4.25) 14.3.2 Control Constraint In this chapter, we consider the control convergence rate Pm as the constraint from the control aspect, where the sampling period is fixed and the inherent parameters of the plant are physical reality and change only when the plants are different. Given a physical control system, the main control variables that can be optimized are state sampling period Sm,n and the control input gain Om, m' which jointly affect the control convergence rate Pm. Then, the control convergence rate further affects the control performance, e.g. control cost or stability. Through constraint on Pm' the state sampling period Sm,n and the control input gain Om are directly constrained. In other words, the relationship between control convergence rate and wireless resource allocation indicates the relationship between the main control variables and wireless resource allocation. Thus, the co-design of this chapter focuses on control convergence rate and wireless resource allocation. To obtain the constraint on the control convergence rate, we consider a Lyapunov-like function for each plant, which can be expressed as [33] Am(5m) = smomsm, (14.26) 14.3 Communication-Control Co-Design where Qm is a given positive definite matrix. The requirement for the Lyapunov-like func- tion is that these functions should decrease at given rates Pm <1 during the control process, which means that the control process guarantees the plant state decreasing to the preset point. However, affected by the control perturbation and stochastic communication coef- ficients, the Lyapunov-like function is random. Thus, for any possible value of the current plant states Em,n' the Lyapunov-like function needs to satisfy [33] E[Am(5m.n+1)/lmm PmAm(sm.n) + Tr(QmR)m), (14.27) where E[.] represents the expectation operator and = 0). The control convergence is guaranteed with Pm<1, which has been discussed in [33]. We conclude it by Lemma 14.1. Lemma 14.1 (Control Convergence Lemma) If (14.27) holds for each time step n = 0, 1, 2, N, then by ta