in a ring and systems with coefficients in the field ~ of real numbers if, in the ring framework, one considers dynamic properties based on the existence of dynamic feed backs. By exploiting this fact, we show that it is possible to extend, in a suitable way, a large number of techniques and results based on the geometric approach. We illustrate this by discussing in particular a class of noninteracting control problems which includes the Disturbance Decoupling Problem and the Block Decoupling Problem with stability or coefficients assignements. The geometric approach is shown to offer satisfactory solutions, as well as feasible procedure for finding them, to those problems for systems with coefficients in quite general rings. Finally, we discuss some example which clarify how systems with coefficients in a ring and the results obtained by means of the geometric approach can be used for studying delay-differential systems and related control problems.
x(t) = Ax(t) { y(t) = Cx(t)
+ Bu(t)
(1)
or, in discrete time, by a set of equations of the form x(t {
+ 1) = y(t)
Ax(t)
= Cx(t)
+ Bu(t)
(2)
where t represents the time variable, x is a vector ~n, U is a vector in the in the state space X input space U = ~m, y is a vector in the output space Y = ~P, and A, B, C are real matrices of suitable dimensions. In both cases, forgetting the physical meaning in terms of states, inputs and outputs of equations (1) and (2), we can say that E is described abstractly by the ordered triple of real matrices (A, B, C). Generalizing this notion, it is then possible, as in (Sontag, 1976), to give the following abstract definition of a dynamical system with coefficients in a commutative ring R.
=
Definition 1. Given a commutative ring R, a linear, dynamical system E with coefficients in R, briefly a system over R, is given by an ordered triple of matrices (A, E, C) with entries in R, whose dimensions are, respectively, n x n, n x rn, p x n. The literature about systems over rings is quite large. Example of problems and applications of systems over rings are found, in particular, in (Kamen, 1976), (Sontag, 1976), (Sontag, 1981), (Conte and Perdon, 1982), (Conte and Perdon, 1995b), (Habets, 1994), (Brewer et al., 1986) and in the references therein. In order to help the intuition, it is convenient to associate to the abstract system E (A, E, C) over the ring R a set of equations of the form
=
x(t {
+ 1) y(t)
= Ax(t) + Eu(t) = Cx(t)
(3)
where t represents an integer variable, x is an element in the free R-module X = Rn, u is an element in the free R-module U = R m and y is an element in the free R-module Y = RP. For systems over rings, free modules play the role of vector spaces in the description of classical dynamical systems and, although a physical interpretation of equations (3) is generally missing, referring to them we can speak, in an obvious way, of states, inputs and outputs and of a discrete-time dynamics for E. Then, it is clear that all notions and problems that concern classical dynamical systems with coefficients in the field of real numbers
2. PRELIMINARIES In control theory, the state space representation of a linear, time-invariant, finite dimensional, dy176
1R can be stated, possibly in an abstract way, and investigated for systems over rings. Although from a technical point of view the study of systems over rings may be more difficult, they are akin to linear systems with real coeficients and, for this reason, the basic ideas of control theory find a natural extension to their framework. This makes them a versatile tool for approaching a number of physically meaningful problems and motivates a concrete interest in them.
the commutative ring R by a set of equations of the form (3), can be defined using the classical discrete-time notion of reach ability and, since the Cayley-Hamilton Theorem holds over a commutative ring, the module of reachable states < A.IImB > turns out to be the submodule of the state module X = Rn spanned by the column of the reach ability matrix
Example 1. Let us consider a parameter dependent, linear dynamical system ~, described by equations of the form
The submodule ImRr; is invariant with respect to the dynamics, but it may happen that no maximal, linearly indepents subset of columns of Rr; can be completed to a basis of X. In such case, a decomposition of I; with respect to reachability is not possible. When, in particular, RE is full rank, it is important to distinguish between the possible situations, as pointed out in the following Definition.
x(t) = A(~)x(t) { y(t) = C(Ox(t)
+ B(~)u(t)
Rr;
(4)
where the entries of A(~), B(~), C(~) are real, continuous functions of a real parameter ~. In spite of the fact that systems of this kind may frequently be necessary for modeling uncertainty or variability of a physical parameter, classical control theory does not offer specific analysis and synthesis tools for dealing with them. However, we can associate to I; the system E', defined by the triple (A(~), B(~), C(~)) over the ring C of real, continous functions of a real variable, and, in this way, we can apply to E the techniques which are avaible for systems over rings. For instance, the problem of stabilizing E for all values of ~ by means of a parameter dependent feedback can be approached by studying the possibility of stabilizing, in a suitable abstract sense, E'.
= (B
AB ... A. n - 1 B).
Definition 2. Let E be a system over the commutative ring R described by a set of equations of the form (3). Then, I; is said:
i) weakly reachable if dim(ImRr;) = dim(X); ii) reachable if ImRE = X. Weak reachability occurs if RE has full rank, while reachability requires the stronger condition that Rr; has a right inverse over R. Reachability, instead of being generic as for systems with coefficients in the field of real numbers ~, is therefore a quite strong property and only weak reachability . . IS genenc.
In dealing with systems over rings, in particular in extending the approaches developed for systems with coefficients in the field of real numbers ~, one has to pay attention to the fact that ring and module algebra is more rich and complicated than linear algebra (see (Atiyha and Macdonald, 1969), (Lang, 1984)). Since nonzero elements in a ring are not necessarily invertible, a linear dependency relation like 2:7=1 ajXj 0 between elements of a free module over a ring R does not imply that each Xi is a linear combination of the remaining ones. So, differently from what happens in the case of vector spaces, we can find sets of generators of a free module from which no basis can be extracted, as well as sets of linearly independent elements of maximal cardinality which are not sets of generators. In particular, we may have submodules of a free module which are not direct summands. From the point of view of dynamical properties, this fact has remarkable consequences, since it may prevent from decomposing a system with respect to dynamically invariant submodules. The situation concerning the submodule of reachable states deserves to be analysed in greater detail. Reachable states for a system E, described over
Example 2. Let us consider a delay-differential system E whose state dynamics is defined by the equation
x(t)
= u(t -
<5)
(5)
where x E ~, u E ~ and <5 is a fixed time delay. Introducing the delay operator b., whose action on a function oftime f(t) is described by b.f(t) = f(t - <5), we can rewrite (5) as x(t) = b.u(t) and associate to E a system E', over the ring ~[b.] of real polynomials in the indeterminate b., whose state dynamics is defined by the equation
=
X(t
+ 1) = b.u(t)
(6)
with x E 1R[~], u E ~[~]. As b. is nonzero but not invertible in ~[b.], I;' is weakly reachable and not reachable. This reflects the fact that a state of E can be reached from 0 only at a time t ~ <5. Example 3. Let us consider a parameter dependent system E defined by equations of the form (4), where the entries of (A(O, B(~), C(~)) are real polynomials in the indeterminate ~. If the 177
corresponding system E' over the ring of real polynomials :R[~l is reachable, so is E for any value of ~. If E' is only weakly reachable, i.e. the reachability matrix R!;, is full rank but not invertible, then E is reachable for almost all, i.e. all but a finite number, values of~.
that all nondecreasing sequences of submodules of a finite dimensional free module converge in a finite number of steps is satisfied by a large class of rings, called l"oetherian rings, which includes, e. g, the ring of integer numbers Z, the rings of polynomials with real coefficients :R[Xl, ... ' Xk], the Principal Ideal Domains (see (Atiyha and Macdonald, 1969), (Lang, 1984)).
In case R is an integral domain, any system over R can also be viewed as a a system over the quotient field 1\ of R. Clearly, weak reachability over R implies reachability over 1\. A concept which turns out ot be usefull in dealing with the phenomena described above is that illustrated in the next Definition.
3. BASIC GEO~IETRIC CONCEPTS
The basic notions of the geometric approach to linear dynamical systems described in (Wonham, 1985) and in (Basile and Marro, 1969) concern subspaces of the state space which have specific invariance properties. Assuming, in the rest of the Section, that R is an integral domain, we can state the Definitions and results that follow.
Definition 3. (Conte and Perdon, 1995b) Let R be an integral domain. Given an R-module X and a submodule V ~ X, the closure of V in X is the submodule V defined by V = {x E X, such that ax E V for some nonzero a ER}. If V ~ X coincides with its closure V, then V is said closed in .\.
Definition 4. (Hautus, 1982) Given a system E, defined over R by equations of the form (3), a submodule V of its state module X is said to be i) (A, B)-invariant, or controlled invariant, if and only if AV ~ V + 1mB; ii) (A, B)-invariant of feedback type if and only if there exists an R-linear map F : X -+ U such that (A + BF)V ~
The closure V of V is the smallest closed submodule of X containing V and dimR V = dimR V. If X has no torsion and V is closed, the quotient module X/V has no torsion. If R is a Principal Ideal Domain, this implies that a closed submodule V is a direct summand of X, i.e. its basis can be completed to a basis of X. The converse, namely that direct summands are closed submodules, is true in general. If R is a PID and X is free, the closure of a submodule V can be algorithmically computed. Let V denote a matrix whose columns are a basis of V and let
V = M [diag{vl
Any feedback F as in ii) above is called a friend of v. For systems with coefficients in a ring, an (A, B)invariant submodule V is not necessarily of feedback type and it cannot be made invariant with respect to a closed loop dynamics, as it happens in the case of systems with coefficients in the field of real numbers !Jr. The geometric notion of (A, B)invariance is weaker than the dynamic notion of feedback type invariance, which is the most important in applications and the most difficolt to check, and, actually, the equivalence between them holds only for direct summands of X (see (Conte and Perdon, 1998)). If R is a PID, (A, B)-invariant submodules of feedback type can be characterized in a quite simple way using the notion of closure.
o·· .,vd] N
be a Smith decomposition. Then, by the columns of M
v.
V is
generated
[~] N.
A further basic difference between ring and module algebra and linear algebra is seen in the behavior of, respectively, nondecreasing or nonincreasing sequences of submodules of a finite dimensional free module and nondecreasing or nonincreasing sequences of subspaces of a finite dimensional vector space. In particular, for a ring R, a sequence {Vdk>o of submodules of, say, Rn, with Vi ~ Vi+l (n-;ndecreasing sequence) or, respectively, Vi 2 Vi+l (non increasing sequence), does not always converge in a finite number of steps, as a sequence of subspaces of a vector space actually does. The requirement that all nonincreasing sequences of submodules of a finite dimensional free module converge in a finite number of steps is quite restrictive and practically unrealistic. On the other hand, the requirement
P1'Oposition 1. (Hautus, 1982) Let R be a PID and let E be a system over R described by equation of the form (3). An (A, B)-invariant submodule V of the state module X of E is of feedback type only if its closure V is (A, B)invariant. Given a submodule K ~ X, there exists a maximum (A, B)-invariant submodule of X contained in K, usually denoted by ).r (K), but there may not be a maximum (A: B)-invariant submodule 178
'R. =< (A + BF)
of feedback type contained in K . The computation of V· (K) is not difficult for systems with coefficients in the field of real numbers !R, since it coincides with the limit of the sequence {Vd defined by
Vo = K Vk+1 = K n A- 1(Vk
and the limit itself is reached in a number of steps lesser than or equal to the dimension of the state space. For systems with coefficients in a ring, the sequence (7), which is not increasing, may not converge in a finite number of steps and, in such case, an algorithm for computing V· (K) is not available. A result useful in the computation of V·(K), in a restrictive situation, is that given by the following Proposition.
'R. =< (A
n=
=
=
i) ii)
A further important geometric concept scribed in the following Definition.
minimum element s.(n) of the family defined by (10) for W =
A pre-controllability submodule n turns out to be a controllability submodule if and only if it is and an (A, B)-invariant of feedback type. As in the situation concerning (A, B)-invariance and invariance of feedback type, we have therefore a geometric notion, namely pre-controllability submodule, which is weaker than the corresponding dynamic notion of controllabilitysubmodule. The family of all pre-controllability submodules of X which are contained in a given submodule K is closed with respect to the sum of submodules. Therefore, it has a maximum element, which is denoted by n·(K). Letting V· denote the maximum (A, B)-invariant submodule of X contained in K, we have that n· (1{) is the minimum element of the family Sv', defined by (10) for }V V·.
(8) 1S
n is (A, B)-invariant; n coincides with the n.
Given a submodule K ~ X, there exists a smallest (C, A)-invariant submodule of X containing K, usually denoted by S·(K). If R is a Noetherian ring, nondecreasing sequences of submodules of X converges in a finite number of steps and S. (K) can be found as the limit of the sequence{ Sk h>o d~Mdby -
So = K
(10)
Definition 7. (Conte and Perdon, 1995a) Given a system E, defined over R by equation of the form (3), a submodule n of its state module X is said to be a pre-controllability submodule if
S).
Sk+1 = K + A(Sk n l{el·G).
nn > .
Since Sw is closed with respect to intersection, it has a unique minimum element, denoted by S. (W), and therefore the following definition makes sense.
Definition 5. Given a system E, defined over R by equations of the form (3), a submodule S of its state module X is said to be (G, A)-invariant, or conditionally invariant, if and only if A(S n ~
lImB
Sw = {S ~ X, such that S = wn (AS + 1mB)}.
Together with the notion of (A, B)-invariance it may be useful to consider that described in the following Definition.
A'erG)
+ BF)
A controllability submodule n is obviously an (A, B)-invariant submodule offeedback type and, in particular, equation (9) holds for every friend F of n. For this reason, the property of being a controllability submodule is quite restrictive and, due to its characterization in dynamic terms, it is very difficult to be checked. A weaker notion, that captures the geometric aspects of the previous one without implying the existence of any friend, turns therefore to be useful. In order to introduce such notion, given a system I:, defined over the Noetherian ring R by equations of the form (3), and a submodule n of its state module X, let us consider the family Sw defined by
Proposition 2. (Conte and Perdon, 1995b) Let R be a PID and let I: be a system over R described by equation of the form (3). Assume that K is closed, and denote by V = i 1k Vi the limit of the sequence {Vdk~o defined by (7). Then, if Vk Vk+l for some k, V Vk V·(J{erG) and {Vk h~o converges in a finite number of stesps to the maximum (A, B)-invariant submodule of K. Otherwise, if Vk :/; Vk+1 for all k, we have dim CV) < dim(K).
=
(9)
Controllability submodules have been considered in (Inaba et al., 1987) for systems over PID's. Following the line of (Wonham, 1985), it can be proved that n is a controllability submodule if and only if there exists a map F : .\ -+ U such that
(7)
+ 1mB)
I ImBG >
de-
Definition 6. Given a system E, defined over R by equations of the form (3), a submodule n of its state module X is said to be a controllability submodule if there exist maps F : X -+ U and G : U -+ U such that
=
179
Assuming that R is a Noetherian ring, we have the following results which allows us to check if a given submodule is a pre-controllability submodule and to compute the maximum pre-controllability submodule contained in a given one.
4.1 Disturbance Decoupling Problem
The noninteracting control problem we consider here is described by the following statement. Problem 1. Let us consider a system E, defined over the Noetherian ring R by equations of the form
Proposition 3. (Conte and Perdon, 19950) Given
a system E, defined over the Noetherian ring R by equations of the form (3), the sequence {Skh>o of submodules of X defined recursively by So = 0 Sk+l = n
n (.4.Sk + 1mB)
x(t + 1) = .4.x(t) { y(t) = Gx(t)
(11)
4. (Assan
xa(t
et al., 1998b) Given a sys-
tem E, defined over the Noetherian ring R by equations of the form (3), the sequence {nk h>o of submodules of X defined recursively by no = S· nKn.4.- l (ImB) nk = S· n K n .4.- l (nk_l + 1mB),
{
+ 1) =
u(t) = Fx(t)
+ H xa(t)
(14)
+G 2 q(t)
A necessary condition for the solution of the DDP is given (see (Wonham, 1985)) by ImD
It is very important to remark that, since the sequence {Rk h>o is nondecreasing, this last Proposition gives a practical, algorithmic way for computing n"(I{erG) if R is a Noetherian ring.
NO~INTERACTI~G
.4. l x(t) + .4. 2 x a(t) +Glq(t)
where X a E X a = Rn., .4. 1 , .4. 2 , F, H, G l and G 2 are matrices of suitable dimensions with entries R, such that the output of the compensated system EF,G does not depend on q.
(12)
where S· = S·(ImB) is the smallest (G,.4.)invariant submodule of X containing 1mB, converges in a finite number of steps to the maximum pre-controllability submodule n·u:) in K.
4.
(13)
where q E Q = Rn is a disturbance. The Disturbance Decoupling Problem (DDP) for E consists in finding an integer n a and a feedback law of the form
converges in a finite numer of steps to S. (R). Proposition
+ Bu(t) + Dq(t)
~
V"
+ 1mB
(15)
where V· denote the maximum (.4., B)-invariant submodule contained in [{erG. Condition (15) is also sufficient for the existence of a static feedback solution, namely a solution of the form (14) with n a = 0, if V· turns out to be of feedback type. Then, (15) characterizes completely the solvability of the DDP for systems with coefficients in the field of real numbers ai and, in turn, this show that the use of dynamic feedback is actually not necessary. For systems over rings, it appears difficult to characterize in geometric terms the existence of a static feedback solution, since, as already remarked, a maximum (.4., B)-invariant of feddback type in [{erG, which, in this case, would be needed, may not exist. The most complete result in this direction is obtained by imposing a restrictive condition on E, namely by requiring that
CONTROL
PROBLEMS The geometric approach is particularly suitable for dealing with a number of problems known as noninteracting control problems (Wonham, 1985), (Basile and Marro, 1969). Basically, they consist in finding a feedback which decouples specific components of the input from specific components of the output of a given system. Additional requirements include in general stability or coefficients assignement. For systems with coefficients in the field of real numbers ai, static feedback solutions exist for several noninteracting control problems. This is not the case for systems with coefficients in a rings, mainly because geometric invariance properties does not imply the existence of appropriate static feed backs. As we will see, it is therefore necessary to employ dynamic feed backs , whose action can be viewed as the action of a static feedback on a suitable dynamic extension of the original system.
V· n 1mB = {O}.
(16)
Condition (16) is verified if different input sequences for E produce different output sequences. Regarding the DDP, the importance of (16) is due to the fact that, if (15) holds, it assures the existence of a smallest element V~ in the lattice [, of the (.4., B)-invariant submodules V of l{ el'G such that ImD ~ V + 1mB. Then, we have the following result. 180
Proposition 5. (Conte and Perdon, 1995b) Given a system ~, defined over the ring R by equations of the form (13), let V· n 1mB = {O}, where V· is the maximum (A, B)-invariant submodule contained in J( erG, be satisfied. Then, the DDP for ~ is solvable with a static feedback, namely a feedback of the form (14) with n a = 0, if and only if 1mD ~ V· + 1mB and v'c, namely the smallest element in the lattice of the (A, B)-invariant submodules V of J( e7'G such that ImD ~ V + 1mB, is of feedback type.
characterization of the solvability conditions of the DDP, they cannot be easily used in practice due to the difficulty in computing V· (I{ erG) we have already mentioned. A solution to this problem comes from a recent results which is related to Proposition 4.
Proposition 8. (Assan et al., 1998a), (Assan et al., 1998b) Given a system ~, defined over the Noetherian ring R by equations of the form (13), denote by RiBD). the ma:imum pre-controlla?ility submodule 111 li. erG With respect to the Input matrix (B D). Then, the DDP for ~ is solvable if and only if ImD ~ 'R(BD)'
.c
The remark made in the previous Section about the properties of the closure of a submodule over a PID R allows us to state also the following result.
It is important to remark that, together with Proposition 4, the above Proposition provides not only a complete characterization of the solvability conditions but also an algorithmic way for computing solutions to the DDP for systems over Noetherian rings.
Proposition 6. (Conte and Perdon, 1995b) Given a system ~, defined over the PIDR by equations of the form (13), let V· n 1mB = {O}, where V· is the maximum (A, B)-invariant submodule contained in f{ erG, be satisfied. Then, the DDP for ~ is solvable with a static feedback, namely a feedback of the form (14) with n a = 0, if and only if 1mD ~ V' + 1mB and the closure V~ of Vc' namely of the smallest element in the lattice .c of the (A, B)-invariant submodules V of J( erG such that ImD ~ V + 1mB, is (A, B)-invariant.
4.2 Block Decoupling Problem The noninteracting control problem we consider here is described by the following statement.
Concerning the computation of Vc, it is interesting to note that, if (16) holds, it coincides with the module V·(J(erG) nS'(ImB + ImD). Further results along this line, obtained relaxing condition (16), are contained in (Assan et al., 1998c). Extending our interest to more general solutions than those consisting of static feed backs, we can circumvent the problems which arise if V' is not of feedback type. By allowing dynamic feedback solutions, we have in facts the following Proposition.
Problem 2. Let us consider a system ~, defined over the Noetherian ring R by equations of the form (3), and assume that the output of E is splitted into k blocks, k ;::: 2. Denoting by Yi a vector in RP', for i = 1, ... , k with L:7=1 Pi = P, the output equation of ~ becomes
Yi(t)
= GiX(t),
i
= 1, ... , k,
(17)
where Gi , for i = 1, ... , k is a Pi x n matrix with entries in R. The Block DecQupling Problem (BDP) for ~ consists in finding an integer n a and a feedback law of the form
Proposition 7. (Conte and Perdon, 1995b) Given a system E, defined over the Noetherian ring R by equations of the form (13), the DDP for ~ is solvable if and only if ImD ~ V· + 1mB, where V' is the maximum (A, B)-invariant submodule contained in J( erG.
Xa(t + 1) = Alx(t) + A:!xa(t) k
+ LGaiVi(t)
i=l
u(t) = Fx(t)
+ Hxa(t)
(18)
k
+ LGiVi(t)
i=1
The key tool in the proof of Proposition 7 consists in finding a dynamic extension ~e of ~, with state module X e = X + X a , in which the original V· gives rise to an (A, B)-invariant submodule of feedback type. In general, to do so, one has to take n a = dimV'. The static feedback which decouples the disturbance from the output of ~e, then, defines a dynamic feedback of the form (14) which decouples the disturbance from the output of ~. Although the above results provide a complete
X a E X a := Rn., Vi E Rnl',i = 1, ... ,k, AI, A:!, F, H, Gi and Gai are matrices of suitable dimensions with entries R, such that in the compensated system ~F,G each block input Vi completely controls the output Yi, but has no influence on the outputs Yj for j =/: i, i = 1, ... , k.
where
In order to state the solvability conditions for the BDP, let us define the suhmodules Ki
=
181
nj=l,j# I{ erCj and let us denote by Ri the maximum pre-controllability submodule contained in k
rankC Rr; = dim(Y)
K;. Then, a necessary condition for the solvability of the BDP is given (see (Wonham, 1985)) by R;+f{erC;=X,
i=I, ... ,k.
or, equivalently, by the fact that C R'f; 1S full row rank, and we modify accordingly the BDP (compare with (Conte et al., 1997)).
(19)
Problem 3. In the same situation as in Problem 2, the Block Decoupling Problem with Weak Output Controllability for E consists in finding an integer n a and a dynamic state feedback law of the form (18) such that in the compensated system EF,G each block input V; weakly controls the output Yi, but has no influence on the outputs Yj for j :j:. i, i = 1, ... , k.
For systems with coefficiens in the field of real numbers lR, under the restrictive hypoyhesis
n k
I{ el'C; = {O},
(20)
;=1
condition (19) implies the existence of a static feedback which solves the problem. Essentially, (19) and (20) guarantee that the various Ri have at least one common friend. Sufficiency of (19), under (20), for systems with coefficients in a ring was proved in (Conte and Perdon, 1995a) (previous partial results were given in (Inaba et a1., 1987)), exploiting the fact that (20) also says that the 'Ri are direct summands of X, and hence they are (A, B)-invariant submodules of feedback type. If (20) is removed, the situation becomes more difficult, no matter if the system has coefficients in a field or in a ring. However, in the last case, the n; cannot be assumed to be (A, B)invariant submodules of feedback type and, in principle, this complicates the problem. Actually, pre-controllability submodules play, in the BDP, a role similar to that played by (A, B)-invariant submodules in the DDP. In facts, if (19) holds, we can construct a suitable dynamic extension E e of E in which the original ni give rise to controllability submodules having a static feedback as common friend. This defines a dynamic feedback of the form (18) which solves the BDP for E. More precisely, the general result we have is the following.
This second formulation of the BDP is more natural in the abstract context of systems over rings and also when these are used for modelling delay-differential systems (see (Sename and Lafay, 1996)) or parameter depending systems. Solvability conditions for the BDP with Weak Output Controllability are given in the following Proposition. Proposition 10. (Conte et al., 1997) Given a system E, defined over the Noetherian ring R by equations of the form (3), (17), let ni, for i = 1, ... , k denote the maximum pre-controllability submodule contained in Ki= n~=l,j;I!i f{ erCj' Then, the BDP with Weak Output Controllability is solvable for E if and only if dim(n;
= dim(X),
i
= 1, ... , k.
An inportant requirement in noninteracting control problem concerns stability or coefficients assignement. For a system E, described over the ring R by equation of the form (3), an abstract notion of stability can be given in terms of the characteristic polyomial p(z) = det(zI - A) of its dynamic matrix, which belongs to the ring R[z] of polynomials in the indeterminate z with coefficients in R. Namely, chosen a multiplicatively closed subset 1£ of R[z] which contains polynomials of all degrees, we say that E is 1£-stable if p(z) belongs to 1£. In this way, 1£, which is called the set of 1£-stable polynomials, play the role of the set of Hurwitz polynomials for systems with cofficients in the field of real numbers lR. In very general terms, the system E is 1£-stabilizable if there exists a feedback that makes the compensated system 1£-stable. Analogously, the system E is coefficients assignable if there exists a feedback that arbitrarily assigns the coefficients in the characteristic polynomial of the dynamic matrix of the compensated system. Coefficients assignability implies stabilizability for any set 1£ of stable polynomials, and it plays therefore a key role in stabilization problems.
Proposition 9. (Conte et al., 1998b) Given a system E, defined over the i\oetherian ring R by equations of the form (3),( 17), let Ri, for i 1, ... , k denote the maximum pre-controllability submodule contained in K;= n~=l,j;I!i f{el·Cj . Then, the BDP for E is solvable if and only if
=
for i
+ f{ erC;)
= 1, ... , k.
It has to be remarked that, in the formulation of the Block Decoupling Problem, controllability of the output implies that the submodule of reachable states I mRr; maps onto the output module. As for reachability, this is a very restrictive condition for systems over rings, since generically, when n 2: p, one has only dim(C( < AllmB > dim(Y). In the same spirit of Definition 2 we will speak, in this last case, of l!'Eak output controllability, characterized by
)) =
182
Coefficients assignability requires reachability and it is implied by that property if dynamic feed backs are allowed. Clearly, this means to have a compensated system whose dynamic matrix is larger than that of the original system E. For this reason, coefficients assignability is better intended as the possibility of finding a suitable dynamic extension E e of E which is coefficients assignable by means of a static feedback. The basic result in this direction is given by the following Proposition.
5. APPLICATIONS TO DELAY-DIFFERENTIAL SYSTEMS System with coefficients in a ring are used for studying control problems which concern delaydifferential systems. This is done, as in Example 2, by associating to a given delay-differential system a system with coefficients in a ring in such a way that the two systems have the same signal flow graph. Then, noninteracting control problems concerning the original delay-differential system can be translated into corresponding problems for the other system and the solutions found for these can be used for deriving solutions to the original problems. In this process, different situations may be originated by different choices of the ring of coefficients. Let us illustrate this facts by means of examples.
Proposition 11. (Emre and Khargonekar, 1982) Given a system E, defined over the ring R by equations of the form (3), it is possible to find a dynamic extension E e of dimension lesser than or equal to n 2 which is coefficient assignable by a static feedback if and only if E is reachable.
The BDP with the additional requirement of coefficients assignement can be stated as follows.
Example 4. Let us consider the delay differential system E defined by the following equations
Problem 4. In the same situation as in Problem 2, the Block Decoupling Problem with Coefficients Assigllement for E consists in finding an integer n a and a dynamic state feedback law of the form (18) such that in the compensated system EF,G each block input Vi controls (or weakly controls, in the case of BDP with weak output controllability) the output Yi, but has no influence on the outputs Yj for j ::j:. i, i = 1, ... , k, and the coefficients of the characteristic polynomial of EF,G are arbitrarily assigned.
xdt) = Xl(t udt X2(t) = xdt udt ydt) = xdt Y2(t) = X2(t)
=
X2(t
+ 1) = .6.1xdt)+ .6. 2Ul(t) + U2(t) ydt) = .6. 1 xdt-) + X2(t)
(22)
with coefficients in the ring of real polynomials in two indeterminates 1R[.6. 1 , .6. 2], We want to investigate if it is possible to decouple line by line the system E and to assign arbitrarily the coefficients of the decoupled system. Since E and E' have the same signal flow graph, this can be done by solving the corresponding problem for E' over the ring 1R[.6. 1 ,.6. 2 ]. On the basis of the results shown in the previous Sections, we are therefore interested in the maximum precontrollability submodules Ri contained, respectively in K i !{ e7'Dj, j::j:. i, for i, j 1,2, with
(or if and only if i
+ 1) = .6. 1 .6. 2xdt) + X2(t)+
Y2(t) = X2(t)
fori=I, ... ,k
+ !\'erCi ) = dim(.l:'),
xdt
.6. 1udt)
Proposition 12. (Conte et al., 1998a) Given a system E, defined over the Noetherian ring R by equations of the form (3), (17), let Ri, for i 1, ... , k denote the maximum pre-controllability submodule contained in Ki= n~:;::l,#i J{ erCj. Assume that E is reachable.Then, the BDP with Coefficients Assignement for E is solvable if and only if
dim(Ri
(21)
Introducing the delay operators .6.i, defined for any function f(t) by !l;J(t) = f(t - oil, we can formally associate to I: the system E' defined by the equations
Combining the above results, it is possible to state the following Proposition.
Ri+J{erCi=X
01 - 02) + X2(t)+ od od+ 02) + U2(t) od + X2(t)
= 1, ... , k
in tha case of BDP with weak output controllability) .
=
=
It is important to remark that, together with
Proposition 4, the above Propositions provides not only a complete characterization of the soh'ability conditions but also an algorithmic way for computing solutions to the BDP for systems over i\oetherian rings.
We have that the submodules Ri, for i = 1,2, are generated, respectively, by the columns of the matrices 183
ticular, the dynamic matrix of the extended compensated system is the matrix Condition Ri + I\ erD; = X, for i = 1,2, is not satisfied, but dimR[~l ,~21 (Ri + I\ er D;) equals dimR[~l '~21X 2 for i 1,2. Therefore the Block Decoupling Problem with Weak Output Controllability and Coefficients Assignment is solvable for the system E'. Since the subm~dules Ri are not (A, B)-invariant submodules of feedback type, they are not controllability submodules. Hence, a solution of the considered BDP requires, in order to define a dynamic feedback, to consider a dynamic extension E~ of E'. Explicit computations, reported in (Conte et al., 2 1998a), show that an extension with n a suffices for finding a solution of the considered BDP. Moreover, E~ can be defined in such a way that it turns out to be coefficients assignable by means of a static state fedback, so that no further extension, as possibly indicated in Proposition 11 is needed. Then, assuming that the desired characteristic polynomial in the extended compensated system is (z + 2)4, further explicit computations (see (Conte et al., 1998a)) allow us to find a feedback matrix F~ of the form
=
=
whose characteristic polynomial is the desired one, and the transfer function matrix of the ex-
tende[de:'~~fensated sy].stem is the diagonal ma-
,
tnx
o - s+2 --
.
(24)
Introducing the delay operator ~, defined for any function f(t) by ~f(t) = f(t - (5), we can formally associate to E the system El defined by the equations
xdt+ 1) X2(t + 1) { y(t)
0 0 113 114]
[
0 e- 262 '
Example 5. Let us consider the delay differential system E defined by the following equations
=
121 fn 123 124 o 0 -2 0 o 0 0 -2
-s+ 2
'
= q(t) = xdt) + ~u(t) = X2(t)
(25)
u(t)
with coefficients in the ring lR[~] of real polynomials in one variable, We want to investigate if it is possible to decouple the disturbance q from the output of 1:. Since 1: and 1: 1 have the same signal flow graph, this can be done by solving the corresponding problem for El over the ring R[~]. Denoting the defining matrices of El by
with
A
= [~ ~] , (B
D)
= [~~ 1 ~]
,
C=[o1], and matrices
[~2]'
G
Gal
=
U],
Gal
=
[n,
G1
=
since the maximum (-4, B)-invariant submodule in I\ erG is V· = span { [ ~ ~ 1 ] }, we have
2= [_~21_ ~i]' These yield a dy-
that ImD is not contained in V· + 1mB. Hence, condition (15) is not satisfied and the DDP for 1: 1 is not solvable with a feedback having coefficients in R. On the other hand, we can see the triple of matrices (.4, (B D), G) as defining a system E 2 over the ring ~[~]r of realisable rational functions in one variable ((Sename and Lafay, 1996)), namely the ring whose elements are of the form :i~j, with p(~), q(~) E lR[~] and q(O) = qo f. O. The systems El and E 2 are different objects, although they are formally defined by the same matrices,
namic feedback law acting on E of the form
xadt) Xa2(t) U1 (t)
= -2x adt) + vdt) = -2X a2(t) + V2(t) = 113xadt) + f14Xa2(t)+ V1 (t)
U2(t)
+ V2(t)
= h1xdt) + h2J.'2(t)+
(23)
h3 Xa1(t) + h4 x a2(t)~2t'1(t) - (~2 + ~i)V2(t)
which decouples the system and assigns the coefficients of the characteristic polynomial. In par184
since the basis rings are different. However, the signal flow graph of L2 is the same as that of ~. Now, for L2 , the maximum (A, B)-invariant submodule in /{ er C is
Basile, G. and G. Marro (1969). Controlled and conditioned invariant subspace in linear system theory. Journal Optimization Theory and Applications, 3. Brewer, J. \V., J.W. Bunce and F.S. Van Vleck (1986). Linear systems over commutative rings. In: Lecture Notes in Pure and Applied Mathematics. Vo!. 104. Marcel Dekker. Conte, G., A.M. Perdon and A. Lombardo (1996). The static and dynamic decoupling problem for systems over a ring. Proceedings of IFAC IVorld Conference, San Francisco, CA. Conte, G., A.M. Perdon and A. Lombardo (1997). The decoupling problem with weak output controllability for systems over a ring. Proceedings IEEE-CDC '97, San Diego,CA. Conte, G., A.M. Perdon and A. Lombardo (19980). Block decoupling problem with coefficient assignment and stability for linear systems over a noetherian ring. Proceedings IFAC Conference on System Structure and Control, Nantes, France. Conte, G., A.M. Perdon and A. Lombardo (1998b). Dynamic feedback decoupling problem for delay-differential systems via systems over rings. IMACS Journal "Mathematics and Computers in Simulation ", special issue "Delay Systems. Conte, G. and A.M. Perdon (1982). Systems over principal ideal domains. a polynomial model approach. SIAM Journal on Control and Optimization, 20. Conte, G. and A.M. Perdon (1995a). The decoupling problem for systems over a ring. Proc. 33rd IEEE CDC '95, New Orleans,LO. Conte, G. and A.M. Perdon (1995b). The disturbance decoupling problem for systems over a ring. SIAM Journal on Control and Optimization. Conte, G. and A.M. Perdon (1998). The block decoupling problem for systems over a ring. to appear on IEEE Trans. AC. Emre, E. and P. Khargonekar (1982). Regulation of split linear systems over rings; coefficient assignment and observers. IEEE Trans. Aut. Control, A C-27. Habets, 1. (1994). Algebraic and computational aspects of time-delay systems. Ph.D. Thesis, Eindhoven University of Technology. Hautus, 11.1.J. (1982). Controlled invariance for systems over a ring. In: Lecture Notes in Control and Inf. Sci .. Vo!. 39. Springer. Inaba, H. and W. \Vaag (1996). Block decoupling for linear systems over rings. Linear Algebra and Its Applications. Inaba, H., N. Ito and T. Munaka (1987). Decoupling and pole assignment for linear systems defined over a principal ideal domain. Linear Circuits, Systems and Signal Processing: theory and applications.
~
V' = span { [ ] } . The difference with the previous sistuation is due to the fact that in ?R[~lr the element ~ - 1 is invertible. So, the solvability condition (15) for the DDP is satisfied and, in particular, u(t) = ~u(t)
is a solution to the DDP for It is now easy to see that
+ xdt)
~2'
u(t) = u(t - 6)
+ xdt)
is a solution for the DDP concerning the original delay-differential system E.
6. CONCLUSION Althoug some problems, mainly concerning the possibility of defining algorithmic procedures for constructing submodules with specific geometric properties, are still to be solved, the geometric approach to systems with coefficients in a ring appears capable of providing satisfactory solutions to many interesting control problems. In turn, this offers valid tools for dealing with delay-differential systems and with families of systems depending on parameters using algebraic methods. Further developments, besides adding new technical results, should concentrate on the construction of suitable algorithms, based on computer algebra, for implementing design procedures involving systems with coefficients in a ring.
7. REFERENCES Assan, J., J.F. Lafay and A.M. Perdon (19980). An algorithm to compute the maximum controllability submodule over a principal ideal domain. Proceedings IFAC Workshop on Linear Time Delay Systems, Grenoble, France. Assan, J., J.F. Lafay and A.~1. Perdon (1998b). Computing the maximum pre-controllability submodule over a noetherian ring. submit to Systems and Control Letters. Assan, J., J.F. Lafay and A.M. Perdon (1998c). On feedback invariance properties for systems over a principal ideal domain. in press on IEEE Transactions on Automatic Control. Atiyha, M.F. and I.G. i'.'lacdonald (1969). Introduction to commutatil'e algebra. AddisonWesley. 185
hamen, E. (1976). Lectures on algebraic system theory. linear systems over rings. NASA Contractor Report 3016. Lang, S. (1984). Algebra. Addison-Wesley. Sename, O. and J.F. Lafay (1996). Decoupling of square linear systems with delays. to appear in IEEE Automatic Control. Sontag, E. (1976). Linear systems over rings control : a survey. Ricerche di A utomatica, 7. Sontag, E. (1981). Linear systems over rings: a (partial) update survey. Proc. IFA C/81 , Eyoto. Wonham, t\L (1985). Linear multit'ariable control: a geometric approach. Springer Verlag.
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