## 3 Multi-Model Modeling of the CO\({}_{2}\) Capture Process
### Derivation of the Multi-Linear Model  
The investigation result shows that the nonlinearity of the CO\({}_{2}\) capture process is dependent on the CO\({}_{2}\) capture rate rather than the flue gas mass flow; therefore, the CO\({}_{2}\) capture rate was selected as the scheduling variable to select the local linear models for constructing an advanced controller. In general, to increase the accuracy of the multi-linear model, as many local linear models as possible should be used for prediction. However, this also increases the complexity of the multi-linear model  
Figure 5: Gap metric of the neighboring operating points with different flue gas flow rates. and the resulting advanced controller. For instance, for multi-model model predictive control methods, the computational cost can be proportional to the number of local models. On the other hand, the effectiveness of increasing local models on improving the prediction accuracy can be minor when the process nonlinearity is weak. Hence, the number of local models in the multi-linear model should be determined according to the nonlinearity analysis such that the model complexity and accuracy can be balanced. Since the operating range from 80% to 95% CO\({}_{2}\) capture rate shows stronger nonlinearity than the operating range, from 50% to 80% CO\({}_{2}\) capture rate, the local linear model with 80% CO\({}_{2}\) capture rate was selected as the intermediate point for the two operating ranges. The local models with 50% and 95% CO\({}_{2}\) capture rate were selected to cover the full operating range. Note that the local linear models can represent the dynamic relationship between the key controlled variables and manipulated variables around specific operating points, while they cannot describe the dynamics of the whole CO\({}_{2}\) capture plant. Hence, they can only be employed to construct multi-linear models for designing model-based controllers but are not suitable for simulating the whole plant. Generally, the controllers designed based on a multi-linear model will have better performance than the single-model controllers, because the multi-linear model can predict the process outputs in a more accurate manner. Trapezoidal scheduling functions were then designed, as shown in Figure 6, to link the three local linear models together and form the integrated multi-model system. As seen in Figure 6, when CO\({}_{2}\) capture rate (CR) is between 45% and 60%, \(\psi_{1}=0\) and \(\psi_{2}=\psi_{3}=0\), which denotes the dynamics of the CO\({}_{2}\) capture process, is described solely by the local model with 50% CR. When CR is between 60% and 70%, \(\psi_{1}\
eq 0\), \(\psi_{2}=1-\psi_{1}\), and \(\psi_{3}=0\), which denotes the dynamics of the CO\({}_{2}\) capture process, is described by combining the local models with 50% CR and 80% CR. The scheduling function in the remaining operating ranges can be explained similarly. The prediction function of the multi-linear model is derived as follows:  
Suppose at time \(k\), the initial state of the \(j\)-th local model is \(x_{j}(k)\). The \(i\)-step-ahead predictions of the \(j\)-th local model can be expressed as:  
\[\left\{ \begin{array}{cc}x_{j}(k+i+1)=A_{j}x_{j}(k+i)+B_{j}u(k+i)\\ y_{j}(k+i)=C_{j}x_{j}(k+i)+D_{j}u(k+i)\end{array} \right.\;i=0,\;1,\ldots,N \tag{14}\]  
Then the outputs of the \(j\)-th local linear model can be derived as follows:  
Figure 6: Trapezoidal scheduling functions (the blue line represents the scheduling function of the 50% capture rate model, the red line represents the scheduling function of the 80% capture rate model, the purple line represents the scheduling function of the 95% capture rate model). \[\left\{