Optimal fuzzy controller design in continuous fuzzy system: Global concept approach

Abstract

In this paper, we propose a systematic and theoretically sound way to design a global optimal fuzzy controller to control and stabilize a continuous fuzzy system with free- or fixed-end point under finite or infinite horizon (time), A linear-like global system representation of continuous fuzzy system is first proposed by viewing a continuous fuzzy system in global concept and unifying the individual matrices into synthetical matrices. Based on this, the optimal control law which can achieve global minimum effect is developed theoretically. The nonlinear segmental two-point boundary-value problem (TPBVP) is derived for the finite-horizon problem and a forward Riccati-like differential equation (DE) for the infinite-horizon problem. To further simplify the computation, a segmental Riccati-like DE is derived in solving the finite- or infinite-horizon issues. Moreover, in the case of time-invariant fuzzy systems, we show that the optimal controller can be obtained by Just solving algebraic Riccati-like equations. Grounding on this, several fascinating characteristics of the resultant closed-loop fuzzy system can be elicited easily. The stability of the closed-loop fuzzy system can be ensured by the designed optimal fuzzy controller. The optimal closed-loop fuzzy system cannot only be guaranteed to be exponentially stable, but also be stabilized to any desired degree. Also, the total energy of system output is absolutely finite. Moreover, the resultant closed-loop fuzzy system possesses an infinite gain margin; that is, its stability is guaranteed no matter how large the feedback gain becomes. An example is given to illustrate the proposed optimal fuzzy controller design approach and to demonstrate the proved stability properties.