Further results on the optimal rejection of persistent bounded disturbances

Abstract

The problem of designing controllers that optimally reject persistent disturbances is studied. The focus is on the case where the plant to be controlled has either zeros or poles on the stability boundary, i.e. the unit circle in the discrete-time case and the extended jω axis in the continuous-time case. For the discrete-time case, the problem of minimizing a cost functional of the form ||f-rg||₁, where the transform g^˜ of g^~ has some unit circle zeros is studied. A previously published dual problem formulation is extended, and it is shown that an optimal controller need not exist. The construction of a sequence of suboptimal controllers whose performance approaches the unattainable infimum of the cost function is studied. It is shown that two results which hold in the case of H_∞ optimization do not hold in the presented situation. Specifically, the introduction of unit circle zeros can increase the value of the infimum, even when every unit circle zero of g^~ is also a zero of f^~, and a sequence of controllers constructed in an obvious fashion fails to be an optimizing sequence. Similar results are obtained for the continuous-time case.