Optimal Vibration Control of a Plate in Presence of Colored Noise Disturbances

Abstract

In this paper a modified discrete-time linear quadratic Gaussian (LQG) controller is presented in which the optimal control law is computed based on the dynamic programming (DP) algorithm. In derivation of the governing equations, it is assumed that the noise is colored and it has a nonzero expectation value, whereas in standard form of LQG equations, the system is perturbed by white noise. Similar to LQG controller, a quadratic cost (energy) function is defined and is minimized with optimal controlling input. After presenting the constitutive equations, the recursive algorithm is applied to a smart plate. In this structure fifteen pairs of PZT actuators, bonded to a flexible plate, are activated by optimal output of controller for vibration control of the plate. A three dimensional finite element method (FEM) analysis is used to model the dynamic behavior of the system. After harmonic analysis of the structure, the displacement of the mid-point of the plate is determined and the results of modeling are exported to MATLAB. The transfer function and state space representation of the system are derived and used for control objective. The goal of the optimal controller is to minimize the vibration energy of the plate and limit the amplitude of displacement in the middle of the structure. First, the derivation of DP-based controller equations is discussed and then it is applied to the system to control the vibration of the plate in the presence of the colored noise with known properties. The results of the modified and standard LQG are compared and discussed in the presence of colored noise. The modified LQG controller shows better performance in comparison with the standard one, which makes it suitable for similar applications in the presence of colored noise with moderate sampling time.