Game theory is the study of tactical interactions involving conflicts and cooperations among multiple decision makers called players with applications in diverse disciplines such as economics, biology, management, communication networks, electric power systems and control. This dissertation studies a statistical differential game problem where finite N players optimize their system performance by shaping the distribution of their cost function through cost cumulants. This research integrates game theory with statistical optimal control theory and considers a statistical Nash non-cooperative nonzero-sum game for a nonlinear dynamic system with nonquadratic cost functions. The objective of the statistical Nash game is to find the equilibrium solution where no player has the incentive to deviate once other players maintain their equilibrium strategy. The necessary condition for the existence of the Nash equilibrium solution is given for the m-th cumulant cost optimization using the Hamilton-Jacobi-Bellman (HJB) equations. In addition, the sufficient condition which is the verification theorem for the existence of Nash equilibrium solution is given for the m-th cumulant cost optimization using the Hamilton-Jacobi-Bellman (HJB) equations. However, solving the HJB equations even for relatively low dimensional game problem is not trivial, we propose to use neural network approximate method to find the solution of the HJB partial differential equations for the statistical game problem. Convergence proof of the neural network approximate method solution to exact solution is given. In addition, numerical examples are provided for the statistical game to demonstrate the applicability of the proposed theoretical developments. / Electrical and Computer Engineering

The space-based solar power system is an alternative to the ground-based solar power system because of its round-the-clock availability. For the space-based solar power transmission, the accurate pointing of a laser from space to ground poses a challenging control task. A gimbaled laser target system, which is used for pointing laser to a target, is a test bench for such a transmission system. The objective of this research is to determine the optimal controller for the gimbaled laser target system in terms of pointing error and error variation. In order to achieve the objective, we modeled the gimbaled laser target system, simulated the model with the controllers, and tested them on the test bench. In this thesis, we developed a mathematical model of a two-axis gimbaled laser target system. The model consists of a pitch-yaw gimbal for the dynamic laser motion, brushless dc motors for actuating the gimbal, and an image-based position sensor. We used a Proportional-Integral-Derivative (PID) controller as the basis for the performance comparison since it is the most commonly used control method in the industry. Then we compared the PID controller with two statistical control methods - Linear Quadratic Gaussian (LQG), and Minimal Cost Variance (MCV) optimal controllers. We evaluated the pointing performance of the controllers by measuring the mean and the standard deviation of the pointing error. The simulation results indicated that the statistical controllers perform better than the PID controller under Gaussian disturbances. Between the statistical controllers, the LQG method had the smaller pointing error, while the MCV method had the smaller standard deviation of the pointing error. We then implemented the PID, LQG, and MCV controllers in an off-the-shelf dSPACE digital signal processing controller board, and tested the controllers on the test bench in a real time environment. The experimental results showed that the LQG method decreased the mean pointing error by 46.28% compared to the PID method. The LQG method reduced the standard deviation of pointing error by 47.85% compared to the PID method. The MCV method reduced the standard deviation of the pointing error by 53.09% compared to the LQG method. Both the simulation and experimental results showed that the MCV controller improved the pointing error variation performance over the LQG controller significantly, while slightly degrading the pointing error performance of the gimbaled laser target system. Experimental results indicate that the statistical controllers will provide a design parameter either to improve the mean pointing error or the standard deviation of the pointing error for the gimbaled laser target system. Subsequently, we believe that the statistical controllers will improve the space-based solar power transmission efficiency. / Electrical and Computer Engineering