# \(H_{2}/H_{\infty}\) Output Tracking Control with a Performance Compensator for Aeroengines
## 3 Nominal Controller Design  
The controller of \(H_{2}/H_{\infty}\) Tracking Problem in this paper is designed consisting of a nominal controller and a compensator, and the nominal control is introduced in this section. Assume the reference command \(y_{r}\) is a unit step, and we have  
\[\dot{e}(t)=\delta(t)-\dot{y}_{p}(t) \tag{8}\]  
where \(\delta(t)\) denotes a unit pulse. Combining Equations (2) and (8), we have the following augmented system:  
\[\dot{x}=Ax+Bu+B_{w}w \tag{9}\]  
where \(u=\dot{u}_{p}\), \(x=\left[ \begin{array}{cc}\dot{x}_{p}&e\\ \end{array} \right]^{T}\), \(A=\left[ \begin{array}{cc}A_{p}&0\\ -C&0\\ \end{array} \right]\), \(B=\left[ \begin{array}{c}B_{p}\\ -D_{p}\\ \end{array} \right]\), \(B_{w}=\left[ \begin{array}{c}B_{wp}\\ -D_{wp}\\ \end{array} \right]\), and the pulse in Equation (8) is considered as a disturbance on the augmented system. The a state feedback controller \(u=Kx=\left[ \begin{array}{cc}K_{1}&K_{2}\\ \end{array} \right]x\) can be re-expressed as:  
\[\dot{u}_{p}=K_{1}\dot{x}_{p}+K_{2}e \tag{10}\]  
Integrating from both sides of Equation (10) yields the following equivalent controller:  
\[\Delta u_{p}= K_{1}\Delta x_{p}+K_{2}\int_{0}^{t_{f}}ed\tau \tag{11}\]The diagram of the closed loop control system is shown in Figure 1.  
Before we discuss the sufficient condition for achieving the solution for the \(H_{2}/H_{\infty}\) Tracking Problem, we first introduce the following lemma. Lemma 1: _[_24_]__. Consider the system as follows:_  
\[\left\{ \begin{array}{l}\dot{x}=\tilde{A}x+\bar{B}_{w}w\\ z=\tilde{C}x\end{array} \right. \tag{12}\]  
_The following conditions are equivalent:_  
1. \(J_{2}<\gamma_{2}\)_;_
2. _there is_ \(P=P^{T}>0\) _such that_ \(\tilde{A}P+P\tilde{A}^{T}+\tilde{C}^{T}\tilde{C}<0\) _and_ \(trace(\bar{B}_{w}^{T}P\bar{B}_{w})<\gamma_{2}^{2}\)_._  
Then, based on the lemma, we obtain the following theorem:  
Theorem 1: _Consider the system (9). If there is a symmetric positive define matrix \(Y\) satisfying the following linear matrix inequalities (LMIs)_  
\[\left[ \begin{array}{cc}YA^{T}+AY-BR^{-1}B^{T}+\gamma_{\infty}^{2}B_{w}B_{w} ^{T}&Y\\ Y&-Q^{-1}\end{array} \right]<0 \tag{13}\]  
\[\left[ \begin{array}{cc}\gamma_{2}^{2}&B_{w}^{T}\\ B_{w}&Y\end{array} \right]>0 \tag{14}\]  
_where \(Q=\left[ \begin{array}{cc}Q_{1}&0\\ 0&Q_{2}\end{array} \right]\), the state feedback controller represented in Equation (10) with \([K_{1},K_{2}]=-R^{-1}B^{T}Y^{-1}\) is a solution for the \(H_{2}/H_{\infty}\) Tracking Problem._  
Proof: (a) Assume the Lyapunov function as \(V=x^{T}Px=x^{T}Y^{-1}x\). By applying the Schur Complement Lemma into Inequality (13), we have  
\[YA^{T}+AY+YQY-BR^{-1}B^{T}+\gamma_{\infty}^{-2}B_{w}B_{w}^{T}<0 \tag{15}\]  
which is equivalent to  
\[A^{T}P+PA+Q-PBR^{-1}B^{T}P+\gamma_{\infty}^{-2}PB_{w}B_{w}^{T}P<0 \tag{16}\]  
Figure 1: Diagram of output tracking control with a step reference. if \(w=0\), then  
\[\dot{V}=x^{T}\left[(A-BR^{-1}BP)^{T}P+P(A-BR^{-1}BP)\right]x<-x^{T}\left[Q+PBR^{-1 }B^{T}P+\gamma_{\infty}^{-2}PB_{w}B_{w}^{T}P\right]x<=0 \tag{17}\]  
Therefore, \(\dot{V}<0\), which means the closed system is asymptotically stable. (b) Let \(M=-[A^{T}P+PA+Q-PBR^{-1}B^{T}P]>0\) and \(\tilde{Q}=Q+M\), and the following game algebraic Riccati equation (GARE) is obtained:  
\[A^{T}P+PA+\tilde{Q}-PBR^{-1}B^{T}P+\gamma_{\infty}^{-2}PB_{w}B_{w}^{T}P=0 \tag{18}\]  
According to [18], the closed-loop is asymptotically stable with the controller  
\[\dot{u}_{p}=u=-R^{-1}B^{T}Px\]  
and  
\[\tilde{J}_{\infty}=\sup\frac{\int_{0}^{\infty}\left(x^{T}\tilde{Q}x+u^{T}Ru \right)d\tau}{\int_{0}^{\infty}w^{T}wdt}=\sup\frac{\int_{QR}+\int_{0}^{\infty} x^{T}Mxd\tau}{\int_{0}^{t_{f}}w^{T}wdt}<\gamma_{\infty} \tag{19}\]  
Because \(M\geq 0\), \(J_{\infty}\leq\tilde{J}_{\infty}<\gamma_{\infty}\) holds. Meanwhile, because \(\gamma_{\infty}^{-2}B_{w}B_{w}^{T}\geq 0\), the following inequality is obtained:  
\[A^{T}P+PA+Q-PBR^{-1}B^{T}P<0 \tag{20}\]  
Let \(\tilde{A}=A-BR^{-1}B^{T}P,\tilde{B}_{w}=B_{w},\tilde{C}=\left[