# \(H_{2}/H_{\infty}\) Output Tracking Control with a Performance Compensator for Aeroengines
## 4 Compensator Design
### Compensator Design for Stable Reference  
If \(h\) is a constant matrix, it seems that a robust controller for system (24) can be easily obtained according to Theorem 1. However, in practical application, \(h\) is time-variant and usually unknown in advance. According to Equation (23), we can obtain \(h(t)\) online as follows:  
\[h(t)=\frac{\dot{y}_{r}(t)}{\dot{y}_{r}(t)} \tag{26}\]  
With this real-time \(h(t)\), it is still difficult to calculate the controller (25) online because the solvability of LMIs (13) and (14) in **Theorem 1** is hard to be guaranteed. To deal with this problem, a real-time compensator for the proposed nominal controller is introduced in the following theorem:  
**Theorem 2**. : _Suppose there exist a matrix \(Y\) satisfying LMIs (13) and (14), and the controller for system (9) is \(u=Kx=\left[ \begin{array}{cc}K_{1}&K_{2}\end{array} \right]x\). Consider the system (24) with \(h=h(t)<0\). A real-time compensator_  
\[K_{3}(t)=-R^{-1}B^{T}P_{2}(t)\]  
_makes the controller \(u=\left[ \begin{array}{cc}K_{1}&K_{2}&K_{3}(t)\end{array} \right]x_{a}\) a solution for the \(H_{2}/H_{\infty}\)**Tracking Problem**, where \(P_{2}(t)\) is calculated by:_  
\[A_{I}^{T}Y^{-1}+h(t)P_{2}(t)^{T}+P_{2}(t)^{T}\left(A+BK\right)=0 \tag{27}\]  
Proof. : Let a symmetric positive define matrix \(P=\left[ \begin{array}{cc}P_{1}&P_{2}\\ P_{2}^{T}&P_{3}\end{array} \right]\) be a solution for the GARE as follows:  
\[\left[ \begin{array}{cc}A^{T}&0\\ A_{I}^{T}&h(t)\end{array} \right]P+P\left[ \begin{array}{cc}A&A_{I}\\ 0&h(t)\end{array} \right]-P\left[ \begin{array}{cc}B\\ 0\end{array} \right]R^{-1}\left[ \begin{array}{cc}B\\ 0\end{array} \right]^{T}P+\left[ \begin{array}{cc}Q+M_{1}&0\\ 0&0\end{array} \right]+\gamma_{\infty}^{-2}P\left[ \begin{array}{cc}B_{w}\\ 0\end{array} \right]\left[ \begin{array}{cc}B_{w}\\ 0\end{array} \right]^{T}P=0 \tag{28}\]  
where \(\tilde{Q}=Q+M_{1}\) and \(M_{1}=-[A^{T}P_{1}+P_{1}A+Q-P_{1}BR^{-1}B^{T}P_{1}]>0\). Equation (28) is equivalent to three equations as follows:  
\[A^{T}P_{1}+P_{1}A-P_{1}BR^{-1}B^{T}P_{1}+\tilde{Q}+\gamma_{\infty}^{-2}P_{1} B_{w}B_{w}^{T}P_{1}=0 \tag{29}\]  
\[A_{I}^{T}P_{1}+h(t)P_{2}^{T}+P_{2}^{T}\left(A-BR^{-1}B^{T}P_{1}\right)=0 \tag{30}\]  
\[A_{I}^{T}P_{2}+P_{2}A_{I}+h(t)P_{3}+P_{3}h(t)-P_{2}^{T}BR^{-1}B^{T}P_{2}=0. \tag{31}\]  
If Equations (29)-(31) are solvable, then the controller \(u=-R^{-1}B_{a}^{T}Px_{a}\) is a solution for the \(H_{2}/H_{\infty}\) Tracking Problem according to the proof of Theorem 1. Because \(Y=P_{1}^{-1}\) satisfies LMIs (13) and (14), Equation (29) is apparently solvable, and the system (9) is asymptotically stable, which indicates \(A+BK=A-BR^{-1}B^{T}P_{1}<0\). Equations (30) and (31) are equivalent to the following two equations:  
\[\left(h(t)\otimes I+I\otimes\left(A-BR^{-1}B^{T}P_{1}\right)^{T}\right)vec \left(P_{2}^{T}\right)=-vec\left(A_{I}P_{1}\right) \tag{32}\]  
\[\left(h(t)\otimes I+I\otimes h(t)\right)vec\left(P_{3}\right)=P_{2}^{T}BR^{-1} B^{T}P_{2}-A_{I}^{T}P_{2}+P_{2}A_{I} \tag{33}\]where \(vec(\cdot)\) denotes the column vector obtained by stacking the columns of the matrix, e.g., \(vec(\left[ \begin{array}{cc}x_{11}&x_{12}\\ x_{21}&x_{22}\end{array} \right])=\left[ \begin{array}{cc}x_{11}&x_{21}&x_{12} &x_{22}\end{array} \right]^{T}\), and \(\otimes\) denotes the Kronecker product multiplicator. Due to \(h<0\) and \(A-BR^{-1}B^{T}P_{1}<0\), we have \(\left(h\otimes I+I\otimes\left(A-BR^{-1}B^{T}P_{1}\right)^{T}\right)<0\) and \((h\otimes I+I\otimes h)<0\), and thereby Equations (30) and (31) are solvable. Therefore, the controller \(u=\left[ \begin{array}{cc}K_{1}&K_{2}&K_{3}(t)\end{array} \right]x_{a}\) is a solution for for the \(H_{2}/H_{\infty}\) Tracking Problem of system (24). \[ \begin{array}{l}\mbox{The controller }u=\left[\begin{array}{cc}K_{1}&K_{2}&K_{3}(t) \end{array}\right]x_{a}\mbox{ can be re-expressed as:}\\ \\ \Delta u_{p}{=}\int_{0}^{t_{f}}\dot{u}_{p}d\tau=K_{1}\Delta x_{p}+K_{2}\int_{ 0}^{t_{f}}ed\tau+\int_{0}^{t_{f}}K_{3}(\tau)\dot{y}_{r}d\tau.\end{array}