h plant and Maximum of transmission power the BS Total number of sensors in the coverage E m,n Packet error probability of the uplink of the BS for the m-th sensor Index of the sensors (1 <m M M) Maximum packet error probability in communications Variance of the AWGN on each Spectral efficiency of the uplink for the subcarrier at the BS m-th sensor Transmission power spectral density Payload information of the m-th sensor at each sample time Plant m Plant state Actuator Sensor Perfect Imperfect wireless wireless networks networks Linear state State feedback estimator Remote controller Figure 14.2 Wireless control system model for a single plant. Example 14.1 The m-th controlled inverted pendulum system consists of an inverted pendulum and a motorized cart. We have the plant state x(t) = (Ct, Ct, At, (if), where Ct repre- sents the cart's position, represents the cart's velocity, At represents the pendulum's angle, and , represents the pendulum's angular velocity. The expression of Am m and Bm consists of the pendulum length 21, the inertia of the pendulum Y, the friction of the cart r, the 14 Optimizing Resource Allocation in URLLC for Real-Time Wireless Control Systems Table 14.2 Summary of notations for control subsystem System parameter on state in Control input calculated by the actuator continuous control function System parameter on input in Feedback parameter calculated by the continuous control function remote controller Control performance constraint on Control parameter for simplification wireless communications Average control cost Xm(t) Plant state in continuous control function of the m-th plant Total sampling time index in control Plant state in discrete time control process function Sample time index Lyapunov-like function nm(t) Disturbance caused by AWGN in System parameter on input in discrete continuous control function control function Disturbance in discrete time control System parameter on input caused by function time delay Generalized disturbance General system parameter during packet losse