# \(H_{2}/H_{\infty}\) Output Tracking Control with a Performance Compensator for Aeroengines
## 5 Simulation Results  
In order to validate the effectiveness of the proposed approach, it was applied to a locally linear system of a turbofan engine in [26]. The linear model is as follows:  
\[\left\{ \begin{array}{l}\left[\begin{array}{c}\Delta\hat{n}_{L}\\ \Delta\hat{n}_{H}\end{array}\right]=\left[\begin{array}{cc}-6.68&4.45\\ -0.82&-2.29\\ \Delta n_{H}\\ \Delta EPR\end{array}\right]=\left[\begin{array}{cc}0&1\\ 0.78&1.04\end{array}\right]\left[\begin{array}{cc}\Delta n_{L}\\ \Delta n_{H}\end{array}\right]+\left[\begin{array}{cc}1.15&0.90\\ 0.72&0.24\\ \Delta n_{L}\end{array}\right]\left[\begin{array}{c}\Delta W_{f}\\ \Delta A_{8}\end{array}\right]+\left[\begin{array}{c}0.1\\ 0.1\\ 0.2\end{array}\right]w_{p}\\ \end{array} \right.\]  
where \(n_{L}\), \(n_{H}\), \(W_{f}\), \(A_{8}\), and \(EPR\) denote the low-pressure speed, high-pressure speed, the fuel flow, the area of the nozzle throat, and the engine pressure ratio, respectively. Here, control strategy 1 (CS1) is \(H_{2}/H_{\infty}\) nominal tracking control with a compensator, control strategy 2 (CS2) is nominal tracking control introduced in MOP (22), and control strategy 3 (CS3) is a robust tracking controller proposed in [21]. Considering both tracking performance and robust performance, we choose \(Q=diag(0.01,0.01,10,14)\), \(R=0.01\cdot I\), \(\gamma_{\infty}=1\). For CS1 and CS2, according to **Theorem 1** and MOP (22), we obtain \(\gamma_{2,min}=0.43\) and  
\[K_{1}=\left[ \begin{array}{cc}-0.03&-6.78\\ 1.03&-1.7\end{array} \right],K_{2}=\left[ \begin{array}{cc}29.01&17.22\\ 14.8&-33.24\end{array} \right]\]  
The compensator \(K_{3}(t)\) in CS1 is calculated by the current output reference in an online mode according to Equation (36). Figure 4 shows the tracking performance of the three controllers for a linear reference with a ramp as follows:  
\[y_{r}(t)=\left\{ \begin{array}{ll}0,&0\leq t<1\\ 0.02t-0.02,&1\leq t\leq 2\\ 0.02,&t>2\end{array} \right.\]  
As can be seen from Figure 4, the tracking performance of CS2 is better than that of CS3. It indicates that the nominal controller derived from GARE is more conservative than CS3 proposed in [21]. However, the system output response of CS1 is closer to the tracking reference than that of CS2 and CS3, which indicates that the compensator designed for the nominal controller CS2 can evidently improve the tracking performance. An instantaneous disturbance \(D_{wp}w_{p}=[0.005,0.005]^{T}\) is added to the output at \(t=3s\). When the disturbance disappears, all of the controllers can drive the output close to reference very quickly, which indicates that the proposed controller CS1 has a strong robustness as well as CS3 does. Figure 4: Output response for the ramp reference. (**a**) \(\Delta n_{H}\) tracking curve; (**b**) \(\Delta EPR\) tracking curve. The next simulation is carried out for an exponential reference to compare the control quality of the three control strategies in Figure 5. The reference is as follows:  
\[y_{r}(t)=\left\{ \begin{array}{ll}0,&0\leq t<1\\ 0.02(1-e^{-2.5(t-1)}),&t\geq 1\end{array} \right.\]  
As is demonstrated in Figure 5, the tracking performance of CS1 is better than those of CS2 and CS3. A continuous disturbance \(D_{wp}w_{p}=[0.005,0.005]^{T}\) is added to the output at \(t=2s\). The proposed controller can still track the reference due to its strong robustness. Figure 6 shows output response for a sine reference:  
\[y_{r}(t)=0.02 sin(\pi t)\]  
A Gaussian white noise \(D_{wp}w_{p}\sim N(0,0.002)\) is added to the output. From Figure 6a,b, we can see that the output curve of CS1 has better tracking performance than those of CS2 and CS3. The variation of the elements in the matrix \(K_{3}(t)\) over time is shown in Figure 7.  
Figure 5: Output response for the exponential reference. (**a**) \(\Delta n_{H}\) tracking curve; (**b**) \(\Delta EPR\) tracking curve. Figure 6: Output response for sine reference. (**a**) \(\Delta n_{H}\) tracking curve; (**b**) \(\Delta EPR\) tracking curve.