Semiglobal stabilization of linear systems using constrained controls: a parametric optimization approach

A bounded feedback control for asymptotic stabilization of linear systems is derived. The designed control law increases the feedback gain as the controlled trajectory converges towards the origin. A sequence of invariant sets of decreasing size, associated with a (quadratic) Lyapunov function, are defined and related to each of them, the corresponding possible highest gain is chosen, while maintaining the input bounded. Gains as functions of the position are designed by explicitly solving a c-parameterized programming problem. The proposed method allows global asymptotic stabilization of open-loop stable systems, with inputs subject to magnitude bounds and globally bounded rates. In the general case of linear systems that are asymptotic null controllable with bounded input, the semiglobal stabilization is also addressed taking into account the problem of semiglobal rate-limited actuators. The method is illustrated with the global stabilization of an inertial navigator, and the stabilization of a nonlinear model of a crane with hanging load. Copyright © 1999 John Wiley & Sons, Ltd.