Stability analysis of interval type-2 polynomial fuzzy-model-based control systems

Student thesis: Doctoral ThesisDoctor of Philosophy


This thesis investigates the stability problem of controlling nonlinear systems. Nonlinear systems are hard to be controlled due to its nonlinear terms and sometimes it is hard to find feedback gains for the controller even with the application of advanced nonlinear control methods due to the conservativeness of these methods. Consequently, some of the nonlinear systems cannot be controlled due to the limitation of control methods. To well control these nonlinear systems, the research of interval type-2 (IT2) polynomial fuzzy-model-based (PFMB) control systems is conducted.

For IT2 PFMB control systems, the nonlinear plant is represented by the polynomial fuzzy model as the PFMB control scheme demonstrates higher potential to represent the nonlinearity of the nonlinear system than the traditional Takagi-Sugeno (T-S) fuzzymodel-based (FMB) control method when polynomials are used in local systems. Meanwhile, the IT2 fuzzy logic is used to tackle the uncertainties because of its property that can incorporate uncertainties into the membership functions. Hence, both of system nonlinearity and uncertainties can be handled by using the IT2 PFMB control method. Then, aims of this thesis are to investigate the stability analysis and to relax the stability conditions of the IT2 PFMB control system.

Firstly, this thesis investigates the stability conditions, in terms of sum-of-squares (SOS), for PFMB control systems with mismatched IT2 membership functions based on Lyapunov stability theory. To relax the stability conditions, the information of the membership functions is introduced into the analysis. By introducing different information of the membership functions, three approaches, namely membership functions approximated by piecewise linear membership functions (PLMFs), linear functions in sub-domains and polynomial functions in sub-domains, are proposed to conduct the stability analysis. All of these methods have feasible control results and can achieve larger feasible regions comparing with that of membership function independent (MFI) method, which is the main stream of analysis techniques in the literature.

Secondly, this thesis employs a switching control scheme to further relax the stability conditions. An IT2 polynomial switching controller is introduced into the IT2 PFMB control system. With the application of the switching controller, the whole operation domain is divided into several sub-domains, and the stability analysis is conducted in each sub-domain. Although this method introduces a more complex controller, it relatively relaxes the stability conditions when the same information of membership functions is used.

Thirdly, this thesis introduces a mean controller of the IT2 fuzzy controller to investigate the system stability. The mean controller is the average sum of some type-1 polynomial fuzzy controllers, whose type-1 membership functions are embedded into the original IT2 membership functions. When more embedded type-1 membership function information is considered the larger feasible region can be obtained and the stability conditions are more relaxed.
Date of Award1 Jul 2020
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorHak-Keung Lam (Supervisor)

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