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Learning Stabilizing Controllers for Unstable Linear Quadratic Regulators from a Single Trajectory

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Document pages: 38 pages

Abstract: The principal task to control dynamical systems is to ensure their stability.When the system is unknown, robust approaches are promising since they aim tostabilize a large set of plausible systems simultaneously. We study linearcontrollers under quadratic costs model also known as linear quadraticregulators (LQR). We present two different semi-definite programs (SDP) whichresults in a controller that stabilizes all systems within an ellipsoiduncertainty set. We further show that the feasibility conditions of theproposed SDPs are emph{equivalent}. Using the derived robust controllersyntheses, we propose an efficient data dependent algorithm -- textsc{eXploration} -- that with high probability quickly identifies astabilizing controller. Our approach can be used to initialize existingalgorithms that require a stabilizing controller as an input while addingconstant to the regret. We further propose different heuristics whichempirically reduce the number of steps taken by textsc{eXploration} and reducethe suffered cost while searching for a stabilizing controller.

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