# Robust Adaptive Control for Nonlinear Aircraft System with Uncertainties
## 3 Controller Design
### Controller Design
#### 3.3.3 Robust Controller  
The standard MRAC is non-robust in that the boundedness of estimated parameters \(\hat{\lambda}\) and \(\hat{\theta}\) cannot be guaranteed. Therefore, a robust compensator was designed to improve system robustness, which was defined as  
\[u_{rc}=K_{rc}e(t) \tag{11}\]  
where \(K_{rc}\in\mathbb{R}^{m\times m}\) is the robust feedback gain. Substituting Equation (6) into Equation (4) allows the closed-loop system to be written as  
\[\left\{ \begin{array}{l}\dot{x}(t)=Ax(t)+B\Big{(}\Lambda\Big{(}K_{x}^{T}x(t) +\hat{\lambda}^{T}x(t)+\hat{\theta}^{T}\Theta(x(t))+K_{rc}e(t)\Big{)}+\theta^{ T}\Theta(x(t))\Big{)}+B_{r}r(t)+d(t)\\ y(t)=Cx(t)\end{array} \right. \tag{12}\]Due to the matching condition in Equation (9), the uncertain system can be obtained as  
\[\left\{ \begin{array}{l}\dot{x}(t)=A_{m}x(t)+B_{r}r(t)+B\Lambda\Big{(}\tilde{ \lambda}^{T}(t)x(t)+\tilde{\theta}^{T}(t)\Theta(x(t))\Big{)}+B\Lambda K_{rc}e(t )+d(t)\\ y(t)=Cx(t)\end{array} \right. \tag{13}\]  
where \(\tilde{\lambda}(t)\) and \(\tilde{\theta}(t)\) are estimated parameter errors, which are defined as \(\tilde{\lambda}(t)=\tilde{\lambda}(t)-\lambda^{*}\), \(\tilde{\theta}(t)=\tilde{\theta}(t)-\theta^{*}\). The error dynamic system can be obtained by subtracting Equation (5) from Equation (13):  
\[\left\{ \begin{array}{l}\dot{e}(t)=A_{m}e(t)+B\Lambda\Big{(}\tilde{\lambda}^ {T}(t)x(t)+\tilde{\theta}^{T}(t)\Theta(x(t))\Big{)}+B\Lambda K_{rc}e(t)+d(t)\\ e_{y}(t)=Ce(t)\end{array} \right. \tag{14}\]  
**Remark 2**. : _Estimation parameter errors may introduce high-frequency components into the control response. Because of this, a robust feedback gain \(K_{rc}\) was designed to resist the effects of \(\tilde{\lambda}\) and \(\tilde{\theta}\) on the control signal._  
The state error dynamics were rewritten without unmatched disturbance as  
\[\left\{ \begin{array}{l}\dot{e}_{m}(t)=(A_{m}+B\Lambda K_{rc})e_{m}(t)+Bw(t) \\ e_{y}(t)=C_{e}e_{m}(t)\end{array} \right. \tag{15}\]  
where \(w(t)=\Lambda\Big{(}\tilde{\lambda}^{T}(t)x(t)+\tilde{\theta}^{T}(t)\Theta(x(t ))\Big{)}\) and \(C_{m}=I_{n}\). Therefore, the key to improving system robustness was to minimize the \(H_{\infty}\) norm of the transfer function from \(w(t)\) to \(e_{m}(t)\). The procedure used to determine the robust feedback matrix \(K_{rc}\) is introduced as follows. **Theorem 1**. : _The system described in Equation (15) is asymptotically stable and \(\|C_{e}(sl-A_{m}-B\Lambda K_{rc})^{-1}B\|_{\infty}<\gamma\), if there is a symmetric matrix \(P>0\) satisfying the inequality expressed as_  
\[\left[ \begin{array}{cccc}A_{m}^{T}P+PA_{m}&PB&C_{e}^{T}&K_{rc}^{T}\\ *&-v^{-1}I&0&0\\ *&*&-I&0\\ *&*&*&-\varepsilon I\end{array} \right]<0 \tag{16}\]  
**Proof of Theorem 1. ** With the Schur complement, Equation (16) is equivalent to the following inequality  
\[A_{m}^{T}P+PA_{m}+vPBB^{T}P+C_{e}^{T}C_{e}+\varepsilon^{-1}K_{rc}^{T}K_{rc}<0 \tag{17}\]  
In addition, a positive \(\varepsilon\) exists, such that [28]  
\[PB\Lambda K_{rc}+(PB\Lambda K_{rc})^{T}\leq\varepsilon\sigma^{2}PBB^{T}P+ \varepsilon^{-1}K_{rc}^{T}K_{rc} \tag{18}\]  
Defining \(\gamma^{-2}=\
u-\varepsilon\sigma^{2}\), we have  
\[(A_{m}+B\Lambda K_{rc})^{T}P+P(A_{m}+B\Lambda K_{rc})+\gamma^{-2}PBB^{T}P+C_{e} ^{T}C_{e}<0 \tag{19}\]  
which is equivalent to  
\[\left[ \begin{array}{cccc}(A_{m}+B\Lambda K_{rc})^{T}P+P(A_{m}+B\Lambda K_{ rc})&PB&C_{e}^{T}\\ *&-\gamma^{2}&0\\ *&*&-I\end{array} \right]<0 \tag{20}\]Therefore, the dynamic system in Equation (15) is asymptotically stable and \(\|C_{e}(sI-A_{m}-B\Lambda K_{rc})^{-1}B\|_{\infty}<\gamma\)[29]. \(\square\)  
Inequality (16) indicates a state feedback \(\gamma\)-suboptimal control issue based on a linear matrix inequality (LMI). By solving the convex constrained optimization problem as shown below, the \(\gamma\)-suboptimal \(H_{\infty}\) controller can be obtained. \[