The Duffing oscillator is a useful model for the non-linear behavior of structural systems. This paper studies the applications of two control strategies to Duffing oscillators; namely, an optimal polynomial control and robust sliding mode control. An advantage of the polynomial controller investigated is that, for a given appropriate weighting matrices, the gain matrices for different orders of the controller can be computed easily by solving Riccati and Lyapunov equations. It is demonstrated through numerical simulation results that the stability region of the softening Duffing system can be expanded rapidly by the optimal polynomial controller and the entire state space of the closed-loop system becomes asymptotically stable when the gain matrix reaches a certain level. On the other hand, the closed-loop softening system is always asymptotically stable in the entire state space for the robust sliding-mode controller. The performances of both controllers, in terms of the system response reduction and the required control effort, have been studied through numerical simulations. Simulation results indicate that the performance of the optimal polynomial controller presented in this paper in reducing the response of the Duffing systems is comparable to a polynomial controller proposed recently in the literature. In comparison with polynomial controllers, the performance of robust sliding-mode control is quite remarkable.
International Journal of Non-Linear Mechanics 33(5), pp.829-841