On asymptotic eigenstructure design for LQR with amplitude constraint

In many circumstances, especially for structure systems, it is important to bound the oscillation amplitude. For a system subject to external disturbances, when the oscillation amplitude is small and well below the tolerance limit of the system, an ordinary controller with small control effort might be adequate for the purpose of stabilization or adding additional damping. However, in some extreme situations, the oscillation amplitude may approach the limit of system tolerance. Then, more powerful control effort must be applied to the system to guarantee that the oscillation amplitude is below a prescribed limit. There are many works on vibration control with amplitude constraints against external disturbances. Chuang and Wang (1992) proposed a two-stage LQR control scheme for the control of oscillation amplitude and the scheme has been further studied for more general MIMO cases. When the amplitude is below a prescribed threshold, a low-gain LQR is used. When the amplitude is crossing the threshold, a high-gain LQR is activated to make the oscillation amplitude below system tolerance limit. The asymptotic eigenstructure of the closed-loop system determines uniquely linear quadratic optimal control, and then can be used to design the two-stage LQR controller. In this paper, an optimization procedure is proposed for the design of the asymptotic eigenstructure of the closed-loop system such that the two-stage LQR controller is more efficient and less conservative