## 5 Results and Discussion
### Estimation of Sensor Values
#### 5.1.2. Throttle Estimation  
The use of PSO, GA, and NLR requires a fitness function. To formulate the fitness function for the throttle estimation, the sum of the least-squares method is used. A 6 th-degree polynomial given by (62) is used to predict the values of throttle sensors in the case of a fault:  
\[y_{e}=a_{6}x^{6}+a_{5}x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0} \tag{62}\]  
Here, \(y_{e}\) is the estimated throttle value; \(x\) is MAP value; and \(a_{6}\), \(a_{5}\), \(a_{4}\), \(a_{3}\), \(a_{2}\), \(a_{1}\), and \(a_{0}\) are unknown constants. The difference between estimated throttle value (\(y_{e}\)) and actual throttle value (\(y\)) is calculated and the sum of the difference over the entire range of \(x\) is computed to find the fitness function\(f_{M}\). The expression for the fitness function is given as:  
\[f_{M}=\frac{1}{2n}\sum_{i=1}^{n}(y_{i}-y_{ei})^{2} \tag{63}\]  
For throttle estimation, \(n=19\), and the values of \(x\) and \(y\) are given in Table 6.  
The fitness function given by (63) is then minimized by using PSO, GA, and NLR to find the best possible values of the unknown constants \(a_{0}\) through \(a_{6}\). All three optimization techniques were implemented in MATLAB and executed on a Core i7 processor with a RAM of 8 GB. PSO was executed with a swarm size of 1800, maximum iterations were kept at 500, and the values of \(\varphi_{1}\) and \(\varphi_{2}\) were set at 2.9 and 1.5, respectively; the values of kappa (\(k\)) and damping ratio of inertia coefficient were assumed to be 1.5 and 0.2. While implementing the GA, the simulated binary crossover is used instead of blend crossover with parameters \(\beta=3.8\) and distribution index \(\eta_{C}=2.1\). The other parameters \(\gamma\), \(\mu\), and \(\sigma\) were set at 0.25, 0.02, and 0.5, respectively. The population size was set at 400 with 100 iterations. To implement the NLR, the residual sum of squares (RSS) is computed and then the unknown constants were found for which the RSS is minimum. With the  
Figure 4: Estimated MAP values with PSO, GA, and NLR. abovementioned settings, all three estimation techniques returned the unknown constants and the estimated throttle values, which are given in Tables 7 and 8, respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(x\) (MAP Value) & 0.05 & 0.10 & 0.15 & 0.20 & 0.25 & 0.30 & 0.35 \\ \hline \(y\) (Throttle Angle) & \(0^{\circ}\) & \(1.98^{\circ}\) & \(4.69^{\circ}\) & \(6.26^{\circ}\) & \(7.47^{\circ}\) & \(8.48^{\circ}\) & \(9.36^{\circ}\) \\ \hline \(x\) (MAP Value) & 0.40 & 0.45 & 0.50 & 0.55 & 0.60 & 0.65 & 0.70 \\ \hline \(y\) (Throttle Angle) & \(10.13^{\circ}\) & \(10.82^{\circ}\) & \(11.45^{\circ}\) & \(12.06^{\circ}\) & \(12.70^{\circ}\) & \(13.40^{\circ}\) & \(14.19^{\circ}\) \\ \hline \(x\) (MAP Value) & 0.75 & 0.80 & 0.85 & 0.90 & 0.95 & \\ \hline \(y\) (Throttle Angle) & \(15.11^{\circ}\) & \(16.24^{\circ}\) & \(17.75^{\circ}\) & \(20.03^{\circ}\) & \(24.50^{\circ}\) & \\ \hline \end{tabular}
\end{table} 
Table 6: MAP value (\(x\)) and throttle angles (\(y\)) at 300 r/min [27]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**MAP Value**} & \multirow{2}{*}{**Actual Throttle Value**} & \multicolumn{3}{c}{**Estimated Throttle Values**} \\ \cline{3-4}  & & **PSO** & **GA** & **NLR** \\ \hline
0.05 & \(0^{\circ}\) & \(0.72^{\circ}\) & \(0.52^{\circ}\) & \(-0.08^{\circ}\) \\ \hline
0.10 & \(1.98^{\circ}\) & \(2.43  
To examine the performance of the individual algorithms, the line fit plots were made, and then the statistics such as residual sum of squares (RSS), mean square error (MSE), R\({}^{2}\), adjusted R\({}^{2}\), standard error of estimation (SEE), and optimization times were recorded and given in Table 9. The line fit plots are shown in Figure 5.  
It is evident from Figure 5 that the estimated values generated by NLR are very close to the actual values of the throttle sensor.