Realization and elimination in rational representations of behaviors

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Abstract

This article deals with the relationship between rational representations of linear differential systems and their state representations. In particular we study the relationship between rational representations on the one hand, and output nulling and driving variable representations on the other. In the input–output framework it is well-known that every controllable and observable realization of the transfer matrix of the system yields a minimal input/state/output representation. If a proper rational matrix is used for a rational kernel or image representation of the system, then the question arises under what conditions realizations of this rational matrix give rise to state representations. We will establish conditions under which realizations of the proper rational matrix appearing in a rational image representation yield minimal driving variable representations of the system. Likewise, we find conditions under which realization leads from rational kernel representations to minimal output nulling representations. We also study the converse problem, namely state elimination from driving variable and output nulling representations. We will establish closed form expressions for kernel and image representations of the external behaviors associated with these particular state representations.

Introduction

A behavior, say B, of a linear time invariant differential systems (LTIDS) admits many different kinds of representations such as polynomial representations, rational representations, state representations etc. Also, a behavior admits different kinds of state representations such as classical input/state/output (I/S/O) representations, driving variable representations, and output nulling representations (see  [1], [2]). In this paper we deal with the relation between rational representations and output nulling/driving variable representations.

Among other things, in this paper, we address elimination of the state variable from a given output nulling or driving variable representation, thereby attaining a kernel representation of its external behavior. Further, given a rational kernel or image representation of a behavior, a natural question is whether a given realization of the rational matrix appearing in it yields a state representation of the behavior. More precisely, given a behavior B and a rational kernel representation with a proper real rational matrix, does a realization of the rational matrix yield an output nulling representation of this behavior? Similarly, given a controllable behavior B and a rational image representation with a proper real rational matrix, does a realization of the rational matrix yield a driving variable representation of this behavior? We show under which conditions the answer to the above posed questions is positive.

We also look into the aspect of minimality of these state representations. Minimality is defined differently for each of the state representations that a behavior admits, see  [1]. Minimal output nulling and driving variable representations were treated earlier in  [3], [4]. It is well-known that every proper real rational matrix admits a realization such that the underlying constant real matrices form a controllable and an observable pair and that the I/S/O representation obtained from such realization is minimal. However, this is not the case with driving variable and output nulling representations. The controllability and observability conditions on the underlying constant real matrices are not sufficient to ensure minimality of the output nulling and driving variable representations. In fact, not every rational matrix appearing in a rational kernel (image) representation yields a minimal output nulling (driving variable) representation via realization. We therefore want to answer the following question: Given a behavior B, a rational kernel representation with a proper real rational matrix, and a realization of this matrix such that the underlying constant real matrices form a controllable and an observable pair, find conditions under which this realization yields a minimal output nulling representation of the behavior. A second question that we want to address is: Given a controllable behavior B, a rational image representation with a proper real rational matrix, and a realization of this matrix such that the underlying constant real matrices form a controllable and an observable pair, find conditions under which this realization yields a minimal driving variable representation of the behavior.

The outline of this paper is as follows. In the remainder of this section we will introduce some notation. In Section  2, we will review various representations that a behavior of a LTIDS admits. In Section  3, we deal with elimination of the state variable from a given output nulling representation, thereby attaining a kernel representation of its external behavior. In the case of driving variable representations, we eliminate both the state and the driving variable. In Section  4, we deal with realizations of the rational matrix appearing in a rational representation, and the state representations it yields. Finally, in Section  5, we address the questions posed above on minimality of output nulling and driving variable representations. We conclude the paper with remarks in Section  6.

We use the standard symbols R and C for the fields of real and complex numbers. We use Rn,Rn×m, etc. for the real linear spaces of vectors and matrices with components in R. C(R,Rw) denotes the set of infinitely often differentiable functions from R to Rw. R(ξ) will denote the field of real rational functions in the indeterminate ξ. R[ξ] will denote the ring of polynomials with real coefficients in the indeterminate ξ. R(ξ)P will denote the ring of proper rational functions in the indeterminate ξ with real coefficients. We will use R(ξ)n,R(ξ)n×m,R[ξ]n,R[ξ]n×m, etc. for the spaces of vectors and matrices with components in R(ξ) and R[ξ], respectively. If dimensionality is clear from the context, we will use the notation R(ξ)×m,R(ξ)n×, R[ξ]× or R(ξ)×, etc. A square non-singular polynomial matrix U is called unimodular if the determinant of U is a non-zero constant. FR(ξ)n×n is biproper if det(F)0 and F,F1 are both proper. A polynomial matrix PR[ξ]× is called left prime over R[ξ] if P(λ) has full row rank for all λC. For constant real matrices A,B,C,D of appropriate dimensions, the quadruple (A,B,C,D) is called strongly controllable if the pair (A+GC,B+GD) is controllable for every G of appropriate dimensions. Similarly, the quadruple (A,B,C,D) is called strongly observable if the pair (C+DF,A+BF) is observable for every F of appropriate dimensions.

Section snippets

Behaviors and their representations

In this section we review linear time invariant differential systems and representations of their behavior. A linear time invariant differential system (LTIDS) is defined as a system Σ=(R,Rw,B) whose behavior B is a solution space of finite set of higher order constant coefficient linear differential equations. Obviously, for any LTIDS Σ=(R,Rw,B) there exists a real polynomial matrix R with w columns, i.e.  RR[ξ]×w, such thatB={wC(R,Rw)R(ddt)w=0}. The representation (1) is called a

External behavior induced by state representations

In this section we investigate how to obtain the external behavior induced by a given state representation. We limit attention to output nulling and driving variable representations. Elimination of the state in I/S/O representations was dealt with, for example, in  [5], [11]. Here, we investigate two main problems. Firstly, if BON(A,B,C,D) is the full behavior induced by the output nulling representation ddtx=Ax+Bw,0=Cx+Dw, how can we obtain a kernel representation of its external behavior BON(A

Realizations and state representations

For linear differential systems represented by polynomial kernel and image representations, realization theory in the behavioral framework has been dealt with in  [1], [3], [11]. In  [1], an algorithm to realize a minimal ISO representation from a given polynomial kernel representation was given. For a given state representation of the system, conditions under which this representation is minimal were given in  [3].

In this section we deal with realizations of the rational matrix appearing in a

Realization and minimal state representations

In the context of I/S/O representations, it is well known that a representation obtained from a realization is minimal if and only if (A,B) is a controllable pair and (C,A) is an observable pair. As mentioned in the introduction, in the case of output nulling and driving variable representations these conditions on (A,B,C) are not sufficient for minimality. This can be clearly seen in the example below:

Example 5.1

Let G(ξ)=ξ1. Consider a realization of G(ξ) in which A=0,B=1,C=1,D=0. Then we have ONR given

Conclusions

In this paper we have dealt with rational representations and their connection with driving variable and output nulling representations of linear differential systems. We have attained kernel representations of the external behaviors induced by output nulling and driving variable representations. We have also established conditions under which a given realization of a proper real rational matrix appearing in a rational kernel (image) representation of a behavior yields an output nulling

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