Nonperiodic Multirate Sampled-Data Fuzzy Control of Singularly Perturbed Nonlinear Systems
Jing Xu, Yugang Niu, Hak‐Keung Lam
Abstract
Choosing adequate sampling frequencies in sensors has a considerably positive impact on the two time scale fuzzy logic controller design. Motivated by this concept, this article addresses the fuzzy-parallel distribution compensation (PDC)-based control synthesis for a singularly perturbed nonlinear systems (SPNS) under a nonperiodic multirate sampling mechanism, which also provides guidance on the reasonable choice of maximum allowable sampling time intervals (MASTIs) for multirate sensors. First, the sampled SPNS is converted into a continuous-time Takagi-Sugeno fuzzy singularly perturbed model (TSFSPM) with slow and fast time-varying delays. Then, an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -dependent Lyapunov-Krasovskii functional of order <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 proposed to derive the sufficient conditions for stabilizing a multirate sampled TSFSPM under a two time scale PDC control. Given the slow MASTI, an efficient linear-matrix-inequality-based design is proposed to recast the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -dependent stabilization conditions as a set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -independent linear matrix inequalities that are easily solved. On this basis, the upper bound of singular perturbation parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> , i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon ^*$</tex-math></inline-formula> , should be determined to compute the fast MASTI for the possibly slow sampling of fast states. The optimal match of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(\epsilon ^*,n)$</tex-math></inline-formula> is detected for a tradeoff among the closed-loop stability, the controller performance, and the sensor cost. The superiority of the obtained results is shown in an example of a flexible joint inverted pendulum system.