## 5 Implementation and Results
\[\mathrm{T}=\frac{\mathrm{W_{air}H+W_{fuel}LHHV}}{(\mathrm{W_{air}+W_{fuel}}) \mathrm{C_{P}}}=\frac{\mathrm{H}}{\mathrm{C_{P}}}\frac{\mathrm{W_{air}}}{ \mathrm{W_{air}+W_{fuel}}}+\frac{\mathrm{\eta\cdot LHHV}}{\mathrm{C_{P}}}\frac{ \mathrm{W_{fuel}}}{\mathrm{W_{air}+W_{fuel}}} \tag{22}\]  
As an additional performance specification, the SFC value can be included to improve the engine fuel economy or to limit the emission. So, the final state-space model of Equation (1) can be rewritten in the form of Equation (23), with nine state variables, one control variable, and one output variable. \[\left[ \begin{array}{c}\mathrm{CN_{L}}\\ \mathrm{CN_{H}}\\ \mathrm{PR_{H}}\\ \mathrm{GTR_{bypass}}\\ \mathrm{GTR_{Core}}\\ \mathrm{TR_{TET}}\\ \mathrm{FAR}\\ \mathrm{SFC}\end{array} \right]_{\mathrm{k+1}}=\left[ \begin{array}{ccccccccc} \mathrm{a_{11}}&\mathrm{a_{12}}&0&0&0&0&0&0&0&0\\ \mathrm{a_{21}}&\mathrm{a_{22}}&0&0&0&0&0&0&0\\ 0&0&\mathrm{a_{33}}&\mathrm{a_{34}}&0&0&0&0&0\\ 0&0&\mathrm{a_{43}}&\mathrm{a_{44}}&0&0&0&0&0\\ 0&0&0&\mathrm{a_{55}}&\mathrm{a_{56}}&0&0&0\\ 0&0&0&0&\mathrm{a_{65}}&\mathrm{a_{66}}&0&0&0\\ 0&0&0&0&0&\mathrm{a_{77}}&\mathrm{a_{78}}&0\\ 0&0&0&0&0&\mathrm{a_{87}}&\mathrm{a_{88}}&0\\ 0&0&0&0&0&0&0&\mathrm{a_{99}}\end{array} \right]\left[ \begin{array}{c} \mathrm{CN_{L}}\\ \mathrm{CN_{H}}\\ \mathrm{PR_{L}}\\ \mathrm{CTR_{TET}}\\ \mathrm{FAR}\\ \mathrm{SFC}\end{array} \right]_{\mathrm{k}}=\left[ \begin{array}{ccccccccc} \mathrm{CN_{L}}\\ \mathrm{CN_{H}}\\ \mathrm{PR_{L}}\\ \mathrm{PR_{H}}\\ \mathrm{TR_{TET}}\\ \mathrm{FAR}\\ \mathrm{SFC}\end{array} \right]_{\mathrm{k}} \tag{23}\]  
Table 2 shows the constraints boundaries, which have been implemented to this engine. The first seven constraints are designed for safety protection, and the last two constraints are designed for performance optimization. To set up the engine constraints, the constraint inequality equation can be written as Equation (24). The surge margin on a high-pressure compressor is approximated from a linear function from the high-pressure shaft speed. According to Table 3, \"aHPC_max\" is 9.5625, \"bHPC_min\" is -2.1096, \"aHPC_min\" is 3.0198, and \"bHPC_min\" is 0.0180, which are the constants for the surge margin. \[\underbrace{ \begin{bmatrix}\text{CN}_{\text{min}}\\ \text{a}_{\text{HPC}\_\text{min}}\cdot\text{CN}_{\text{H}}+\text{b}_{\text{HPC}\_ \text{min}}\\ 0.95\cdot\text{GTR}_{\text{Min}\_\text{Target}}\\ \text{TET}_{\text{min}}\\ \text{N/A}\end{bmatrix} }_{\text{Mcon\_MIN}}\leq\underbrace{ \begin{bmatrix} \text{CN}_{\text{H}}\\ \text{PR}_{\text{HPC}}\\ \text{GTR}_{\text{overall}}\\ \text{TET}\\ \text{SFC}\end{bmatrix} }_{\text{Y}_{\text{con}}}\leq\underbrace{ \begin{bmatrix} \text{CN}_{\text{max}}\\ \text{a}_{\text{HPC}\_\text{max}}\cdot\text{CN}_{\text{H}}+\text{b}_{\text{HPC} \_\text{max}}\\ 1.05\cdot\text{GTR}_{\text{Max}\_\text{Target}}\\ \text{TET}_{\text{max}}\\ 0\end{bmatrix} }_{\text{Y}_{\text{con}}} \tag{24}\]  
The constraints in Equation (24) can be separated into factor and constant terms, as shown in Equation (25). \[\underbrace{ \begin{bmatrix}\text{CN}_{\text{H}}\\ \text{PR}_{\text{HPC}}\\ \text{TR}_{\text{overall}}\\ \text{TET}\\ \text{SFC}\end{bmatrix} }_{\text{Y}_{\text{con}}}\leq\underbrace{ \begin{bmatrix} \text{0}&\text{0}&\text{\cdots}&\text{\cdots}&\text{0}\\ 0&\text{a}_{\text{HPC}\_\text{max}}&\text{0}&\text{\cdots}&\text{0}\\ 0&0&\text{\cdots}&\text{0}\\ 0&0&\text{\cdots}&\text{\cdots}&\text{0}\\ 0&0&\text{\cdots}&\text{\cdots}&\text{0}\end{bmatrix} }_{\text{A}_{\text{max}}} \underbrace{ \begin{bmatrix}\text{CN}_{\text{L}}\\ \text{CN}_{\text{H}}\\ \text{PR}_{\text{LPC}}\\ \text{PR}_{\text{HPC}}\\ \text{TR}_{\text{Bypass}}\\ \text{TR}_{\text{Core}}\\ \text{FAR}\\ \text{STC}_{\text{max}}\end{bmatrix} }_{\text{K}}+\underbrace{ \begin{bmatrix} \text{CN}_{\text{max}}\\ \text{b}_{\text{HPC}\_\text{max}}\\ 1.05\cdot\text{TR}_{\text{Max}\_\text{Target}}\\ \text{STC}_{\text{max}}\\ \text{Mcon\_MAX}\end{bmatrix} }_{\text{T}_{\text{max}}} \tag{25}\] \[-1\cdot\underbrace{