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Online Optimization of Switched LTI Systems Using Continuous-Time and Hybrid Accelerated Gradient Flows

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Document pages: 18 pages

Abstract: This paper studies the design of feedback controllers that steer the outputof a switched linear time-invariant system to the solution of a possiblytime-varying optimization problem. The design of the feedback controllers isbased on an online gradient descent method, and an online hybrid controllerthat can be seen as a regularized Nesterov s accelerated gradient method. Bothof the proposed approaches accommodate output measurements of the plant, andare implemented in closed-loop with the switched dynamical system. By design,the controllers continuously steer the system output to an optimal trajectoryimplicitly defined by the time-varying optimization problem without requiringknowledge of exogenous inputs and disturbances. For cost functions that aresmooth and satisfy the Polyak-Lojasiewicz inequality, we demonstrate that theonline gradient descent controller ensures uniform global exponential stabilitywhen the time-scales of the plant and the controller are sufficiently separatedand the switching signal of the plant is slow on the average. Under a strongconvexity assumption, we also show that the online hybrid Nesterov s methodguarantees tracking of optimal trajectories, and outperforms online controllersbased on gradient descent. Interestingly, the proposed hybrid acceleratedcontroller resolves the potential lack of robustness suffered by standardcontinuous-time accelerated gradient methods when coupled with a dynamicalsystem. When the function is not strongly convex, we establish global practicalasymptotic stability results for the accelerated method, and we unveil theexistence of a trade-off between acceleration and exact convergence in onlineoptimization problems with controllers using dynamic momentum. Our theoreticalresults are illustrated via different numerical examples.

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