# \(H_{2}/H_{\infty}\) Output Tracking Control with a Performance Compensator for Aeroengines
## 2 Problem Formulation  
In this section, a new formulation for \(H_{2}/H_{\infty}\) tracking control is presented, and the basic problem is described.  
Consider the locally linearized system of an aeroengine in the following form:  
\[\left\{\begin{array}{l}\Delta\dot{x}_{p}=A_{p}\Delta x_{p}+B_{p}\Delta u_{p} +B_{wp}w_{p}\\ \Delta y_{p}=C_{p}\Delta x_{p}+D_{p}\Delta u_{p}+D_{wp}w_{p}\end{array}\right. \tag{1}\]  
where \(A_{p}\in\ \mathbb{R}^{n_{x}\times n_{x}}\), \(B_{p}\in\ \mathbb{R}^{n_{x}\times n_{u}}\), \(C_{p}\in\ \mathbb{R}^{n_{y}\times n_{x}}\) and \(D_{p}\in\ \mathbb{R}^{n_{y}\times n_{u}}\) are known constant matrices; \(w_{p}\in\ \mathbb{R}^{n_{w}}\) is the disturbance with the influence matrices \(B_{wp}\in\ \mathbb{R}^{n_{x}\times n_{w}}\), \(D_{wp}\in\ \mathbb{R}^{n_{y}\times n_{w}}\); \(u_{p}\in\ \mathbb{R}^{n_{u}}\) is the input, \(x_{p}\in\ \mathbb{R}^{n_{x}}\) is the state, and \(y_{p}\in\ \mathbb{R}^{n_{y}}\) is the output; \(\Delta\) denotes the deviation between the present operating point and the initial point.  
Similar to [22], in order to make it easy to augment the system with the reference, an equivalent representation of system (1) is obtained by making a differentiation in time as follows:  
\[\left\{\begin{array}{l}\ddot{x}_{p}=A_{p}\dot{x}_{p}+B_{p}\dot{u}_{p}+B_{wp}w \\ \dot{y}_{p}=C_{p}\dot{x}_{p}+D_{p}\dot{u}_{p}+D_{wp}w\end{array}\right. \tag{2}\]  
where \(w=\dot{w}_{p}\).  
The output tracking error is as follows:  
\[e(t)=y_{r}(t)-y_{p}(t) \tag{3}\]  
where \(y_{r}\) is the desired output reference.  
Define a performance output \(z\) and a quadratic performance index \(J_{QR}\) as follows:  
\[z=\left(\dot{x}_{p}^{T}Q_{1}\dot{x}_{p}+e^{T}Q_{2}e+\dot{u}_{p}^{T}R\dot{u}_{p} \right)^{1/2} \tag{4}\]  
\[J_{QR}=\int_{0}^{\infty}z^{T}zd\tau=\int_{0}^{\infty}\left(\dot{x}_{p}^{T}Q_{1} \dot{x}_{p}+e^{T}Q_{2}e+\dot{u}_{p}^{T}R\dot{u}_{p}\right)d\tau \tag{5}\]  
where \(Q_{1}^{T}=Q_{1}\in\mathbb{R}^{n_{x}\times n_{x}}\), \(Q_{2}^{T}=Q_{2}\in\mathbb{R}^{n_{y}\times n_{y}}\), and \(R^{T}=R\in\mathbb{R}^{n_{u}\times n_{u}}\) are the given weighting matrices for \(\dot{x}_{p}\), \(e\) and \(\dot{u}_{p}\), respectively. If \(\|Q_{2}\|\gg\|Q_{1}\|\), \(\|Q_{1}\|\approx 0\) and \(\|R\|\approx 0\), we obtain \(J_{QR}\approx\int_{0}^{\infty}e^{T}Q_{2}ed\tau\), which stands for the tracking performance.  
Consider the following \(H_{2}\) and \(H_{\infty}\) control performance index:  
\[J_{2}=\quad\sup_{w=\delta(t)}\int_{0}^{\infty}z^{T}zdt=\sup_{w=\delta(t)}J_{QR} \tag{6}\]  
\[J_{\infty}=\quad\sup\frac{J_{0}^{\infty}z^{T}zdt}{\int_{0}^{\infty}w^{T}wdt}= \sup\frac{J_{QR}}{\int_{0}^{\infty}w^{T}zndt} \tag{7}\]  
Equations (6) and (7) indicate that the tracking performance \(J_{QR}\) is bounded when \(J_{2}\) and \(J_{\infty}\) are bounded. Similar to the definition of \(H_{\infty}\) tracking problem in [23], the \(H_{2}/H_{\infty}\) robust tracking control problem can be defined as follows:  
**Definition 1**.: \((H_{2}/H_{\infty}\) _Tracking Problem) Find a controller such that the following holds. Numbered lists can be added as follows:_  
1. _the controlled system with_ \(w=0\) _is asymptotically stable;_
2. _are constant variables_ \(\gamma_{2}\) _and_ \(\gamma_{\infty}\) _such that_ \(J_{2}<\gamma_{2}\) _and_ \(J_{\infty}<\gamma_{\infty}\) _hold._