Bounded real property of time‐delay time‐varying systems and its application to learning control

Abstract

AbstractLearning control is one of the effective control methods to achieve high‐speed motion and accurate positioning of mechanical systems which repeat the same motion. In this method, the control input is improved in each trial based on the error between the desired trajectory and the system output. the learning control system is stable if the error transition system defined in learning control is bounded real (BR).Kawamura et al. have shown the stability condition of the learning control for time‐invariant continuous‐time systems based on BR property. Takahashi has studied the stability of learning control in the case of time‐varying continuous‐time systems and time‐varying discrete‐time systems.This paper derives the stability condition of the learning control for time‐delay time‐varying (TDTV) systems which have not been reported so far because of its complexity. First, the sufficient condition for TDTV systems to be BR independent of delay is derived and the circuit's theoretical interpretation for the BR condition is given. Using this result, the stability condition for learning control for TDTV systems is given. the error of learning control converges to zero by increasing the number of the trials if the corresponding error transition system, which is defined from the error at the i‐th trial and error at i + 1‐th trial in the learning control, is BR. Finally, a numerical example of the learning control for the TDTV system which satisfies the obtained stability condition is demonstrated, showing that the derived stability condition is valid.