Global Convergence of Policy Gradient Primal-dual Methods for Risk-constrained LQRs

While the techniques in optimal control theory are often model-based, the policy optimization (PO) approach can directly optimize the performance metric of interest without explicit dynamical models, and is an essential approach for reinforcement learning problems. However, it usually leads to a non-convex optimization problem in most cases, where there is little theoretical understanding on its performance. In this paper, we focus on the risk-constrained Linear Quadratic Regulator (LQR) problem with noisy input via the PO approach, which results in a challenging non-convex problem. To this end, we first build on our earlier result that the optimal policy has an affine structure to show that the associated Lagrangian function is locally gradient dominated with respect to the policy, based on which we establish strong duality. Then, we design policy gradient primal-dual methods with global convergence guarantees to find an optimal policy-multiplier pair in both model-based and sample-based settings. Finally, we use samples of system trajectories in simulations to validate our policy gradient primal-dual methods.

Infinite-horizon Risk-constrained Linear Quadratic Regulator with Average Cost

The behaviour of a stochastic dynamical system may be largely influenced by those low-probability, yet extreme events. To address such occurrences, this paper proposes an infinite-horizon risk-constrained Linear Quadratic Regulator (LQR) framework with time-average cost. In addition to the standard LQR objective, the average one-stage predictive variance of the state penalty is constrained to lie within a user-specified level. By leveraging the duality, its optimal solution is first shown to be stationary and affine in the state, i.e., $u(x,lambda^*) = -K(lambda^*)x + l(lambda^*)$, where $lambda^*$ is an optimal multiplier, used to address the risk constraint. Then, we establish the stability of the resulting closed-loop system. Furthermore, we propose a primal-dual method with sublinear convergence rate to find an optimal policy $u(x,lambda^*)$. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed framework and the primal-dual method.

Primal-dual Learning for the Model-free Risk-constrained Linear Quadratic Regulator

Risk-aware control, though with promise to tackle unexpected events, requires a known exact dynamical model. In this work, we propose a model-free framework to learn a risk-aware controller with a focus on the linear system. We formulate it as a discrete-time infinite-horizon LQR problem with a state predictive variance constraint. To solve it, we parameterize the policy with a feedback gain pair and leverage primal-dual methods to optimize it by solely using data. We first study the optimization landscape of the Lagrangian function and establish the strong duality in spite of its non-convex nature. Alongside, we find that the Lagrangian function enjoys an important local gradient dominance property, which is then exploited to develop a convergent random search algorithm to learn the dual function. Furthermore, we propose a primal-dual algorithm with global convergence to learn the optimal policy-multiplier pair. Finally, we validate our results via simulations.

Data-driven Distributionally Robust Optimal Stochastic Control Using the Wasserstein Metric

Optimal control of a stochastic dynamical system usually requires a good dynamical model with probability distributions, which is difficult to obtain due to limited measurements and/or complicated dynamics. To solve it, this work proposes a data-driven distributionally robust control framework with the Wasserstein metric via a constrained two-player zero-sum Markov game, where the adversarial player selects the probability distribution from a Wasserstein ball centered at an empirical distribution. Then, the game is approached by its penalized version, an optimal stabilizing solution of which is derived explicitly in a linear structure under the Riccati-type iterations. Moreover, we design a model-free Q-learning algorithm with global convergence to learn the optimal controller. Finally, we verify the effectiveness of the proposed learning algorithm and demonstrate its robustness to the probability distribution errors via numerical examples.

Suspension Regulation of Medium-low-speed Maglev Trains via Deep Reinforcement Learning

The suspension regulation is critical to the operation of medium-low-speed maglev trains (mlsMTs). Due to uncertain environment, strong disturbances and high nonlinearity of the system dynamics, this problem cannot be well solved by most of the model-based controllers. In this paper, we propose a model-free controller by reformulating it as a continuous-state, continuous-action Markov decision process (MDP) with unknown transition probabilities. With the deterministic policy gradient and neural network approximation, we design reinforcement learning (RL) algorithms to solve the MDP and obtain a state-feedback controller by using sampled data from the suspension system. To further improve its performance, we adopt a double Q-learning scheme for learning the regulation controller. We illustrate that the proposed controllers outperform the existing PID controller with a real dataset from the mlsMT in Changsha, China and is even comparable to model-based controllers, which assume that the complete information of the model is known, via simulations.