This paper deals with the use of matrix inequalities for the aim of multi-modal piezoelectric shunt damping. The paper shows that the shunt impedance can be seen as a controller in a state space model of the electro-mechanical system; this makes it possible to use the mentioned approach to find the layout of the impedance for different kinds of control problems. The particular focus is on passive multi-mode vibration control with the aim of finding the optimal shunt impedance among the passive and realizable candidates. The proposed method overcomes most of the problems related to the development of the optimal shunt electrical network, which arise when using the most common shunt design strategies for multi-mode control. The results were validated experimentally and compared to well-established methods for multi-mode shunt damping. The proposed method proved to be effective, and the results demonstrate the capability of the matrix inequality approach to provide attenuation levels that are usually higher than those from the reference methods.
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