Mixed H2/H∞ Control for Nonlinear Closed-Loop Stackelberg Games With Application to Power Systems

This paper studies the mixed $H_{2}/H_{\infty} $ control problem for nonlinear closed-loop Stackelberg games via adaptive dynamic programming (ADP) technology. Firstly, by constructing a cost function with the Lagrange multiplier, the hierarchical Stackelberg game problem is transformed into a coupled Hamilton-Jacobi-Isaacs (HJI) equations problem. In the second place, the critic-actor neural networks (NNs) framework is established to approximate the cost functions and control strategies of Stackelberg game. The corresponding algorithm flow is given. This is a novel idea that NNs are used to approximate the control strategies in Stackelberg game that cannot be given a specific form. Finally, the algorithm is applied to the load frequency control (LFC) problem of single-area power system. The optimal neural network weights are obtained by the designed ADP algorithm. And the comparison results of the three control schemes show that the mixed $H_{2}/H_{\infty} $ based on Stackelberg game can not only achieve better frequency response, but also have better performance indicators for leader. Note to Practitioners— $H_{\infty} $ control studies the anti-interference ability of the system, and the designed controller should attenuate the suppression coefficient of adjusting error and external disturbance to a given minimum level. $H_{2}$ control is optimal control, and the controller is designed to minimize the cost function of system state and system input. In general, in $H_{\infty} $ or $H_{2}$ control, only a single control performance can be achieved. In the control system design, if the control system satisfies both $H_{\infty} $ and $H_{2}$ performance, the overall performance of the system will be greatly improved. Therefore, in engineering practice, in order to achieve satisfactory system performance, the mixed $H_{2}/H_{\infty} $ control is more attractive, because it integrates the advantages of $H_{\infty} $ control and $H_{2}$ control. On the other hand, in a real game, Stackelberg game is the best choice if two players are not on the same level. This paper takes Stackelberg game as the main research object. Stackelberg game is a kind of game problem in which the participants have different decision-making status, and there is one or more decision makers higher than other decision makers. As a traditional control strategy, PID control and SMC control are not uncommon in the research of LFC. With the development of control theory and application, state space and optimal control strategy are also applied to the design of load frequency control system. Intelligent control is the current development trend of LFC research.