Robust Stability Analysis of Sampled-Data Systems With Uncertainties Characterized by the ${\mathcal {L}}_\infty$-Induced Norm: Gridding Treatment With Convergence Rate Analysis

Abstract

This paper tackles the problem of stability analysis for uncertain sampled-data systems consisting of the feedback interconnection between a nominal linear time-invariant (LTI) sampled-data system and a structured linear time-varying (LTV) model uncertainty. When the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathcal {L}}_\infty$</tex-math></inline-formula> norm and its induced norm are taken into account to characterize such uncertain systems, it has been shown that the necessary and sufficient condition for robust stability of such uncertain sampled-data systems is described through the supremum of the spectral radii of a certain nonnegative matrix defined on the sampling interval <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[0,h)$</tex-math></inline-formula> . However, a direct computation of the supremum for practical uncertain sampled-data systems is quite difficult since it intrinsically involves an infinite number of computations of spectral radii over the continuous-time domain <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[0,h)$</tex-math></inline-formula> . To solve this problem, we introduce a gridding approach, by which an upper bound, as well as a lower bound on the supremum of the spectral radii, are obtained. The upper and lower bounds also lead to a sufficient condition and a necessary condition for the stability of uncertain sampled-data systems, respectively. Furthermore, the gap between the original supremum of the spectral radii and the upper bound (as well as the lower bound) is shown to converge to 0 at the rate no smaller than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(1/M)^{1/n}$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> is the gridding parameter and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of perturbation blocks in the structured model uncertainty. To put it another way, both the upper bound used in the sufficient condition and the lower bound taken in the necessary condition tend to the exact supremum of the spectral radii by taking <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> larger. Finally, a numerical example is given to demonstrate the effectiveness of the overall arguments.