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$\mathcal{H}_2$ optimal model reduction of linear systems with multiple quadratic outputs

Sean Reiter, Igor Pontes Duff, Ion Victor Gosea, Serkan Gugercin

TL;DR

The paper develops an $ ext{H}_2$-optimal model reduction framework for linear-state, quadratic-output (LQO) systems by deriving gradients of the squared $ ext{H}_2$ error and establishing Gramian-based first-order necessary conditions (FONCs) for local optimality. It extends the Wilson Gramian framework from purely linear systems to the LQO setting, showing that optimal reductions are realized via Petrov–Galerkin projection with projection matrices determined by Sylvester equations. The authors formulate and analyze an iteratively corrected two-sided algorithm, LQO-TSIA, which enforces the optimality conditions and outputs reduced-order models that closely approximate the full system's input-to-output behavior, verified on a quadratic-output advection–diffusion example. The work provides a principled, computationally practical route to accurate MOR for systems where outputs are quadratic in the state, with potential impact on high-dimensional control and estimation problems involving weak nonlinear observables.

Abstract

In this work, we consider the $\mathcal{H}_2$ optimal model reduction of dynamical systems that are linear in the state equation and up to quadratic nonlinearity in the output equation. As our primary theoretical contributions, we derive gradients of the squared $\mathcal{H}_2$ system error with respect to the reduced model quantities and, from the stationary points of these gradients, introduce Gramian-based first-order necessary conditions for the $\mathcal{H}_2$ optimal approximation of a linear quadratic output (LQO) system. The resulting $\mathcal{H}_2$ optimality framework neatly generalizes the analogous Gramian-based optimality framework for purely linear systems. Computationally, we show how to enforce the necessary optimality conditions using Petrov-Galerkin projection; the corresponding projection matrices are obtained from a pair of Sylvester equations. Based on this result, we propose an iteratively corrected algorithm for the $\mathcal{H}_2$ model reduction of LQO systems, which we refer to as LQO-TSIA (linear quadratic output two-sided iteration algorithm). Numerical examples are included to illustrate the effectiveness of the proposed computational method against other existing approaches.

$\mathcal{H}_2$ optimal model reduction of linear systems with multiple quadratic outputs

TL;DR

The paper develops an -optimal model reduction framework for linear-state, quadratic-output (LQO) systems by deriving gradients of the squared error and establishing Gramian-based first-order necessary conditions (FONCs) for local optimality. It extends the Wilson Gramian framework from purely linear systems to the LQO setting, showing that optimal reductions are realized via Petrov–Galerkin projection with projection matrices determined by Sylvester equations. The authors formulate and analyze an iteratively corrected two-sided algorithm, LQO-TSIA, which enforces the optimality conditions and outputs reduced-order models that closely approximate the full system's input-to-output behavior, verified on a quadratic-output advection–diffusion example. The work provides a principled, computationally practical route to accurate MOR for systems where outputs are quadratic in the state, with potential impact on high-dimensional control and estimation problems involving weak nonlinear observables.

Abstract

In this work, we consider the optimal model reduction of dynamical systems that are linear in the state equation and up to quadratic nonlinearity in the output equation. As our primary theoretical contributions, we derive gradients of the squared system error with respect to the reduced model quantities and, from the stationary points of these gradients, introduce Gramian-based first-order necessary conditions for the optimal approximation of a linear quadratic output (LQO) system. The resulting optimality framework neatly generalizes the analogous Gramian-based optimality framework for purely linear systems. Computationally, we show how to enforce the necessary optimality conditions using Petrov-Galerkin projection; the corresponding projection matrices are obtained from a pair of Sylvester equations. Based on this result, we propose an iteratively corrected algorithm for the model reduction of LQO systems, which we refer to as LQO-TSIA (linear quadratic output two-sided iteration algorithm). Numerical examples are included to illustrate the effectiveness of the proposed computational method against other existing approaches.
Paper Structure (12 sections, 4 theorems, 96 equations, 4 figures, 1 algorithm)

This paper contains 12 sections, 4 theorems, 96 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. Linear systems with quadratic outputs
  4. The $\mathcal{H}_2$ system norm
  5. The quadratic output observability Gramian
  6. Relating the $\mathcal{H}_2$ system error and $\mathcal{L}_\infty$ output error
  7. Optimal $\mathcal{H}_2$ model reduction
  8. Optimal $\mathcal{H}_2$ model reduction of purely linear systems
  9. Optimal $\mathcal{H}_2$ model reduction of linear quadratic output systems
  10. A two-sided iteration algorithm for LQO model reduction
  11. Numerical Results
  12. Conclusions

Key Result

Theorem 2.1

Let $\mathcal{S}$ and $\mathcal{S}_r$ be asymptotically stable LQO systems as in eq:lqosys and eq:lqosys_red, respectively. Let ${\bf X}\in{\mathbbm R}^{n\times r}$ and ${\bf Z}\in{\mathbbm R}^{n\times r}$ denote the unique solutions to the following Sylvester equations Then, the $\mathcal{H}_2$ inner product of $\mathcal{S}$ and $\mathcal{S}_r$ is given by Moreover, if instead $\mathcal{S}=\mat

Figures (4)

  • Figure 1: Output response of the FOM and order $r=30$ROMs computed by LQO-TSIA and LQO-BT driven by the inputs $u_0(t)=0$ and $u_1(t)=.5\cos(\pi t)+1$.
  • Figure 2: Output response of the FOM and order $r=30$ROMs computed by LQO-TSIA and LQO-BT driven by the inputs $u_0(t)=0$ and $u_1(t) = t^2 e^{-t/5}$.
  • Figure 3: Convergence of the LQO-TSIA algorithm for $r=30$. The relative changes in $\eta^{(j)}$ and $\tau^{(j)}$ according to \ref{['eq:err_conv']} and \ref{['eq:err_tails']} are measured throughout the iteration.
  • Figure 4: Relative $\mathcal{H}_2$ errors for hierarchy of LQO-TSIA and LQO-BT reduced models for orders of reduction $r = 2, 4, \ldots, 30$.

Theorems & Definitions (12)

  • Remark 2.1
  • Definition 2.1
  • Theorem 2.1
  • proof
  • Lemma 3.1: YanJ99
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.1
  • ...and 2 more