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Learning Nonlinear Reduced Order Models using State-Space Neural Networks with Ordered State Variance

Midhun T. Augustine, Mani Bhushan, Sharad Bhartiya

TL;DR

The paper addresses nonlinear system identification with unknown order by introducing SSNNO, a state-space neural network that imposes an ordered variance structure on state variables to enable automatic reduction to a reduced-order SSNNO (R-SSNNO). It introduces a variance-based training objective that combines prediction error with a variance-regularization term, proving existence of SSNNO with bounded $J_y$ and deriving an R-SSNNO by discarding low-variance states while adjusting the first-layer connections. Through a nonlinear CSTR simulation, the approach achieves accurate short- and long-horizon predictions while yielding a smaller, data-driven predictor; an EKF-MPC scheme using the reduced predictor demonstrates practical control performance. Overall, the work provides a principled, data-driven pathway to compact nonlinear state-space representations suitable for MPC and state estimation, with theoreticalJustifications for variance ordering and empirical validation on a challenging process.

Abstract

A novel State-Space Neural Network with Ordered variance (SSNNO) is presented in which the state variables are ordered in decreasing variance. A systematic way of model order reduction with SSNNO is proposed, which leads to a Reduced order SSNNO (R-SSNNO). Theoretical results for the existence of an SSNNO with arbitrary bounds on the output prediction error are presented. The application of SSNNO in control: Model Predictive Control (MPC) and state estimation: Extended Kalman Filter (EKF) is discussed. The effectiveness of SSNNO in system identification and control is illustrated using simulations on a nonlinear continuous reactor process example.

Learning Nonlinear Reduced Order Models using State-Space Neural Networks with Ordered State Variance

TL;DR

The paper addresses nonlinear system identification with unknown order by introducing SSNNO, a state-space neural network that imposes an ordered variance structure on state variables to enable automatic reduction to a reduced-order SSNNO (R-SSNNO). It introduces a variance-based training objective that combines prediction error with a variance-regularization term, proving existence of SSNNO with bounded and deriving an R-SSNNO by discarding low-variance states while adjusting the first-layer connections. Through a nonlinear CSTR simulation, the approach achieves accurate short- and long-horizon predictions while yielding a smaller, data-driven predictor; an EKF-MPC scheme using the reduced predictor demonstrates practical control performance. Overall, the work provides a principled, data-driven pathway to compact nonlinear state-space representations suitable for MPC and state estimation, with theoreticalJustifications for variance ordering and empirical validation on a challenging process.

Abstract

A novel State-Space Neural Network with Ordered variance (SSNNO) is presented in which the state variables are ordered in decreasing variance. A systematic way of model order reduction with SSNNO is proposed, which leads to a Reduced order SSNNO (R-SSNNO). Theoretical results for the existence of an SSNNO with arbitrary bounds on the output prediction error are presented. The application of SSNNO in control: Model Predictive Control (MPC) and state estimation: Extended Kalman Filter (EKF) is discussed. The effectiveness of SSNNO in system identification and control is illustrated using simulations on a nonlinear continuous reactor process example.
Paper Structure (16 sections, 3 theorems, 51 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 16 sections, 3 theorems, 51 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. State-Space Neural Network (SSNN)
  4. Determination of Model Order in SSNN
  5. State-Space Neural Network with Ordered variance (SSNNO)
  6. SSNNO Training: Variance-based State Regularization
  7. Reduced order SSNNO (R-SSNNO)
  8. Properties of SSNNO and R-SSNNO
  9. Simulation Results
  10. EKF based state space MPC design Example
  11. Conclusions
  12. Appendices
  13. Proof of Lemma \ref{['thssnn']}
  14. Prrof of Lemma \ref{['lemmared']}
  15. Proof of Lemma \ref{['lemmassnno']}
  16. ...and 1 more sections

Key Result

Lemma 1

Consider the training data sequence in Eq. equy, generated by applying bounded inputs $\textbf{U}$ to the nonlinear system Eq. (eqss), which satisfies Assumption lipz. Then for any $\epsilon>0$, there exists a trained SSNN Eq. (eqssnn) under Assumption order, with an initial condition $\hat{\textbf{

Figures (3)

  • Figure 1: SSNN Block diagram.
  • Figure 2: Response of SSNNO for CSTR with white noise: (a) Training (b) Testing (c) Step test.
  • Figure 3: CSTR with SSNNO-EKF-MPC (a) Output (b) Control input.

Theorems & Definitions (15)

  • Lemma 1: bKK18
  • proof
  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Remark 2
  • Remark 3
  • Lemma 2
  • proof
  • ...and 5 more