Predictive control of an on-off system with two control variables

Abstract

This paper studies the optimum control of multi-variable systems by considering the behaviour of the two-variable on-off system described by the equation x ¨ + A x ˙ = u where u 1 , u 2 = ± 1. By the application of Pontryagin's maximum principle, it is shown how the number of switchings and the drive ratio u 1 /u 2 , for an optimum trajectory (minimum settling time criterion), depend on the nature of the matrix A. For the case when the interaction terms a 12 and a 21 have the same sign, the maximum number of switchings for an optimum trajectory is three, and the drive ratio u 1 /u 2 near the origin of the error phase space has the same sign as a 12 and a 21 . A predictive controller described in this paper finds the three switching times by an iterative process which involves the application of logical rules to the predicted behaviour of the system, computed in a fast analogue model. This realizable controller is optimum when the two-by-two matrix A has a 12 a 21 ≥ 0; and will also control systems where a 12 a 21 < 0, although not optimally. It is also shown that the problem of hunting may be overcome by confining the hunting to the fast model alone.