KLAP: KYP lemma based low-rank approximation for $\mathcal{H}_2$-optimal passivation

TL;DR

KLAP tackles the problem of making a non-passive, asymptotically stable LTI system passive while minimizing the $H_2$ distance to the original system. It leverages the Kalman–Yakubovich–Popov lemma in a low-rank parameterization derived from L and M matrices (with $D+D^\top=MM^\top$ and $C= B^\top \mathcal{L}^{-1}(-LL^\top) + ML^\top$) to explicitly describe all passive realizations. The $H_2$ optimization becomes an unconstrained problem in the decision variable $L$, with a computable gradient obtained via a Lyapunov-based auxiliary variable, ensuring differentiability and solvability; in the special case $M=0$ global optimality is guaranteed, while for $M\neq 0$ a practical criterion using extremal ARE solutions helps distinguish global from local minima. To cope with potential non-global minima, the authors introduce an initialization strategy based on ARE perturbations and a restart mechanism that robustly guides convergence to a near-global optimum. Numerical experiments on benchmarks up to size $n=800$ demonstrate substantial speedups over standard LMI-based passivation while achieving comparable or better $H_2$ accuracy.

Abstract

We present a novel passivity enforcement (passivation) method, called KLAP, for linear time-invariant systems based on the Kalman-Yakubovich-Popov (KYP) lemma and the closely related Lur'e equations. The passivation problem in our framework corresponds to finding a perturbation to a given non-passive system that renders the system passive while minimizing the $\mathcal{H}_2$ or frequency-weighted $\mathcal{H}_2$ distance between the original non-passive and the resulting passive system. We show that this problem can be formulated as an unconstrained optimization problem whose objective function can be differentiated efficiently even in large-scale settings. We show that any minimizer of the unconstrained problem yields the same passive system. Furthermore, we prove that, in the absence of a feedthrough term, every local minimizer is also a global minimizer. For cases involving a non-trivial feedthrough term, we analyze global minimizers in relation to the extremal solutions of the Lur'e equations, which can serve as tools for identifying local minima. To solve the resulting numerical optimization problem efficiently, we propose an initialization strategy based on modifying the feedthrough term and a restart strategy when it is likely that the optimization has converged to a non-global local minimum. Numerical examples illustrate the effectiveness of the proposed method.

Figures (7)

• Figure 1: Squared $\mathcal{H}_2$-error for all $L$ and passivating $\widehat{C}$, respectively, for \ref{['ex:Meq0']} with $M=0$. The blue and orange dot correspond to the local minimizers, which are global minimizers due to \ref{['prop:locGlobMin']}.
• Figure 2: Squared $\mathcal{H}_2$-error for all $L$ respectively passivating $\widehat{C}$ for \ref{['ex:Mneq0']} with $M\neq 0$. The blue dot is a non-global local minimizer and the orange dot is the global minimizer.
• Figure 3: Visualization of the $\mathcal{H}_2$-error during the optimization for \ref{['ex:Mneq0']} with $M\neq 0$. The path line corresponds to the path taken by the optimizer, the solid parts represent the standard L-BFGS optimization and the dashed part represents the gradient step performed after a local minimum is detected.
• Figure 4: Sampled modified Popov function $\Phi(\imath \omega)$ for the system in \ref{['ex:Mneq0']} and the perturbed system.
• Figure 5: Visualization of the $\mathcal{H}_2$-error during the optimization for \ref{['ex:init']}. In reduced opacity, we show the $\mathcal{H}_2$-error for the perturbed system with (enlarged) $D_{\mathrm{pert}}$. The green dot represents the $C$ matrix of the original system, which provides a passive realization for the perturbed system. Blue dots indicate the initial guesses for the optimization, while orange dots denote the global minimizer. The path line illustrates the trajectory taken by the optimizer.

Theorems & Definitions (23)

• Definition 2.1: Positive realness, passivity, storage function
• Theorem 2.2
• Theorem 2.3: KYP lemma BLME20, CGH24
• Proposition 2.4
• proof
• Remark 2.5
• Remark 3.1
• Theorem 3.2
• Lemma 3.3: WL99
• proof : Proof of \ref{['thm:H2passivation']}