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RegretOptimal FullInformation Control
We consider the infinitehorizon, discretetime fullinformation control problem. Motivated by learning theory, as a criterion for controller design we focus on regret, defined as the difference between the LQR cost of a causal controller (that has only access to past and current disturbances) and the LQR cost of a clairvoyant one (that has also access to future disturbances). In the fullinformation setting, there is a unique optimal noncausal controller that in terms of LQR cost dominates all other controllers. Since the regret itself is a function of the disturbances, we consider the worstcase regret over all possible bounded energy disturbances, and propose to find a causal controller that minimizes this worstcase regret. The resulting controller has the interpretation of guaranteeing the smallest possible regret compared to the best noncausal controller, no matter what the future disturbances are. We show that the regretoptimal control problem can be reduced to a Nehari problem, i.e., to approximate an anticausal operator with a causal one in the operator norm. In the statespace setting, explicit formulas for the optimal regret and for the regretoptimal controller (in both the causal and the strictly causal settings) are derived. The regretoptimal controller is the sum of the classical H_2 statefeedback law and a finitedimensional controller obtained from the Nehari problem. The controller construction simply requires the solution to the standard LQR Riccati equation, in addition to two Lyapunov equations. Simulations over a range of plants demonstrates that the regretoptimal controller interpolates nicely between the H_2 and the H_∞ optimal controllers, and generally has H_2 and H_∞ costs that are simultaneously close to their optimal values. The regretoptimal controller thus presents itself as a viable option for control system design.
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