## 6 Computational Study
### Experimental Results  
Table 1 shows the ISE results and CPU execution times (in milliseconds) that were obtained over the course of the experiments; the ISE increase between nominal and error case is quoted as a percentage. Note that in the execution times, only the time required to design and apply the controller is reported; an additional 1.486 ms execution time was required for plant identification in both methods, and is not added to these figures. Figure 5 shows a snapshot of the comparative performance of behaviors for the two methods for the first 100 s of each experiment; the upper section of the figure shows the process outputs whilst the lower sections are indicators of burst errors, the lines having a value of 1 when a packet loss occurs and 0 otherwise. Please note that in both cases, the controller was disabled for the first 10 s to allow initial parameter convergence and a pulse input was applied to the process. This is a typical strategy for an adaptive controller to help prevent large initial swings in the control input and aid model identification. The upper section of this indicator represents losses in the sensor-controller link, while the lower represents losses in the controller-actuator link. As may be observed, the closed loop responses are very similar (with the RC method having an additional 1 second delay), and all bursts are well-tolerated during this initial period. The evolution of the estimated process parameters for the PC method is shown in Figure 6 (the estimated parameters for the RC method follows an almost identical trajectory). Digitizing the transfer function given by Equation (18) with a sampling time of \(T=0.1\) s yields the discrete equivalent transfer function shown in Equation (19), where an extra one-sample delay has been added due to the synchronized sampling-actuation in the control workflow (Figure 2). The two dominant identified parameters (\(b_{2}\) and \(a_{1}\)) are shown bolded in the Figure (note that due to the identification mechanism, the \"\(a\)\" parameters have reversed signs); the remaining parameters have values in the range [\(-0.01\), \(0.01\)] and most are close to zero. Such residual effects are common in the presence of noise when using an over-parameterized \(B\) polynomial. Comparing the parameter values observable in Figure 6 with the parameters in the model given by Equation (19), one sees a stable and rapid convergence to appropriate values for the two key parameters. \[G(z)=\frac{0.3625z^{-2}}{1-0.8187z^{-1}} \tag{19}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **ISE Ratio (\%)** & **Exec Time (ms)** \\ \hline RC Method & 677.024 & 61.345 \\ PC Method & 319.087 & 0.268 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparative evaluation of Integral of Squared Error (ISE) ratios and CPU Execution Times. Applying the design formulate represented by Equation (14) using the model given by Equation (19) as the process \(G(z)\) gives an ideal controller design \(D(z)\) to be employed in the PC method given as  
Figure 5: (**A**) effects of burst errors with RC (solid) _vs._ PC Method (dashed); (**B**): sensor-controller node link state; (**C**): controller-actuator node link state. Equation (20). Examining Figure 5, one can see that allowing for the impact of packet losses, the reference is tracked very well, and the effective controller designed on-line by the PC method implementation (using the parameters identified in as shown in Figure 6) gives a controller very close to the ideal. The residual effects of noise on the parameter identification have had little impact, as expected [28]. \[G(z)=\frac{0.009(1-0.8187z^{-1})}{1-1.9198z^{-1}+0.9198z^{-2}} \tag{20}\]  
From these data and recorded responses, it can be observed that both methods experienced a deterioration of performance during the course of the experiments. This deterioration is to be expected, since both methods rely on buffering schemes in the path from controller to actuator and identical buffer sizes were employed.