Closed-Form Solutions for Feedback Control with Terminal Constraints

The problem of closed-loop control of maneuvers between two states for linear dynamical systems, subject to an arbitrarily specified terminal state, is considered. The feedback controller design is based on finite-time quadratic regulator theory. Closed-form expressions for the optimal control law are developed. Solutions are presented for both conventional and smoothed control profiles with fixed and/or free end condition problems. In the maneuvers using control-rate penalties, smooth profiles are generated throughout the maneuvers in the sense that the initial condition jump discontinuities have been eliminated. Several examples involving large-angle maneuvers of a spacecraft are demonstrated. Results include control maneuvers from one state to another, such as rest to rest and spin to rest, which effectively justify the solutions developed in this paper.