The H ∞ control method is an effective approach for attenuating the effect of disturbances on practical systems, but it is difficult to obtain the H ∞ controller due to the nonlinear Hamilton—Jacobi—Isaacs equation, even for linear systems. This study deals with the design of an H ∞ controller for linear discrete-time systems. To solve the related game algebraic Riccati equation (GARE), a novel model-free minimax Q -learning method is developed, on the basis of an offline policy iteration algorithm, which is shown to be Newton’s method for solving the GARE. The proposed minimax Q -learning method, which employs off-policy reinforcement learning, learns the optimal control policies for the controller and the disturbance online, using only the state samples generated by the implemented behavior policies. Different from existing Q -learning methods, a novel gradient-based policy improvement scheme is proposed. We prove that the minimax Q -learning method converges to the saddle solution under initially admissible control policies and an appropriate positive learning rate, provided that certain persistence of excitation (PE) conditions are satisfied. In addition, the PE conditions can be easily met by choosing appropriate behavior policies containing certain excitation noises, without causing any excitation noise bias. In the simulation study, we apply the proposed minimax Q -learning method to design an H ∞ load-frequency controller for an electrical power system generator that suffers from load disturbance, and the simulation results indicate that the obtained H ∞ load-frequency controller has good disturbance rejection performance.

In this paper, a novel synchronous Q-learning method is proposed for solving discrete-time linear quadratic regulator (LQR) problems. To begin with, the Bellman equation corresponding to the optimal Q-function is reformulated into a consistency equation on the parameters of the optimal Q-function and the optimal controller. Then an actor-critic structure is introduced to learn the optimal Q-function and the optimal controller online in real time by using the state samples generated by the behavior policy. Particularly, the proposed synchronous Q-learning scheme simultaneously updates the Q-function approximation and the optimal controller approximation, rather than iterating between policy evaluation and policy improvement. The proposed control scheme is proved to be uniformly ultimately bounded (UUB) under appropriate learning rates, provided that certain persistence of excitation (PE) conditions are satisfied. Besides, the PE conditions can be easily met by injecting appropriate exploration noise into the behavior policy without causing any excitation noise bias. Finally, one simulation example is provided to verify the effectiveness of the proposed synchronous Q-learning method.

This paper addresses a particular pursuit-evasion game, called as "fishing game" where a faster evader attempts to pass the gap between two pursuers. We are concerned with the conditions under which the evader or pursuers can win the game. This is a game of kind in which an essential aspect, barrier, separates the state space into disjoint parts associated with each player's winning region. We present a method of explicit policy to construct the barrier. This method divides the fishing game into two subgames related to the included angle and the relative distances between the evader and the pursuers, respectively, and then analyzes the possibility of capture or escape for each subgame to ascertain the analytical forms of the barrier. Furthermore, we fuse the games of kind and degree by solving the optimal control strategies in the minimum time for each player when the initial state lies in their winning regions. Along with the optimal strategies, the trajectories of the players are delineated and the upper bounds of their winning times are also derived.

Generally, the sample data of Multiple Attributes Decision Making (MADM) problems is incomplete because of variety of factors such as noise in data, compactness of representation, prediction capability and randomness of experiment. Rough set theory is a useful mathematical tool for this incomplete decision systems, while the fuzziness of relation-based classification and uncertainty of attribute reduction always exist in traditional extended rough sets model. In order to classify the incomplete decision systems effectively, a new refined classification method with tolerance relation-based rough sets was presented in this paper. Considering the randomness of missing value, this method used attribute importance to replace attribute reduction to establish refined classification rules directly. Not only it can reduce the computational complexity, but also can increase classification accuracy. From the analysis and comparison of examples about classification problems of air weapon targets, the effectiveness and stability of this method for incomplete decision systems were verified.