Stability Analysis and Control Design of TSK Systems with Applications in Power Systems

Abstract

In this dissertation, we propose a new approach for the stability analysis of discrete-time and continuous-time, type-1 and type-2 Takagi-Sugano-Kang (TSK) fuzzy systems. A major advantage of the new exponential stability conditions is that they do not require the existence of a common Lyapunov function and are therefore applicable to systems with unstable consequents. Our results compare favorably with results available in the literature and provide stability tests where other approaches fail.New control system design methodologies are proposed based on the proposed stability results. The new methodologies do not require a common Lyapunov function and are therefore applicable to systems with non-stabilizable consequents. Our controllers include fuzzy type-1 proportional and PI controllers as well as constant state feedback controllers. The controllers result in exponentially stable systems and the designer can specify the rate of exponential convergence. The controller designs can be tested using linear matrix inequalities (LMIs). The design methodology is demonstrated using examples where methods based on a common Lyapunov function fail and the new methodologies provide better performance.This dissertation also introduces four sufficient instability tests for type-1 and type-2 continuous-time TSK fuzzy systems. The first test is developed using a common Lyapunov function. The second test is derived using the variable gradient method. The third test is based on the comparison principle with a non-differentiable Lyapunov function candidate and the upper right hand derivative. Finally, the fourth test is obtained using Chetaev's theorem. We provide numerical examples to demonstrate and compare our instability tests.In addition to the theoretical results, this dissertation uses the proposed stability and instability conditions to investigate the stability of energy markets and their implications for the deregulated market environment. We formulate a TSK model for energy markets to account for their uncertainty and nonlinearity and obtain more realistic stability results. Simulation results demonstrate the application of the stability and instability tests to electric power markets under different market conditions. The tests also demonstrate the importance of including nonlinearities and uncertainty in the energy market when investigating its stability.This dissertation also presents a TSK model for the generator voltage control system including its saturation nonlinearities. A TSK proportional controller using our control design is designed for the generator voltage control system. The TSK modeling and TSK control strategy are shown to provide a good dynamic response for the system, including saturation nonlinearities.