Constrained optimization and control of nonlinear systems: new results in optimal control

The main goal of this paper is to outline and present a straightforward constrained optimization framework as well as to develop a feasible analytical method to enable the designer to solve optimization problems for nonlinear time-varying systems with state and control bounds. To establish these results, we present a new design methodology which leads to innovative developments. For control problems of a general nature, the most efficient methodologies of attack, yet devised, are the Hamilton-Jacobi theory, maximum principle and Lyapunov's concept. This paper elaborates a general procedure to solve the constrained optimization problem using the dynamic programming method. Instead of encountering difficulties via the calculus of variations or Pontryagin's maximum principle, we develop a feasible and computationally efficient optimization algorithm. This involves the application of a new nonquadratic functional as well as modified mappings of control and state bounds. A particular class of nonquadratic, sufficiently smooth and real-valued positive-definite performance integrands is selected. The functional depends on the state and control variables and associated bounds. These constraints limit a set of solutions as well as a class of control structures from which an optimal algorithm can be found. The nonquadratic, continuously differentiable return functions are used. Our approach simplifies optimization issues and allows one to solve the bounded control problem for high-order dynamical systems. The approach is presented through illustrations. Analytical and numerical results are given.