CONTROLLER REDUCTION FOR NONLINEAR PLANTS—ANL₂APPROACH

Model Reduction by Generalized Differential Balancing

“…Proof of item 1): By using Lemma 4.8, Corollary 4.9, and the counterpart of Theorem 4.2 to the reduced-order closedloop system Γ(G r , K r ), item 1) can be shown by following a similar procedure as the proof of [20,Theorem 4], where ε in [20] is ρ r /β of this paper.…”

“…In contrast to linear systems [16]- [19], controller reduction has been rarely studied for nonlinear systems. A few papers [20], [21] proceed with theoretical analysis based on solutions to Hamilton-Jacobi equations/inequalities (HJE/HJI). As well recognized, solving HJE/HJI is a computationally challenging problem.…”

“…For stabilizing reduced-order control design, H ∞ -balancing has been proposed for linear systems [18] and extended to nonlinear systems [20], [21] with HJE/HJI. The last and main objective of this paper is to develop GD H ∞ -balancing based on GD balancing and GD LQG balancing.…”

“…The last and main objective of this paper is to develop GD H ∞ -balancing based on GD balancing and GD LQG balancing. The main difference from [20], [21] is that the proposed method can be achieved by solving sets of linear matrix inequalities instead of nonlinear partial differential equations/inequalities. In other words, this paper can be viewed as the first paper that provides a computationally tractable method for controller reduction of nonlinear systems while guaranteeing the stability of the closed-loop systems consisting of the full-order systems and reduced-order controllers.…”

“…τ ) − u ′ (τ )| 2 + |y(τ ) − y ′ (τ )| 2 )dτ.This inequality holds for arbitrary pair (u, u ′ ). By restricting the class of the pairs into ones achieving lim t→∞ (x(−t) − x ′ (−t)) = 0, we obtain(20) for any(x 0 , x ′ 0 ) ∈ R n × R n .C. Proof of Theorem 3.3Proof: First, (17) can be rearranged asP (∂f (x) − BB ⊤ P ) + (∂f (x) − BB ⊤ P ) ⊤ P − C −B ⊤ P ⊤ C −B ⊤ P − εP, ∀x ∈ R n .…”