Robust design and control of piezoelectric laminate beams using a simultaneous optimization method

In this paper, robust control of piezoelectric laminate beams by simultaneously optimizing the smart material distribution and the closed-loop control system is implemented. Through topological optimization of smart material, using a homogenization approach and a linear quadratic regulator (LQR), a new type of sensor with the ability to increase the stability margin is obtained. The method is applied to a pinned-pinned beam where two cost functions are considered, both focus on increasing the stability margin of the closed-loop system. The first one is based on the observability gramian and the second one 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 to uniform density distributions. Although not explicitly considered in the cost function design, the sensor distribution based on the LQR control weighting parameter was consistently smoother than those based on the observability gramian. This is an important practical consideration for sensor fabrication.