## 5 Results and Discussion
### Estimation of Sensor Values
#### 5.1.1 MAP Estimation  
The use of PSO, GA, and NLR requires a fitness function. To formulate the fitness function for the MAP estimation, the sum of the least-squares method is used. An exponential equation given by (60) is used to predict the values of MAP sensors in the case of a fault:  
\[y_{e}=1-e^{-ax^{b}}+ce^{-x}+\frac{d}{x^{2}-1} \tag{60}\]  
Here, \(y_{e}\) is the estimated MAP value; \(x\) is throttle angle; and \(a\), \(b\), \(c\), and \(d\) are unknown constants. The difference between the estimated MAP value (\(y_{e}\)) and the actual MAP 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:  
\[f_{M}=\frac{1}{2n}\sum_{i=1}^{n}(y_{i}-y_{ei})^{2} \tag{61}\]  
Here, \(n=17\) and the values of \(x\) and \(y\) are given in Table 2.  
The fitness function given by (61) is then minimized by using PSO, GA, and NLR to find the best possible values of the unknown constants \(a\), \(b\), \(c\), and \(d\). 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 1500, maximum iterations were kept at 200, and the values of \(\varphi_{1}\) and \(\varphi_{2}\) were set at 2.1 each; the values of kappa (\(k\)) and damping ratio of inertia coefficient were assumed to be 1.1 and 0.9 respectively. While implementing the GA, the simulated binary crossover is used instead of blend crossover with parameters \(\beta=2.1\) and distribution index \(\eta_{C}=2.1\). The other parameters \(\gamma\), \(\mu\), and \(\sigma\) were set at 0.1, 0.015, and 1, respectively. The population size was set at 500 with 150 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 abovementioned settings, all three estimations techniques returned the unknown constants and the estimated MAP values, which are given in Tables 3 and 4, respectively. 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), \(\mathrm{R}^{2}\), adjusted \(\mathrm{R}^{2}\), standard error of estimation (SEE), and optimization times were recorded and given in Table 5. The line fit plots are shown in Figure 4. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(x\) (Throttle Angle) & \(0^{\circ}\) & \(3^{\circ}\) & \(6^{\circ}\) & \(9^{\circ}\) & \(12^{\circ}\) & \(15^{\circ}\) & \(18^{\circ}\) & \(21^{\circ}\) & \(24^{\circ}\) \\ \hline \(y\) (MAP Value) & 0.091 & 0.114 & 0.191 & 0.329 & 0.545 & 0.745 & 0.857 & 0.915 & 0.946 \\ \hline \(x\) (Throttle Angle) & \(27^{\circ}\) & \(30^{\circ}\) & \(35^{\circ}\) & \(46^{\circ}\) & \(57^{\circ}\) & \(68^{\circ}\) & \(79^{\circ}\) & \(90^{\circ}\) & \\ \hline \(y\) (MAP Value) & 0.964 & 0.975 & 0.985 & 0.994 & 0.997 & 0.998 & 0.999 & 0.999 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Throttle angle (\(x\)) and MAP values (\(y\)) at 300 r/min [27]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Constant** & **PSO** & **GA** & **NLR** \\ \hline a & 0.0048 & 0.0106 & 0.0042 \\ \hline b & 2.0909 & 1.7589 & 2.1109 \\ \hline c & 0.4797 & 0.2606 & 0.4897 \\ \hline d & 0.4084 & 0.1690 & 0.3883 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of unknown constants found by PSO, GA, and NLR for MAP estimation. It is evident from Figure 4 that the estimated values generated by NLR are very close to the actual values of the MAP sensor. It is further supported by entries in Table 4. From Table 5, it can be noted that RSS, MSE, R\({}^{2}\), and adjusted R\({}^{2}\) values for NLR are much better than PSO and GA. It can be seen in Table 5 that the R\({}^{2}\) values for all three techniques are the same (0.99), but the adjusted R\({}^{2}\) of NLR is very impressive (0.98) as compared to GA (0.92) and PSO (0.96).