Statistical Control Paradigms for Structural Vibration Supression

An optimal k-cost-cumulant (kCC) control problem is formulated, in which the objective is minimization of a finite linear combination of the first k cost cumulants of a finite-horizon integral quadratic cost associated with a linear stochastic system, when the controller measures the states. This problem not only defines a very general linear-quadratic Gaussian problem class, but also may be seen as an approximation in some sense to the theory of risk sensitive control. The solution is obtained by a more direct dynamic programming approach to the kCC initial-cost problem. Moreover, the research continues with a development of cost cumulants per unit time in infinite horizon control for the state-feedback kCC problem. A constant controller is obtained by using a Lagrange multiplier technique. The performance and stability properties of kCC controllers are discussed at length. A theory of output feedback for the linear-quadratic kCC problem class is also developed. Under linearity and Gaussian assumptions, it is reasonable to conclude that Kalman-state estimates contain all statistical information. Henceforth, the finite-horizon kCC control problem with Kalman state-estimate feedback laws is formulated. Solutions to the output-feedback kCC control problem having both standard and generalized finite-horizon integral quadratic costs are then obtained by adapting a dynamic programming technique. Furthermore, constant controllers measuring Kalman state-estimates for the output-feedback kCC problem in infinite horizon control are also derived by a Lagrange multiplier theory. Some kCC control strategies having perfect state measurements for linear stochastic differential games and the worst-case model scenarios are subsequently proposed. Performance sensitivity with respect to system parameter variations of optimal kCC systems over finite horizon is examined as well. The research finally extends the findings to a class of uncertain linear systems and problems of tracking, servo, and model-following. The use of these statistical control paradigms in the American Society of Civil Engineers first, second and third generation benchmarks for response control of buildings and cable-stayed bridges is successfully illustrated. Simulation results indicate that kCC controller designs offer both robust performance and stability improvements for structural vibration suppression.