Sub-optimal Control Of A Class Of Nonlinear Singularly Perturbed Systems Based On Adaptive Dynamic Programming

In some complex physical systems and industrial processes, there always exist some small parameters such as small inertia, conductance, capacitance or some multi time scales, which cause high-order or ill-conditioned differential equations for the systems. The systems with small parameters are called singularly perturbed systems. Based on adaptive dynamic programming (ADP) algorithm, the optimal control problem for singularly perturbed systems is discussed. The main contents can be described as follows:(1). The adaptive dynamic programming algorithm solving the optimal control problem for normal nonlinear affine systems is studied. This iterative algorithm starts from an initial performance index and converges to its optimal solution by updating the control and performance. It avoids solving the complex HJB equation directly. Besides, rigorous proof of its convergence and realizations for nonlinear systems are provided. To improve the approximate accuracy, neural network structures are used to approximate the performance index, and the ADP algorithm based on neural network is proposed.(2). The characteristics of singularly perturbed systems are discussed. Firstly, boundary layer adjustment is provided to calculate the solution of singular perturbation. Secondly, from the view of multi time scales, it is cognized that in the singularly perturbed system with a smallε, there are two classes of eigenvalues:one isεtimes bigger than the other. Therefore, by variable transformation, the optimal control problem of singularly perturbed systems is decomposed into two independent slow-fast problems. According to composite control principle, the sub-optimal of the whole system can be represented by the optimal control of the two problems.(3). The sub-optimal control for a class of nonlinear singularly perturbed systems is obtained by adaptive dynamic programming. First, the nonlinear system is decomposed into two slow-fast subsystems. Then the adaptive dynamic programming algorithms for slow-fast subsystems based on neural network are proposed to acquire its optimal control respectively. Finally the composite control for the whole system is obtained and proven to be a sub-optimal control for the singularly perturbed systems.(4). For a class of nonlinear singularly perturbed systems the generalized HJB (GHJB) equation is applied to solve the optimal control problem for two subsystems. This method starts from an initial stabilizing control law. By Galerkin's method, the corresponding performance index can be calculated by the GHJB equation and then the control law updates according to the obtained performance. The iteration goes on until it converges to its optimal solution.