Co-design of Optimal Transmission Power and Controller for Networked Control Systems Under State-dependent Markovian Channels

Bin Hu, Tua A. Tamba

This paper considers a co-design problem for industrial networked control systems to ensure both the stability and efficiency properties of such systems. The assurance of such properties is particularly challenging due to the fact that wireless communications in industrial environments are not only subject to shadow fading but also stochastically correlated with their surrounding environments. To address such challenges, this paper first introduces a novel state-dependent Markov channel (SD-MC) model that explicitly captures the state-dependent features of industrial wireless communication systems by defining the proposed model's transition probabilities as a function of both its environment's states and transmission power. Under the proposed channel model, sufficient conditions on Maximum Allowable Transmission Interval (MATI) are presented to ensure both asymptotic stability in expectation and almost sure asymptotic stability properties of a continuous nonlinear control system with state-dependent fading channels. Based on such conditions, the co-design problem is then formulated as a constrained polynomial optimization problem (CPOP), which can be efficiently solved using semidefinite programming methods for the case of a two-state state dependent Markovian channel. The solutions to such a CPOP represent optimal control and power strategies that optimize the average expected joint costs in an infinite time horizon while still respect the stability constraints. For a general SD-MC model, this paper further shows that sub-optimal solutions can be obtained from linear programming formulations of the considered CPOP. Simulation results are given to illustrate the efficacy of the proposed co-design scheme.

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