Probabilistic criteria of structural stochastic optimal controls

A family of probabilistic criteria for stochastic optimal controls is developed in the context of physical stochastic optimal control scheme of structures. A physical form of the control policy is firstly conducted in conjunction with the classical optimal control theory, specifically, an LQR control. In order to obtain the optimal weighting matrices included in the control gain, two classes of probabilistic criteria, in accordance with the objective structural performance, are then proposed, including single-objective criteria, of which the statistics and tail details of probability density of equivalent extreme-value vectors of interest are involved, and multi-objective criteria, of which the ensemble-expectation and exceedance probability of equivalent extreme-value processes in the sense of performance and energy trade-off are involved. A linear single-degree-of-freedom structural system subjected to random ground motion is investigated for illustrative purpose. The results indicate that the effectiveness of response control hinges on the physical meanings of the probabilistic criteria. The implementation of control criteria related to the exceedance probability with multiple constraints results in a more economic and more accurate control effectiveness than that of control criteria related to statistics with single constraint. The exceedance probability criterion in energy trade-off sense accommodates system performance to a better trade-off between response reductions and control requirements, which is also included in the comparative study against other control criteria currently in use. The former, meanwhile, provides accurate reliabilities of system quantities simultaneously, while other control criteria fail to do so. It is thus the primary criterion of structural performance controls. Following that, a randomly base-excited eight-storey shear frame controlled by active tendons is investigated as a numerical example. The numerical results reveal that using the advocated probabilistic criterion, the structural stochastic optimal control operates efficiently and achieves a desirable objective performance.