A Study On Structural Stochastic Optimal Control Based On Explicit Time-Domain Method

Natural hazards,e.g.,earthquakes and typhoons,will cause severe vibration of structures,and even lead to structural damage and casualties.Active control technique is an effective means for mitigating structural responses,of which the critical issue is the reasonable design of the control system according to the characteristics of hazards and structures.It is noted that the external loadings of the structure and the disturbance,such as measurement and control noises,are essentially random,and the external loadings,e.g.,earthquake and strong wind,are generally non-stationary.Therefore,it is necessary to carry out research on stochastic optimal control of structures subject to non-stationary random excitations.The conventional deterministic optimal control methods,which need to solve the optimization problem with the constraint imposed by the equation of motion of structures,are difficult to be implemented for a dimension-reduced structural control.On the other hand,the classical stochastic optimal control methods are developed based on the Ito stochastic differential equation with white-noise and filtered-white-noise excitation assumption,and therefore inapplicable to structures subject to non-stationary non-white-noise excitations,e.g.,earthquake and strong wind.In view of this,this paper will first conduct research on the optimal design of control law and the optimal selection of weighting parameters involved in the control law by employing time-domain explicit formulation of dynamic responses,and finally,achieve the stochastic optimal control of structures subject to non-stationary non-white-noise excitations.The main work in this paper is described as follows:(1)The literature review on the problem of structural optimal control is conducted.Brief introduction for the development of structural control is first carried out.Then,the deterministic and stochastic optimal control theories are emphasized,and their advantages and disadvantages are summarized.(2)The research on the structural optimal control method is conducted.By use of the explicit formulation of structural responses of controlled structures and its dimension-reduced capability,a new explicit optimal control(EOC)method of structures is developed.On the one hand,the constrained optimization model of linear quadratic regulator(LQR)problem is transformed into an unconstrained optimization problem,and the optimal control law can be derived analytically.On the other hand,a dimension-reduced optimal control scheme is further developed for the control of critical responses of complex structures.(3)The research on the random vibration analysis of controlled structures is conducted.On the basis of the proposed EOC method,considering the influences of randomness existing in external loadings,measurement noises,and control noise,the explicit expressions of control forces and structural responses for the controlled structure are derived depending on the external loadings,measurement noises,and control noises.The derived explicit expressions are then incorporated into the moment operation and the Monte Carlo simulation(MCS)for the analysis of standard deviations and dynamic reliabilities of controlled structures.(4)The research on stochastic optimal design of weighting parameters involved in the control law of the controlled structures is carried out.Based on the explicit time-domain method(ETDM)for random vibration analyses of controlled structures,the explicit expressions for the sensitivities of control forces and structural responses with respect to the weighting parameters are derived.Then the sensitivities of the standard deviations and dynamic reliabilities of controlled responses with respect to the weighting parameters are further achieved.On this basis,combined with the gradient-based optimization algorithm,the stochastic optimal design of weighting parameters is conducted for the minimization of standard deviations of controlled responses and the minimization of failure probabilities of the controlled structure,respectively.Results of numerical examples show that the proposed stochastic optimal control method based on ETDM can achieve ideal efficiency and accuracy in many scenarios,including the optimal design of control law,the random vibration analysis of controlled structures,and the optimal selection of weighting parameters involved in the control law,and has good performance in the stochastic optimal control of structures.