Path-Following-Based Design for Guaranteed Cost Control of Polynomial Fuzzy Systems
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Path-Following-Based Design for Guaranteed Cost Control of Polynomial Fuzzy Systems Kai-Yi Wong1 • Motoyasu Tanaka1 • Kazuo Tanaka1
Received: 12 June 2020 / Revised: 17 July 2020 / Accepted: 27 July 2020 Ó Taiwan Fuzzy Systems Association 2020
Abstract This paper presents a path-following-based design for guaranteed cost control of a class of nonlinear systems represented by polynomial fuzzy systems. First, this paper proposes a polynomial Lyapunov function approach to guaranteed cost control for the feedback system consisting of a polynomial fuzzy system and a polynomial fuzzy controller. In particular, we introduce a new type of polynomial fuzzy controllers based on an approximate solution for the Hamilton–Jacobi–Bellman inequality. To design a guaranteed cost polynomial fuzzy controller effectively, a path-following-based design algorithm is newly developed by formulating as a sum-ofsquares (SOS) stabilization problem. Two new relaxations are provided by bringing a peculiar benefit of the SOS framework. One is an S-procedure relaxation for the considered outmost Lyapunov function level set that is contractively invariant set. The other is an S-procedure relaxation for design conditions obtained for polynomial membership functions redefined by variable replacements in considered ranges. Furthermore, this paper provides a practical and reasonable way for estimating lower upperbounds of a given performance function by increasing the order of a considered polynomial function. Finally, a complicated nonlinear system design example is employed
& Kai-Yi Wong [email protected] Motoyasu Tanaka [email protected] Kazuo Tanaka [email protected] 1
Department of Mechanical and Intelligent Systems Engineering, The University of Electro-Communications, Tokyo 182-8585, Japan
to illustrate the validity of the proposed design algorithm and the lower upper-bound estimation. Keywords Guaranteed cost control Hamilton–Jacobi– Bellman inequality Lower upper-bound estimation Pathfollowing-based design Polynomial Lyapunov function S-procedure Sum-of-squares
1 Introduction In the recent quarter century, linear matrix inequality (LMI) approaches to Takagi–Sugeno (T–S) fuzzy systems have played a central role in fuzzy control research [1] since the pioneering works [2–4] were published. A huge number of studies [5] on the topic have paid a lot of effort to formulation as numerically feasibility design problems within the LMI framework. In [6–8], as a new attempt beyond the LMI framework, a number of extended ideas regarding polynomialization, such as polynomial fuzzy systems, polynomial fuzzy controllers, polynomial Lyapunov functions, etc., were considered by introducing a sum-of-squares (SOS) framework. In fact, the works [6–8] first brought the SOS framework into fuzzy control systems design and analysis. A remarkable observation of the works is that polynomial fuzzy systems and polynomial fuzzy controllers can be regarded as a natural generalization of T–S fuzzy systems and T–S fuzzy controllers, respectively.
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