(257d) Constrained Control Lyapunov Function Construction Via Approximation of Static Hamilton-Jacobi-Bellman Equations

The control of complex chemical processes is challenged by the presence of model nonlinearities and input constraints. As such, Lyapunov functions are often used to determine how to stabilize such systems, but only if the states lie within the chosen Lyapunov function's stability region. If it lies outside, the Lyapunov function provides no information about the stabilizing control, so the controller may fail to avert catastrophic destabilization even when it is possible to do so. Thus, designing controllers based on Lyapunov functions with large stability regions is an important safety consideration. Therefore, in this conference presentation, we consider the problem of constructing Lyapunov functions for nonlinear input-constrained systems with the largest possible stability regions by solving the associated Hamilton-Jacobi-Bellman PDE. To solve this equation, we employ a finite difference approximation and novel boundary conditions based on a recently-developed algorithmic construction of the boundary of the system's null controllable region, which efficiently determines all reachable states. Furthermore, since even smooth H-J-B PDEs are observed to contain discontinuities, the artificial viscosity perturbation method is used to improve the quality of the approximation. The sub-problem of determining the optimal input at each node is reduced to finding the roots of a certain nonlinear polynomial. Lastly, we illustrate the results via simulation examples.