Optimal switching for Networked Control Systems with information multiplexing

In this article, we examine a Networked Control System (NCS) in which the plant and the controller communicate over a network subject to a certain communication constraint. The plant is described by discrete-time nonlinear dynamics subject to bounded disturbances. Due to an overloaded communication network, we assume that the control signal and the information from the plant (the measured output signal) cannot be transmitted simultaneously and are subject to a multiplexing constraint. The goal is to design a switching strategy that allows us to sequentially communicate given these constraints while optimizing a quadratic cost over a finite horizon. Consequently, we proceed by emulation and assume that a controller that satisfies performance requirements is already provided. The resulting optimization problem is observed to be an integer programming problem that is generally NP-complete, i.e., the complexity is exponential in the time horizon considered. To overcome this issue, we provide a different perspective on this problem than what has been presented by the community before. Our main contribution is to reformulate the problem with all its constraints to a form that renders it amenable to apply the discrete-time Pontryagin Maximum Principle to get the necessary conditions for the optimality of the control action sequence. These necessary conditions are then solved numerically by a multiple-shooting method. To validate the approach, we present some illustrative numerical experiments on an inverted pendulum. Different setups are considered and numerically analyzed: usage of a predictor when the output is not transmitted and usage of the previous value of the output when the new value is not transmitted, with or without the choice of non-transmission.