Adaptive optimal synchronization control of two linear systems and its applications

For the outputs of two nth-order linear control systems to track their command signals respectively, and in mean while to work in equal synchronization status as soon as possible, an adaptive optimal synchronization control scheme, consisting of optimal feedback control, and adaptive constant disturbance cancellation, is presented. The two nth-order linear systems are written into tracking error state vector equations respectively and then combined together. To achieve synchronization between the two systems in both dynamic and static periods, the integral of synchronization error signal is selected as an independent state variable and then integrated with the tracking error state vectors to form an augmented state vector equation. Due to the specific selection of the state variable denoting synchronization error information, the entire augmented system remains controllable. The linear quadratic optimal control theory is then used to design a feedback controller for the augmented system. In this design process, we can arbitrarily adjust the weights for the two tracking error state vectors and the weight for the independent synchronization error state variable, so that perfect dynamic and static synchronization properties could be achieved, as well as good tracking performance could be realized in each tracking system. In order to cancel the influence of constant external disturbances, an adaptive scheme is proposed. While designing the update law for the adaptive control, system stability is proved by use of the second method of Lyapunov. Eventually, the presented method is applied to a position synchronization system, and simulation results show its effectiveness.