NCO Tracking for Singular Control Problems using Neighboring Extremals

A powerful approach for dynamic optimization in the presence of uncertainty is to incorporate measurements into the optimization framework so as to track the necessary conditions of optimality (NCO), the so-called NCO-tracking approach. For nonsingular control problems, this can be done by tracking active constraints along boundary arcs, and using neighboring-extremal (NE) control along interior arcs to force the first-order variation of the NCO to zero. In this paper, an extension of NE control to singular control problems is proposed. The idea is to design NE controllers from successive time differentiations of the first-order variation of the NCO. Based on these results, a NCO-tracking controller that is easily tractable from a real-time optimization perspective is proposed, whose application guarantees that the first-order variation of the NCO converges to zero exponentially. The performance of this NCO-tracking controller is illustrated via the case study of a steered car, a 5th-order two-input dynamical system.