Simple and weak delta-invariant psolyhedral sets for discrete-time singular systems

Abstracts

In this paper, necessary and sufficient conditions for the positive invariance of convex polyhedra with respect to linear discrete-time singular systems subject to bounded additive disturbances are established. New notions of delta-invariance under different assumptions on the initial conditions are defined. Specifically, the notions of simple and weak delta-invariance are considered. They can be seen as extensions of the delta-positive invariance concept used for the regular linear systems with additive disturbances. The results are presented by considering classical equivalent system representations for linear singular systems.

Singular systems; Disturbances; Convex polyhedra; Invariance; Initial conditions

Apresentam-se condições necessárias e suficientes para a invariância positiva de poliédros convexos relativamente a um sistema singular, linear e em tempo discreto, sujeito a perturbações aditivas e limitadas. Introduz-se as noções de delta-invariância simples e de delta-invariância fraca, associadas a diferentes hipóteses a serem verificadas pelas condições iniciais. Estas noções podem ser consideradas como extensões, para o caso de sistemas singulares, do conceito de delta-invariância utilizado no caso de sistemas regulares sujeitos a perturbações. Os resultados são desenvolvidos a partir de duas formas usuais de representação de sistemas singulares lineares.

Sistemas singulares; Perturbações; Po-liédros convexos; Invariância; Condições iniciais

Simple and weak D-invariant psolyhedral sets for discrete-time singular systems

Eugênio B. Castelan^I; Sophie Tarbouriech^II

^IDAS / CTC / UFSC, Departamento de Automação e Sistemas, 88.040-900 - Florianópolis (S.C.), Brazil, [e-mail]

^IILAAS - CNRS, 7, Avenue du Colonel Roche, 31077 - Toulouse Cedex 4, France, [e-mail]

ABSTRACT

In this paper, necessary and sufficient conditions for the positive invariance of convex polyhedra with respect to linear discrete-time singular systems subject to bounded additive disturbances are established. New notions of D-invariance under different assumptions on the initial conditions are defined. Specifically, the notions of simple and weak D-invariance are considered. They can be seen as extensions of the D-positive invariance concept used for the regular linear systems with additive disturbances. The results are presented by considering classical equivalent system representations for linear singular systems.

Keywords: Singular systems, Disturbances, Convex polyhedra, Invariance, Initial conditions.

RESUMO

Apresentam-se condições necessárias e suficientes para a invariância positiva de poliédros convexos relativamente a um sistema singular, linear e em tempo discreto, sujeito a perturbações aditivas e limitadas. Introduz-se as noções de D-invariância simples e de D-invariância fraca, associadas a diferentes hipóteses a serem verificadas pelas condições iniciais. Estas noções podem ser consideradas como extensões, para o caso de sistemas singulares, do conceito de D-invariância utilizado no caso de sistemas regulares sujeitos a perturbações. Os resultados são desenvolvidos a partir de duas formas usuais de representação de sistemas singulares lineares.

Palavras-chave: Sistemas singulares, Perturbações, Po-liédros convexos, Invariância, Condições iniciais.

1 INTRODUCTION

The use of the positive invariance property in the control of constrained dynamical systems has been receiving much attention in the last years (Blanchini, 1990; Blanchini, 1994; De Santis, 1994; Georgiou and Krikelis, 1991; Gilbert and Tan, 1991; Hennet, 1989; Hennet and Béziat, 1991; Kol-manovski and Gilbert, 1995; Milani and Dórea, 1996; Tar-bouriech and Gomes da Silva Jr., 1997; Tarbouriech and Castelan, 1993; Tarbouriech and Castelan, 1995). This property is used, for instance, to guarantee the maintenance of the state trajectories of a controlled system in the interior of some prescribed sets of admissible states determined from some sets of control or state constraints. External disturbances and/or parametric perturbations can also be considered. The positive invariance in presence of disturbances is commonly referred in the literature as D-invariance (Blanchini, 1990; De Santis, 1994; Kolmanovski and Gilbert, 1995).

However, few works exist dealing with the positive invariance property in the case of linear singular systems (Georgiou and Krikelis, 1991; Tarbouriech and Castelan, 1993; Tarbouriech and Castelan, 1995). Furthermore, these works do not consider the presence of external disturbances in the considered model.

The objective of this paper is to present necessary and sufficient algebraic conditions to guarantee the positive invariance property of convex polyhedra with respect to singular discrete-time systems subject to additive disturbances belonging to a convex set D. The general notion of D-invariance of a domain Ì Âⁿ with respect to a dynamical system is associated to the maintenance of the system trajectories within the domain for any initial condition belonging to and for any sequence of admissible disturbances belonging to D. Thus, a major objective of this paper consists in extending the D-invariance concept used for the regular linear systems to the linear singular systems.

Due to the specificities of singular systems in terms of initial conditions (initial conditions may be consistent or not), some different notions have to be considered. Hence, from the assumptions on the initial conditions with respect to the domain , we define the notions of simple and weak D-invariance.

Considering convex polyhedra, the algebraic characterization of both the simple and weak D-invariance properties are obtained for an equivalent representation of special interest in the theory of linear singular systems, called Differential-Algebraic Form.

The paper is organized as follows. The problem to be treated and the simple and weak D-invariance properties are formulated in section 2. Section 3 presents the main results in terms of the Differential-Algebraic Representation. Some general comments about the proposed results are given in section 4. An illustrative example in section 5 allows to show an application of the results to a constrained control problem. Section 6 ends the paper with some concluding remarks.

2 PROBLEM PRESENTATION

Consider a linear discrete-time singular perturbed system described by:

where E Î Â^n×n with rank(E) = q < n, A₀Î Â^n×n, D Î Â ^n×d. x_k Î Âⁿ and w_k Î Â ^d represent respectively the state and additive disturbance vectors. System (1) can, for instance, represent the closed-loop dynamic behavior of a linear singular system

controlled by a static full state feedback or static output feedback. In these cases, we have A₀ = A + BF or A₀ = A + BKC, respectively.

Since for control purposes system (1) represents controlled systems, we shall consider that it has a unique solution for all initial condition x₀Î Â ⁿ and for any admissible sequence {w_k}, with w_k Î Â ^d , "k > 0, and also that the system is causal. Thus, the following assumption is supposed to hold throughout this note (for details see, for instance, Dai (1989) and Lewis (1986)).

Assumption 1 The pair (E, A₀) is assumed to be regular and system (1) is supposed to be impulse free.

However, the dynamic behavior of system (1) can present discontinuous behavior at k = 0 if the associated initial condition x₀ is not consistent. For any given initial condition x₀, the actual state at k = 0 is denoted x_k|_{k = 0} = x₀+. The set of consistent initial conditions (Dai 1989) is the set of x₀ which prevents the system of discontinuous behavior:

It is well-known that in the case of the unperturbed system (w_k = 0, "k), ₀ corresponds to the subspace of Âⁿ spanned by the finite eigenvectors of the pair (E, A₀). But, in the case of system (1), ₀ depends also on w₀. In general, if w₀¹ 0, a finite jump with amplitude |x₀ – x₀+| may occur at k = 0 for any initial condition x₀.

Thus, let us now introduce different notions of D-invariance depending on certain assumptions made about the initial conditions. The first definition is the closest to the practical situation of hard constraints.

Definition 1 A nonempty set Ì Âⁿ is a simple D-invariant domain with respect to system (1) if for any initial condition x₀Î and sequence {w_k}, with w_k Î D, "k > 0, it follows that x₀+ Î and x_k Î , "k > 1.

The second definition is weaker in terms of hypothesis because it assumes that only the initial condition x₀ belongs to the invariant domain. It may be used, for instance, in practical situations of soft constraints or, as indicated by the comments in section 4, for stability and disturbance rejection purposes.

Definition 2 A nonempty set Ì Âⁿ is a weak D-invariant domain with respect to system (1) if for any initial condition x₀Î and sequence {w_k}, with w_k Î D, "k > 0, it follows that x_k Î , "k > 1.

Other two notions of D-invariance have been considered in and . They are omitted in the present paper since the notions of simple and weak D-invariance better cope with realistic practical situations.

In this work, we are mainly concerned with the application of Definitions 1 and 2 to the case of closed polyhedral domains. In practical control problems, polyhedral domains can represent linear constraints on the state and limits for the allowed disturbances. The polyhedral sets of states and disturbances to be considered in the sequel are defined by:

and

Any nonempty convex polyhedron of Âⁿ or Â^p can be characterized by (4) or (5), respectively. By convention the inequalities between vectors are component-wise.

Thus, the primary objective of this work is to give algebraic necessary and sufficient conditions for both the simple and weak R(T, m)-invariance of convex polyhedron R(G, r) with respect to system (1). To accomplish the stated objectives, we shall consider an equivalent representation of system (1), called Differential-Algebraic form, and the corresponding representation of the polyhedral set R(G, r). Another particular equivalent representation of system (1), under a Standard form, is also used in the proofs of the proposed results.

In general, an equivalent representation of system (1) can be obtained as follows (see Dai (1989)). Let and be two nonsingular n-order matrices. Then, by considering the change of coordinates x = , the following equivalent representation of system (1) can be defined:

where: = E , ₀ = A₀, and = D. The corresponding representation of the polyhedral set R(G,r) is given by: R(, r) = {Î Â ⁿ ;

< r} , with = G.

3 MAIN RESULTS

Since rank(E) = q, there exist nonsingular n-order matrices Q = and P = [ P₁ P₂] such that QEP = (Dai, 1989). Thus, by considering the change of coordinates x = [ P₁ P₂], with x¹Î  ^q and x²Î  ^(n–q), system (1) can be rewritten as

where: QA₀P = , QD = , with

A₁Î Â ^q×q , A₂Î Â^q×(n–q) , A₃Î Â^(n–q)×q ,

A₄Î Â^{(n–q)×(n-q)} , D₁Î Â^q×d , D₂Î >Â^{(n–q) ×d} .

Since the singular system is supposed to be impulse free, matrix A₄ is nonsingular (Dai, 1989; Lewis, 1986). The corresponding representation of the polyhedron R(G, r) is given by

where: G₁ = GP₁Î Â^g×q and G₂ = GP₂Î Â^g×(n-q).

The equivalent representation (7) gives a good meaning to singular systems: the system is composed of dynamic subsystems and an algebraic part which represents the connection between subsystems. Thus, descriptions of dynamical systems under the Differential-Algebraic Form arise naturally when systems are formed from interconnected systems (Dai, 1989). Otherwise, a representation of any linear singular system under the form (7) can be generally obtained from a singular value decomposition of matrix E (Dai, 1989).

The following result presents necessary and sufficient algebraic conditions for the simple D-invariance property of convex polyhedra.

Proposition 1 The polyhedral set R(G₁, G₂, r) is simply R(T, m)-invariant with respect to system (7) if and only if there exist nonnegative matrices S₁Î Â^g×g, S₂Î Â^g×p, S₃Î Â^g×p, S₄Î Â^g×g, S₅Î Â^g×p and S₆Î Â^g×p such that:

where, by definition:

₁ = G₁ – G₂

A₃ and ₂ = G₂.

Proof: To develop the proof, we shall represent the system under a standard form. This equivalent representation decomposes the system into slow and fast subsystems related, respectively, to the finite and infinite eigenvalues of the (impulse free) singular system (Dai, 1989). Thus, by considering the nonsingular n-order matrices = , = [₁

₂] = and the change of coordinates = [₁

₂] = , system (7) can be rewritten as

where:

₁ = (A₁– A₂

A₃) , ₁ = (D₁- A₂

D₂) and ₂ = D₂.

The corresponding representation of the polyhedron R(G, r) is given by

Notice that with respect to system (19), we always have = –₂w_k, "k. In particular, if k = 0 and an initial condition ₀ = is considered, it follows that the actual sub-state |_{k = 0} = -₂w₀ and, hence, ₀+ = . Thus, in general a jump occurs at k = 0, which is now considered.

>From Definition 1, the simple R(T, m)-invariance of R(₁, ₂,r) with respect to system (19) corresponds to:

and

for all and w_k such that

From (19), this condition also writes:

for all , w_k and w_k+1 such that

By using the extended Farkas' lemma (Hennet, 1989), it follows that a necessary and sufficient condition for the weak D-invariance of R(₁, ₂, r) is the existence of a nonnegative matrix satisfying both:

and

Therefore relations (9)-(18) of Proposition 1 follow.

>

Relative to the weak D-invariance property we get the following result.

Proposition 2 The polyhedral set R(G₁, G₂, r) is weakly R(T, m)-invariant with respect to system (7) if and only if there exist nonnegative matrices W₁Î Â^g×g, W₂Î Â^g×p and W₃Î Â^g×p such that

where, by definition:

₁ = G₁ -G₂

A₃ and ₂ = G₂.

Proof: As in the previous proof, the standard form (19) is used to obtain the desired invariance relations. From Definition 2, a necessary and sufficient condition for the weakR(T, m)-invariance of R(₁, ₂, r) with respect to system (19) is

for all and w_k such that

For all

_k Î R(₁, ₂, r) and for all admissible disturbances w_k Î R(T, m), "k, one can also write:

In the same way, since from (19) we get

₊₁ = - ₂w_k+1, (27) is equivalent to

Hence the weak R(T, m)-invariance of R(₁, ₂, r), expressed by (27) and (28), is obtained when every solution of (29) is also solution of (30), which corresponds to the inclusion of a polyhedral convex set into another polyhedral convex set. Then by applying the extended Farkas' lemma (Hennet, 1989), it follows that a necessary and sufficient condition for the weak D-invariance of R(₁, ₂, r) is the existence of a nonnegative matrix [ W₁ W₂ W₃] satisfying both:

and

Therefore, relations (22)-(26) of Proposition 2 follow.

We finish this section with the following remarks related to the Standard form (19). By recalling the general procedure described in section 2 to obtain an equivalent representation (6), we first remark that (19) can be obtained from (1) by considering = Q, < = P and the change of coordinates x = , where = . Next, let us recall that the finite eigenvalues of the considered regular and impulse-free singular system are given by the zeros of the characteristic polynomial

Thus, the q finite eigenvalues of pair (E, A₀) correspond to the eigenvalues of matrix ₁Î Â^q×q that are the roots of the characteristic equation (Kailath, 1980):

det(lI_q – (A₁ – A₂

A₃) ) = 0 .

4 GENERAL COMMENTS

The concept of D-invariance reduces to the classical concept of positive invariance in the case of unperturbed (regular and impulse free) singular system Ex_k+1 = A₀x_k. Hence, algebraic characterizations of the positive invariance property of R(G, r), as presented by , can be easily obtained from Propositions 1 and 2 by considering D = 0, T = 0 and m = 0.

The case of regular linear systems can be viewed as a particular case of system (19), by considering only the slow subsystem

Thus, the classical R(T, m)-invariance of R(₁, r) with respect to system (31) can be characterized by using Proposition 2 with ₂ = 0, ₂ = 0 and W₃ = 0.

Furthermore, notice that no assumption on the signs of the elements of vectors r and m is considered in the presented results. However, in most control applications the zero-state has to be feasible and we can consider that both vectors r and m are nonnegative, that is, r Î and m Î , where (resp. ) denotes the positive orthant of Â^g (resp. Â^d).

Under the assumption of non-negativeness of r and m, some stability properties of the eigenvalues of matrix W₁ can be deduced from relation (26) by using the theory of nonnegative matrices and M-matrices (Tarbouriech and Castelan, 1993). Thus, relation (22) can be thought as expressing a certain intersection between the spectra of W₁ and ₁; relation (23) implies the existence of some null eigenvalues in the spectrum of W₁. According to this intersection, some stability properties of the singular system can be deduced since its set of finite eigenvalues is given by the spectrum of ₁ (Tarbouriech and Castelan, 1993).

As shown by , the existence of admissible matrices W₂ and W₃ satisfying relations (24)-(25) is related to the following null-space intersections: ₁Ker(T) Í Ker(₁) and – ₂Ker(T) Í Ker(₂).

Also, consider both the set of relations (9)-(18) of Proposition 1 and (22)-(26) of Proposition 2. It is clear that the conditions of Proposition 1 are harder to satisfy than those of Proposition 2. If relations (9)-(18) hold then relations (22)-(26) also hold for W₁ = S₁, W₂ = S₂, W₃ = S₃. Hence, if R(G, r) is a simple R(T, m)-invariant domain, then it is also a weak R(T, m)-invariant domain, but the converse is not generally true. Therefore relative to R(G, r) the following implication holds :

simple R(T,m)-invariant Þ weak R(T,m)-invariant.

This fact follows from the different definitions of every notion. The notion of weak D-invariance requires that only the initial condition x₀ belongs to , whereas the notion of simple D-invariance requires that both the initial condition x₀ and the associated discontinuous state x₀+ belong to . However, a particular case exists where the two studied notions of D-invariance become equivalent. This case occurs whenever R(

₂, r) is unbounded in the directions of sub-state ². In such a case, we necessarily have G₂ = ₂ = 0 and it means that any jump due to non-consistent initial condition is admitted. Thus, if R(G₁, r) = R(G₁, 0, r) is a weak R(T, m)-invariant set it is also a simple R(T, m)-invariant set. This can also be verified from the conditions of Propositions 1 and 2.

Since the singular system can describe a kind of interconnected system, its sub-state

² can be considered as a pseudo-state and represents the interconnection between subsystems at initial time k = 0 (Dai, 1989). In this context, non-consistent initial conditions would represent topological changes in the interconnections' level. Therefore, in practical situations, both the choice of matrix ₂ = G₂ and the used D-invariance concept (simple and/or weak) has to be associated to the possible topological changes in the interconnections between subsystems.

5 ILLUSTRATIVE EXAMPLE

Consider the open-loop discrete-time singular system (2) borrowed from (Tarbouriech and Castelan, 1993), described by the following matrices:

The set R(T, m) of allowed persistent disturbances is described by

where g is some positive scalar.

The vector of admissible controls is constrained to belong to a compact set W

Thus, assuming that a saturated state-feedback control law is applied

where sat(u_i) = sign(u_i)min{|u_i|, _i} , for i = 1,2, the closed-loop system is given by:

This closed-loop system is non-linear but, for any state x_k inside the polyhedral set S(F, ), defined from W by (37), the state at k + 1 is determined by the linear model (2):

Let us now consider the state feedback matrix F Î Â^2×3 computed by Tarbouriech and Castelan (1993):

The corresponding closed-loop linear model under the form (1) is determined by matrices E and D given before and by:

Since the corresponding set S(F, ) is not an invariant domain with respect to (39), saturations may occur for trajectories emanating from S(F, ). Hence, one can be interested in verifying the invariance property in the presence of disturbances from some subset of S(F, ). In this way, we are primarily interested in verifying the R(T, m)-invariance property of the set R(G, r) defined in (4), where G and r are given by:

Notice that the considered set R(G, r) verifies R(G, r) Í S(F, ). In the unperturbed case (w_k = 0, "k), the set R(G, r) is a weak positively invariant set of system (39). In the perturbed case, the objective is to determine the maximal scalar g = g_max such that R(G, r) is weakly R(T, m)-invariant with respect to (39). By using Linear Programming to evaluate the relations of Proposition 2, we obtain g_max = 0.208815 with:

Let us now consider the polyhedral set R(, ), where:

This set corresponds to the intersection R(G, r) Ç ₀, where ₀ is the set of consistent initial conditions of system (39) in the unperturbed case. Since both R(G, r) and ₀ are invariant domains with respect to (39), the unbounded set R(, ) is also an invariant domain in the disturbance-free case (Tarbouriech and Castelan, 1993). In the perturbed case, it can be verified that R(, ) is both simply and weakly R(T, m)-invariant with respect to system (39), for g = _max = 0.6667. In particular, the following matrices can be used to verify the relations of Proposition 2:

Let us now show some system trajectories emanating from the above considered D-invariant sets. To simplify the analysis, we shall consider that the control bounds, represented by the elements of > 0, are sufficiently large so that no saturations occur, i.e., any considered trajectory evolves inside S(F, r) and is determined by the linear model (39).

Fig.1 shows the time-response of the first three components of vector Gx_k (due to symmetry property) by considering the set R(G, r) and g_max = 0.208815. The considered disturbances are randomly generated so that the corresponding sequence w_k belongs to the interval [-g_max, g_max]. By choosing the initial condition x₀ = [ -2.1164 -1.9318 1 ]¢, one gets x₀+ = [ -2.1164 -1.9318 0.2160 ]¢. Notice that Gx₀< r whereas Gx₀+ r. But, since R(G, r) is a weak R(T, m)-invariant set, one gets Gx_k < r, "k > 1.

Fig.2 shows the time-response of the first two components of vector x_k (due to symmetry property) and the time- response of the last component of vector x_k by considering the set R(G, r) Ç ₀ and g_max = 0.6667. By choosing the initial condition x₀ = [ -2 -1 0 ]¢, one gets x₀+ = [ 2 1 0.6896 ]¢. Notice that both x₀< r and x₀+< r.

6 CONCLUSION

We have presented some results on the invariance property of convex polyhedra with respect to linear singular systems with additive disturbances. In this way, we have defined the notions of simple and weak D-invariance according to the assumptions on the initial conditions. Algebraic necessary and sufficient conditions for characterizing these properties have been proposed relative to two classical equivalent system representations.

As for regular systems, the presented results may be used for solving constrained control problems (Castelan and Tarbouriech, 1996). To this end, the computation of the maximal admissible weakly D-invariant set contained in a given polyhedron of constraints is considered in . Finally, we remark that a similar technique can be proposed to compute the maximal simply D-invariant set.

ACKNOWLEDGEMENTS

This work has been partially supported by CNPq (Brazil) and CNRS (France). The authors would like to thank the reviewers for their comments and suggestions.

Artigo submetido em 20/12/2000

1a. Revisão em 10/9/2002; 2a. Revisão 30/4/2003 Aceito sob recomendação do Ed. Assoc. Prof. Liu Hsu

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Publication Dates

• Publication in this collection
  19 Apr 2004
• Date of issue
  Dec 2003

History

• Received
  20 Dec 2000
• Reviewed
  10 Sept 2002
• Accepted
  30 Apr 2003