Optimal Control of a Flexible Beam with Multiple Time Delays

Abstract

In this paper, active control of a flexible cantilever beam with multiple time delays is studied, where a controller is designed using a discrete optimal control method and piezoelectric patches are used as actuators. Firstly, the state equation of controlled modes of the system with multiple time delays is obtained using modal orthogonality. Then, the state equation is discretized and transformed into a standard form without any explicit time delay by a particular augmenting for state variables. A continuous performance index is also discretized into a standard discrete form that is a function of the augmented states. So, the discrete controller may be designed based on the augmented system. In every step of computation for the controller, it contains not only the current step of state feedback but also a linear combination of some former steps of controls. Simulation results indicate that the system may suffer from instability at a small time delay if the time delay is not treated. It may be well compensated by the proposed controller and good control effectiveness may be obtained simultaneously. In addition, the proposed controller may possibly obtain better control effectiveness than the controller with no time delay.