Optimal self-powered control of dynamic systems: Duality techniques

We consider the control of physical systems in which the control actions are constrained to be self-powered. In self-powered control technologies, the energy available to impose control inputs on an exogenously-excited system is limited exclusively to energy that has been previously harvested by the technology. As such, for a self-powered control input to be feasible, it must satisfy a certain constraint similar to, but more restrictive than, a passivity condition. Here we consider the use of self-powered control to minimize a quadratic performance objective, for a linear discrete-time system. We restrict our attention here to the full-information open-loop control context, in which it can be shown that the optimization problem is nonconvex due to the self-powered constraint. For this problem, we illustrate a technique to determine a lower bound on the optimal performance by transformation to a (convex) dual optimization, which may be solved uniquely and efficiently. We illustrate that if a certain easy-to-check condition holds for the obtained dual-optimum, then there is no duality gap, and consequently the dual and primal optima are coincident. In this situation, it follows that this duality technique can be used as a convex means of solving the optimal self-powered control problem exactly. We illustrate the technique on a simple example related to vibration suppression, in which we illustrate that the solutions obtained adhere to this condition.