## 3 WSAN System Models
### Performance Metrics
#### 4.1.2 Control Performance Problem  
Figure 4 shows the system output from the plant. This plot represents a value of the room temperature sample _versus_ the simulation time. The desirable temperature in the room is set at 21 \(\,\mathrm{\SIUnitSymbolC}\). Figure 3: Jitter duration histograms. Figure 4(a) shows the system output when the CBR rate is increased from 0 to 2.00 Mbps. The temperature response converges slowly when the CBR rate is getting higher. These results agree on the facts formerly discussed. In Figure 4(b), the control system is completely uncontrollable at the CBR rate of 2.25 Mbps, and the time response of the room temperature starts to recover at the CBR rates of 3.00 and 3.25 Mbps. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**CBR Rate (Mbps)** & **000 to 125** & **1.50** & **1.75** & **2.00** & **2.25** & **2.50** & **2.75** & **3.00** & **3.25** \\ \hline Settling time (s) & 4,100 & 4,300 & 4,650 & \(>\)5,400 & \(>\)5,400 & \(>\)5,400 & \(>\)5,400 & 4,100 & 4,100 \\ \hline \end{tabular}
\end{table} 
Table 4: Settling time _vs_. CBR rate. Figure 4: The system output at the plant. (**a**) Vary the CBR rates from 0 to 2.00 Mbps. (**b**) Vary the CBR rates from 2.25 to 3.25 Mbps. This is due to the fact that the route selection is almost completely changed to the upper path as previously discussed in Table 3. Table 4 demonstrates the control performance in term of the settling time. The results confirm that the CBR rates influence the communication performance and the resulting control output. Four control response cases at the CBR rates between 2.00 and 2.75 cannot converge to the desirable target at the end of simulation time. Why the CBR rates influence the resulting control output is further discussed here. The chances of success in route establishment and temperature-data transmissions are decreased when the CBR rate is high in this scenario. The results of these consecutive communication failures often return worse control performance. When the sink doesn't receive any new data packets for a period of time, the most recent temperature can't be updated at the controller. Consequently, the controller and the actuator cannot feed an appropriate supply air temperature to the room. Thus, the room temperature can't converge to the desired setting point. Consider the results of varying the CBR rate at 2.25 Mbps in Table 3 and Figure 3, there are only nine data packets successfully received at the sink before the beginning of the CBR-traffic flow at the simulation time of 500 s. After that there is no successful data transmission at all. The new control input is not updated for a long period of time. Accordingly, the control response measured by the zone temperature reaches higher than 30 \(\ \mathrm{\SIUnitSymbolCelsius}\) as shown in Figure 4(b). To guarantee an acceptable control performance of the closed loop control system, the sampling period should be set to 4 to 10 times per rise time ( \(T_{r}\) ) [38, 39, 40]. The rise time is the time required for the system response to rise from 0% to 100% of its desired value. Thus, the maximum sampling period or the maximum allowable loop delay [38, 39] is equal to _Tr_/4. In our work, the rise time and the maximum allowable loop delay are 900 s and 225 s respectively. Thus, at the CBR rates 0-1.75 Mb/s and 3.00-3.25 Mb/s as in Figure 3, the control system is completely controllable. This is because the jitter durations of successfully received data packets are almost less than 225 s.  
Figure 5 shows the control performance in term of the IAE. The results as in Figures 4 and 5 are directly correlated. This means that when the temperature in the room cannot be controlled to the desired level, the sum of absolute value of the error between the measured temperature and the set-point values is getting bigger. This is because when the measured temperature packets are not reached to the PID controller/the actuator, the controller will not update a suitable supply air temperature and respond to the plant. This will lead to increase in the IAE.