Stochastic Averaging-Based Direct Method for Response Control of Nonlinear Vibrating System

Abstract

Superficially, the optimal control problem of a general system may be handled by the stochastic dynamic programming or stochastic maximum principles. The practical application of these two principles, however, is confined to the low-dimensional or weakly nonlinear system due to the difficulty of solving the dynamic programming equation and forward–backward stochastic differential equations with high dimension or strong nonlinearity. To attenuate the solving difficulty, a nonlinear stochastic optimal control method in the Hamiltonian framework has been conceived by taking the low-dimensional averaged system as the controlled system. This method reduces the difficulty of solving equations to some extent, while not including the displacement feedback mechanism in the control force. In this manuscript, a new procedure is proposed to establish the control strategy for vibrating systems with relatively high dimension and strong nonlinearity. By directly separating the control force to conservative and dissipative components and combining the stochastic averaging technique, the functional extreme value problem associated with the original optimal control problem is converted to an extremum value problem of multivariable function. Numerical results on a single-degree-of-freedom Duffing oscillator, a single-degree-of-freedom Van der Pol oscillator, and a two-degree-of-freedom nonlinear system demonstrate the high control effectiveness and efficiency by being comparing with the linear quadratic Gaussian control and nonlinear stochastic optimal control. This direct method not only avoids the difficulty of solving complex differential equations, but also includes the whole information of system displacement and velocity.