Input-to-State Stability Certification via Projection Residuals for Koopman Learning Control of Nonlinear Repetitive Systems
This paper studies input-to-state stability (ISS) certification for data-driven Koopman learning control of unknown discrete-time nonlinear repetitive systems over finite trial horizons. Rather than proposing a new learning law, we certify when a fixed Koopman-assisted constrained update yields practical stability of the selected tracking error along the trial axis. Prediction accuracy alone is insufficient for this purpose: the selected finite-horizon input-output channel must have a positive margin, and the unreachable component of the requested output increment must be accounted for through a projection residual. Thus, a Koopman predictor with small held-out prediction residuals may still fail the learning-stability certificate if its selected channel is weak. We formulate the selected stacked tracking error as the state of a discrete-time learning-axis system and treat Koopman residuals, reset mismatch, channel uncertainty, projection residuals, deployment shifts, and numerical tolerances as ISS inputs. The deterministic result gives a practical ISS estimate from the initial learning error to an explicit ultimate band. A finite-sample implementation constructs an episode-level residual bound under a fixed controller and combines it with reported channel, projection, shift, and numerical margins. Numerical checks on nonlinear repetitive systems support the predicted residual-to-band scaling, weak-channel rejection, projection closure, and ultimate-band coverage.