Discrete-Time, Performance-Guaranteed Control of Vibratory Systems with Power Directionality Constraints

We consider the design of feedback controllers for physical systems in which the actuators are constrained, such that they cannot inject power into the plant. Such constraints are called power directionality constraints (PDCs), and control technologies exhibiting them arise in both vibration suppression and energy harvesting applications. We consider the case in which the plant is linear, disturbances are stochastic, the response is stationary, and the performance is assessed via a multi-objective linear-quadratic measure. For this case, we develop a control design technique which consists of two design stages. In the first stage, a time-invariant static linear feedback law is optimized for performance, while also adhering to the PDC. In the second stage, a nonlinear feedback law is designed which guarantees to improve upon the optimized performance obtained in the first design stage. This nonlinear controller requires a small-scale semidefinite program to be solved in real-time. To accommodate the finite computational time required with this online optimization, the control methodology is developed entirely in a discrete-time setting. The technique is illustrated on a numerical example pertaining to a structural control system for a seismically-excited building.