Overcoming Singularity and Degeneracy in Neighboring Extremal Solutions of Discrete-Time Optimal Control Problem with Mixed Input-State Constraints

Neighboring Extremal Optimal approach is effective in solving optimal control problems through approximation. Under certain conditions, a matrix involved in the calculation of optimal control approximation can become singular, leading to a technical difficulty in the application of the approach. These situations may include the cases when a constraint depends only on the states but not the inputs, or the cases when the inequality constraints outnumber the inputs. In this paper, we propose a solution that can circumvent this technical difficulty. First, by back-propagating the state constraints, we show that input-independent constraints can be recast as the state-input constraints to avoid the matrix singularity. The back-propagation, however, might lead to another problem of “degeneracy,” where a back-propagated constraint is imposed on the initial state, so that no feasible neighboring extremal solution exists for the problem. In the latter case, a linear programming approach is proposed to deal with this degeneracy. A ship maneuvering control problem is used in the paper to illustrate the singularity and degeneracy issues, and to elucidate the mechanics of the proposed scheme.