The small-scale fading hm,n follows Rayleigh distribution with mean zero and variance of = 1. In addition, small-scale fading is constant within coherence time, which is larger than the maximum end-to-end (E2E) time delay. Thus, we consider quasi-static fading channel, which is constant for each uplink subcarrier within a frame [32]. From (14.12), (14.14), (14.16), and (14.15), we can obtain SE of the uplink for the m-th plant at time index n, which can be expressed as (14.18) where Am is the desired payload transmitted by the sensor, and SE means successful decod- ing bits at the BS per bit. In this chapter, we intend to obtain the uplink optimal wireless resource allocation by maximizing SE in (14.18). 14.3 Communication-Control Co-Design Our goal is to maximize the communication SE and maintain good overall system per- formance. Thus, in this section, we first discuss the constraints from the perspectives of communication and control, respectively. Then, we formulate the co-design problem based on the constraints. 14.3.1 Communication Constraint The main constraints from communication are the limited wireless resource and URLLC QoS. We assume that the total available transmission power and bandwidth without reuse 14 Optimizing Resource Allocation in URLLC for Real-Time Wireless Control Systems among sensors for each sensor are pmax and Bmax at each time index n, respectively, where the constraints can be expressed as (14.19) (14.20) Based on the constraint of reliability in URLLC, successful transmission probability can be expressed as Pr{amn=1}=1-Emn1-Eth (14.21) and the failed transmission probability can be expressed as (14.22) where Eth is the packet error probability bounded by the URLLC QoS requirement. Further- more, communication time delay should also be bounded by the URLLC QoS requirement. Then, we have (14.23) We assume that the sizes of payload Am for each sensor are identical, which can be rewrit- ten as A. Then, for a given A, we have = A. Then, (14.16) and (14.18) can be rewritten as (14.24) (1