Elsevier

Automatica

Volume 50, Issue 1, January 2014, Pages 211-217
Automatica

Brief paper
An extension of the prediction scheme to the case of systems with both input and state delay

https://doi.org/10.1016/j.automatica.2013.09.042Get rights and content

Abstract

In this contribution we present an extension of the prediction scheme proposed in Manitius and Olbrot (1979) for the compensation of the input delay to the case of linear systems with both input delay and state delay. For simplicity of the presentation we treat the case of systems with one state delay.

Introduction

In the Introduction Chapter of the recently published book (Krstic, 2009) it is stated that “the area of control design for systems with simultaneous input and state delay is underdeveloped”. At the same time, it is mentioned there that the stabilizing problems for systems with state delay only “are the easiest in our list as they can be solved using finite dimensional feedback laws”. In this contribution we present an extension of the prediction scheme proposed in Manitius and Olbrot (1979) for the compensation of the input delay in the computation of stabilizing controllers for linear systems with both input delay and state delay. For simplicity of the presentation we treat the case of systems with one state delay, but the presented results can be extended to the case of systems with multiple state delays, as well.

In Section  2 we provide basic notations used in the contribution and give the formulation of the stabilization problem for systems with input and state delay. Section  3 is devoted to the computation of the stabilizing control laws. Similar to the case of systems with input delay we start with an explicit expression for the solution of an initial value problem for a time-delay system. Then, we apply this expression for the computation of future states in the form of functionals that depend on the present and past states of the time-delay system. And, finally, we compute the desired stabilizing control law. The stabilizing law is of the form of an integral equation, similar to that obtained in Manitius and Olbrot (1979), with some additional terms due to the presence of the state delay in the system. Section  4 is dedicated to the stability analysis of the closed-loop system. The principal result of the section is an upper exponential estimate for the solutions of the closed-loop system. In Section  5 some basic results concerning the complete type functionals for an exponentially stable system are given. In Section  6 we present a Lyapunov–Krasovskii type stability analysis of the closed-loop system. The key element of the analysis is a simple modification of the backstepping transformation of the control variable proposed in Krstic and Smyshlyaev (2008). This transformation allows us to present the closed-loop system in a form more appropriate for the consequent stability analysis. Here we propose for the transformed system a Lyapunov functional, similar to that of Krstic (2009), with a single modification: we use a complete type functional instead of the quadratic Lyapunov form used in  Krstic (2009). As a result we obtain an upper exponential estimate for the solutions of the transformed system and derive a similar exponential estimate for the original control variable. Several examples illustrating the computation of the stabilizing control laws are given in Section  7.

Section snippets

Problem formulation

Given a time-delay system of the form dx(t)dt=A0x(t)+A1x(th)+Bu(tτ), where Aj, j=0,1, are real n×n matrices, and B is a real n×m matrix. The system delays satisfy the inequalities 0<hτ. The opposite case, τ<h, can be treated similarly with trivial modifications. Let t00 be an initial time instant and φ:[h,0]Rn be an initial function. We assume that the function belongs to the space of piece-wise continuous functions, PC([h,0],Rn), defined on the segment [h,0]. Let x(t,t0,φ) stand for

General scheme

Let us denote by K(t) the fundamental matrix of system (1); see  Bellman & Cooke, 1963. The matrix satisfies the equation ddtK(t)=A0K(t)+A1K(th),t0, and the initial conditions: K(t)=0n×n, t<0, K(0)=I.

Given an initial time instant t00, and an initial function φPC([h,0],Rn), then the corresponding solution of system (1) can be written as x(t,t0,φ)=K(tt0)φ(0)+h0K(tt0θh)A1φ(θ)dθ+t0tK(tξ)Bu(ξτ)dξ,tt0. In particular this means that x(t+τ)=K(τ)x(t)+h0K(τθh)A1x(t+θ)dθ+τ0K(ξ)Bu(t+ξ)

Exponential estimates

Any particular solution of the closed-loop system (1), (5) is defined by the corresponding initial conditionsx(t)=φ(t),t[h,0],φPC([h,0],Rn),u(t)=ψ(t),t[τ,0),ψPC([τ,0],Rm). For t[0,τ) the corresponding solution is x(t,φ,ψ)=K(t)φ(0)+h0K(thθ)A1φ(θ)dθ+0tK(tξ)Bψ(ξτ)dξ,t[0,τ). It follows from the preceding equality that x(t,φ,ψ)α(φh+ψτ),t[0,τ], where α=max{η(1+hA1),ητB} and η=maxt[0,τ]K(t).

Remark 3

As η=maxt[0,τ]K(t)1, then α1, and inequality (6) holds for t[h,τ].

Since

Complete type functionals

The stability analysis performed in the previous section is based on a given exponential estimate of the solutions of system (2). In the following section we are going to provide a Lyapunov type stability analysis of the closed-loop system (1), (5). As the system involves both input and state delay we are not able to apply for this analysis quadratic Lyapunov forms as it has been done in  Krstic (2009), any more. Actually, we have to replace these forms by quadratic Lyapunov–Krasovskii

Lyapunov–Krasovskii approach

In this section we present a Lyapunov type stability analysis of the closed-loop system (1), (5). To this end we apply a simplified version of the backstepping transformation of the control variable proposed in  Krstic and Smyshlyaev (2008), (xu)(xy), where the new variable y(t)=u(t)F0x(t+τ)F1x(t+τh),tτ. It is worth to be mentioned that in  Krstic and Smyshlyaev (2008) the input delay is represented under a transport PDE, and the backstepping transformation is applied to the

Examples

We start with the following example from  Krstic (2009).

Example 1

Let us consider the scalar equation dx(t)dt=x(t1)+u(tτ), where for simplicity we assume 1<τ<2. Here the fundamental solution k(t)={0,t<0,1,t[0,1],and x(t+τ1)=k(τ1)x(t)+10k(τθ2)x(t+θ)dθ+τ1k(1ξ)u(t+ξ)dξ=x(t)+1τ2x(t+θ)dθ+τ1u(t+ξ)dξ. This means that the following control u(t)=fτ1u(t+ξ)dξ+f[x(t)+1τ2x(t+θ)dθ] transforms the original equation to the following one: dx(t)dt=(1+f)x(t1),tτ. The last equation is

Conclusions

In this contribution the classical prediction scheme used for the computation of stabilizing control laws is extended to the case of systems with both input and state delay. It is shown that the presented control laws convert to the standard ones when the state delay h disappears. It is worth mentioning that the new control laws are described by integral equations and belong to the class of retarded type time-delay systems. Nevertheless, similar to the case of systems with only input delay, an

Vladimir L. Kharitonov, received the Candidate of Science Degree in Automatic Control in 1977 and the Doctor of Science Degree in Automatic Control in 1990, both from the Leningrad State University. Dr. Kharitonov is a Professor in the Department of Applied Mathematics and Control Processes, St.-Petersburg State University, Russia. His scientific interests include: control, time-delay systems, stability and robust stability.

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Vladimir L. Kharitonov, received the Candidate of Science Degree in Automatic Control in 1977 and the Doctor of Science Degree in Automatic Control in 1990, both from the Leningrad State University. Dr. Kharitonov is a Professor in the Department of Applied Mathematics and Control Processes, St.-Petersburg State University, Russia. His scientific interests include: control, time-delay systems, stability and robust stability.

The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Nicolas Petit under the direction of Editor Miroslav Krstic.

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