## 3 Controller Design
### The Architecture of the \(\mathcal{L}_{1}\) Adaptive Controller  
In order to ensure that the input and output of uncertain linear systems can track the input and output of the reference system in real time, we can reform the MRAC of the Stewart-Gough flight simulation platform given in Equation (18) into the following control structure:  
\[\dot{x}(t)=\left(A_{p}-B_{p}F^{T}\right)x(t)+B_{p}K(t)u(t)+B_{p}\Big{(}F^{T}-F( t)\Big{)}x(t)+B_{p}\hat{\delta}(t)\]  
\[y(t)=c^{T}x(t) \tag{20}\]  
where the system is divided into a linear time invariant (LTI) and linear time varying (LTV) system. After simplification, the system yields [30]:  
\[\dot{x}(t)=A_{m}x(t)+b(\omega(t)u(t)+\theta(t)x(t)+\sigma(t))\]  
\[y(t)=c^{T}x(t) \tag{21}\]  
where \(x(t)\in\mathbb{R}^{m}\) represents the state of the system, \(u(t)\in\mathbb{R}^{n}\) represents the control input, \(c^{T}\in\mathbb{R}^{n\times m}\) represents the output matrix, \(b\in\mathbb{R}^{m\times n}\) represents the known constant matrix, \(y(t)\in\mathbb{R}^{n}\) denotes the regulated output, \(\omega(t)\in\mathbb{R}^{n\times n}\) denotes the input uncertainly parameter of the model, \(\theta(t)\in\mathbb{R}^{n\times m}\) is the uncertainly parameter of the model itself, and \(\sigma(t)\in\mathbb{R}^{n\times 1}\) represents the unmatched disturbances. \(A_{m}\in\mathbb{R}^{m\times m}\) represents a Hurwitz matrix, and \((A_{m},\ b)\) is controllable and \(A_{m}-A=b\theta(t)\). For the linear time invariant parameterized system in Equation (21), we give the following state observer model:  
\[\dot{\hat{x}}(t)=A_{m}\hat{x}(t)+b\big{(}\hat{\sigma}(t)u(t)+\hat{\theta}(t)x (t)+\hat{\sigma}(t)\big{)}\]  
\[\hat{y}(t)=c^{T}\hat{x}(t) \tag{22}\]  
where \(\hat{\omega}(t),\ \hat{\theta}(t)\) and \(\hat{\sigma}(t)\) denote the parameter estimates of \(\omega(t),\theta(t)\) and \(\sigma(t)\). The block diagram of the closed-loop system is shown in Figure 4.  
Next, we use Lyapunov stability theory to obtain adaptive law, considering the following Lyapunov function candidate [30]:  
\[V=e^{T}(t)Pe(t)+\frac{1}{\Gamma}\left(\widetilde{\omega}^{T}(t)\widetilde{ \omega}(t)+\widetilde{\theta}^{T}(t)\widetilde{\theta}(t)+\widetilde{\sigma}^ {T}(t)\widetilde{\sigma}(t)\right) \tag{23}\]  
where \(e(t)=\hat{x}(t)-x(t),\ \widetilde{\omega}=\hat{\omega}(t)-\omega(t),\ \widetilde{ \theta}=\hat{\theta}(t)-\theta(t),\ \widetilde{\sigma}=\hat{\sigma}(t)-\sigma(t)\) denotes the tracking errors between state predictor and system, \(\Gamma>0\) denotes the adaptation gain, and \(P=P^{T}>0\) denotes the solution of equation \(A_{m}^{T}P+PA_{m}=-Q,Q>0\). The adaptive laws can be obtained by solving algebraic Lyapunov equation. Using projection operator [16]\(Proj(\cdot,\ \cdot)\) to prevent parameter drift, we have:  
\[\dot{\hat{\theta}}(t)=\Gamma Proj\Big{(}\hat{\theta}(t),-b^{T}Pe(t)x^{T}(t) \Big{)}\]The fundamental difference between the MRAC in Figure 3 and the predictor-based of \(\mathcal{L}_{1}\) adaption in Figure 4 is the design of the control law. The design of error signal and controller are not independent in MRAC system, but they are independent in \(\mathcal{L}_{1}\) adaptive control system. Therefore, in order to ensure the stability of the system, besides using the Lyapunov theory to design the stability of the error signal, the stability design of the control law is also needed.