Optimizing a Class of Nonlinear Singularly Perturbed Systems Using SDRE Technique

In this paper, we address the finite-horizon optimization of a class of nonlinear singularly perturbed systems based on the state-dependent Riccati equation (SDRE) technique and singular perturbation theory. In such systems, both slow and fast variables are nonlinear. Moreover, the performance index for the system states is nonlinearly quadratic. In this study, unlike conventional methods, linearization does not occur around the equilibrium point, and it provides a description of the system as state-dependent coefficients (SDCs) in the form f(x) = A(x)x. One of the advantages of the state-dependent Riccati equation method is that no information about the Jacobian of the nonlinear system, just like the Hamilton–Jacobi–Belman (HJB) equation, is required. Thus, the state-dependent Riccati equation has simplicity of the linear quadratic method. On the other hand, one of the advantages of the singular perturbation theory is that it reduces high-order systems into two lower order subsystems due to the interaction between slow and fast variables. In the proposed method, the singularly perturbed state-dependent Riccati equations are first derived for the system under study. Using the singular perturbation theory, the singularly perturbed state and state-dependent Riccati equations are separated into outer layer, initial, and final layer correction equations. These equations are then solved to obtain the optimal control law. Simulation results in comparison with the previous methods indicate the desirable performance and efficiency of the proposed method. However, it should be noted that due to the dependence of the proposed method on the choice of state-dependent matrices and the presence of a nonlinear optimal control problem, the results are generally suboptimal.