# \(L_{1}\) Adaptive Control for a Vertical Rotor Orientation System
## 4 Adaptive Control System Analysis  
### Boundedness of State Error and Parameter Error  
To evaluate the performance of the adaptive control algorithm, first, we need to analyze the error boundedness. Consider the following prediction error equation derived from Equations (11) and (12):  
\[\dot{\mathbf{e}}(t)=\mathbf{A}_{\mathrm{m}}\mathbf{e}(t)+\mathbf{B}\mathbf{\theta}_{\mathbf{e}}(t)\| \mathbf{x}(t)\|_{L_{\infty}}\quad\mathbf{e}(0)=\mathbf{0} \tag{16}\]  
where \(\mathbf{\theta}_{\mathbf{e}}(t)=\hat{\mathbf{K}}_{\theta}(t)-\mathbf{K}_{\theta}(t)\). For the stability of Equation (16), we have the following proposition:  
**Proposition 1**. : _If the adaptive input \(\mathbf{u}_{\mathrm{ad}}\) is defined as Equation (15), then we have the following result:_  
\[\|\mathbf{e}\|_{L_{\infty}}\leq\sqrt{\frac{\theta_{m}}{\lambda_{\mathrm{min}}(\bm {P})\Gamma}} \tag{17}\]  
_where:_  
\[\theta_{m}\triangleq 4\max_{\mathbf{\theta}\in\Theta}\|\mathbf{\theta}\|^{2}+4\frac{ \lambda_{\mathrm{max}}(\mathbf{P})}{\lambda_{\mathrm{min}}(\mathbf{Q})}\left(d_{\theta }\max_{\mathbf{\theta}\in\Theta}\|\mathbf{\theta}\|\right)\]  
Proof. : First, we consider the following positive definite function:  
\[V\left(\mathbf{e}(t),\mathbf{\theta}_{\mathbf{e}}(t)\right)=\mathbf{e}^{\top}(t)\mathbf{P}\mathbf{e}(t )+\frac{1}{\Gamma}\mathbf{\theta}_{\mathbf{e}}^{\top}(t)\mathbf{\theta}_{\mathbf{e}}(t) \tag{18}\]  
Figure 2: \(L_{1}\) adaptive control system algorithm. Obviously,  
\[V(\mathbf{e}(0),\mathbf{\theta}_{e}(0))=\frac{1}{\Gamma}\mathbf{\theta}_{e}^{\top}(0)\mathbf{ \theta}_{e}(0)\leq\frac{1}{\Gamma}\max_{\mathbf{\theta}\in\Theta}\|\mathbf{\theta}\|^{2} \leq\frac{\theta_{m}}{\Gamma} \tag{19}\]  
The derivative of \(V\) about \(t\) is:  
\[\dot{V}(\mathbf{e},\mathbf{\theta}_{e})= \dot{\mathbf{e}}^{\top}\mathbf{P}\mathbf{e}+e^{\top}\mathbf{P}\mathbf{\dot{e}}+\frac{2 }{\Gamma}\mathbf{\theta}_{e}\dot{\hat{K}}_{\theta}-\frac{2}{\Gamma}\mathbf{\theta}_{e} \dot{\hat{K}}_{\theta} \tag{20}\] \[= -\mathbf{e}^{\top}\mathbf{Q}\mathbf{e}+2\mathbf{\theta}_{e}\left(\mathbf{B}^{\top} \mathbf{P}\mathbf{e}\|\mathbf{x}\|_{L_{\infty}}+\mathrm{Proj}(\mathbf{\hat{K}}_{\theta},-\mathbf{ B}^{\top}\mathbf{P}\mathbf{e}\|\mathbf{x}\|_{L_{\infty}})\right)\] \[-\frac{2}{\Gamma}\mathbf{\theta}_{e}\dot{\hat{K}}_{\theta}\]  
According to the property of the projection operator, it follows that:  
\[2\mathbf{\theta}_{e}\left(\mathbf{B}^{\top}\mathbf{P}\mathbf{e}\|\mathbf{x}\|_{L_{\infty}}+ \mathrm{Proj}(\mathbf{\hat{K}}_{\theta},-\mathbf{B}^{\top}\mathbf{P}\mathbf{e}\|\mathbf{x}\|_{L_{ \infty}})\right)\leq 0 \tag{21}\]  
Then, the derivative yields:  
\[\|\mathbf{e}\|_{L_{\infty}}\leq\sqrt{\frac{\theta_{m}}{\lambda_{\mathrm{min}}(\bm {P})\Gamma}} \tag{17}\]0  
Given that \(\mathbf{\hat{K}}_{\theta}\in\Theta,\mathbf{K}_{\theta}\in\Theta\) and \(\|\dot{\mathbf{K}}_{\theta}\|\leq d_{\theta}\), for any \(\tau\geq 0\), the parameter error yields:  
\[\|\mathbf{e}\|_{L_{\infty}}\leq\sqrt{\frac{\theta_{m}}{\lambda_{\mathrm{min}}(\bm {P})\Gamma}} \tag{17}\]1 \[\|\mathbf{e}\|_{L_{\infty}}\leq\sqrt{\frac{\theta_{m}}{\lambda_{\mathrm{min}}(\bm {P})\Gamma}} \tag{17}\]2  
Therefore, if, for any \(\tau\), there exists \(V(\tau)>\theta_{m}/\Gamma\), the quadratic form of the prediction error has to yield:  
\[\|\mathbf{e}\|_{L_{\infty}}\leq\sqrt{\frac{\theta_{m}}{\lambda_{\mathrm{min}}(\bm {P})\Gamma}} \tag{17}\]3  
Hence:  
\[\|\mathbf{e}\|_{L_{\infty}}\leq\sqrt{\frac{\theta_{m}}{\lambda_{\mathrm{min}}(\bm {P})\Gamma}} \tag{17}\]4  
which means that, for any \(\tau>0\), as long as \(V(\tau)>\theta_{m}/\Gamma\), its derivative yields.