Time-limited H2-optimal Model Order Reduction of Linear Systems with Quadratic Outputs

TL;DR

This work advances model order reduction for linear systems with quadratic outputs by formulating a time-limited $\mathcal{H}_2$ norm $||H||_{\mathcal{H}_{2,\tau}}$ and deriving its local optimality conditions for reduced models. It shows that Petrov-Galerkin projection cannot, in general, satisfy all four optimality conditions in the time-limited setting, and then proposes the TLHNOIA algorithm that achieves three of four conditions by iteratively updating projection data and enforcing a Petrov-Galerkin constraint. The approach leverages time-limited Gramians $P_{\tau}$ and $Q_{\tau}$, Sylvester and Lyapunov equations, and Fréchet derivatives to quantify and minimize the output error on a finite interval $[0,\tau]$, with an illustrative 6th-order example demonstrating superior accuracy over TLBT within the time window. This has practical impact for finite-horizon control, transient analysis, and applications where accurate short-time behavior is critical for high-order LQO models. The results provide a principled framework for near-optimal time-limited MOR in systems with quadratic outputs and highlight the trade-offs of projection choices under time constraints.

Abstract

An important class of dynamical systems with several practical applications is linear systems with quadratic outputs. These models have the same state equation as standard linear time-invariant systems but differ in their output equations, which are nonlinear quadratic functions of the system states. When dealing with models of exceptionally high order, the computational demands for simulation and analysis can become overwhelming. In such cases, model order reduction proves to be a useful technique, as it allows for constructing a reduced-order model that accurately represents the essential characteristics of the original high-order system while significantly simplifying its complexity. In time-limited model order reduction, the main goal is to maintain the output response of the original system within a specific time range in the reduced-order model. To assess the error within this time interval, a mathematical expression for the time-limited $\mathcal{H}_2$-norm is derived in this paper. This norm acts as a measure of the accuracy of the reduced-order model within the specified time range. Subsequently, the necessary conditions for achieving a local optimum of the time-limited $\mathcal{H}_2$ norm error are derived. The inherent inability to satisfy these optimality conditions within the Petrov-Galerkin projection framework is also discussed. After that, a stationary point iteration algorithm based on the optimality conditions and Petrov-Galerkin projection is proposed. Upon convergence, this algorithm fulfills three of the four optimality conditions. To demonstrate the effectiveness of the proposed algorithm, a numerical example is provided that showcases its ability to effectively approximate the original high-order model within the desired time interval.

Figures (1)

• Figure 1: Relative Error in the Output Response within $[0,0.5]$ sec

Theorems & Definitions (10)

• Definition 3.1
• Proposition 3.2
• proof
• Corollary 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5
• proof
• Remark 1