Formulation of nonlinear control problems with actuator saturation as linear programs

Abstract

Control of input-affine nonlinear dynamical systems is investigated in this paper. In this regard, a system of linear inequalities (SLI) in the control inputs is developed, and it is proved that if this SLI has a solution at all times, applying it to the system leads to moving its state trajectories towards the desired point in space for all initial conditions. But, since it may happen that this SLI has infinitely many solutions or even no solution, the best (possibly, approximate) solution is obtained by solving a linear programming (LP) problem. Different LP problems with different properties and applications are proposed for this purpose. Formulation of the control problem as an LP makes it possible to take into account the effect of actuator saturation simply by adding bound constraints to the problem. We can also make the resulting control system robust to model uncertainties. Some numerical examples including the output voltage regulation of double-source DC-DC converter and robot path-planning, both by considering the effect of actuators saturation and taking into account the uncertainties in model, are also presented.