## 3 Controller Design
### Design of \(H_{\infty}\)/Leitmann Control for the Uncertain System  
Under the \(H_{\infty}\) controller (22) and (26) of the nominal System (24), we get a basic performance of the system (8). Then, we further investigate the design of the robust control \(\bar{p}(t)\). By introducing (22) and (23) into (6), we have:  
\[\dot{\bar{x}}(t) =\bar{A}\bar{x}(t)+B\bar{p}(t)+Bv(t), \tag{29}\] \[\dot{y}(t) =\bar{C}\bar{x}(t)+D\bar{p}(t)+D_{1}w(t),\] (30) \[\bar{x}(0) =x_{0}-x_{\text{s}0}. \tag{31}\]  
Here, \(K_{1}\) is determined by Theorem 1. Define:  
\[\mu(\bar{x},v,t):=B^{T}(\bar{x},t)\
abla_{\bar{x}}V(\bar{x},t)\rho(\bar{x},v,t). \tag{32}\]  
We present the robust control \(\bar{p}(t)\) as:  
\[\bar{p}(t)=\bar{p}(\bar{x},v,t)=\left\{ \begin{split}&-\frac{\mu(\bar{x},v,t)}{\| \mu(\bar{x},v,t)\|}\rho(\bar{x},v,t),&\text{if}\quad\|\mu(\bar{x},v,t)\|>\epsilon,\\ &-\frac{\mu(\bar{x},v,t)}{\epsilon}\rho(\bar{x},v,t),& \text{if}\quad\|\mu(\bar{x},v,t)\|\leq\epsilon.\end{split} \right. \tag{33}\]Here, \(\epsilon>0\) is a given scalar. Now, by the preparation given above, we are ready to propose our main result as Theorem 2. **Theorem 2**. : _Consider the uncertain dynamic system (29) subject to Assumptions 1-3. Under the control (7), (22) and (31), the solution of the resulting controlled system has the performance of uniform tracking boundedness and uniform ultimate tracking boundedness._  
**Proof of Theorem 2. ** See Appendix B. **4. Application to Turbofan Engines**  
_4.1. Control System of Turbofan Engines_  
Consider an aircraft turbofan engine shown in Figure 1, which consists of the fan, the compressor, the combustor, the high pressure turbine( HPT), the low pressure turbine (LPT), the mixing chamber, the afterburner and the exhaust nozzle. The numbers in Figure 1 denote the main sections of the turbofan, which are taken as the subscript of the related signals, such as the exhaust nozzle area \(A_{8}\). As an air-breath engine, the air is inhaled into the inlet of the turbofan and compressed by the fan and the compressor. The high-pressure air flows into the combustor and gets mixed with fuel. The mixed gas is ignited and burned in the combustor. The burned gas with high temperature and pressure drives the HPT and LPT, which connect to the compressor and the fan through the high-pressure rotation spool and the low-pressure rotation spool, respectively. By this mechanical connection, the HPT and LPT transfer the power to the fan and compressor. The thrust of the turbofan is determined by the velocities of the turbofan input air and output gas and the exhausting pressure of nozzle. For the modern aircraft turbofan, the fuel flow rate \(W_{f}\) and \(A_{8}\) are selected to control \(N_{L}\) and \(\pi_{T}\), the rotational speeds of the low-pressure spool and the high-pressure spool, and to realize the indirect control of the thrust. By employing perturbation techniques at each equilibrium condition, a set of linear state variable models (SVM) that depict the interactions of the engine states with inputs and outputs is established. Consequently, the turbofan can be described as (4), and:  
\[x(t)=\left[ \begin{array}{c}N_{L}\\ \pi_{T}\end{array} \right],\;\;u(t)=\left[ \begin{array}{c}W_{f}\\ A_{8}\end{array} \right],\;\;y(t)=\left[ \begin{array}{c}N_{L}\\ \pi_{T}\end{array} \right], \tag{34}\]  
where \(N_{L}\) is the rotational speed of the low-pressure spool, \(\pi_{T}\) is the decreasing pressure ratio of the turbine, \(W_{f}\) is the main fuel flow and \(A_{8}\) is the exhaust nozzle area. The reference model (5) can be specified by:  
\[x_{ss}(t)=\left[ \begin{array}{c}N_{Ls}\\ \pi_{Ts}\end{array} \right],\;\;u_{ss}(t)=\left[ \begin{array}{c}W_{fss}\\ A_{88}\end{array} \right],\;\;y(t)=\left[ \begin{array}{c}N_{Ls}\\ \pi_{Ts}\end{array} \right], \tag{35}\]  
Figure 1: Structure of a turbofan. where \(N_{Lss}\), \(\pi_{Tss}\), \(W_{fss}\), \(A_{8 ss}\) are corresponding parameters of the reference model.