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This paper considers a single-trajectory system identification problem forlinear systems under general nonlinear and/or time-varying policies with i.i.d.random excitation noises. The problem is motivated by safe learning-basedcontrol for constrained linear systems, where the safe policies during thelearning process are usually nonlinear and time-varying for satisfying thestate and input constraints. In this paper, we provide a non-asymptotic errorbound for least square estimation when the data trajectory is generated by anynonlinear and/or time-varying policies as long as the generated state andaction trajectories are bounded. This significantly generalizes the existingnon-asymptotic guarantees for linear system identification, which usuallyconsider i.i.d. random inputs or linear policies. Interestingly, our errorbound is consistent with that for linear policies with respect to thedependence on the trajectory length, system dimensions, and excitation levels.Lastly, we demonstrate the applications of our results by safe learning withrobust model predictive control and provide numerical analysis.

Non-asymptotic System Identification for Linear Systems with Nonlinear Policies