Non-Saturating Constrained Input Piecewise-Linear LQ Control

Abstract

A Piecewise-linear control (PLC) law, based on LQ theory, is derived. This law raises an LQ gain as the controlled error converges towards the origin. It can be computed off-line and it guarantess that input bounds are never exceeded without causing input saturation. A succession of positively invariant sets are defined of diminishing size and associated with each is the corresponding highest. LQ gain possible in the presence of the input bounds. The formulation gives rise to an iteration function whose solution is a fixed point (FP). The FP defines the optimal gain and the invariant set, which comprise ellipsoids. FP existence, uniqueness and convergence properties are discussed. We explain the algorithms for the generation and subsequent implementation of the PLC law. The notion of a switching function is introduced. The PLC scheme is demonstrated for the linearized dynamics of the simple pendulum and the PUMA 560 robot. A discrele time PLC law call also be derived using the discrete time algebraic Riccati equation.