End-to-end guarantees for indirect data-driven control of bilinear systems with finite stochastic data

In this paper we propose an end-to-end algorithm for indirect data-driven control for bilinear systems with stability guarantees. We consider the case where the collected i.i.d. data is affected by probabilistic noise with possibly unbounded support and leverage tools from statistical learning theory to derive finite sample identification error bounds. To this end, we solve the bilinear identification problem by solving a set of linear and affine identification problems, by a particular choice of a control input during the data collection phase. We provide a priori as well as data-dependent finite sample identification error bounds on the individual matrices as well as ellipsoidal bounds, both of which are structurally suitable for control. Further, we integrate the structure of the derived identification error bounds in a robust controller design to obtain an exponentially stable closed-loop. By means of an extensive numerical study we showcase the interplay between the controller design and the derived identification error bounds. Moreover, we note appealing connections of our results to indirect data-driven control of general nonlinear systems through Koopman operator theory and discuss how our results may be applied in this setup.

💡 Research Summary

This paper addresses the challenging problem of controlling bilinear dynamical systems when only a finite amount of noisy, independently sampled data is available. The authors propose a complete end‑to‑end pipeline that starts from data collection, proceeds through system identification with rigorous finite‑sample error guarantees, and culminates in a robust controller design that guarantees exponential stability of the closed‑loop system despite the identification uncertainty.

System model and data collection
The target system is described by
(x_{k+1}=Ax_k+B_0u_k+\sum_{i=1}^{n_u}u_k^{(i)}A_i x_k+w_k),
where (w_k) is a sub‑Gaussian process noise. To make the nonlinear identification tractable, the authors deliberately excite the plant with a set of simple inputs: the zero input and each canonical basis vector of the input space. This yields (n_u+1) separate experiments. The zero‑input experiment isolates the autonomous linear part (x_{k+1}=Ax_k+w_k). Each basis‑input experiment produces an affine system of the form (x_{k+1}=Ax_k+