A System Level Approach to Regret Optimal Control

We present an optimisation-based method for synthesising a dynamic regret optimal controller for linear systems with potentially adversarial disturbances and known or adversarial initial conditions. The dynamic regret is defined as the difference between the true incurred cost of the system and the cost which could have optimally been achieved under any input sequence having full knowledge of all future disturbances for a given disturbance energy. This problem formulation can be seen as an alternative to classical
${\mathcal {H}_{2}}$
- or
${\mathcal {H}_{\infty }}$
-control. The proposed controller synthesis is based on the system level parametrisation, which allows reformulating the dynamic regret problem as a semi-definite problem. This yields a new framework that allows to consider structured dynamic regret problems, which have not yet been considered in the literature. For known pointwise ellipsoidal bounds on the disturbance, we show that the dynamic regret bound can be improved compared to using only a bounded energy assumption and that the optimal dynamic regret bound differs by at most a factor of
$ {\scriptstyle {}^{\scriptstyle {2}}}\hspace {-0.224em}/\hspace {-0.112em}{\scriptstyle {\pi }}$
from the computed solution. Furthermore, the proposed framework allows guaranteeing state and input constraint satisfaction.