Optimal Piecewise State Feedback Control for Nonlinear Dynamical Systems

In this paper, we consider a class of optimal piecewise state feedback control problems. For this optimal control problem, the time horizon is divided into N subintervals with the end points of each subinterval being referred to as switching times. The control is given by a piecewise state feedback form, where different feedback gain matrices are allowed for different subintervals. Both the switching instants and the feedback gain matrices on the corresponding subintervals are decision variables to be determined such that a given cost functional is minimized. Our approach is to use a time scaling transform to convert this optimal piecewise state feedback control problem into an equivalent optimal parameter selection problem, where the varying time points are being mapped into fixed time points. Then the gradient formulae of the cost functional are derived. On this basis, the equivalent transformed optimal parameter selection problem can be solved as a nonlinear optimization problem. An illustrative example is solved using the proposed method