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

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aligning it more precisely with the time-varying nonlinear system by integrating the updated data and discarding the outdated data.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.

• The paper introduces an online iterative update scheme for deep Koopman lifted system matrices that ensures stability-guaranteed control for unknown nonlinear time-varying systems, showing a 70-90% error reduction over existing methods.
• The approach includes result-oriented update mechanisms that decide whether updates are beneficial and a model predictive control design that ensures closed-loop stability.
• Key components of the scheme will discard outdated data, leading to improved computational efficiency.

Overview

This paper develops a unified framework for modeling, prediction, and control of unknown nonlinear time-varying systems of the form $x_{k+1} = f(x_k, u_k, k)$, where the dynamics are not known a priori. The approach combines three components: (i) an online iterative update scheme for deep Koopman lifted system matrices that explicitly discards outdated data via a sliding window; (ii) result-oriented update mechanisms that decide when updating is beneficial; and (iii) a model predictive control (MPC) design with provable input-to-state stability (ISS) of the closed loop with respect to the Koopman approximation error. The work positions itself against two lines of prior art: online DMD variants with forgetting factors or sliding windows [zhangOnlineDynamicMode2019], which lack feasibility conditions and control integration, and deep Koopman learning for time-varying systems (DKTV) [haoDeepKoopmanLearning2024], which accumulates new data without removing outdated snapshots.

Problem formulation

The system is embedded in a lifted space using observables $g(\cdot,\theta_\tau)$ parameterized by a multilayer perceptron, yielding the approximate linear time-varying model

$$g(\hat{x}_{k+1},\theta_\tau) = A_\tau g(\hat{x}_k,\theta_\tau) + B_\tau u_k, \qquad \hat{x}_{k+1} = C_\tau g(\hat{x}_{k+1},\theta_\tau),$$

where $\tau$ indexes updates. The matrices are obtained by least-squares fitting (EDMD-style) over a current window of $w$ snapshots. Two structural assumptions carry the theory: full row rank of the stacked data matrix $[G_\tau^\top \ U_\tau^\top]^\top$ and of $H_\tau$, and boundedness/continuity of the Koopman operator together with orthonormality of the final hidden-layer activations. The latter allows convergence of the approximate operator as the hidden width $n_h \to \infty$ rather than requiring the observable dimension $r$ to grow unboundedly — a key point, since EDMD convergence would otherwise force the window width $w > r+m$ to grow, which is incompatible with time-varying dynamics where old data must be discarded.

Iterative update of lifted system matrices

The central algorithmic contribution is a rank-one-style recursive update of the lifted matrices based on the Sherman–Morrison–Woodbury formula. When $g(\cdot,\theta_\tau)$0 new snapshots arrive, the oldest $g(\cdot,\theta_\tau)$1 are removed and the newest $g(\cdot,\theta_\tau)$2 added simultaneously; the update takes the form

$g(\cdot,\theta_\tau)$3

with analogous recursions for $g(\cdot,\theta_\tau)$4 and the inverse Gram factors $g(\cdot,\theta_\tau)$5, $g(\cdot,\theta_\tau)$6. The correction term is proportional to the prediction residual of the current model on the exchanged data, giving the update an intuitive error-driven interpretation.

Because discarding data can destroy full row rank, the paper derives a necessary condition for iterative feasibility: invertibility of the low-dimensional matrices $g(\cdot,\theta_\tau)$7 and $g(\cdot,\theta_\tau)$8, where $g(\cdot,\theta_\tau)$9 encodes the simultaneous removal and addition. A Schur-complement argument establishes that this condition is equivalent to Assumption 1 holding at the next update index, so feasibility can be checked in $$g(\hat{x}_{k+1},\theta_\tau) = A_\tau g(\hat{x}_k,\theta_\tau) + B_\tau u_k, \qquad \hat{x}_{k+1} = C_\tau g(\hat{x}_{k+1},\theta_\tau),$$0 instead of $$g(\hat{x}_{k+1},\theta_\tau) = A_\tau g(\hat{x}_k,\theta_\tau) + B_\tau u_k, \qquad \hat{x}_{k+1} = C_\tau g(\hat{x}_{k+1},\theta_\tau),$$1. The per-update cost of the recursion itself is $$g(\hat{x}_{k+1},\theta_\tau) = A_\tau g(\hat{x}_k,\theta_\tau) + B_\tau u_k, \qquad \hat{x}_{k+1} = C_\tau g(\hat{x}_{k+1},\theta_\tau),$$2 versus $$g(\hat{x}_{k+1},\theta_\tau) = A_\tau g(\hat{x}_k,\theta_\tau) + B_\tau u_k, \qquad \hat{x}_{k+1} = C_\tau g(\hat{x}_{k+1},\theta_\tau),$$3 for recomputing the pseudoinverse from scratch.

An estimation-error bound is established under Lipschitz continuity of the network and bounded state/input variation rates ($$g(\hat{x}_{k+1},\theta_\tau) = A_\tau g(\hat{x}_k,\theta_\tau) + B_\tau u_k, \qquad \hat{x}_{k+1} = C_\tau g(\hat{x}_{k+1},\theta_\tau),$$4, $$g(\hat{x}_{k+1},\theta_\tau) = A_\tau g(\hat{x}_k,\theta_\tau) + B_\tau u_k, \qquad \hat{x}_{k+1} = C_\tau g(\hat{x}_{k+1},\theta_\tau),$$5): the asymptotic prediction error is bounded by $$g(\hat{x}_{k+1},\theta_\tau) = A_\tau g(\hat{x}_k,\theta_\tau) + B_\tau u_k, \qquad \hat{x}_{k+1} = C_\tau g(\hat{x}_{k+1},\theta_\tau),$$6. In practice, since infinite width is unattainable, training includes a regularization term enforcing $$g(\hat{x}_{k+1},\theta_\tau) = A_\tau g(\hat{x}_k,\theta_\tau) + B_\tau u_k, \qquad \hat{x}_{k+1} = C_\tau g(\hat{x}_{k+1},\theta_\tau),$$7, under which the bound depends only on the reconstruction error and variation rates as $$g(\hat{x}_{k+1},\theta_\tau) = A_\tau g(\hat{x}_k,\theta_\tau) + B_\tau u_k, \qquad \hat{x}_{k+1} = C_\tau g(\hat{x}_{k+1},\theta_\tau),$$8.

Update mechanisms

A motivating linear time-varying example demonstrates a counterintuitive effect: refitting with newly collected data produced RMSE 0.0274 versus 0.0050 with outdated data over a 30-step horizon, because periodic dynamics make fresh data locally less representative. The authors therefore introduce two gating conditions:

• Reliability skip: if the current model's fitting error on the pending batch falls below a threshold $$g(\hat{x}_{k+1},\theta_\tau) = A_\tau g(\hat{x}_k,\theta_\tau) + B_\tau u_k, \qquad \hat{x}_{k+1} = C_\tau g(\hat{x}_{k+1},\theta_\tau),$$9, no update is computed at all.
• Result-oriented acceptance: even when an update is computed, it is applied only if the candidate model's error on the new data does not exceed that of the incumbent model.

These mechanisms convert unconditional sliding-window updates into event-triggered ones, reducing both computation and the risk of performance-degrading updates.

Stability-guaranteed controller design

Control is performed entirely in the lifted space via MPC on the nominal linear time-varying model, treating the true lifted dynamics as the nominal model plus a deterministic residual $\tau$0. The terminal cost matrix $\tau$1 and feedback gain $\tau$2 are obtained online by solving a semidefinite program (via the standard substitution $\tau$3, $\tau$4) enforcing a terminal Lyapunov decrease condition, an ellipsoidal invariant set constraint, and input bounds; all constraints are LMIs, so the SDP is convex.

The stability analysis proceeds in three steps: the ellipsoid $\tau$5 is shown to be positively invariant and a domain of attraction for the nominal closed loop; recursive feasibility of both the SDP and the MPC problem is proven under an assumption that the lifted state maintains sufficient margin from the ellipsoid boundary relative to the modeling error magnitude; and the MPC optimal value function is shown to be an ISS Lyapunov function, establishing ISS of the lifted closed loop with respect to $\tau$6. When the lifting function concatenates the physical state ($\tau$7), ISS transfers to the original nonlinear system by a corollary. Notably, at update instants the proof absorbs the model-variation discrepancy $\tau$8 into the error signal, so stability holds across model switches. The practical implication is direct: tracking quality is governed by the same quantities appearing in the estimation-error bound, namely hidden width, sampling rate, and update frequency.

Numerical results

Four case studies support the claims, with code released publicly.

Simple time-varying nonlinear system. Against TVDMD and DKTV, the proposed method reduces MAE from 1.865 to 0.1846 (90.1% reduction) and RMSE from 1.980 to 0.2245 (88.7%) relative to TVDMD, and MAE/RMSE by roughly 70% relative to DKTV.

Duffing oscillator. With window $\tau$9, batch $w$0, threshold $w$1, and ten repeated runs, the full method (OTVDKL*) achieves MAE $w$2 and RMSE $w$3 versus DKTV's $w$4 and $w$5 — reductions of 71.5% and 75.5% — while cutting computation time from 4.42 s to 0.89 s, attributable largely to the update-gating mechanisms. The ablation (OTVDKL without gating) yields intermediate errors (MAE 0.222) at higher cost (3.14 s), isolating the benefit of the trigger conditions. Hyperparameter studies show larger $w$6 and smaller $w$7, $w$8 reduce prediction RMSE here.

Serial manipulator. For trajectory tracking of a 3-DOF arm under sinusoidal disturbance torque and measurement noise, the proposed method attains RMSE $w$9, outperforming fixed-model DKO (0.4737), adaptive linear Koopman (0.2358), and simultaneous identification-MPC (0.0942). Interestingly, the hyperparameter trend reverses relative to the prediction task: larger $[G_\tau^\top \ U_\tau^\top]^\top$0 increases tracking error, consistent with stale data being harmful for control.

Synthetic gene regulatory network. On a six-state repressilator-type system with periodic parameter variation, mean prediction errors over 40 experiments are 0.262 (offline deep Koopman), 0.0485 (DKTV), and 0.0150 (proposed); tracking errors for protein concentration are 0.623, 0.269, and 0.106 respectively. The offline model's error grows rapidly beyond its training distribution, underscoring the necessity of online adaptation for time-varying plants.

Limitations and open questions

Several caveats qualify the results. All guarantees rest on Assumption 5, requiring the lifted state to remain sufficiently inside the terminal ellipsoid relative to the modeling error; this implicitly demands high model accuracy, and the paper offers no constructive procedure for verifying it a priori. The estimation-error bound is asymptotic in the hidden width $[G_\tau^\top \ U_\tau^\top]^\top$1, and the finite-width guarantee relies on the auxiliary constraint $[G_\tau^\top \ U_\tau^\top]^\top$2, which constrains the admissible learned dynamics. The orthonormal-basis assumption on final-layer activations is inherited from prior work and is not literally satisfied by ReLU networks used in the simulations. The claim that new data can worsen short-term prediction is demonstrated only on a periodic example, and the update-gating conditions are heuristic rather than accompanied by performance guarantees. Finally, all validation is numerical; no hardware experiments are reported, and the interaction between frequent model switching and MPC feasibility near the invariant-set boundary remains unexamined.

Conclusion

The paper contributes a computationally efficient, feasibility-checked iterative scheme for maintaining a deep Koopman surrogate of an unknown nonlinear time-varying system, gated update rules that avoid both unnecessary and detrimental updates, and an LMI-based robust MPC design with proven ISS of the closed loop with respect to the approximation error. Reported error reductions of 70–90% over existing online Koopman/DMD baselines, together with order-of-magnitude computational savings on the Duffing benchmark, indicate that explicit data expiration and selective updating are effective for time-varying identification, and that these modeling gains translate directly into tracking performance under the proposed stability-guaranteed controller.