Dual approximation between optimal μ and optimal H∞ controllers

The main purpose of this paper is to solve the dual approximation problem: approximation of optimal μ controllers by a sequence of fixed-order optimal H_∞ controllers, and the dual approximation of optimal H_∞ controllers by a sequence of fixed-order optimal μ controllers. It is shown analytically that for a control system described by the linear fractional transformation F_l(P,K), there exists a scalar frequency-shaping function W(s) such that the weighted H_∞-optimization problem inf_K||WF_l(P,K)||_∞ is identical to the μ-optimization problem inf_Ksup_Wμ_Δ(F_l(P,K))(jω); on the other hand, it is also shown that there exists a frequency-shaping function W*(s) such that the weighted μ-optimization problem inf_K*sup_ωμ_Δ(W*F_l(P,K*))(jω) is identical to the H_∞-optimization problem inf_K*l||Fsub l/(P,K*)||_∞. The frequency-shaping functions W(s), W*(s), and the optimal controllers K_opt(s), K_opt*(s) are characterized explicitly in terms of a dual pair of minimizing sequences, and are solved numerically by a dual pair of iteration algorithms.