Input–output behavior for stable linear systems

Abstract

In this paper, a controllable, observable, asymptotically stable, finite-dimensional linear system is considered. The input–output problem considered is whether the input in a fairly general class of Hilbert spaces will produce the output in the same class. The problem in this generality appears to be very difficult and in this paper a large class of Hilbert spaces is determined for which the result is true and a series of counter examples are given to the more obvious conjectures.

Introduction

In this paper, we consider a controllable, observable, asymptotically stable, single-input–single-output finite-dimensional linear system. The problem we want to consider is motivated by a classical result on BIBO stability. It is well known that a controllable and observable linear system of BIBO is stable if and only if it is internally exponentially stable. It means that for such a system, a “bounded” input, namely, a function in the L^p space, where p∈[1,∞], produces a “bounded” output in the same L^p space. However, this result is limited due to the fact that this assumes that the input is either bounded or “small” for large values of t. We are often interested in the response of a system to other inputs — for example, ramp inputs. In this paper, we only consider the case p=2 where the input space is also Hilbert. We will show that this classical result can be extended to other classes of inputs. We consider the case, that the input is in a fairly general class of Hilbert spaces and ask if this implies that the output is in the same class. The problem in this generality appears to be very difficult, but we will determine a large class of Hilbert spaces for which the result is true and give a series of counterexamples to the more obvious conjectures.

We begin by stating the well-known result in the L² case (see, for example, [1], [2], [3], [4]).

Theorem 1.1

The linear, controllable and observable system $\text{x}\text{̇}\text{=Ax+bu,}\text{y=cx}$ is asymptotically stable if and only if for every u(t)∈L²[0,∞) the output y(t)∈L²[0,∞).

We will begin by proving the following theorem which includes many of the inputs of interest.

Theorem 1.2

The linear finite-dimensional controllable observable system $\text{x}\text{̇}\text{=Ax+bu,}\text{y=cx}$ has its spectrum in the closed left half-plane if and only if for every λ>0 and for every control $\text{u(t)∈L}{}^2\text{([0,∞),}\text{e}{}^{\mathrm{-\lambda t}}\text{d}\text{t)}$ $\text{y(t)∈L}{}^2\text{([0,∞),}\text{e}{}^{\mathrm{-\lambda t}}\text{d}\text{t)}$.

It is tempting to believe that the following would be true.

Conjecture 1.1

Let H be a L²([0,∞,dμ(t)) space where dμ(t) is an arbitrary positive measure. Let $\text{x}\text{̇}\text{=Ax+bu,}\text{y=cx}$ be an asymptotically stable finite-dimensional linear system, then u∈H implies that $\text{∫}\text{0}\text{t}\text{c}\text{e}{}^{\mathrm{A(t-s)}}\text{bu(s)}\text{d}\text{s∈H}$.

We will show that this conjecture is false and that it is quite difficult to find a completely suitable substitute.

The main thrust of this paper will be devoted to attempting to find an answer to the following problem.

Problem 1.1

Let f have the following properties:

• f is locally Lebesgue integrable on the positive real numbers.

• f(t)⩾0 for t⩾0.

• f is piecewise continuous.

What are necessary and sufficient conditions on the function f which ensure the following statement:

Let

$$\text{x}\text{̇}\text{=Ax+Bu,y=Cx,}$$

where x∈Rⁿ, y∈R^p and u∈R^m, be a controllable, observable and exponentially stable system. If

$$\text{∫}\text{0}\text{∞}\text{|u(t)|}{}^2\text{f(t)}\text{d}\text{t<∞}$$

then

$$\text{∫}\text{0}\text{∞}\text{|y(t)|}{}^2\text{f(t)}\text{d}\text{t<∞.}$$

In Section 2, we present two theorems with their proofs concerning Problem 1.1. In Section 3, we present a series of counterexamples.