Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Noisy Data to Hierarchical Control: A Model-Order-Reduction Framework

Behrad Samari, Henrik Sandberg, Karl H. Johansson, Abolfazl Lavaei

Abstract

This paper develops a direct data-driven framework for constructing reduced-order models (ROMs) of discrete-time linear dynamical systems with unknown dynamics and process disturbances. The proposed scheme enables controller synthesis on the ROM and its refinement to the original system by an interface function designed using noisy data. To achieve this, the notion of simulation functions (SFs) is employed to establish a formal relation between the original system and its ROM, yielding a quantitative bound on the mismatch between their output trajectories. To construct such relations and interface functions, we rely on data collected from the unknown system. In particular, using noise-corrupted input-state data gathered along a single trajectory of the system, and without identifying the original dynamics, we propose data-dependent conditions, cast as a semidefinite program, for the simultaneous construction of ROMs, SFs, and interface functions. Through a case study, we demonstrate that data-driven controller synthesis on the ROM, combined with controller refinement via the interface function, enables the enforcement of complex specifications beyond stability.

From Noisy Data to Hierarchical Control: A Model-Order-Reduction Framework

Abstract

This paper develops a direct data-driven framework for constructing reduced-order models (ROMs) of discrete-time linear dynamical systems with unknown dynamics and process disturbances. The proposed scheme enables controller synthesis on the ROM and its refinement to the original system by an interface function designed using noisy data. To achieve this, the notion of simulation functions (SFs) is employed to establish a formal relation between the original system and its ROM, yielding a quantitative bound on the mismatch between their output trajectories. To construct such relations and interface functions, we rely on data collected from the unknown system. In particular, using noise-corrupted input-state data gathered along a single trajectory of the system, and without identifying the original dynamics, we propose data-dependent conditions, cast as a semidefinite program, for the simultaneous construction of ROMs, SFs, and interface functions. Through a case study, we demonstrate that data-driven controller synthesis on the ROM, combined with controller refinement via the interface function, enables the enforcement of complex specifications beyond stability.

Paper Structure

This paper contains 13 sections, 3 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System and ROM Descriptions
  4. Simulation Functions
  5. Data-Driven Methodology
  6. Data Collection
  7. Data-Driven ROM and SF Construction
  8. Discussion
  9. Feasibility Analysis of SDP \ref{['eq:SDP']}
  10. Scalability Analysis
  11. Limitations
  12. Simulation Results
  13. Conclusion

Key Result

Theorem II.4

Given $\Sigma = (\mathds{X}, \mathds{U}, \mathds{Y}, \mathds{W}, A, B, {\mathbb{I}}_n)$ and its ROM $\hat{\Sigma} = (\hat{ \mathds{X}}, \hat{ \mathds{U}}, \hat{ \mathds{Y}}, \hat{A}, \hat{B}, \hat{C})$, let $\mathcal{S}$ be an SF from $\hat{\Sigma}$ to $\Sigma$. Then, for any initial states $\mathrm

Figures (1)

  • Figure 1: (a) Trajectories of the dt-LCS for $5$ simulation runs with different initial conditions, all satisfying the complex specification. (b) A representative trajectory after applying a smoothing filter to mitigate the zigzag behavior observed in (a), which arises from the SCOTS synthesis tool due to the chosen discretization parameter. A video of this simulation is available at https://youtu.be/_cp93M1UmTc.

Theorems & Definitions (13)

  • Definition II.1
  • Definition II.2
  • Definition II.3
  • Theorem II.4
  • proof
  • Remark II.5
  • Remark III.1
  • Lemma III.2
  • proof
  • Theorem III.3
  • ...and 3 more