Time domain constrained H_∞‐synthesis

Abstract

SUMMARYStructured H ∞ -synthesis subject to time domain performance specifications is introduced. Our method allows to control trajectories of the linearized system and the underlying nonlinear dynamics simultaneously. A non-smooth bundle optimization method for this class of programs is proposed and discussed. Our approach is tested against two benchmark studies: control of a rotational actuator to attenuate vibration noise, and control of a continuous crystallizer. Our algorithm gives a local convergence certificate and is suited for systems with large state dimension.

Cutting Plane Oracles to Minimize Non-smooth Non-convex Functions

“…What we do here is a fine analysis of the case ρ k < γ. We observed in experiments [1,17,22] that it sometimes happens that after a long series of aggregation and cutting plane steps with frozen τ k , Φ k comes close enough to M k+1 to have ρ k ≥ γ. Then the τ -parameter ultimately has to be increased.…”

“…whose use was in fact established in numerical tests in [1]. Fixing a third threshold γ < 1, we now have the following decision:…”

“…Lemma 4 shows that from some counter k 0 onwards we must have ρ k < γ and ρ k < γ for all k ≥ k 0 . In other words, the third alternative in the update formula of step 6 (third line in table 1) can no longer occur. That means, during the remaining infinitely many inner loop steps k either line 4 or line 5 in table 1 must occur.…”

“…We consider optimization programs of the form min x∈R n f (x), (1) where f : R n → R is non-smooth, non-convex, and locally Lipschitz. We present a bundle algorithm for (1), which converges globally in the sense that every accumulation pointx of the sequence of serious steps is critical, that is, satisfies 0 ∈ ∂f (x), (2) where ∂f (x) is the Clarke subdifferential of f at x. Non-convexity of (1) leads to three major difficulties.…”

“…We present a bundle algorithm for (1), which converges globally in the sense that every accumulation pointx of the sequence of serious steps is critical, that is, satisfies 0 ∈ ∂f (x), (2) where ∂f (x) is the Clarke subdifferential of f at x. Non-convexity of (1) leads to three major difficulties. Firstly, cutting planes can no longer be used as in the convex case, and a substitute has to be found.…”