king the expectation at both sides and by iterating backwards we can obtain E[Am(Sm.N)] (14.28) Since the sum in (14.28) converges with Pm 1, the second moments of the plant states decay exponentially, which is bounded by Tr(QmRm)/(1-Pm 14.3.3 Problem Formulation According to the constraints from the communication and control aspects, we formulate the original communication-control co-design problem as max n=m (14.29a) (14.29b) NmPm.n pmax, (14.29c) Bm.n Bmax, (14.29d) Tm.n Th, (14.29e) Pm-m(5m.n) + Tr(QmR'm). (14.29f) In this problem, we intend to maximize the communication SE with optimal wireless resource allocation in URLLC and control convergence rate. More importantly, we jointly consider the control constraint on the control convergence rate Pm and com- munication constraint on URLLC QoS, which is difficult to deal with in the proposed communication-control co-design since the control constraint (14.29f) seems independent of the rest of the communication constraint terms in (14.29). However, the control constraint on the control convergence rate Pm can be affected and is determined by the communication QoS actually. In Section 14.4, we will deal with the problem in (14.29) in detail. 14 Optimizing Resource Allocation in URLLC for Real-Time Wireless Control Systems 14.4 Optimal Resource Allocation for The Proposed Co-Design In this section, we first explore the relationship between control and communication. Then, we discuss the solution for the problem in (14.29). 14.4.1 Relationship Between Control and Communication From (14.8), we can understand that the plant state 5m,n in (14.8) is determined by the con- trol parameters and packet transmission. Then, the expression depends on the packet transmission probability. From (14.8), we can find that the Lyapunov-like function can be expressed as Pr{am,n + Tr(QmR'm), (14.30) which indicates that the communication reliability can affect the control Lyapunov-like function directly. Substituting (14.30) into (14.29f), we can obtain (14.31) where Sm.n #0. From (14.