On the reachability in any fixed time for positive continuous-time linear systems

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Abstract

This paper deals with the reachability of continuous-time linear positive systems. The reachability of such systems, which we will call here the strong reachability, amounts to the possibility of steering the state in any fixed time to any point of the positive orthant by using nonnegative control functions. The main result of this paper essentially says that the only strongly reachable positive systems are those made of decoupled scalar subsystems. Moreover, the strongly reachable set is also characterized.

Introduction

Positive linear systems appear naturally in the modeling of a lot of practical problems in economics, engineering and biology [8], [1]. The reachability of such systems has been studied in the past but mainly for discrete-time systems [5], [6], [10], see [2], [3], [4] for more recent references in which it appears that this problem is now well solved and understood.

It seems that the reachability of continuous-time systems is harder and was much less studied, see [9] for a classical reference and [7] for a more recent state of the art. A common (and surprising) feature of these works is that they use as a definition of reachability the possibility of steering the state in any point of the positive orthant in finite time. In this paper we consider the possibility of steering the state in any point of the positive orthant in any fixed time. This property is referred to in this paper as the strong reachability. This seems to us to be more in the spirit of the original definition of reachability although the two definitions coincide in the case where the input function is not restricted to be nonnegative. In the case of positive systems, however, the two definitions are not equivalent. The example 13 of [7] shows a system that is reachable in finite time but not reachable in any fixed time (for some parameters choice).

The main result of this paper is essentially that the only strongly reachable positive systems are those made of decoupled scalar subsystems. For nonstrongly reachable positive systems, a complete characterization of the set of strongly reachable states is given.

The paper is organized as follows. In Section 2 we review some well-known material on positive systems and their graph representation. In Section 3 we prove that the scalar systems are strongly reachable and give the necessary and sufficient conditions for strong reachability. Section 4 provides the strongly reachable set.

Section snippets

Linear positive systems: definitions and basic results

In this paper we denote by R+ the set of nonnegative real numbers. A p×q matrix L is said to be positive if Lij0 for i=1,,p,j=1,,q, and there exists indices i,j such that Lij>0.

We study continuous-time linear time-invariant systems of the following form:x˙(t)=Ax(t)+Bu(t),where the matrices A and B are real valued matrices of respective dimensions n×n and n×m. The matrix B is a positive matrix and A is of Metzler type that is, Aij0 for i=1,,n,j=1,,n and ij. The initial state x(0)R+n

Scalar positive systems are strongly reachable

We consider a scalar positive system as in (1) with one input and one state (m=n=1):x˙(t)=αx(t)+βu(t),where β>0.

Denote by H(t) the unit step function. With x(0)=0 and u(t)=H(t) we get for system (2) the response x(t)=(β/α)(1-eαt) for α0 and x(t)=βt for α=0. Then, for any T>0 and any xfR+, using the input functionu(t)=αxfβ(1-eαT)H(t)ifα0andu(t)=xfβTH(t)ifα=0,we get x(T)=xf and the system is therefore strongly reachable. This proves the following.

Proposition 2

The scalar system defined in (2) is strongly

The strongly reachable set

In this section we will use the following notations. . denotes the Euclidian norm, int(.) and cl(.) denote, respectively, the interior and the closure of a given set in the induced topology.

Define the strongly reachable set as R={xfR+n for x(0)=0 and any T>0 there exists a nonnegative piecewise continuous function u(t) such that x(T)=xf}.

A first property of the strongly reachable set is easily obtained.

Proposition 4

R is a convex cone.

Proof

Follows directly from the linearity and the positivity of the system. 

Conclusion

In this paper the problem of reachability for continuous-time positive systems is studied in the case where we ask for this reachability in any fixed time. The condition for such a reachability are given together with the set of states which are reachable in this sense.

The results of this paper may appear to be “intuitively obvious” since reachability in any fixed time induces input functions which are approximates of impulse distributions. Since only the Dirac distribution can be approached by

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