Piezoelectric transducers are commonly used as strain actuators in the control of mechanical vibration. One control strategy, termed piezoelectric shunt damping, involves the connection of an electrical impedance to the terminals of a structurally bonded transducer. Many passive, non-linear, and semi-active impedance designs have been proposed that reduce structural vibration. This paper introduces a new technique for the design and implementation of piezoelectric shunt impedances. By considering the transducer voltage and charge as inputs and outputs, the design problem is reduced to a standard linear regulator problem enabling the application of standard synthesis techniques such as LQG, H2, and Hinf. The resulting impedance is extensible to multi-transducer systems, is unrestricted in structure, and is capable of minimizing an arbitrary performance objective. An experimental comparison to a resonant shunt circuit is carried out on a cantilevered beam. Previous problems such as ad-hoc tuning, limited performance, and sensitivity to variation in structural resonance frequencies are significantly alleviated.