Stability and stabilization conditions for Takagi-Sugeno fuzzy model via polyhedral Lyapunov functions
Esterhuizen, Willem Daniël
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The Takagi-Sugeno (T-S) fuzzy model together with parallel distributed compensation forms a very effective framework for modeling, analysis and control design for nonlinear systems. A large body of theory exists that deals with this framework and most of the fundamental notions, such as stability, stabilizability, controller design, observer design, etc., have been studied extensively. A large number of the well-established results are based on quadratic Lyapunov functions. The main reason is that the stability and design conditions under quadratic Lyapunov functions are in the form of linear matrix inequalities which are easily solvable. However, the class of quadratic Lyapunov functions are conservative, in the sense that there are systems for which their stability cannot be established by quadratic Lyapunov functions. A natural question to ask is: are there other candidate Lyapunov functions that are less conservative? It turns out that the class of polyhedral Lyapunov functions are universal for the T-S fuzzy model stability problem, that is if a T-S fuzzy system is stable, there exists a polyhedral Lyapunov function that proves the stability. This thesis is a first look at the applicability of polyhedral Lyapunov functions to the T-S fuzzy model-based framework for the stability analysis and control design of nonlinear systems. First, two stability theorems are presented in this thesis. It is shown that the stability of a T-S fuzzy system via polyhedral Lyapunov functions can be established through linear programming. Next, the stabilization problem is investigated to find control laws that guarantee the stability of the closed-loop systems. Two stabilization theorems are presented which are derived from the stability theorems. The conditions of the stabilization theorems are in the form of nonconvex inequalities that are not readily solvable. Implementation examples are included in which solutions are found through either brute-force, or making the constraints convex in exchange for a loss of feasible space. Two relaxed stabilization theorems are also derived by taking advantage of certain aspects of the T-S fuzzy model.
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