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Data-Driven Time-Limited h2 Optimal Model Reduction for Linear Discrete-Time Systems

Hiroki Sakamoto, Kazuhiro Sato

TL;DR

The paper addresses finite-horizon model reduction for discrete-time LTI systems when system matrices are unknown, by developing a data-driven gradient-based method that minimizes the time-limited $h^2$ norm using only impulse response data. It derives data-enabled gradient expressions and proposes an Armijo backtracking algorithm that iteratively updates ROM matrices to converge to a stationary point, avoiding explicit system identification. The method is validated on the SLICOT CD player benchmark, showing improved time-limited $h^2$ performance over ERA-based initializations and demonstrating robustness to measurement noise. This work enables accurate, scalable MOR directly from data for finite-horizon performance, with potential extensions to stability-constrained formulations.

Abstract

This paper develops a data-driven h2 model reduction method for discrete-time linear time-invariant systems. Specifically, we solve the h2 model reduction problem defined over a finite horizon using only impulse response data. Furthermore, we show that the proposed data-driven algorithm converges to a stationary point under certain assumptions. Numerical experiments demonstrate that the proposed method constructs a good reduced-order model in terms of the h2 norm defined over the finite horizon using a SLICOT benchmark (the CD player model).

Data-Driven Time-Limited h2 Optimal Model Reduction for Linear Discrete-Time Systems

TL;DR

The paper addresses finite-horizon model reduction for discrete-time LTI systems when system matrices are unknown, by developing a data-driven gradient-based method that minimizes the time-limited norm using only impulse response data. It derives data-enabled gradient expressions and proposes an Armijo backtracking algorithm that iteratively updates ROM matrices to converge to a stationary point, avoiding explicit system identification. The method is validated on the SLICOT CD player benchmark, showing improved time-limited performance over ERA-based initializations and demonstrating robustness to measurement noise. This work enables accurate, scalable MOR directly from data for finite-horizon performance, with potential extensions to stability-constrained formulations.

Abstract

This paper develops a data-driven h2 model reduction method for discrete-time linear time-invariant systems. Specifically, we solve the h2 model reduction problem defined over a finite horizon using only impulse response data. Furthermore, we show that the proposed data-driven algorithm converges to a stationary point under certain assumptions. Numerical experiments demonstrate that the proposed method constructs a good reduced-order model in terms of the h2 norm defined over the finite horizon using a SLICOT benchmark (the CD player model).
Paper Structure (11 sections, 6 theorems, 46 equations, 4 figures, 1 algorithm)

This paper contains 11 sections, 6 theorems, 46 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model-based Time-Limited $h^2$ MOR for LTI systems
  3. Discrete-Time LTI systems over a finite horizon
  4. Time-Limited $h^2$ MOR
  5. Problem Formulation
  6. Gradients from Impulse Response Data
  7. Impulse Response Data-Driven $h^2$ MOR Algorithm
  8. Experiments
  9. Conclusion
  10. Proof
  11. Time-Limited $h^2$ MOR with Stability Constraints

Key Result

Proposition 1

The gradients of $f_{L}(\hat{\theta})$ are where $P_L$, $Q_L$, $R_L$, and $S_L$ are solutions of eq:dt_tlLyap_PL, eq:dt_tlLyap_QL, eq:dt_tlSylve_RL, and eq:dt_tlSylve_SL. $P$ and $R$ are obtained by solving the following discrete-time Lyapunov equation and Sylvester equation Furthermore,

Figures (4)

  • Figure 1: Overview of the proposed data-driven time-limited $h^2$ model reduction framework.
  • Figure 2: Convergence behavior of Algorithm \ref{['alg:ABCunknown']} ($L=20$) initialized by ERA under three noise levels: noise-free, $\sigma=1$, and $\sigma=50$.
  • Figure 3: Convergence behavior of Algorithm \ref{['alg:ABCunknown']} ($L=40$) initialized by ERA under three noise levels: noise-free, $\sigma=1$, and $\sigma=50$.
  • Figure : Gradient Method for Problem \ref{['prob:dt_lti_tlh2_mod_data']}

Theorems & Definitions (20)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Theorem 2
  • proof
  • Proposition 3
  • ...and 10 more