## 5 Simulation Validation
### Simulation Scenarios  
To validate the proposed method comprehensively, 2 scenarios with diffident operating conditions are simulated. Figures that showing the performances of dynamical model based method, the virtual measurement based method, and the data fusion method (data fusion of the dynamical model and the virtual measurement method) with different combinations of uncertainties are summarized in Table 1.  
**Scenario I**, with no gas mass uncertainties and various ignition delay model errors.  
This scenario demonstrates the situations when there are no gas mass uncertainties (perfect knowledge for the dynamical model) and various errors of measuring model (imperfect knowledge for the virtual measurement).  
Figure 9 shows the performance of the dynamical model based method when there are no uncertainties. The 2 main remarks can be found in Figure 9: (1) At cold start state, the estimation results are rather poor (roughly (\(-\)0.05, 0.05)) due to the inevitable pressure and temperature fluctuation on this engine state. (2) At steady state, the estimation results tracks the simulation value well with the error less than 0.01.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Figure No.** & **Gas Mass Uncertainties** & **Measuring Model Errors** \\ \hline Figure 9 (DM) & No & N/A \\ Figure 10 (VM) & No & 10\% drift \\ Figure 11 (DM) & Yes & N/A \\ Figure 12 (DFM) & Yes & 10\% drift \\ \hline \hline \end{tabular} DM = the dynamical model, VM = the virtual measurement method, DFM = proposed method (data fusion of the dynamical model and the virtual measurement method); For the gas mass uncertainties, Yes means the volumetric efficiency is inaccurate and is 10\% larger than real value; for the ignition delay model errors, 10\% drift means the oxygen fraction calculation error from ignition delay model is set as 10\%.  
\end{table}
Table 1: Uncertainty combinations for figures.  
Figure 9: Performance of dynamical model based method (no uncertainties).  
Figure 10 shows the performance of the virtual measurement. Remarks can be found as: (1) At cold start state, the error is improved from (\(-\)0.05, 0.05) to (0, 0.025). (2) At steady state, the stair effect damages the performance to some extent, but the overall error is much less than 0.01 (roughly (\(-\)0.005, 0.005)).  
Compare Figures 9 and 10, the results prove that the virtual measurement can promote the robustness (insensitive to cold start state and steady state) and the overall accuracy even with relatively large model error.  
**Scenario II**, with gas mass uncertainties and various ignition delay model errors.  
This scenario tests the performance of the data fusion method when the common gas mass errors (represented by the inaccurate volumetric efficiency) and large virtual measurement error exist.  
Figures 11 and 12 shows the simulation results for both methods.  
Figure 11 shows that due to the inaccuracy of the volumetric efficiency, the dynamical model cannot estimate the ICOC accurately, the convergence cannot even be guaranteed. The estimation error is about (\(-\)0.05, 0.05) at cold start state, and (0, 0.03) at steady state, which is unacceptable for ICOC with a range from 12% to 23%.  
Although Figure 12 represents the performance of the data fusion method. Even there are 10% volumetric efficiency uncertainty and 10% of model error of the virtual measurement, the ICOC can still be accurately estimated. The estimation error is about (\(-\)0.01, 0.01) at cold start state and is roughly (0, 0.005) at steady state, which is much better than the dynamical model.  
To be noted, the data fusion method is also much better than by only applying the virtual measurement. More specifically: (1) At cold start state, the error is reduced from (0, 0.025) to (\(-\)0.01, 0.01). (2) At steady state, the error is reduced from (\(-\)0.005, 0.005) to (0, 0.005).  
Figure 10: Performance of the virtual measurement (no uncertainties, 10% drift).