## 3 Controller Design
### Numerical Simulation  
For the numerical simulation, generally we extract a linear model from a turbofan nonlinear system by the small perturbation method and the fitting method at some operating condition. This nonlinear system is modeled based on the component thermodynamic and experiment data. In this paper, only the linear model will be discussed for the control design, and the more detailed modeling procedure of the turbofan nonlinear system can be seen in [31]. By the above-mentioned methods, we obtained a linear model at \(Height=0\) km, \(Ma=0\) with:  
\[\begin{gathered} A=\left[\begin{array}{cc}-2.4477&-0.6760\\ -13.5906&-81.2450\end{array}\right],B=\left[\begin{array}{cc}1.3295&1.5096\\ 16.5005&39.5006\end{array}\right],\\ B_{1}=\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right],C=\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right],D=\left[\begin{array}{cc}0&0\\ 0&0\end{array}\right],D_{1}=\left[\begin{array}{cc}0&0\\ 0&0\end{array}\right].\end{gathered} \tag{36}\]  
Let \(\Delta A=\alpha(t)A\) and:  
\[\alpha(t)=0.2[1.8(-0.5+rand(1,1))+0.1 sin(t)], \tag{37}\]  
where \(rand(1,1)\in[0,1]\) is a random number and \(sin(\cdot)\) is a sine function. By simple calculation, we know \(\alpha_{max}=0.2\). Let \(w^{T}(t)=[w_{1}(t)\ w_{2}(t)]\), and \(w_{i}(t)\in[-10^{-5},\ 10^{5}],\ i=1,\ 2\), is a white noise with zero expectation. The noise \(w_{i}(t)\) and the uncertainty \(\alpha(t)\) are shown in Figures 3 and 4.  
Figure 2: Scheme of the turbofan tracking control.  
For the nominal system of the turbofan, we solve the linear matrix inequations (LMIs) in (25) and obtain the following gain matrices:  
\[K_{1}=\left[\begin{array}{cc}-11.9628&5.1980\\ 5.1980&-0.3805\end{array}\right]. \tag{38}\]  
Let \(\epsilon=0.5\), and \(\tilde{p}(t)\) can be written as:  
\[\tilde{p}(t)=\left\{\begin{array}{cc}\dfrac{-2(1.12\|x(t)\|+0.0074)^{2}}{\mu }\left[\begin{array}{cc}1.8346&22.7699\\ 2.0831&54.5090\end{array}\right]\tilde{x}(t),\;\;\;if\;\;\;\|\mu\|>0.5,\\ \dfrac{-2(1.12\|x(t)\|+0.0074)^{2}}{0.02}\left[\begin{array}{cc}1.8346&22.769 \\ 2.0831&54.5090\end{array}\right]\tilde{x}(t),\;\;\;if\;\;\;\|\mu\|\leq 0.5, \end{array}\right. \tag{39}\]  
where:  
Figure 4: Uncertain parameter history, \(\alpha(t)\).  
Figure 3: Noise histories, \(w(t)\).  
\[\mu=2\Bigg{(}\left[\begin{array}{cc}2.0475&25.4121\\ 2.3249&60.8342\end{array}\right]\|x(t)\|\bar{x}(t)+\left[\begin{array}{cc}0. 0136&0.1687\\ 0.0154&0.4039\end{array}\right]\bar{x}(t)\Bigg{)}. \tag{40}\]  
Finally, by (7), (38), (39) and (40), the \(H_{\infty}/\)Leitmann control is:  
\[\bar{u}(t)=K_{1}\bar{x}(t)+\bar{p}(t) \tag{41}\]  
Choose:  
\[\gamma_{1}(\|\tilde{x}_{n}(t)\|) = \lambda_{m}(P)\|\tilde{x}_{n}(t)\|^{2}, \tag{42}\] \[\gamma_{2}(\|\tilde{x}_{n}(t)\|) = \lambda_{M}(P)\|\tilde{x}_{n}(t)\|^{2},\] (43) \[\gamma_{3}(\|\tilde{x}_{n}(t)\|) = \lambda_{m}(\bar{A}^{T}P+P\bar{A})\|\tilde{x}_{n}(t)\|^{2}. \tag{44}\]  
Here, \(\lambda_{m}(\cdot)\) and \(\lambda_{M}(\cdot)\) are the minimum and maximum eigenvalues of a matrix. By solving the LMIs in (25), we have \(\lambda_{m}(P)=0.0479\), \(\lambda_{m}(P)=0.0521\) and \(\lambda_{m}(\bar{A}^{T}P+P\bar{A})=1\).  
At the same time, to compare the robustness between our developed approach and an applied robust control, we designed a controller with the LQR method, which was used to obtain a multivariable control law on an F100 engine system [4]. The LQR control is:  
\[u_{LQR}=K_{2}x(t). \tag{45}\]  
Let \(Q=R=I_{2\times 2}\), and then, the controller gain matrix is:  
\[K_{2}=\left[\begin{array}{cc}-0.2193&-0.0916\\ -0.1798&-0.2248\end{array}\right]. \tag{46}\]  
Apply the \(H_{\infty}/\)_Leitmann_ controller (41), \(H_{\infty}\) controller (38) and LQR controller (42) to the system (4) with (36) and (37). Let \(N_{Lr}\) and \(\pi_{Tr}\), the commands of \(N_{L}\) and \(\pi_{T}\), step in the magnitude of \(\pm 2\%\), \(\pm 3\%\) and \(\pm 5\%\), separately, and the responses of states and control inputs are shown in Figures 5-14 and in Tables 1 and 2.