An Integrative Perspective to LQ and $\ell _\infty$ Control for Delayed and Quantized Systems

Deterministic and stochastic approaches to handle uncertainties may incur very different complexities in computation time and memory usage, in addition to different uncertainty models. For linear systems with delay and rate constrained communications between the observer and the controller, previous work shows that a deterministic approach, the
$\ell _\infty$
control has low complexity but can only handle bounded disturbances. In this article, we take a stochastic approach and propose a linear-quadratic (LQ) controller that can handle arbitrarily large disturbance but has large complexity in time and space. The differences in robustness and complexity of the
$\ell _\infty$
and LQ controllers motivate the design of a hybrid controller that interpolates between the two: The
$\ell _\infty$
controller is applied when the disturbance is not too large (normal mode) and the LQ controller is resorted to otherwise (acute mode). We characterize the switching behavior between the normal and acute modes. Using our theoretical bounds which are supplemented by numerical experiments, we show that the hybrid controller can achieve a sweet spot in the robustness-complexity tradeoff, i.e., reject occasional large disturbance while operating with low complexity most of the time.