Relaxed Stabilization Conditions for TS Fuzzy Systems With Optimal Upper Bounds for the Time Derivative of Fuzzy Lyapunov Functions

Abstract

This paper initially proposes an optimization problem and after presents its optimal solution. Then, this result is applied to obtain relaxed conditions to design controllers for nonlinear plants described by Takagi-Sugeno (TS) models, based on fuzzy Lyapunov function (FLF) and Linear Matrix Inequalities (LMI). The FLF is given by $V(x(t)) = x(t)^{T}P(\alpha (x(t)))x(t)$ , where $x(t)$ is the plant state vector, $P(\alpha (x(t))) = \alpha _{1}(x(t))P_{1} + \alpha _{2}(x(t))P_{2} + \cdots + \alpha _{r}(x(t))P_{r}$ , $P_{i}=P_{i}^{T} > 0$ and $\alpha _{i}(x(t))$ is the weight related to the local model $i$ in the representation of the plant by TS fuzzy models, for $i=1,2,\cdots,r$ . When one calculates the time derivative of this $V(x(t))$ , it appears the term $x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ , that is usually handled using conservative upper bounds, supposing that the bounds of the time derivative of $\alpha _{i}(x(t))$ , $i=1,2,\cdots,r$ , are available. The main result of this paper is a procedure to obtain optimal upper bounds for the term $x(t)^{T}\dot {P}(\alpha (x(t)))x(t)$ , such that they contemplate the maximum value and are always smaller than or equal to the maximum value. It is a relevant result on this subject, because these optimal upper bounds do not add any constraint. With these optimal upper bounds, a relaxed design method for stabilization of TS fuzzy models is proposed. Two numerical examples illustrate the effectiveness of this procedure.

Keywords

• Fuzzy Lyapunov function (FLF)
• Takagi-Sugeno (TS) fuzzy systems
• linear matrix inequalities (LMIs)
• fuzzy control
• stability
• stabilization