Deep Koopman Iterative Learning and Stability-Guaranteed Control for Unknown Nonlinear Time-Varying Systems

TL;DR

The paper develops an online, data-driven Koopman framework using deep observables to model unknown nonlinear time-varying systems via a lifted linear representation $g(x_{k+1})=A_k g(x_k)+B_k u_k$ with decoding $x_{k+1}=C_k g(x_{k+1})$, enabling prediction and control. It introduces a sliding-window online learning algorithm with selective update rules that discard outdated data, reducing computation while maintaining predictive accuracy, and establishes a necessary feasibility condition and an error bound for the Koopman surrogate. A stability-guaranteed MPC controller is designed on the online-updated lifted model, with an SDP-based terminal cost design that decouples gains and costs to guarantee input-to-state stability with respect to approximation error $\epsilon_k$. Numerical experiments on a simple system, Duffing oscillator, serial manipulator, and synthetic biological network demonstrate substantial improvements in prediction accuracy and tracking performance, as well as reduced computation relative to existing online Koopman methods. Overall, the work provides a unified, data-driven approach for modeling, predicting, and stabilizing unknown nonlinear time-varying systems with provable guarantees and practical efficiency. The framework has potential impact across robotics, dynamics identification, and biology where time-varying nonlinear dynamics are common.

Abstract

This paper proposes a Koopman-based framework for modeling, prediction, and control of unknown nonlinear time-varying systems. We present a novel Koopman-based learning method for predicting the state of unknown nonlinear time-varying systems, upon which a robust controller is designed to ensure that the resulting closed-loop system is input-to-state stable with respect to the Koopman approximation error. The error of the lifted system model learned through the Koopman-based method increases over time due to the time-varying nature of the nonlinear time-varying system. To address this issue, an online iterative update scheme is incorporated into the learning process to update the lifted system model, aligning it more precisely with the time-varying nonlinear system by integrating the updated data and discarding the outdated data. A necessary condition for the feasibility of the proposed iterative learning method is derived. In order to reduce unnecessary system updates while ensuring the prediction accuracy of the lifted system, the update mechanism is enhanced to determine whether to update the lifted system and meanwhile to reduce updates that deteriorate the fitting performance. Furthermore, based on the online-updated lifted system, a controller is designed to ensure the closed-loop controlled system be input-to-state stable with respect to the Koopman approximation error. Numerical simulations on the Duffing oscillator, the serial manipulator, and the synthetic biological network system are presented to demonstrate the effectiveness of the proposed method for the approximation and control of unknown nonlinear time-varying systems. The results show that the proposed approach outperforms existing methods in terms of approximation accuracy and computational efficiency, even under significant system variations.

Figures (11)

• Figure 1: Snapshots division and evolution.
• Figure 2: Comparison of prediction performance using new or outdated data. In the figure, the blue solid line represents the true trajectory of the system state $y$, the red dots indicate the snapshots used for fitting the system, and the yellow dashed line represents the predicted system state. A part of the trajectory is enlarged for clearer visualization.
• Figure 3: Control framework for unknown time-varying nonlinear systems. The left part of the figure, Data update iteratively, illustrates that the lifted system matrices are updated online based on the measured trajectories (where the color change reflects the variation in matrix entries). The right part, Forward prediction, shows the iterative forward prediction of the unknown nonlinear system state using the lifted linear model. During trajectory tracking, the Controller module computes the control input $u_k$.
• Figure 4: Prediction results and errors of the three methods. Subfigures (a) (b) and (c) represent the trajectory predictions of $x_1$, the trajectory predictions of $x_2$, and the prediction error plots for the three methods, respectively.
• Figure 5: Prediction results and errors for the Duffing oscillator. (a) The prediction results of the DKTV. (b) The prediction results of the OTVDKL. (c) The prediction results of the OTVDKL*. In these three subfigures, the blue curves represent the ground truth, while the orange curves correspond to the estimates produced by the respective methods. (d) The absolute error over time for DKTV (Blue), OTVDKL (Orange), and OTVDKL* (Green). (e) The MAE of OTVDKL* at each evaluation step. Blue dots represent the MAE of the current model on new data; Orange dots represent the MAE after a potential model update. The vertical red dashed lines indicate the time instances where the update condition \ref{['eq:event_trigger']} was met, and the model was updated. (f) The impact of hyperparameters on the RMSE of prediction error. (g) Evolution of Training Data: Visualization of the iterative update process. The blue arrows indicate the sliding window mechanism. The blue solid curve and the orange solid curve represent the true values and the predicted values, respectively, while the blue scatter points denote the data samples in $S_{\tau}^{\text{cur}}$. As time progresses, the dataset $\mathcal{S}_{\tau}^{\text{cur}}$ selectively shifts to incorporate new data while discarding outdated snapshots.

Theorems & Definitions (12)

• Remark 1
• Lemma 1
• Proposition 1
• Remark 2
• Lemma 2
• Corollary 1
• Lemma 3
• Lemma 4
• Theorem 1
• Lemma 5