Optimization of partial-state feedback for vibratory energy harvesters subjected to broadband stochastic disturbances

In many applications of vibratory energy harvesting, the external disturbance is most appropriately modeled as a broadband stochastic process. Optimization of the average power generated from such disturbances is a feedback control problem, and solvable via LQG (linear-quadratic-Gaussian) control theory. Implementing the optimal feedback controller requires a power electronic drive capable of two-way power flow, which can impose dynamic relationships between the voltage and current of the transducer. Determining the optimal energy harvesting current control is accomplished by solving a nonstandard Riccati equation. In this paper we show that appropriate tuning of the passive parameters in the harvesting system results in a decoupled solution to the Riccati equation and a corresponding controller that only requires half of the states for feedback. However, even when such tuning methods are not used and the solution to the Riccati equation does not decouple, it is possible to determine the states in the feedback law that contribute the most to the average power generated by the harvester. As such, partial-state feedback gains can be optimized using a gradient descent method. Two energy harvesting examples are presented, including a single-degree-of-freedom oscillator with an electromagnetic actuator and a piezoelectric bimorph cantilever beam, to demonstrate these concepts.