Comment on "Discrete-time optimal fuzzy controller design: global concept approach"

In this paper, we propose a systematic and theoretically sound way to design a global optimal discrete-time fuzzy controller to control and stabilize a nonlinear discrete-time fuzzy system with finite or infinite horizon (time). A linear-like global system representation of discrete-time fuzzy system is first proposed by viewing a discrete-time fuzzy system in global concept and unifying the individual matrices into synthetical matrices. Then, based on this kind of system representation, the discrete-time optimal fuzzy control law which can achieve global minimum effect is developed theoretically. A nonlinear two-point-boundary-value-problem (TPBVP) is derived as the necessary and sufficient condition for the nonlinear quadratic optimal control problem. To simplify the computation, a multistage decomposition of optimization scheme is proposed and then a segmental recursive Riccati-like equation is derived. Moreover, in the case of time-invariant fuzzy systems, we show that the optimal controller can be obtained by just solving discrete-time 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 can not 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.