[Paper Review] Numerical computation and new output bounds for time-limited balanced truncation of discrete-time systems

Igor Pontes Duff, Patrick Kürschner|arXiv (Cornell University)|Feb 5, 2019

Model Reduction and Neural Networks32 references5 citations

TL;DR

This paper presents a novel error bound for time-limited balanced truncation (TLBT) in discrete-time systems, derived via a time-limited ℋ₂ norm formulation, and proposes efficient low-rank rational Krylov methods for solving large-scale TL Stein equations. The key contribution is a computable output error bound that enables adaptive model order reduction, showing TLBT achieves higher accuracy than standard BT within finite time horizons while maintaining computational efficiency.

ABSTRACT

In this paper, balancing based model order reduction (MOR) for large-scale linear discrete-time time-invariant systems in prescribed finite time intervals is studied. The first main topic is the development of error bounds regarding the approximated output vector within the time limits. The influence of different components in the established bounds will be highlighted. After that, the second part of the article proposes strategies that enable an efficient numerical execution of time-limited balanced truncation for large-scale systems. Numerical experiments illustrate the performance of the proposed techniques.

Motivation & Objective

• To develop computable error bounds for the output of time-limited balanced truncation (TLBT) in discrete-time systems.
• To enable adaptive model order reduction by leveraging neglected Hankel singular values as error indicators.
• To design efficient numerical algorithms for solving large-scale time-limited Stein equations arising in TLBT.
• To compare the performance of TLBT against standard balanced truncation in terms of accuracy and computational cost.

Proposed method

• Introduces a time-limited ℋ₂ norm via matrix equations to characterize output error in finite time intervals.
• Derives a new output error bound using the time-limited ℋ₂ norm, expressed as a sum of neglected Hankel singular values.
• Applies rational Krylov subspace methods to compute low-rank factors of time-limited Gramians, enabling large-scale computation.
• Proposes shift parameter selection strategies and residual-based error estimation for improved accuracy in the reduced model.
• Uses the sum of neglected Hankel singular values (σr) as a stopping criterion for adaptive order reduction.
• Validates the method on large-scale benchmark examples, comparing TLBT with standard BT in accuracy and efficiency.

Experimental results

Research questions

• RQ1 Can a computable output error bound be derived for time-limited balanced truncation in discrete-time systems?
• RQ2 How does the performance of TLBT compare to standard balanced truncation in terms of output accuracy within a finite time horizon?
• RQ3 Can the sum of neglected Hankel singular values be used as a reliable indicator for adaptive model order reduction in TLBT?
• RQ4 What numerical methods are most effective for solving large-scale time-limited Stein equations in the context of model order reduction?

Key findings

• The proposed output error bound based on the time-limited ℋ₂ norm accurately predicts the actual output error, with tight agreement observed in numerical experiments.
• Adaptive order reduction using the sum of neglected Hankel singular values (σr) successfully controls error within user-defined tolerances, matching the performance of standard BT.
• TLBT consistently produces more accurate reduced-order models than standard BT within the time horizon τ, as evidenced by lower output error bounds and actual errors.
• The rational Krylov method with optimized shift parameters efficiently computes low-rank solutions of time-limited Stein equations, achieving good accuracy and scalability on large-scale systems.
• For the same accuracy, TLBT yields reduced models of lower order than standard BT, indicating improved efficiency in time-limited settings.
• The method occasionally produces unstable reduced models, suggesting a need for stabilization techniques in future work, similar to findings in continuous-time TLBT.