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

A study is made of the problem of determining when a stabilizing controller is l₁-optimal for a given plant for some stable weighting function. This problem belongs to the class of inverse problems in optimal control introduced by R.E. Kalman (1964). Only SISO discrete-time plants are considered. The authors give a characterization of all the possible l₁-optimal compensators for a given plant with different weights under some assumptions on the plant and the allowable weights. A few results are also obtained in the general case (i.e. without making overly restrictive assumptions on the plant and allowable weights). In particular, 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. The authors also show 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. They characterize some of these weighting functions