Adaptive composite suboptimal control for linear singularly perturbed systems with unknown slow dynamics

In this article, using singular perturbation theory and adaptive dynamic programming (ADP) approach, an adaptive composite suboptimal control method is proposed for linear singularly perturbed systems (SPSs) with unknown slow dynamics. First, the system is decomposed into fast- and slow-subsystems and the original optimal control problem is reduced to two subproblems in different time-scales. Afterward, the fast subproblem is solved based on the known model of the fast-subsystem and a fast optimal control law is designed by solving the algebraic Riccati equation corresponding to the fast-subsystem. Then, the slow subproblem is reformulated by introducing a system transformation for the slow-subsystem. An online learning algorithm is proposed to design a slow optimal control law by using the information of the original system state in the framework of ADP. As a result, the obtained fast and slow optimal control laws constitute the adaptive composite suboptimal control law for the original SPSs. Furthermore, convergence of the learning algorithm, suboptimality of the adaptive composite suboptimal control law and stability of the whole closed-loop system are analyzed by singular perturbation theory. Finally, a numerical example is given to show the feasibility and effectiveness of the proposed methods.