Online Linear Quadratic Tracking with Regret Guarantees

Abstract

Online learning algorithms for dynamical systems provide finite time guarantees for control in the presence of sequentially revealed cost functions. We pose the classical linear quadratic tracking problem in the framework of online optimization where the time-varying reference state is unknown a priori and is revealed after the applied control input. We show the equivalence of this problem to the control of linear systems subject to adversarial disturbances and propose a novel online gradient descent based algorithm to achieve efficient tracking in finite time. We provide a dynamic regret upper bound scaling linearly with the path length of the reference trajectory and a numerical example to corroborate the theoretical guarantees.

Online Convex Optimization for Data-Driven Control of Dynamical Systems

“…The upper bound in Theorem 1 depends linearly on T t=1 ∥ζ t − ζ t−1 ∥, commonly termed path length in the literature [11], [37], which can be interpreted as a measure for the variation of the cost functions. In [5], it was shown for the nominal setting (i.e., without disturbances) that an upper bound that depends linearly on the path length is optimal.…”

“…Next, the predicted input sequence ût is updated in step [S5] by a convex combination and will be used for prediction at the next time step t + 1 (trajectory 3 in Figure 1). The convex combination ensures i) constraint satisfaction due to convexity of the constraint sets, and ii) that the predicted input sequence is not updated anymore if it already reaches the estimated optimal steady state θt , since then ẑµ t − ζt = 0, which implies λ t = 0 by (11). The latter avoids degrading closedloop performance in case the virtual input sequence g t in (10) is chosen poorly, e.g., when xt = θt , but g t is chosen such that the state trajectory resulting from application of g t deviates from θt and only returns to the estimated optimal steady θt at the end of the prediction horizon µ.…”

“…Such time-varying cost functions arise in a range of relevant applications, for example due to renewable energy generation and a priori unknown consumption in energy grids [3] or in tracking control, when the desired trajectory is computed online itself [4]. Therefore, various algorithms for control of dynamical systems based on the OCO framework have recently been proposed in the literature, see, e.g., [5]- [11] and the references therein. These algorithms typically aim to track the optimal steady states of the system, which are a priori unknown and time-varying due to their dependence on the cost functions as well.…”