Adaptive tracking control of linear systems to periodic target with set-valued information

This paper studies the adaptive control for linear systems with set-valued observations to track given periodic target. Based on the system parameters, accessorial parameters with the same order as tracking target is introduced and estimated. Consider the system parameters are unknown and set-valued observations can supply only limit information each time, we construct a two-scale adaptive control algorithm. Each control input is designed at the large time scale and last for a holding time (small scale), during which the parameter estimation algorithm is constructed. From the estimate of accessorial parameters, the control signal is updated at the large time scale by the certainty equivalence principle. As the holding time goes to infinity, the algorithm can be proved to be asymptotically efficient in a certain sense. Meanwhile, the adaptive tracking algorithm is shown to be asymptotically optimal. A numerical example is given to demonstrate the effectiveness of the algorithms and the main results obtained.