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Robust reduced-order model predictive control using peak-to-peak analysis of filtered signals

Johannes Köhler, Carlo Scholz, Melanie Zeilinger

TL;DR

This work develops a robust reduced-order model predictive control framework for large-scale linear systems by introducing a scalar, input-dependent error-bounding system that quantifies the ROM–full-order deviation via peak-to-peak analysis. A dynamic filter is incorporated to tighten the bound when the excitation is high-frequency, and the resulting ROM-OCP guarantees constraint satisfaction for the full-order system while reducing computation to a lower-dimensional ODE system. The approach delivers up to four orders of magnitude less conservatism than existing ROM-based bounds and is demonstrated on a $100$-dimensional mass-spring-damper, with scalable implementation and an IQC-adapted appendix. The method offers a practical path to safe, efficient MPC for large-scale systems, enabling tighter performance while maintaining robust constraint guarantees.

Abstract

We address the design of a model predictive control (MPC) scheme for large-scale linear systems using reduced-order models (ROMs). Our approach uses a ROM, leverages tools from robust control, and integrates them into an MPC framework to achieve computational tractability with robust constraint satisfaction. Our key contribution is a method to obtain guaranteed bounds on the predicted outputs of the full-order system by predicting a (scalar) error-bounding system alongside the ROM. This bound is then used to formulate a robust ROM-based MPC that guarantees constraint satisfaction and robust performance. Our method is developed step-by-step by (i) analysing the error, (ii) bounding the peak-to-peak gain, an (iii) using filtered signals. We demonstrate our method on a 100-dimensional mass-spring-damper system, achieving over four orders of magnitude reduction in conservatism relative to existing approaches.

Robust reduced-order model predictive control using peak-to-peak analysis of filtered signals

TL;DR

This work develops a robust reduced-order model predictive control framework for large-scale linear systems by introducing a scalar, input-dependent error-bounding system that quantifies the ROM–full-order deviation via peak-to-peak analysis. A dynamic filter is incorporated to tighten the bound when the excitation is high-frequency, and the resulting ROM-OCP guarantees constraint satisfaction for the full-order system while reducing computation to a lower-dimensional ODE system. The approach delivers up to four orders of magnitude less conservatism than existing ROM-based bounds and is demonstrated on a -dimensional mass-spring-damper, with scalable implementation and an IQC-adapted appendix. The method offers a practical path to safe, efficient MPC for large-scale systems, enabling tighter performance while maintaining robust constraint guarantees.

Abstract

We address the design of a model predictive control (MPC) scheme for large-scale linear systems using reduced-order models (ROMs). Our approach uses a ROM, leverages tools from robust control, and integrates them into an MPC framework to achieve computational tractability with robust constraint satisfaction. Our key contribution is a method to obtain guaranteed bounds on the predicted outputs of the full-order system by predicting a (scalar) error-bounding system alongside the ROM. This bound is then used to formulate a robust ROM-based MPC that guarantees constraint satisfaction and robust performance. Our method is developed step-by-step by (i) analysing the error, (ii) bounding the peak-to-peak gain, an (iii) using filtered signals. We demonstrate our method on a 100-dimensional mass-spring-damper system, achieving over four orders of magnitude reduction in conservatism relative to existing approaches.

Paper Structure

This paper contains 12 sections, 7 theorems, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution
  4. Problem setup
  5. Reduced-order models and dynamic uncertainty
  6. Peak-to-peak analysis & robust prediction
  7. Robust predictions using peak-to-peak analysis of filtered signals
  8. Robust predictive control with ROMs
  9. Discussion & qualitative comparison
  10. Numerical comparisons
  11. Conclusion
  12. IQC-based peak-to-peak reachability analysis

Key Result

Proposition 1

The full-order system $\Sigma$eq:sys is equivalent to the interconnection between the reduced-order system $\Sigma_{\mathrm{r}}$eq:sys_ROM and the following error dynamics with $e_0=(I-VW^\top)x_0$ and $z(t)=z_{\mathrm{r}}(t)+z_{\mathrm{e}}(t)$.

Figures (4)

  • Figure 1: Numerical example: chain of $N=50$ masses connected by spring and damper elements with first position $z$ and control input $u$.
  • Figure 2: Naïve ROM-based MPC: Trajectory optimized with nominal ROM (blue) shows significant deviation from full-order trajectory (red), and violates the constraints (black), since the structural model mismatch is not accounted for.
  • Figure 3: Proposed robust ROM-based MPC \ref{['eq:OCP_ROM']}: Full-order simulation (red) contained in robust ROM-based prediction (grey shaded area with blue boundary), and below the constraint (black, dashed).
  • Figure 4: Comparison of prediction error bounds of ROM. The dashed line indicates the size of the constraint set $\mathcal{Z}$, i.e., alternative methods provide bounds that are significantly larger than the constraint set.

Theorems & Definitions (16)

  • Proposition 1: Error dynamics
  • proof
  • Remark 1: Choice of decomposition
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 6 more