Recently, it has been shown that some Lyapunov-based stability conditions for precisely known time-delay systems are equivalent to the robust stability of a delay-free comparison system through the small gain theorem with constant scales. The extension of those previous results to cope with uncertain time-delay systems in polytopic domains is the main contribution of this paper. From the definition of a generic system realization, linear matrix inequalities that are equivalent to the scaled small-gain conditions but have extra matrix variables are given. Thanks to these extra matrices, the delay-dependent stability conditions for the analysis of time-delay systems in polytopic domains through parameter-dependent Lyapunov matrices derived from the proposed conditions are less conservative than the standard ones, as illustrated by means of numerical examples.
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