## 3 Design of SIRAC
### Design of RAC
\right]<0 \tag{48}\]  
Let Inequality (48) left multiplies \(diag(F_{1}^{T},I,I)\) and right multiplies \(diag(F_{1},I,I)\) and substitute \(A_{cl}\), \(B_{d}\), \(C_{cl}\), and \(D_{d}\) into Inequality (48) yields  
\[\left\{ \begin{array}{ccc}\left(AR+RA^{T}+B_{2}\hat{C}+\left(B_{2}\hat{C} \right)^{T}\right)&*&*&*\\ \hat{A}+\left(A+B_{2}\hat{D}C_{2}\right)^{T}&A^{T}S+SA+\hat{B}C_{2}+\left(\hat {B}C_{2}\right)^{T}&*&*\\ B_{1}^{T}+\left(B_{2}\hat{D}D_{21}\right)^{T}&\left(SB_{1}+\hat{B}D_{21} \right)^{T}&-\gamma I&*\\ C_{1}R+D_{12}\hat{C}&C_{1}+D_{12}\hat{D}C_{2}&D_{11}+D_{12}\hat{D}D_{21}&- \gamma I\end{array} \right\}<0 \tag{49}\]  
where \(diag(\cdot)\) represents diagonal matrix and  
\[ \begin{array}{l}\hat{A}=S(A+B_{2}D_{k}C_{2})R+NB_{k}C_{2}R+SB_{2}C_{k}M^{T}+ NA_{k}M^{T}\\ \hat{B}=SB_{2}D_{k}+NB_{k}\\ \hat{C}=D_{k}C_{2}R+C_{k}M^{T}\\ \hat{D}=D_{k}\end{array}  \tag{50}\]  
Now, the controller \(K(s)\) can be constructed, as follows:  
\[ \begin{array}{l}D_{k}=\hat{D}\\ C_{k}=(\hat{C}-D_{k}C_{2}R)(M^{T})^{-1}\\ B_{k}=N^{-1}(\hat{B}-SB_{2}D_{k})\\ A_{k}=N^{-1}[\hat{A}-S(A+B_{2}D_{k}C_{2})R](M^{T})^{-1}-B_{k}C_{2}R(M^{T})^{-1 }-N^{-1}SB_{2}C_{k}\end{array}  \tag{51}\]  
The above method has been expanded into RAC design. Figure 5 shows the interconnected structure of the general plant for the RAC. Where \(n\) is the sensors noises, \(W_{in}\), \(W_{p}\), \(W_{e}\), \(W_{act}\), and \(W_{n}\) are the weighting functions. In this case, \(h\) and \(V\) are the variables to be controlled. The weighting functions are selected, as follows, according to the specifications to achieve the desired performances. There are two main tuning criteria that are used for selecting the weighting functions of the \(H_{\infty}\) controller: one is the weighting functions of some components of a vector signal should be bigger if they are more important than others, the other is  
Figure 5: Interconnected structure of the general plant. each component of the signal should be scaled to the same units according to the weighting functions to make these components comparable. The altitude \(h_{d}\) and speed \(V_{d}\) commands are scaled by \(W_{in}\) in order to normalize the reference inputs. Select the average altitude \(h_{ave}=250\) m and average speed \(V_{ave}=67.4\) m/s, and yields:  
\[W_{in}=diag(h_{ave},V_{ave})=diag(250,67.4) \tag{52}\]  
The gain of \(W_{e}\) should be big enough at low frequency to enable the controlled system to track ramp commands with a very small steady state error, as the altitude error \(e_{h}\) and the speed error \(e_{V}\) should be small, and the altitude and speed do not change quickly. Hence, the \(W_{e}\) is selected as:  
\[W_{e}=diag\Big{(}20000\frac{1}{s+0.007},12000\frac{1}{s+0.002}\Big{)} \tag{53}\]  
where 20,000 and 12,000 are proportional gains and 1/0.007 and 1/0.002 are time constants. It is obvious that the magnitude of \(W_{e}\) is big at low frequency, which is similar with a proportional integral (PI) element, as the integral element can eliminate the steady state error. As the maximum pitch rate is \(\pm 0.52\) rad/s [25], the weighting function of \(W_{p}\) is selected as:  
\[W_{p}=\frac{1}{0.052} \tag{54}\]  
The weighting function of \(W_{act}\) is applied to confine the control deflections and their rates. According to the maximum deflections and rates of elevator and engine in Table 3, the \(W_{act}\) is selected as:  
\[W_{act}=diag\Big{(}\frac{1}{0.35},\frac{1}{0.26},\frac{1}{0.088},\frac{1}{0.01 7}\Big{)} \tag{55}\]  
As measurements are often corrupted with some noises, according to the measurement noise error [25], the measurement noise weighting function \(W_{n}\) is selected, as:  
\[W_{n}=diag\Big{(}W_{n\!h},W_{n\!h},W_{nV},W_{n\!i^{\prime}},W_{n\!n},W_{n\!n} \Big{)}=diag\Big{(}0.01,0.025,0.015,0.02,\frac{0.05}{57.3},\frac{0.1}{57.3} \Big{)} \tag{56}\]  
Based on the above weighting functions, RAC is constructed following the preceding three steps and the matrices are given, as follows:  
\[A_{k}=\left[