Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs

Abstract

This paper presents new relaxed stability conditions and LMI-(linear matrix inequality) based designs for both continuous and discrete fuzzy control systems. They are applied to design problems of fuzzy regulators and fuzzy observers. First, Takagi and Sugeno's fuzzy models and some stability results are recalled. To design fuzzy regulators and fuzzy observers, nonlinear systems are represented by Takagi-Sugeno's (T-S) fuzzy models. The concept of parallel distributed compensation is employed to design fuzzy regulators and fuzzy observers from the T-S fuzzy models. New stability conditions are obtained by relaxing the stability conditions derived in previous papers. LMIbased design procedures for fuzzy regulators and fuzzy observers are constructed using the parallel distributed compensation and the relaxed stability conditions. Other LMI's with respect to decay rate and constraints on control input and output are also derived and utilized in the design procedures. Design examples for nonlinear systems demonstrate the utility of the relaxed stability conditions and the LMI-based design procedures.

Membership-Function-Dependent Stability Analysis and Control Synthesis of Guaranteed Cost Fuzzy-Model-Based Control Systems

“…Since then, the T-S FMB control systems have drawn the attention of fuzzy control researchers for more than 20 years due to its effectiveness on handling nonlinear control systems [7,8]. In particular, the issues of stability analysis and control synthesis have been investigated extensively and fruitful results can be found in [2,[9][10][11][12][13][14][15][16][17][18][19] and the references therein.…”

“…With the parallel distribution compensation (PDC) [9] design approach, the stability conditions can be relaxed and some further related works can be found in [2,[9][10][11][12][13][14][15][16]. The work in [10] used the symmetry property of the membership functions of the T-S fuzzy model and fuzzy controller in the analysis and then managed to relax the LMI-based stability conditions. Inspired by the work in [10], various techniques have been proposed to gather the membership functions in the stability analysis [2,[11][12][13][14][15].…”

“…Inspired by the work in [10], various techniques have been proposed to gather the membership functions in the stability analysis [2,[11][12][13][14][15]. The work in [11] combined all the LMIs used in [10] to form a large symmetric matrix resulting in further reducing the conservativeness of stability conditions. The work in [16] generalized the stability conditions with the consideration of the permutations of membership functions using the Pólya theorem.…”

“…If there exists a common solution of a group of Lyapunov inequalities in terms of LMIs which can be solved effectively by convex optimization methods such as interior point method [2], the FMB control system is guaranteed to be asymptotically stable [9]. With the parallel distribution compensation (PDC) [9] design approach, the stability conditions can be relaxed and some further related works can be found in [2,[9][10][11][12][13][14][15][16]. The work in [10] used the symmetry property of the membership functions of the T-S fuzzy model and fuzzy controller in the analysis and then managed to relax the LMI-based stability conditions.…”

“…The work in [10] used the symmetry property of the membership functions of the T-S fuzzy model and fuzzy controller in the analysis and then managed to relax the LMI-based stability conditions. Inspired by the work in [10], various techniques have been proposed to gather the membership functions in the stability analysis [2,[11][12][13][14][15]. The work in [11] combined all the LMIs used in [10] to form a large symmetric matrix resulting in further reducing the conservativeness of stability conditions.…”