New results for the analysis of linear systems with time‐invariant delays

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Abstract
This paper presents a comparison system approach for the analysis of stability and ℋ︁∞performance of linear time‐invariant systems with unknown delays. The comparison system is developed by replacing the delay elements with certain parameter‐dependent Padé approximations. It is shown using the special properties of the Padé approximation to e−sthat the value sets of these approximations provide outer and inner coverings for that of each delay element and that the robust stability of the outer covering system is a sufficient condition for the stability of the original time delay system. The inner covering system, in turn, is used to provide an upper bound on the degree of conservatism of the delay margin established by the sufficient condition. This upper bound is dependent only upon the Padé approximation order and may be made arbitrarily small. In the single delay case, the delay margin can be calculated explicitly without incurring any additional conservatism. In the general case, this condition can be reduced with some (typically small) conservatism to finite‐dimensional LMIs. Finally, this approach is also extended to the analysis of ℋ︁∞performance for linear time‐delay systems with an exogenous disturbance. Copyright © 2003 John Wiley & Sons, Ltd.

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