Approximate Finite-Horizon Optimal Control for Input-Affine Nonlinear Systems with Input Constraints

The finite-horizon optimal control problem with input constraints consists in controlling the state of a dynamical system over a finite time interval (possibly unknown) minimizing a cost functional, while satisfying hard constraints on the input. For linear systems the solution of the problem often relies upon the use of bang-bang control signals. For nonlinear systems the “shape” of the optimal input is in general not known. The control input can be found solving an Hamilton-Jacobi-Bellman (HJB) partial differential equation (pde): it typically consists of a combination of bang-bang arcs and singular arcs. In the paper a methodology to approximate the solution of the HJB pde arising in the finite-horizon optimal control problem with input constraints is proposed. This approximation yields a dynamic state feedback law. The theory is illustrated by means of an example: the minimum time optimal control problem for an industrial wastewater treatment plant.