Time-optimal and saturating controls with application to flexible structures

This dissertation is concerned with developing new time-optimal control techniques for higher-order linear and weakly nonlinear systems. As an application, we consider the simultaneous slewing and vibration suppression of a flexible beam, possibly with a tip mass. This application arises in the design of large space structures and flexible lightweight and accurate robotic arms.

The solution of the soft-constrained time-optimal control problem is expressed in terms of the controllability Grammian. The properties of the open-loop solution are studied. A closed-loop control algorithm, which takes into account the mUltiplicity of extremal solutions, is then developed. The algorithm is based on the concept of continuation and reduces the computational complexity by as much as two orders of magnitude when compared to the brute-force approach.

The amplitude of the soft-constrained time-optimal control is found to saturate as the state norm becomes large, thus suggesting a simpler but suboptimal feedback implementation. We develop and discuss the concept of saturating controls for linear systems, and we develop a design approach that generates a family of saturating control laws in which the speed of the response and amount of available control action can be explicitly traded off.

The soft-constrained time-optimal cheap-control problem is formulated and solved using singular-perturbation theory. The solution procedures are illustrated with an example solved using the MACSYMA symbolic manipulation language.

Regular-perturbation theory is then used to find the open-loop hard-constrained time-optimal control for a class of weakly nonlinear systems. The control is found by solving a nonlinear two-point boundary value problem (TPBVP) characterizing the control of the linearized system, and a second linear TPBVP.