Technical Summary: Input-to-State Stability Certification via Projection Residuals for Koopman Learning Control

1. Problem Statement

The paper addresses the stability certification of data-driven Koopman learning control for unknown discrete-time nonlinear repetitive systems operating over finite trial horizons. Unlike classical Iterative Learning Control (ILC) which often seeks zero-error convergence or nominal monotonic convergence, this work focuses on practical Input-to-State Stability (ISS) along the learning-trial axis ($k$).

The core challenge identified is that in data-driven settings, a small prediction residual (model fit) does not guarantee stability. Stability depends on:

1. Channel Informativeness: The selected finite-horizon input-output channel must have a positive margin (invertibility) to generate the requested output correction.
2. Reachability: The requested output increment must lie within the constrained reachable set of the learned channel. If not, a projection residual arises, acting as a persistent disturbance.
3. Persistent Perturbations: Factors such as reset mismatch, non-repetitive disturbances, channel identification errors, deployment shifts, and numerical tolerances do not vanish across trials.

The goal is to construct a deterministic certificate that bounds the learning error from an initial state to an explicit ultimate band, treating Koopman residuals, projection errors, and uncertainties as ISS inputs.

2. Methodology

System Formulation

The system is modeled as a nonlinear repetitive task where the state $x_k(t)$, input $u_k(t)$, and output $y_k(t)$ evolve over a within-trial horizon $t \in \{0, \dots, N-1\}$ and a trial index $k$. The controller uses a frozen Koopman operator approximation (via Extended Dynamic Mode Decomposition with Control, EDMDc) to update inputs trial-by-trial.

The learning dynamics are formulated as a discrete-time system on the trial axis:
$$\bar{E}_k = \Lambda \bar{E}_{k-1} + r_k$$
where $\bar{E}_k$ is the selected stacked tracking error, $\Lambda$ is a learning matrix with $\|\Lambda\|_\infty < 1$, and $r_k$ aggregates all perturbation sources.

Key Components of the Certificate

The authors decompose the perturbation term $r_k$ (the ISS input) into specific, auditable components:

1. Koopman Residuals ($\omega_k$): Model mismatch and non-repetitive disturbances.
2. Projection Residuals ($p_k$): The distance between the requested output increment and the constrained reachable set of the learned channel. This is a structural limitation, not just a numerical error.
3. Channel Uncertainty ($\Delta G_k \Delta U_k$): Errors in the learned finite-horizon channel matrix.
4. Reset/Observability Perturbations ($\Delta O_k \Delta z_{k,0}$): Mismatches in initial conditions and observability.
5. Deployment Shifts and Numerical Tolerances: Differences between calibration and deployment distributions, and solver errors.

Certification Protocol

The paper proposes a rejection-capable certification procedure (Algorithm 1) that validates a candidate controller before deployment:

1. Model Fitting: Learn Koopman matrices from identification data.
2. Margin Verification: Check if the selected channel has a positive lower singular value margin ($\sigma_{\min} - \epsilon > 0$). If not, the candidate is rejected (weak-channel non-certifiability).
3. Projection Audit: Compute the worst-case projection budget over the certified request set.
4. Invariance Check: Verify that the contraction condition $\lambda R + P(R, \bar{Z}_0) + \bar{\xi} \leq R$ holds, ensuring the error remains within a certified ball.
5. Finite-Sample Calibration: Use split-conformal prediction on a calibration dataset to bound the complete-episode residual score with a specified risk level.

Theoretical Results

• Theorem 1 (Deterministic Practical ISS): Establishes that if the channel margin is positive and perturbations are bounded, the learning error converges geometrically to an ultimate band $\Delta_{ISS} = \bar{w} / (1-\lambda)$, where $\bar{w}$ is the aggregate perturbation budget.
• Theorem 2 (Finite-Sample ISS Budget): Provides a probabilistic bound on $\bar{w}$ by combining the conformal residual score with deterministic bounds on channel uncertainty, projection residuals, and deployment shifts.

3. Key Contributions

1. Practical ISS Formulation: The paper formulates Koopman learning control stability as a practical ISS problem along the trial axis, explicitly separating model fit from learning certifiability.
2. Projection Residuals as Stability Objects: It identifies projection residuals and selected-channel margins as necessary certificate objects. It demonstrates that a predictor with small residuals can still fail stability certification if the channel is weak or the request is unreachable.
3. Computable Certificate Budget: The authors provide a method to construct an explicit ultimate band by combining:
   • Complete-episode residual calibration (via split-conformal methods).
   • Finite-horizon channel and reset margins.
   • Projection residuals.
   • Implementation margins (shift, numerical).
4. Audit Framework: A numerical audit framework is developed to verify each component of the theory (residual-to-band scaling, weak-channel rejection, projection closure, and ultimate-band coverage) rather than merely comparing tracking performance.

4. Numerical Results and Validation

The paper validates the theory on four nonlinear repetitive systems: a Duffing oscillator (closed-loop and budget audits), a cubic damping oscillator, and a nonlinear servo.

• Residual-to-Band Scaling: Experiments confirm that increasing disturbance, noise, or model mismatch linearly increases the certified ultimate band, consistent with the ISS gain $\gamma(s) = s/(1-\lambda)$.
• Weak-Channel Rejection: Cases with "weak" selected channels (low singular values) are correctly rejected by the certificate, even if their prediction residuals are small. This highlights that prediction accuracy alone is insufficient.
• Projection Closure: The audit verifies that the requested output corrections remain within the constrained reachable set for certified candidates.
• Ultimate Band Coverage: In successful trials, the observed tracking errors enter and remain within the theoretically predicted ultimate band after a certified finite-entry index.
• Ablation Studies: The paper demonstrates that "nominal" controllers (zero residual budget) fail certification, while the same controllers become certifiable when the calibrated residual budget is included. Conversely, inflating the budget conservatively expands the ultimate band, maintaining coverage.

5. Significance and Claims

The paper claims to provide a practical, data-driven stability certificate for Koopman learning control that is distinct from traditional convergence analysis.

• Separation of Concerns: It rigorously separates model fit (prediction residuals) from control certifiability (channel margins and reachability). A good predictor is not automatically a stabilizing controller.
• Explicit Bounds: Unlike asymptotic results, this method provides an explicit, computable ultimate bound on the tracking error before deployment, accounting for finite data, constraints, and persistent disturbances.
• Robustness to Non-Idealities: The framework explicitly accounts for reset mismatch, deployment shifts, and numerical tolerances, treating them as ISS inputs rather than assuming they vanish.
• Modest Scope: The authors emphasize that the result is a finite-horizon, task-protocol dependent certificate. It does not claim universal stability for arbitrary learned controllers but provides a rigorous audit trail for specific, frozen control configurations under defined protocols.

The work concludes that Koopman learning control can be certified as stable only when model residuals, projection residuals, and channel margins are reported as distinct, auditable stability quantities.