Optimal Control of Self-Powered Systems Using Primal-Dual Techniques

Abstract

Vibration-based self-powered systems are electromechanical technologies that are mechanically coupled to vibratory phenomena, and have the capability to convert this mechanical energy into electrical energy to power their operations. These systems are fully energy-autonomous because they derive all the energy needed for operation directly from the vibratory disturbances to which they are subjected. Examples include (i) a wireless sensor node that powers its sensing, computing, and transmission tasks by converting low-level structural vibrations into electrical energy, (ii) an ocean wave energy converter that transforms the oscillating motion of ocean waves into electrical energy and uses a portion of this converted energy to power its control operations, and (iii) a structural vibration suppression control system that powers its operation by storing and recycling the energy it extracts from the vibrating structure.

In this thesis, we consider the general problem of control design for vibration-based self-powered systems in the context of discrete-time optimal control theory, and realize the optimal control solution in real-time using Model Predictive Control (MPC). The functionality of a self-powered system is constrained due to the limited availability of the vibratory energy resource, and also due to the finite bounds of its on-board energy storage subsystem. In addition, there are parasitic losses associated with harvesting energy and running intelligence, as well as decay of stored energy. These effects further restrict the functionality of the system. Consequently, the main challenge associated with control design for these systems relates to carefully managing energy harvesting, usage, and storage.

First, we develop a general model for self-powered systems and provide conditions on the model parameters for stability and feasibility. We restrict our attention to linear, time-varying, discrete-time systems, and assume the exogenous disturbances are known exactly. We then formulate the discrete-time optimal control problem to minimize a quadratic performance measure subject to constraints on the on-board energy storage, which is, in general, a nonconvex quadratically constrained quadratic program. We formulate the dual relaxation of the self-powered optimal control problem, which may be solved uniquely and efficiently. Its solution provides a lower bound on the optimal primal performance measure. The duality gap is the difference between the optimal primal and optimal dual objectives. 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 primal (nonconvex) optimal control problem exactly. If there is a nonzero duality gap, the resulting trajectory does not satisfy the constraints of the original optimal control problem. In this case, we introduce an algorithm to guarantee that the first time-step of the trajectory is feasible and can be implemented in real-time via MPC.