Digital Stabilization of Fuzzy Systems with Time-Delay and Its Application to Backing up Control of a Truck-Trailer

The control problems for delayed systems have attention over the last few decades since the time-delay is frequently a source of instability and encountered in various engineering systems. Extensive research has already been done in the conventional control to find the solutions [1][2]. However, for fuzzy control systems, there are few studies on the stabilization problem for especially systems with time-delay[3][4]. A linear controller like PID controllers has a short time-delay in calculating the output since its algorithm is so simple. However, in the case of a complex algorithm like fuzzy or neural networks, a considerable time-delay can occur because so many calculations are needed to get the output. Nevertheless, the most conventional discrete time fuzzy controllers are the ideal controllers in which the time-delay is not considered. Recently, to deal with the time-delay, the design methods of the fuzzy control systems with higher order have been proposed in [5]. However the structure of the control system is very complex because the design of higher order fuzzy rule-base is highly difficult. In this chapter, the digital fuzzy control system considering a time delay is developed and its stability analysis and design method are proposed. We use the discrete TakagiSugeno(TS) fuzzy model and parallel distributed compensation(PDC) conception for the controller[6-9]. And we follow the linear matrix inequality(LMI) approach to formulate and solve the problem of stabilization for the fuzzy controlled systems with time-delay. The analysis and the design of the discrete time fuzzy control systems by LMI theory are considered in [10-12]. If the system has a considerable time-delay the analysis and the design of the controller are very difficult since the time-delay makes the output of the controller not synchronized with the sampling time. We propose the PDC-type fuzzy feedback controller whose output is delayed with unit sampling period and predicted using current states and the control input to the plant at previous sampling time. The analysis and the design of the controller are very easy because the output of the proposed controller is synchronized with the sampling time. Therefore, the proposed control system can be designed using the conventional methods for stabilizing the discrete time fuzzy systems and the feedback gains of the controller can be obtained using the concept of the LMI feasibility problem.