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In this paper, we study a wireless networked control system (WNCS) with $N\ge 2$ sub-systems sharing a common wireless channel. Each sub-system consistsof a plant and a controller and the control message must be delivered from thecontroller to the plant through the shared wireless channel. The wirelesschannel is unreliable due to interference and fading. As a result, a packet canbe successfully delivered in a slot with a certain probability. A networkscheduling policy determines how to transmit those control messages generatedby such $N$ sub-systems and directly influences the transmission delay ofcontrol messages. We first consider the case that all sub-systems have the samesampling period. We characterize the stability condition of such a WNCS underthe joint design of the control policy and the network scheduling policy bymeans of $2^N$ linear inequalities. We further simplify the stability conditioninto only one linear inequality for two special cases: the perfect-channel casewhere the wireless channel can successfully deliver a control message withcertainty in each slot, and the symmetric-structure case where all sub-systemshave identical system parameters. We then consider the case that differentsub-systems can have different sampling periods, where we characterize asufficient condition for stability.

On Stability Condition of Wireless Networked Control Systems under Joint Design of Control Policy and Network Scheduling Policy