Optimal sensor design and control of piezoelectric laminate beams

In this paper, a new smart structure optimal design strategy is introduced and applied to robust vibration control of a piezoelectric laminate beam. The optimization of the smart material layout and the control law are performed simultaneously to extract maximum performance from the system. A homogenization approach is used to allocate sensor material, while a linear quadratic regulator (LQR) is used for the control law. The method is applied to a pinned-pinned beam where two cost functions are considered, both focusing on increasing the stability margin of the closed-loop system. The first is based on the observability gramian and the second on the control weighting parameter of the LQR cost function. Both cost functions yield optimal sensor distributions that improve the closed-loop performance as compared with uniform density distributions. In addition to investigating the effect of the cost function on the design, two different sensor design domains are considered. The first consists of five isolated patches of sensor material (a discrete sensor domain), while the second assumes the five patches cover the entire beam, but are electrically isolated (segmented distributed sensor domain). In all cases considered, the cost function based on the LQR control weighting parameter generates smoother sensor distributions that are always positively polled, an important fabrication consideration. The segmented distributed sensor approach yields the overall best performance.