Stochastic Optimal Control of Structural Systems

The stochastic optimal control is an important research subject in structural engineering. Recently, a stochastic optimal nonlinear control method has been proposed based on the stochastic dynamical programming principle and stochastic averaging method. The active and semi-active stochastic optimal control methods have been further developed for structural systems. The control saturation or bound, partial state observation, etc. have been taken into account by the stochastic optimal control. The present paper surveys these research developments. INTRODUCTION The strong nonlinear stochastic vibration of engineering structures such as tall buildings and large bridges is frequently caused by severe loadings such as earthquake and storm. The stochastic optimal control is an important research subject in structural engineering [1]. The mathematics theory of stochastic optimal control has been developed [24]. However, the linear-quadratic-Gauss (LQG) control method has mostly used for structural vibration control [5]. Recently, some stochastic optimal nonlinear control methods [6, 7] have been proposed, in particular, the method based on the stochastic dynamical programming principle and stochastic averaging method [8-17]. This control method can achieve better control effectiveness and efficiency than the LQG control method. These research developments are surveyed as follows: (1) stochastic optimal control law; (2) optimal active control; (3) optimal semi-active control; (4) developments of stochastic optimal control method. STOCHASTIC OPTIMAL CONTROL LAW [8, 9] A controlled and stochastically excited nonlinear structural system with multi-degree-of-freedom can be expressed as BU W X X F X X X X X C X M + = ! ! + + ) ( ) , ( ) ( ) , ( t V s & & & & & (1) where X is the n-dimensional structural displacement vector, Vs(X) is the structural potential, W(t) is the stochastic excitation vector, U is the control force vector. Rewrite Eq. (1) in the following form of quasi Hamiltonian equation: