Robust stabilization of infinite dimensional systems by finite dimensional controllers

Abstract The problem of robustly stabilizing an infinite dimensional system with transfer function G, subject to an additive perturbation Δ is considered. It is assumed that: G ϵ B 0(σ) of systems introduced by Callier and Desoer [3]; the perturbation satisfies |W1ΔW2| C +. Now write W1GW2=G1 + G1, where G1 is rational and totally unstable and G2 is stable. Generalizing the finite dimensional results of Glover [12] this family of perturbed systems is shown to be stabilizable if and only if ϵ ⩽ σ min (G ∗ 1 ) ( = the smallest Hankel singular value of G ∗ 1 ). A finite dimensional stabilizing controller is then given by K=W 2 K 1 (I+ G 2 K 1 ) −1 W 1 where Ĝ2 is a rational approximation of G2 such that |G 2 − G 2 | ⩽ δ min (G ∗ 1 ) and K1 robustly stabilizes G1 to margin ϵ. The feedback system (G, K) will then be stable if |W1ΔW2| ∞

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