An $l_{1}$ formulation is often appropriate for feedback systems subject to persistent bounded disturbances, or in situations where plant uncertainty is best described in terms of the $l_{\infty}$ norm. Any discrete-time $l_{1}$ optimization problem can be converted to a (usually infinite-dimensional) linear program, to which modern techniques such as interior point methods can be applied. A sub-optimal controller giving performance arbitrarily close to optimal can then be constructed. The special structure implied by exact solutions can however easily be missed with this approach. Although $l_{1}$ optimal controllers can be of arbitrarily high order, it has long been suspected that, subject to mild restrictions, the exact optimal controller always has a rational $Z-$transform. In this talk we shall present some new exact solutions to a class of $l_{1}$ problems.
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