Neural network-based identification of state-space switching nonlinear systems

TL;DR

This work tackles the identification of switching nonlinear systems in state-space form by pairing neural-network models with an EM framework. A moving-window E-step estimates the switching sequence while an EKF-based M-step trains two RNN groups that represent the state and output maps for each subsystem, enabling accurate parameter estimation and reliable switching identification. The authors prove that EKF updates are equivalent to maximizing the EM objective and demonstrate quadratic convergence under mild conditions, with extensive experiments on academic switching systems and a battery SOC case showing superior parameter fit, switching sequence recovery, and predictive accuracy. The approach reduces computational burden via the moving-window strategy and offers a practical pathway for real-world switching systems where internal mode dynamics are nonlinear and unknown.

Abstract

We design specific neural networks (NNs) for the identification of switching nonlinear systems in the state-space form, which explicitly model the switching behavior and address the inherent coupling between system parameters and switching modes. This coupling is specifically addressed by leveraging the expectation-maximization (EM) framework. In particular, our technique will combine a moving window approach in the E-step to efficiently estimate the switching sequence, together with an extended Kalman filter (EKF) in the M-step to train the NNs with a quadratic convergence rate. Extensive numerical simulations, involving both academic examples and a battery charge management system case study, illustrate that our technique outperforms available ones in terms of parameter estimation accuracy, model fitting, and switching sequence identification.

Figures (6)

• Figure 1: Top: The true output (solid blue line with asterisks) and the estimated one by our method (solid red line with circles) over the time window $t\in[200,300]$. Middle: The resulting MSE. Bottom: The true switching sequence (blue crosses) and the estimated one (red circles).
• Figure 2: Top: The true (solid blue line with asterisks) and the estimated output by using our RNN-based method (solid red line with circles) over the time window $t\in[100,300]$. Bottom: The true switching sequence (blue crosses) and the one estimated by our method (red circles).
• Figure 3: The MSE obtained by proposed method (red), the kernel-based method (yellow) Anna2018 and the Bayesian ensemble learning (blue) mlp under different noise conditions.
• Figure 4: The BFR obtained by proposed method (red), the kernel-based method (yellow) Anna2018 and the Bayesian ensemble learning (blue) mlp under different noise conditions.
• Figure 5: The training data collected in the battery platform where $U$ and $I$ respectively represent the voltage and current in the circuit, and SOC represents the battery state of charge.

Theorems & Definitions (9)

• Lemma 1
• Remark 1
• Proposition 1
• Remark 2
• Theorem 1
• Remark 3
• Remark 4
• Proposition 2
• Theorem 2