Suboptimal control of nonlinear systems under restrictions in the manipulated variable

This paper presents a numerical scheme to approximate the solution of the optimal control problem for nonlinear systems with restrictions on the manipulated variable. In there exists a unique regular time-interval, it has been theoretically shown that there is a related unrestricted nonlinear problem, with unknown final state and costate, whose saturated optimal control leads to the solution of the original problem with restrictions. The method proposed here systematically reduces the cost of a seed control trajectory by using the solution of the differential Riccati equation (DRE) corresponding to the linearization of the system around the seed data. The final condition for the DRE is constructed from the variations of the final state ρ_Δ and costate μ_Δ. The updating of the control strategy is performed through the gradient method applied to (ρ_Δ, μ_Δ)and to the time instants τ_i when the control becomes saturated. This allows to avoid working with the Hamiltonian equations, usually unstable, and also to construct a feedback control law, robust under disturbances. The results are illustrated by optimizing a quadratic cost imposed to a two-dimensional non-linear system with bounded control values.