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 ∥W F_l(P, K)∥_∞ is identical to the μ- optimization problem inf_K sup$-ω/μΔ(F_l(P, K))(jω); on the other hand, it is also shown that there exists a frequency-shaping function W^*(s) such that the weighted μ-optimization problem inf_K(*) sup$-ω/μΔ(W^*F_l(P, K^*)) (jω) is identical to the H_∞-optimization problem inf_K(*) ∥F_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.