Data-Driven Safe Output Regulation of Strict-Feedback Linear Systems with Input Delay

This paper develops a data-driven safe control framework for linear systems possessing a known strict-feedback structure, but with most plant parameters, external disturbances, and input delay being unknown. By leveraging Koopman operator theory, we utilize Krylov dynamic mode decomposition (DMD) to extract the system dynamics from measured data, enabling the reconstruction of the system and disturbance matrices. Concurrently, the batch least-squares identification (BaLSI) method is employed to identify other unknown parameters in the input channel. Using control barrier functions (CBFs) and backstepping, we first develop a full-state safe controller. Based on this, we build an output-feedback controller by performing system identification using only the output data and actuation signals as well as constructing an observer to estimate the unmeasured plant states. The proposed approach achieves: 1) finite-time identification of a substantial set of unknown system quantities, and 2) exponential convergence of the output state (the state furthest from the control input) to a reference trajectory while rigorously ensuring safety constraints. The effectiveness of the proposed method is demonstrated through a safe vehicle platooning application.

💡 Research Summary

The paper tackles the challenging problem of safe output regulation for strict‑feedback linear systems that suffer from unknown plant parameters, external disturbances, and an unknown input delay. The authors propose a comprehensive data‑driven control framework that integrates modern system identification tools—Koopman operator theory with Krylov Dynamic Mode Decomposition (DMD) and a batch least‑squares identifier (BaLSI)—with safety‑oriented control design based on Control Barrier Functions (CBFs) and backstepping.

System model and assumptions
The plant is described by (\dot X(t)=A X(t)+B U(t-D)+G d(t)), (Y(t)=C X(t)) where (A), (B), (G), the delay (D), and the disturbance (d(t)) are unknown. The output matrix (C) selects the first state, which is the farthest state from the control input, thereby imposing an output‑feedback limitation. Disturbances and the reference signal are generated by a finite‑dimensional exosystem (\dot V=S V) with eigenvalues on the imaginary axis, allowing constant or sinusoidal signals. The delay is modeled as a transport PDE, converting the time‑delay problem into a spatial‑domain PDE that can be handled by predictor‑based techniques.

Data‑driven identification

1. Koopman‑Krylov DMD: By constructing Hankel (Krylov) matrices from measured input–output data, the authors approximate the infinite‑dimensional Koopman operator with a finite‑dimensional matrix. This yields estimates of the system matrix (A) and the disturbance‑injection matrix (\bar G) without requiring full‑state measurements. The Krylov approach improves robustness to noise and reduces the need for rich excitation.
2. Batch Least‑Squares Identification (BaLSI): Simultaneously, BaLSI identifies the scalar input gain (b) (sign known) and the unknown delay (D). The algorithm guarantees exact recovery of these parameters in a finite identification horizon (T_{\text{id}}). Importantly, the required bounds on (b) and (D) are arbitrary and can be chosen conservatively, eliminating the need for precise prior knowledge.

The identification stage produces a set of estimated parameters (\hat\Theta={\hat A,\hat{\bar G},\hat b,\hat D}) that belong to known compact sets, enabling the subsequent control design to be performed with rigorous guarantees.

Full‑state safe controller
Assuming full state availability, the authors employ a three‑step backstepping transformation that converts the strict‑feedback plant into a chain of integrators with an augmented exosystem. A safety barrier function (h(e,t)) defines the safe set (\mathcal C={e\mid h(e,t)\ge0}). To handle potentially unsafe initial conditions, a recovery term (\varsigma(t)) is introduced, which drives the system back into (\mathcal C) within a user‑specified time (\bar t). The backstepping steps generate intermediate variables (z_i) and barrier variables (h_i) such that the dynamics of the barrier vector (H=