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SIMBa: System Identification Methods leveraging Backpropagation

Loris Di Natale, Muhammad Zakwan, Philipp Heer, Giancarlo Ferrari-Trecate, Colin N. Jones

TL;DR

SIMBa tackles stable linear system identification by integrating prior knowledge into state-space models while guaranteeing stability by design via free Schur parametrizations. It leverages backpropagation and multi-step prediction losses to learn $(A,B,C,D)$ (or states) under prescribed sparsity or known entries, ensuring $|\,\lambda_i(A) |<1$. The paper introduces several LMIs- and scaling-based parametrizations that preserve stability, demonstrates substantial performance gains over traditional stable SI methods across simulated and real data, and provides an open-source Python toolbox. Despite higher computational demands and sensitivity to initialization, SIMBa consistently delivers accurate models when structure or physical constraints are important, opening pathways to structured nonlinear SI and Koopman-based extensions. Overall, SIMBa represents a flexible, knowledge-grounded approach that extends stable SI capabilities to incorporate prior information without sacrificing stability, with significant practical impact for high-precision modeling of networked or physically constrained systems.

Abstract

This manuscript details and extends the SIMBa toolbox (System Identification Methods leveraging Backpropagation) presented in previous work, which uses well-established Machine Learning tools for discrete-time linear multi-step-ahead state-space System Identification (SI). SIMBa leverages linear-matrix-inequality-based free parametrizations of Schur matrices to guarantee the stability of the identified model by design. In this paper, backed up by novel free parametrizations of Schur matrices, we extend the toolbox to show how SIMBa can incorporate known sparsity patterns or true values of the state-space matrices to identify without jeopardizing stability. We extensively investigate SIMBa's behavior when identifying diverse systems with various properties from both simulated and real-world data. Overall, we find it consistently outperforms traditional stable subspace identification methods, and sometimes significantly, especially when enforcing desired model properties. These results hint at the potential of SIMBa to pave the way for generic structured nonlinear SI. The toolbox is open-sourced on https://github.com/Cemempamoi/simba.

SIMBa: System Identification Methods leveraging Backpropagation

TL;DR

SIMBa tackles stable linear system identification by integrating prior knowledge into state-space models while guaranteeing stability by design via free Schur parametrizations. It leverages backpropagation and multi-step prediction losses to learn (or states) under prescribed sparsity or known entries, ensuring . The paper introduces several LMIs- and scaling-based parametrizations that preserve stability, demonstrates substantial performance gains over traditional stable SI methods across simulated and real data, and provides an open-source Python toolbox. Despite higher computational demands and sensitivity to initialization, SIMBa consistently delivers accurate models when structure or physical constraints are important, opening pathways to structured nonlinear SI and Koopman-based extensions. Overall, SIMBa represents a flexible, knowledge-grounded approach that extends stable SI capabilities to incorporate prior information without sacrificing stability, with significant practical impact for high-precision modeling of networked or physically constrained systems.

Abstract

This manuscript details and extends the SIMBa toolbox (System Identification Methods leveraging Backpropagation) presented in previous work, which uses well-established Machine Learning tools for discrete-time linear multi-step-ahead state-space System Identification (SI). SIMBa leverages linear-matrix-inequality-based free parametrizations of Schur matrices to guarantee the stability of the identified model by design. In this paper, backed up by novel free parametrizations of Schur matrices, we extend the toolbox to show how SIMBa can incorporate known sparsity patterns or true values of the state-space matrices to identify without jeopardizing stability. We extensively investigate SIMBa's behavior when identifying diverse systems with various properties from both simulated and real-world data. Overall, we find it consistently outperforms traditional stable subspace identification methods, and sometimes significantly, especially when enforcing desired model properties. These results hint at the potential of SIMBa to pave the way for generic structured nonlinear SI. The toolbox is open-sourced on https://github.com/Cemempamoi/simba.
Paper Structure (35 sections, 7 theorems, 49 equations, 9 figures, 2 tables)

This paper contains 35 sections, 7 theorems, 49 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Traditional linear system identification
  3. Prior knowledge integration
  4. Contribution
  5. Preliminaries
  6. Free Parametrizations of Schur matrices
  7. Dense Schur matrices
  8. Discretized continuous-time systems
  9. Sparse Schur matrices
  10. An alternative general parametrization
  11. Using free parametrizations
  12. The SIMBa toolbox
  13. Optimization framework
  14. Input-state identification
  15. Training procedure
  16. ...and 20 more sections

Key Result

Proposition 1

For any $W\in\mathbb{R}^{2n \times 2n}$, $V\in\mathbb{R}^{n \times n}$, $0< \gamma \leq 1$, and $\epsilon>0$, let Then is Schur with $|\lambda_i(A)|<\gamma, \forall i=1,...,n$.

Figures (9)

  • Figure 1: Main steps of running SIMBa in Python.
  • Figure 2: Performance of input-output state-space identification methods on $30$ randomly generated systems, where the MSEs have been normalized by the best-obtained error for each system. The performance of SIMBa (ours) --- either relying on the parametrization proposed in Proposition \ref{['prop:generic']} or \ref{['prop:naive']} --- is plotted in green, other stable SI methods in blue, and unstable ones in red.
  • Figure 3: Performance of input-output state-space identification methods on $20$ randomly generated systems, where the data has been standardized and the MSEs have been normalized by the best-obtained error for each system. The performance of SIMBa (ours) --- either relying on the parametrization proposed in Proposition \ref{['prop:generic']} or \ref{['prop:naive']} --- is plotted in green, other stable SI methods in blue, and potentially unstable ones in red.
  • Figure 4: Normalized MSE of each method on test input-output data from $10$ randomly generated systems with sparse matrices $A$, $B$, $C$, and $D$. The letters in square brackets encode which matrices $X$ or sparsity pattern $m_X$, respectively, are assumed to be known and fixed. Both plots show the same data with a different zoom to appreciate the difference between SIMBa (ours) --- either relying on Proposition \ref{['prop:generic']}, \ref{['prop:sparse']} or \ref{['prop:naive']} --- in green and the ssest function in the MATLAB SI toolbox in blue. SIMBa-4 was run with eight random seeds on each system, and we report both the median and minimum error.
  • Figure 5: Performance of each method, normalized by the best one, on test input-output data from $10$ randomly generated systems with sparse matrices $A$, $B$, $C$, and $D$. The performance of SIMBa (ours) --- relying on Proposition \ref{['prop:naive']} --- is plotted in green, the one of the ssest function in the MATLAB SI toolbox in blue, and the bottom plot is a zoomed-in version of the top one for better visualization. Note that SIMBa was run with $10$ different random seeds on each system, and we report both the median and minimum error.
  • ...and 4 more figures

Theorems & Definitions (19)

  • Proposition 1
  • Corollary 1
  • proof
  • Remark 1
  • Proposition 2
  • proof
  • Corollary 2
  • proof
  • Remark 2
  • Proposition 3
  • ...and 9 more