Generalization and optimization of switching controller to control free vibration of a multi-linear state space system

This paper studies the optimized quadratic integral of a generalized multi-linear state space system under free vibration. The general formulation of the optimization problem is presented and can be solved by the standard numerical procedure. For demonstration, two examples are examined thoroughly and analytically. First, the minimized integral of mechanical energy for a single-dof semi-active damping or stiffness controlled system are shown analytically and compared with the best active and passive ones. The Pontryagin Maximum Principle is used to prove the true optimality of the presented semi-active damping control. The Linear Quadratic Regulator on/off or continuous clipping controller is shown to be truly optimized in the first example. In the second example, the maximized dissipation energy of the single-dof on-off damping system is studied and verified by a sensorless experiment.