Balanced Truncation of Linear Systems with Quadratic Outputs in Limited Time and Frequency Intervals

TL;DR

The paper tackles time-limited and frequency-limited model order reduction for linear systems with quadratic outputs by extending structure-preserving balanced truncation via time- and frequency-limited system Gramians. It derives Lyapunov equations for these limited Gramians, develops low-rank solution strategies including LDL^T-ADI and truncated Laguerre expansions, and presents TLBT and FLBT algorithms that preserve the quadratic-output structure. The approach is validated on benchmark models, showing superior accuracy within specified time or frequency intervals and highlighting computational benefits of the Laguerre-based low-rank methods over traditional ADI in many scenarios. These results underscore the practicality of interval-specific MOR for large-scale LTI-QO systems, enabling efficient, accurate simulations where limited-domain performance is critical.

Abstract

Model order reduction involves constructing a reduced-order approximation of a high-order model while retaining its essential characteristics. This reduced-order model serves as a substitute for the original one in various applications such as simulation, analysis, and design. Often, there's a need to maintain high accuracy within a specific time or frequency interval, while errors beyond this limit can be tolerated. This paper addresses time-limited and frequency-limited model order reduction scenarios for linear systems with quadratic outputs, by generalizing the recently introduced structure-preserving balanced truncation algorithm. To that end, limited interval system Gramians are defined, and the corresponding generalized Lyapunov equations governing their computation are derived. Additionally, low-rank solutions for these equations are investigated. Next, balanced truncation algorithms are proposed for time-limited and frequency-limited scenarios, each utilizing its corresponding limited-interval system Gramians. The proposed algorithms ensure accurate results within specified time and frequency intervals while preserving the quadratic-output structure. Two benchmark numerical examples are presented to demonstrate the effectiveness of the algorithms, showcasing their ability to achieve superior accuracy within the desired time or frequency interval.

Figures (6)

• Figure 1: Decay in eigenvalues of $P$, $P_\tau$, and $P_\Omega$
• Figure 2: Decay in eigenvalues of $Q$, $Q_\tau$, and $Q_\Omega$
• Figure 3: Relative Error in the Output Response within $[0,1]$ sec
• Figure 4: Relative Error in the Output Response
• Figure 5: Relative Error in the Output-I Response within $[0,2]$ sec

Theorems & Definitions (24)

• Theorem 4.1
• proof
• Proposition 4.2
• proof
• Corollary 4.3
• proof
• Theorem 4.4
• proof
• Proposition 4.5
• proof