Phase Lag and Lead Weighting in Linear-quadratic Control

The use of frequency-weighted performance indices in linear quadratic (LQ) optimal control has been the subject of recent r e ~ e a c h ' ~ . The primary motivation for frequency-weighting is the possibility of enhancing the robustness of LQ designs in critical frequency bands. A preliminary study3 of the case where high frequencies in the control are weighted more heavily has shown that the optimal controller in this case comprises a stable compensator, nominally introducing phase lag, in tandem with the state feedback term. In this paper, further properties of the optimal dynamic compensator are derived with a view to developing design guidelines for such compensators. A specific question addressed is the determination of conditions under which the optimal compensator is indeed a lag, and not a lead. Attention is also focused on the optimal (nominally) phase lead compensator case which arises when !ow frequencies are weighted more ht-avily in the performance index. A detailed analysis of a first-order single-input plant with both lag and lead compensation is undertaken in order to gain insight and to help in the formulation of design rules. An alternative to frequency-weighting the control term in the qaudratic performance index is to apply an (inverse) frequency-weighting to the state term5. Further relations between these two types of weighting are explored in this paper, especially as regards to their robustness properties. The generalization of the result to the multi-input case is investigated in the final section of the paper.