We study the problem of adaptive control of a high dimensional linear quadratic (LQ) system. Previous work established the asymptotic convergence to an optimal controller for various adaptive control schemes. More recently, an asymptotic regret bound of \tilde{O}(\sqrt{T}) was shown for T \gg p where p is the dimension of the state space. In this work we consider the case where the matrices describing the dynamic of the LQ system are sparse and their dimensions are large. We present an adaptive control scheme that for p \gg 1 and T \gg \polylog(p) achieves a regret bound of \tilde{O}(p \sqrt{T}) .

Aerial robots are required to remain operational even in the event of system disturbances, damages, or failures to ensure resilient and robust task completion and safety. One common failure case is propeller damage, which presents a significant challenge in both quantification and compensation. We propose a novel adaptive control scheme capable of detecting and compensating for multi-rotor propeller damages, ensuring safe and robust flight performances. Our control scheme includes an L1 adaptive controller for damage inference and compensation of single or dual propellers, with the capability to seamlessly transition to a fault-tolerant solution in case the damage becomes severe. We experimentally identify the conditions under which the L1 adaptive solution remains preferable over a fault-tolerant alternative. Experimental results validate the proposed approach, demonstrating its effectiveness in running the adaptive strategy in real time on a quadrotor even in case of damage to multiple propellers.

We study the problem of adaptive control of a high dimensional linear quadratic (LQ) system. Previous work established the asymptotic convergence to an optimal controller for various adaptive control schemes. More recently, an asymptotic regret bound of $\tilde{O}(\sqrt{T})$ was shown for $T \gg p$ where $p$ is the dimension of the state space. In this work we consider the case where the matrices describing the dynamic of the LQ system are sparse and their dimensions are large. We present an adaptive control scheme that for $p \gg 1$ and $T \gg \polylog(p)$ achieves a regret bound of $\tilde{O}(p \sqrt{T})$.