Nonlinear Optimal Control via Finite Time Horizon State-dependent Riccati Equation

In this article, the finite time horizon state-dependent Riccati equation (FTSDRE) is used as an extension to the state-dependent Riccati equation (SDRE) for controlling a class of nonlinear systems. The derivations of the SDRE for two classes of systems are presented in the finite time horizon. First class, systems with states and control nonlinearities which results nonlinear differential optimal control equation (NDOCE) and a simplification of the NDOCE which is state-dependent differential Riccati equation like (SDDRE-like). Next, second class, the formulation for nonlinear systems without nonlinearities in control is presented to reach the structure of FTSDRE. On top of that, state transition matrix (STM) approach is presented to solve the problem, as an alternative way to verify the solution to NDOCE and FTSDRE. An illustrative example for the first case, systems with control nonlinearities, and a two-links planar robot for the second case are simulated to support the idea.