l1-optimality of feedback control systems: the SISO discrete-time case

Controllers that optimally reject a class of disturbances are considered. The problem of determining when a stabilizing control is l₁-optimal for a given plant is studied for some stable weighting function. This problem belongs to the class of inverse problems in optimal control introduced by Kalman. It is shown that for a given plant, the set of all the H_∞-optimal controllers (obtained by considering all stable weighting functions with no zeros on the unit circle) is actually contained in the corresponding set of l₁-optimal controllers. It is also demonstrated that an l₁-optimal controller, unlike an H_∞-optimal controller, can remain l₁-optimal for the same plant for a wide range of nontrivially different weighting functions