Optimization of the ℓ^∞-induced norm under full state feedback

This paper considers the ℓ¹-optimal control problem, i.e., minimization of the effects of disturbances as measured by the ℓ^∞-induced norm. Earlier work showed that even in the case of full state feedback, optimal and near-optimal linear controllers may be dynamic and of arbitrarily high order. However, previous work by the author derived the existence of near-optimal nonlinear controllers which are static. This paper presents a constructive algorithm for such nonlinear controllers. The main idea is to construct a certain subset of the state space such that achieving disturbance rejection is equivalent to restricting the state dynamics to this set. A construction of this subset requires the solution of several finite, linear programs with the number of variables being at least the number of states but less than the number of states plus the number of controls. The concept of controlled invariance plays a central role throughout.