An Interaction-Coordination Algorithm with Time- and State-Decompositions for Nonlinear Optimal Control Problems

Abstract

This paper proposes a computational algorithm for the multilevel control of a composite nonlinear dynamical system with the performance index of a quadratic type. First, for the solution of a linear two-point boundary-value problem (TPBVP), a computational technique, termed time-decomposition, is proposed. The time-decomposition implies decomposition of the system equations along the subintervals of the independent variable. The boundary conditions of each subinterval are determined algebraically in such a way that the continuity condition of the variables at the boundary points is satisfied. This technique plays an important role in the subsequent discussions. Second, a nonlinear optimal control problem is considered. The objective is to minimize the performance index of a quadratic type. For this problem, the authors have previously presented ‘the interaction-coordination algorithm with modified performance index.’ The basic idea of this algorithm is to decompose the overall nonlinear problem into a number of smaller linear subproblems. Here this kind of decomposition is termed state-decomposition. In the present paper, a computational algorithm by use of both the time- and the state-decomposition is proposed. The algorithm essentially constructs a three-level computational structure, and results in a fast convergence even for problems with strong nonlinearities and/or a long control interval.