## 5 Case Study: NCS Performance over WirelessHART
### Stability of the Benchmark NCS  
Firstly, we investigate the effects of different channel models on the stability of the batch reactor setup. The uplink graph is shown in Figure 8, with Node 7 being the sensor node; thus, the gateway is reachable within three wireless hops. The simulation parameters are summarized in Table 1. We compare the successful packet delivery ratios and the ratio of the stable outcomes for the batch reactor, with three different Ricean \(K\)-factors and three different frequency correlation values (the independent block fading channel, the frequency-correlated channel with low correlation, \(\sigma_{\tau}=2\) ns, and the frequency-correlated channel with high correlation, \(\sigma_{\tau}=100\) ns.) We run every 10 s-long independent simulation trial 500 times. In order to assess the simulation efficiency, each run takes approximately one second on an Intel Xeon CPU running at 3 GHz.  
Figure 7: Step response of the batch reactor with the model predictive controller (MPC) controller.  
To assess the NCS performance, we show two figures of merit for the experiments. The probability of successful delivery is the ratio of packets that arrive to the intended recipient (plant or controller) within the deadline that we specify in advance. This is a traditional, purely network-centric quantity. The ratio of the stable outcomes shows the probability for the NCS to remain stable within the investigated time horizon. The two quantities are clearly related, as a high delivery probability within a well-chosen deadline would result in a well-behaved control loop.  
We consider a non-redundant routing scenario, _i.e._, only Nodes 7, 5, 3 and 2 are activated. The deadline is set to six time slots (60 ms), such that every link can be activated twice (e.g., \(7\to 5\), \(5\to 3\), \(3\to 2\), \(7\to 5\), \(5\to 3\), \(3\to 2\)). The successful delivery ratios are shown in Figure 9. As expected, the low correlation channel and the i.i.d. channel perform similarly, whereas the delivery ratios are adversely impacted in the highly correlated channel, with diminishing impact as the Ricean \(K\) factor increases. Figure 10 shows the ratio of stable outcomes under the same assumptions. The stable outcome ratio closely follows the successful delivery ratios.  
Finally, we compare the effects of the shadow fading on the success rates and the stable outcomes by disabling the shadow fading compared to the parameters in Table 1. We only show the outcomes for the highly frequency-correlated channel. The success rates and the ratio of stable outcomes are shown in Figures 11 and 12, respectively. Increasing the standard deviation of the shadow fading deteriorates the  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Reference distance \(d_{0}\) & 15 m \\ Reference path loss \(L_{0}(d_{0})\) & 65 dB \\ Path loss exponent \(n\) & 2.5 \\ Ricean \(K\) factor & \{0, 5, 10\} dB \\ Shadow fading \(\sigma_{S}\) & 6 dB \\ Inter-node distance & 130 m \\ Sample size & 166 bits \\ Number of drops & 500 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation assumptions for the batch rector trials.  
Figure 8: Example uplink graph. Node 0 is the gateway node (GW); Nodes 1–2 are access points (APs); Nodes 3–7 are field devices. Node 7 is the sensor.