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This paper considers the linear-quadratic dual control problem where thesystem parameters need to be identified and the control objective needs to beoptimized in the meantime. Contrary to existing works on data-drivenlinear-quadratic regulation, which typically provide error or regret boundswithin a certain probability, we propose an online algorithm that guaranteesthe asymptotic optimality of the controller in the almost sure sense. Our dualcontrol strategy consists of two parts: a switched controller withtime-decaying exploration noise and Markov parameter inference based on thecross-correlation between the exploration noise and system output. Central tothe almost sure performance guarantee is a safe switched control strategy thatfalls back to a known conservative but stable controller when the actual statedeviates significantly from the target state. We prove that this switchingstrategy rules out any potential destabilizing controllers from being applied,while the performance gap between our switching strategy and the optimal linearstate feedback is exponentially small. Under our dual control scheme, theparameter inference error scales as $O(T^{-1/4+\epsilon})$, while thesuboptimality gap of control performance scales as $O(T^{-1/2+\epsilon})$,where $T$ is the number of time steps, and $\epsilon$ is an arbitrarily smallpositive number. Simulation results on an industrial process example areprovided to illustrate the effectiveness of our proposed strategy.

Safe Linear-Quadratic Dual Control with Almost Sure Performance Guarantee