This paper presents a memory-augmented control solution to the optimal reference tracking problem for linear systems subject to adversarial disturbances. We assume that the dynamics of the linear system are known and that the reference signal is generated by a linear system with unknown dynamics. Under these assumptions, finding the optimal tracking controller is formalized as an online convex optimization problem that leverages memory of past disturbance and reference values to capture their temporal effects on the performance. That is, a (disturbance, reference)-action control policy is formalized, which selects the control actions as a linear map of the past disturbance and reference values. The online convex optimization is then formulated over the parameters of the policy on its past disturbance and reference values to optimize general convex costs. It is shown that our approach outperforms robust control methods and achieves a tight regret bound O(√T) where in our regret analysis, we have benchmarked against the best linear policy.

In this article, we consider linear quadratic (LQ) control problem with process and measurement noises. We analyze the LQ problem in terms of the average cost and the structure of the value function. We assume that the dynamics of the linear system is unknown and only noisy measurements of the state variable are available. Using noisy measurements of the state variable, we propose two model-free iterative algorithms to solve the LQ problem. The proposed algorithms are variants of policy iteration routine where the policy is greedy with respect to the average of all previous iterations. We rigorously analyze the properties of the proposed algorithms, including stability of the generated controllers and convergence. We analyze the effect of measurement noise on the performance of the proposed algorithms, the classical off-policy, and the classical Q-learning routines. We also investigate a model-building approach, inspired by adaptive control, where a model of the dynamical system is estimated and the optimal control problem is solved assuming that the estimated model is the true model. We use a benchmark to evaluate and compare our proposed algorithms with the classical off-policy, the classical Q-learning, and the policy gradient. We show that our model-building approach performs nearly identical to the analytical solution and our proposed policy iteration based algorithms outperform the classical off-policy and the classical Q-learning algorithms on this benchmark but do not outperform the model-building approach.

Differential graphical games have been introduced in the literature to solve state synchronization problem for linear homogeneous agents. When the agents are heterogeneous, the previous notion of graphical games cannot be used anymore and a new definition is required. In this paper, we define a novel concept of differential graphical games for linear heterogeneous agents subject to external unmodeled disturbances, which contain the previously introduced graphical game for homogeneous agents as a special case. Using our new formulation, we can solve both the output regulation and H-infinity output regulation problems. Our graphical game framework yields coupled Hamilton-Jacobi-Bellman equations, which are, in general, impossible to solve analytically. Therefore, we propose a new actor-critic algorithm to solve these coupled equations numerically in real time. Moreover, we find an explicit upper bound for the overall L2-gain of the output synchronization error with respect to disturbance. We demonstrate our developments by a simulation example.

In this paper, we analyze a Linear Quadratic (LQ) control problem in terms of the average cost and the structure of the value function. We develop a completely model-free reinforcement learning algorithm to solve the LQ problem. Our algorithm is an off-policy routine where each policy is greedy with respect to all previous value functions. We prove that the algorithm produces stable policies given that the estimation errors remain small. Empirically, our algorithm outperforms the classical Q and off-policy learning routines.