Robust Stability of Linear Delay Systems with Unknown Parameters

Abstract

This note is concerned with the stability of linear delay systems. We investigate the robust stability of the delay systems with unknown parameters. By means of the argument principle, sufficient conditions are derived for the robust stability of the delay systems. It is easy to check the sufficient conditions. Numerical examples are given to illustrate the main results.

APPENDIX

Proof of Lemma 1. As we are discussing an unstable root, we assume $\Re s \geqslant 0$ throughout the proof. Let

$$W(s) = {{A}_{0}} + \sum\limits_{j = 1}^m {{A}_{j}}\exp ( - {{\tau }_{j}}s).$$

We have

$$P(s) = \det \left[ {sI - {{A}_{0}} - \sum\limits_{j = 1}^m {{A}_{j}}\exp ( - {{\tau }_{j}}s)} \right] = \det [sI - W(s)] = 0.$$

This implies the s is an eigenvalue of the matrix W(s) and there exists an integer $j\;(1 \leqslant j \leqslant n)$ such that

$$s = {{\lambda }_{j}}(W(s)).$$

From $\Re s \geqslant 0,$ we have $\left| {\exp ( - {{\tau }_{j}}s)} \right| \leqslant 1$ for $j = 1, \cdots ,m.$ Then we obtain

$${\text{|}}s{\text{|}} = {\text{|}}{{\lambda }_{j}}(W(s)){\text{|}} \leqslant {\text{||}}W(s){\text{||}} = \left\| {{{A}_{0}} + \sum\limits_{j = 1}^m {{A}_{j}}\exp ( - {{\tau }_{j}}s)} \right\|$$

$$ \leqslant {\text{||}}{{A}_{0}}{\text{||}} + \left\| {\sum\limits_{j = 1}^m {{A}_{j}}\exp ( - {{\tau }_{j}}s)} \right\| \leqslant {\text{||}}{{A}_{0}}{\text{||}} + \sum\limits_{j = 1}^m {\text{||}}{{A}_{j}}{\text{||}} = {{r}_{1}}.$$

Thus the proof is completed.

Proof of Lemma 2. Suppose the system is asymptotically stable. All zeros of $P(s)$ are located on the left half complex plane. It means that $P(s) \ne 0$ when $\Re s \geqslant 0$. By the argument principle [11], we have that (7) and (8) hold.

Conversely, assume that the conditions (7) and (8) hold. According to Lemma 2.1, it means that $P(s)$ never vanishes for $\Re s \geqslant 0$. Hence (7) and (8) imply system (3) is asymptotically stable. Thus the proof is completed.

Proof of Lemma 3. It is similar to the proof of Lemma 2.