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.
Engineers are often required to design mechanical structures to meet specific loading conditions. Topological optimization automates the process of finding an optimal structural design, allowing for size, shape and topological variations. For a given set of boundary conditions and design specifications (constraints), a structure, optimal in terms of a formulated cost function, is computed. As the cost function, static properties such as the mean compliance, as well as dynamic properties such as the eigenfrequencies of the structure can be chosen. In this paper, the optimization considers not only the placement of conventional material, but also, simultaneously, the placement of smart PZT material. In the formulation of the topology optimization problem, a microstructure consisting of smart as well as conventional material is used. Instances of the microstructure occur in a rectangular grid and cover the design domain. Since the microstructure is defined parametrically, the density of each material can be controlled independently at each point. This enables one to formulate the problem of finding an optimal material distribution as a parameter optimization problem. A homogenization approach is used to find and use effective material properties for the limiting case of an infinitely small microstructure. Several numerical examples demonstrate the use of this method.
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