Cost of Sensing in Optimal Control: Basic Formulations, Examples, and Applications

Incorporating a notion of cost of sensing, or sensing-cost, within the optimal control framework is beneficial in controlling systems where the duration of sensing, and/or the cost of sensors themselves, have a considerable impact on the overall cost. In this regard, this paper presents multiple methods for incorporating an integral sensing-cost into the optimal control framework for Linear Time-Invariant (LTI) systems. Sensing-cost is traded off against the conventional costs of control and stabilization. Optimal sensing intervals are derived by applying the Pontryagin's Minimum Principle. Other formulations of the sensing-cost problem, and extension to nonlinear systems, are possible. The theoretical developments of this paper are validated through numerical solutions and demonstrated through simulations. A reduced-form expression for the infinite-horizon multi-dimensional case with single switching point is derived, and a closed-form solution is obtained for the infinite-horizon first-order case. Additionally, a Shrinking Horizon method is demonstrated for practical implementation of the proposed theory and as a means to address uncertainties. A practical case study of a wastewater treatment plant is introduced to examine the applicability of sensing-cost considerations in a real-world setting.