Optimal feedback design for nonlinear stochastic systems using the pseudospectral method

• Propose an approach to obtain optimal feedback control of nonlinear stochastic system.
• Discretize the HJB equation using the pseudospectral method for the first time.
• Solve the discretized problem using the successive approximation method.
• Verify the proposed approach with closed-form solution of linear system.
• Demonstrate the proposed approach with Van der Pol, Duffing, Bouc–Wen systems.

The development of smart structures involves the use of various types of controllable damping devices by integrating them into structural systems for energy dissipation. An optimal feedback control law that can consider the nonlinearity embedded in the systems and the random characteristics of the external inputs is desirable to control such dissipation mechanism. This paper proposes an innovative numerical technique to obtain the optimal feedback control strategy of nonlinear stochastic systems. The systems under investigation are modeled as mechanical oscillators and a damping device. The numerical approach proposed to obtain the optimal control strategy involves solving a nonlinear partial differential equation—the Hamilton–Jacobi–Bellman equation. Because civil engineering structural systems may exhibit nonlinear hysteretic behavior under extreme loading conditions, the application of the obtained control strategy could provide an optimal feedback control law to reduce the system response under random excitations (such as earthquakes, wind load and sea waves). Several numerical examples are presented to verify optimality and demonstrate the efficacy of the proposed optimal control solution. First, a linear oscillator is used to verify that the obtained solution is indeed the optimal solution by comparing it to the closed-form optimal solution. Then the proposed method is applied to several nonlinear systems. In each nonlinear example, control optimality is demonstrated by comparing the system responses and control costs obtained using the proposed optimal controls with those obtained using corresponding linearized optimal controls.