Ronald W. Diersing

Satellite attitude control using statistical game theory

We consider the application of multi-objective statistical game theory to a remote sensing satellite attitude control. Statistical game theory is a generalization of mixed H₂/H_infin control, where we have two objective functions and we optimize the higher order cumulants of these objective functions. We use previously developed satellite attitude model with thrusters, gravity torquers and a reaction wheel cluster. Then we control the satellite attitude using the statistical game (Minimal Cost Variance/H_infin) control, H_infin control, and mixed H₂/H_infin control. Throughout the simulations, statistical game control has an extra degree of freedom to improve the performance and reduce the overshoot and undershoot compared to either H_infin control and H₂/H_infin control. The simulations show that the performance of Minimal Cost Variance/H_infin is 8.8% and 5.4% faster than H₂/H_infin and H_infin control, respectively. Moreover, the control actions of MCV/H_infin is reduced by 67% compared to H_infin control and 55% compared to H₂/H_infin control. So, we achieve both performance improvement while saving the control energy. In the case of the stability margin, MCV/H_infin control has the highest stability margin, H_infin control has the lowest, and H₂/H_infin control an intermediate value between those two.

Statistical control of control-affine nonlinear systems with nonquadratic cost functions: HJB and verification theorems

In statistical control, the cost function is viewed as a random variable and one optimizes the distribution of the cost function through the cost cumulants. We consider a statistical control problem for a control-affine nonlinear system with a nonquadratic cost function. Using the Dynkin formula, the Hamilton-Jacobi-Bellman equation for the nth cost moment case is derived as a necessary condition for optimality and corresponding sufficient conditions are also derived. Utilizing the nth moment results, the higher order cost cumulant Hamilton-Jacobi-Bellman equations are derived. In particular, we derive HJB equations for the second, third, and fourth cost cumulants. Even though moments and cumulants are similar mathematically, in control engineering higher order cumulant control shows a greater promise in contrast to cost moment control. We present the solution for a control-affine nonlinear system using the derived Hamilton-Jacobi-Bellman equation, which we solve numerically using a neural network method.

Cumulant Control Systems: The Cost-Variance, Discrete-Time Case

The expected value of a random cost may be viewed either as its first moment or as its first cumulant. Recently, the Kalman control gain formulas have been generalized to finite linear combinations of cost cumulants, when the systems are described in continuous time. This paper initiates the investigation of cost cumulant control for discrete-time systems. The cost variance is minimized, subject to a cost mean constraint. A new version of Bellman’s optimal cost recursion equation is obtained and solved for the case of full-state measurement. Application is made to the First Generation Structural Benchmark for seismically excited buildings.

Nash and Minimax Bi-Cumulant Games

Cumulants have been gaining interest in control lately. This paper applies cumulants to the area of game theory and gives sufficient conditions for nonzero and zero sum games to a class of nonlinear systems. With a linear system and quadratic costs, Nash and minimax equilibrium solutions are found. The games consider the finite time horizon, state feedback problem. Special cases of these methods reduce to the H2/Hinfin and Hinfin control problems, so that a generalization is achieved

Weighted least-squares, cost density-shaping, stochastic optimal control: A step towards total probabilistic control design

To date, both moment-based and cumulant-based stochastic control schemes have been proposed to constrain statistics of the random cost functional in the stochastic optimal control formalism. However, existing methodologies do not enable the designer to deliberately shape the probability density function of the random cost according to a pre-specified target density characterized by a finite number of cost statistics. Since the mean, variance, skew, and kurtosis of a variate are strongly associated with the appearance of its density function, a cumulant-based control paradigm that can steer four or more cost cumulants towards nominal target values would be ideal for cost density-shaping objectives. Bearing this idea in mind, we propose a novel weighted least-squares optimization problem of minimizing a weighted sum of squared differences between initial cost cumulants and target initial cost cumulants, for arbitrarily-many terms. The problem is solved using dynamic programming techniques adapted for the cost cumulant-generating equations of the Linear Quadratic Gaussian (LQG) framework. The Minimum Weighted Least-Squares, Cost Density-Shaping (MWLS-CDS) optimal controller results and is applied in a building protection problem. It is shown that MWLS-CDS controls can achieve target cost cumulants resultant from a family of 3CC controls to within a 0.5% margin of normalized error.

Infinite-horizon, Multiple-Cumulant Cost Density-Shaping for stochastic optimal control

Recently for the LQG framework, cost density-shaping control paradigms for stochastic optimizations have been proposed. These new control methods have enabled the shape of a target cost density to be transformed into a linear control law. However, the theory developed so far pertains exclusively to the finite-horizon. The purpose of this work is to develop a Multiple-Cumulant Cost Density-Shaping (MCCDS) control solution as the terminal time approaches infinity. The first-generation benchmark for seismically-excited buildings will be used to validate the infinite-horizon MCCDS control law.

Multiplayer nash solution for noncooperative cost density-shaping games

Finite-horizon, LQ cost density-shaping has been achieved through several different control paradigms that are based on the Multiple-Cumulant Cost Density-Shaping (MCCDS) theory. With these cost density-shaping control methods, the shape of a target cost density can be transformed into a linear control law. However, the existing MCCDS theory does not permit control design that accounts for competing objectives among multiple noncooperative agents. The aim of this work is to derive the Nash equilibrium solution to an N-Player MCCDS game posed for the LQG framework. Simulation results are provided to support the new theory.

Discounted cost infinite time horizon cumulant control

Cumulants are gaining in popularity for use in stochastic control and game theory. They also have been effective in application to building and vibration control problems. Much of the work has been done for the finite time horizon case. In this paper, cost cumulants will be used on a discounted cost function. The control will be concerned with the first two cumulants, the mean and variance. The approach will initially be done for a nonlinear system with non-quadratic costs and sufficient conditions are determined. With the sufficient conditions in place, attention will be turned to the linear quadratic special case. A coupled Riccati equation will be seen to give an optimal cumulant control law.

Discrete-time, Bi-cumulant minimax and Nash games

Continuous time, cumulant games have gathered interest, recently, but not much has been done for the discrete time case. In this paper, this problem will be addressed. Discrete time, two player, Nash and minimax cumulant games will be formulated and developed for a nonlinear system with non-quadratic costs. A recursion equation for the determining the equilibrium solutions is derived. For the linear quadratic case, equilibrium solutions are determined. Furthermore, through the use of cumulants, generalizations of H2/Hinfin and Hinfin control are shown.

Output Feedback Multiobjective Cumulant Control with Structural Applications

Recently, a cumulant generalization of H2/Hinfin control has been given for the state feedback problem. This paper extends those results to the output feedback case. The Nash game approach to the H2/Hinfin problem is used. Sufficient conditions for two problems are determined. In the first problem the one player, the control, has only partial state information, while the other player, the disturbance, has full state information. In the other problem both players only have information based upon estimates of the state. Coupled Riccati equations for both cases are given, along with equilibrium solutions. The results are also applied to the first generation structural benchmark for buildings under seismic excitation.