Optimal Fuzzy Control Of The Continuous-Time Nonlinear Stochastic Fuzzy Systems

As the parameters of a nonlinear Takagi-Sugeno(T-S) fuzzy control system are perturbed by random noises with known covariance matrix, the system turns to be a stochastic T-S system. Essentially speaking, it is a nonlinear stochastic system. Since the continuous-time stochastic T-S fuzzy system is a nonlinear Ito stochastic differential equation, the related optimal control issue reduces to solving a second order Hamilton-Jacobi-Bellman (HJB) partial differential equation. But in most cases, this HJB equation can be solved neither analytically, nor numerically with satisfactory convergence and stability.The main purpose of this thesis is combining the theory of stochastic optimal control and the method of fuzzy control, to attack the design issue on optimal fuzzy control of continuous-time nonlinear stochastic T-S fuzzy systems with additive noise or multiplicative noise, and apply them to process control and financial engineering. Three sub-optimalapproaches are utilized: local-concept optimal control, inverse optimal control and optimal guaranteed-cost control.With regard to continuous-time stochastic T-S fuzzy system with additive noise, its well-posedness issue is studied in Ito stochastic calculus sense and the theorem of the existence and uniqueness of solutions is proved. An example is given to show the well-posedness of some models addressed in the references is invalid in mean-square sense, and then demonstrate the importance of the Ito integral in the definition of the model of stochastic T-S fuzzy system with additive noise. Furthermore, base on Wu and Lin's (2000) local concept approach to the optimal fuzzy controller design for the T-S fuzzy system, we present an optimal fuzzy controller in local concept design method for the continuous-time stochastic T-S fuzzy system with additive noise in finite horizon case. An illustrative example shows the usage of the proposed approach.As far as continuous-time nonlinear stochastic system with multiplicative noise is concerned, two inverse optimal control theorems in mean square exponential stability are deduced firstly by using the Lyapunov theory of stochastic stability. Then, the inverse optimal control theorems are employed to deal with optimal controller design for continuous-time stochastic T-S fuzzy system. The corresponding linear matrix inequality (LMI) design for the inverse optimal controller in bothcommon control input matrices case and different control input matrices case, respectively. The design algorithms are explained by numerical examples.Moreover, guaranteed-cost control and optimal guaranteed-cost control are also studied for continuous-time stochastic T-S fuzzy system, with respect to quadratic performance index. The related LMI design skills are put forward to achieve the desired performance and illustrated via examples. Finally, the stability issue of the exchange rate as the application of the proposed control technique is studied in financial engineering.