## 3 Multi-Model Modeling of the CO\({}_{2}\) Capture Process
### Nonlinearity Analysis of the Monoethanolamine-Based Post-Combustion CO\({}_{2}\) Capture Process  
In this section, the nonlinearity distribution of the MEA-based post-combustion CO\({}_{2}\) capture process is analyzed using the gap metric. As mentioned before, the CO\({}_{2}\) capture plant must operate flexibly under the change of the flue gas mass flow rate and desired CO\({}_{2}\) capture rate; therefore, the nonlinearity distribution is analyzed for these two scenarios. The gap metric can be regarded as the \"distance\" between two linear models, which is originally proposed as a tool to analyze the robust stability for closed-loop linear systems [43; 44]. In later studies, the gap metric was found especially useful in the evaluation of process nonlinearity and thus it is applied in the model bank selection of multi-linear models [45; 46; 47]. The gap metric of two linear models is defined as follows. Denote two linear models as \(P_{1}\) and \(P_{2}\), and perform the normalized right coprime factorization on \(P_{1}\) and \(P_{2}\):  
\[P_{1} = N_{1}M_{1}^{-1} \tag{11}\] \[P_{2} = N_{2}M_{2}^{-1} \tag{12}\]  
Then the gap metric between \(P_{1}\) and \(P_{2}\) is calculated as:  
\[\delta(P_{1},P_{2})=max\Bigg{\{}inf_{Q\in H_{\infty}}\parallel\left[ \begin{array} []{c}M_{1}\\ N_{1}\end{array} \right]-\left[ \begin{array}{c}M_{2}\\ N_{2}\end{array} \right]Q\parallel_{\infty},inf_{Q\in H_{\infty}}\parallel\left[ \begin{array}{c}M_{2}\\ N_{2}\end{array} \right]-\left[ \begin{array}{c}M_{1}\\ N_{1}\end{array} \right]Q\parallel_{\infty}\Bigg{\}} \tag{13}\]  
where \(\delta(P_{1},P_{2})\) is bounded between 0 and 1. If the gap metric between two local linear models is close to one, it means their dynamic behavior is quite different and thus the process nonlinearity is strong between the two operating points. On the contrary, if the gap metric is close to zero, it means the dynamics of the two models are similar and the process nonlinearity is weak. The gap metric of the neighboring local models of the CO\({}_{2}\) capture rate (Group I) and flue gas mass flow (Group II) are plotted in Figures 4 and 5, respectively. Figure 4 shows that there is an increasing trend in the gap metric value as the capture rate increases from 50% to 95%. The peak value of the gap metric is observed at a CO\({}_{2}\) capture rate of 95%; however, at other operating points the gap metric is much smaller. This indicates that the process nonlinearity is extremely strong around 95% CO\({}_{2}\) capture rate but relatively weak from 50% to 90%. An intuitive explanation for the strong nonlinearity at 95% CO\({}_{2}\) capture rate is that: as most of the CO\({}_{2}\) gas is already captured, it becomes much harder to extract the remaining CO\({}_{2}\) from the flue gas, which  
Figure 4: Gap metric of the neighboring operating points with different CO\({}_{2}\) capture rates. significantly alters the dynamic behavior of the CO\({}_{2}\) capture process. However, as seen in Figure 5, the operating points with different flue gas mass flow rates have small and similar gap metric values, which means the local models with different flue gas mass flows but the same CO\({}_{2}\) capture rate have similar dynamics. Therefore, the flue gas mass flow rate's influence on the distribution of process nonlinearity is very small. These results can be further explained using the first principle models of the mass balance, energy balance, and chemical reactions in the packed column. As seen in Equation (8), the reaction rate of CO\({}_{2}\) is directly determined by the concentration of reactants and reaction constants. Since in practice the reaction temperature is generally well controlled to the economic operating point, the temperature has little influence on the reaction constants and on the energy balance in the packed column. Hence, the reaction rate of the CO\({}_{2}\) mainly depends on the concentration of reactants.