## 2 Dynamics and Nonlinearity Analysis of MGT-CCP
### Nonlinear Analysis via Gap-Metric  
Vinnicombe gap (V-gap) metric has been successfully applied to multi-model modeling and control of power plant boilers-turbine ([PERSON], 2016) and spacecraft attitude control ([PERSON], 2016), providing an appropriate partition of the operating region and the selection of local models, making it possible to use a minimum number of linear models to approximate the nonlinear behavior of the system.  
The value of V-gap is a measure of the difference between two linear models. Suppose \(P_{1}\) and \(P_{2}\) are the transfer functions of linear models at any two operating points of the MGT-CCP system. The V-gap between \(P_{1}\) and \(P_{2}\) can be computed by:  
\[v_{g}(P_{1},P_{2})=\max\Biggl{\{}\inf_{Q\in H_{\infty}}\parallel\left[\begin{array} []{c}M_{1}\\ N_{1}\end{array}\right]-\left[\begin{array}{c}M_{2}\\ N_{2}\end{array}\right]Q\parallel\,\inf_{Q\in H_{\infty}}\parallel\left[\begin{array}{c}M_{2}\\ N_{2}\end{array}\right]-\left[\begin{array}{c}M_{1}\\ N_{1}\end{array}\right]Q\parallel\Biggr{\}} \tag{1}\]  
where \((N_{1},M_{1})\) and \((N_{2},M_{2})\) are the elements of the normalized right coprime factorization of \(P_{1}\) and \(P_{2}\) as: \(P_{1}=N_{1}M_{1}^{-1}\) and \(P_{2}=N_{2}M_{2}^{-1}\), and \(Q\) is a matrix parameter which has finite H-infinity norm. The detailed process of gap calculation is referenced in ([PERSON], 2016), which has been programmed as a \(gap(P_{1},P_{2})\) function in the MATLAB toolbox. The value of V-gap is between 0 and 1. The greater the value of V-gap, the greater the difference between the two linear models, that is, the stronger the nonlinearity of the region.  
This section proposes the use of V-gap to quantitatively analyze the nonlinearity distribution of the MGT-CCP process over the operating range (power output: 65.18 kW-80 kW; chilled water temperature: 8 \({}^{\circ}\)C-9.1 \({}^{\circ}\)C).  
A local linear model is first developed for a specific operating point for gap metric calculations. At each operating point, a random identification signal with a steady-state value of \(\pm 2\%\) for 3000 s is applied to the fuel and the refrigerant valve opening path, respectively, to generate a corresponding output signals. A linear model for each operating point in Table 1 is built using the System Identification Toolbox in MATLAB based on the resulting input-output data. The gap metric between adjacent linear models is then calculated to indicate the level of non-linearity along the corresponding region as shown in Figure 3.  
Figure 3a is a nonlinearity distribution of the MGT-CCP process under varying fuel. The results show that the gaps between adjacent operating points are small and uniform, implying that one linear model is sufficient for operating points #1 through #5. Therefore, in Section 4, a linear controller is designed for the control loop of the power output.  
Figure 3b shows the nonlinearity distribution of the MGT-CCP process under varying refrigerant valve opening. The gaps between the operating points #2 and #3 and #4 and #5 show extreme nonlinearity, and the gap between the operating points #3 and #4 also shows nonlinearity. The gap between the operating points #1 and #2 shows the least nonlinearity. The physical processes of the chiller include desorption and absorption of the lithium bromide solution, phase change of the liquid, and various heat-transferring processes. Its physical structure is complex, which leads to its inevitable nonlinear dynamics. Therefore, in Section 4, a nonlinear controller is designed for the control loop of the cold-water temperature.