Optimal rejection of persistent and bounded disturbances: continuity properties and adaptation

Continuity properties of the problem of optimal rejection of bounded persistent disturbances, otherwise known as l¹ -optimal control, are furnished for discrete-time, single-input, single-output systems. It is shown that for any plant with no zeros on the unit circle, the l¹-optimal design is continuous as a map from the plant to the optimal closed-loop solution if the plant has a unique optimal solution. In the case of nonuniqueness, a superoptimal criterion is proposed that will always lead to a unique solution; however, it still fails to supply continuity. A characterization of a general class of adaptive controllers in the presence of bounded disturbances is given. Using this characterization, an adaptive/robust scheme based on the l¹ design methodology is proposed, and it is shown to be globally convergent