# \(L_{1}\) Adaptive Control for a Vertical Rotor Orientation System
## 4 Adaptive Control System Analysis
### Boundedness of State Variables  
To analyze the boundedness of the system state \(\boldsymbol{x}(t)\), we first consider the following ideal system:  
\[\dot{\boldsymbol{x}}_{\text{id}}(t)=\boldsymbol{A}_{\text{m}} \boldsymbol{x}_{\text{id}}(t)+\boldsymbol{B}\left(\boldsymbol{K}_{\boldsymbol {\theta}}\|\boldsymbol{x}_{\text{id}}(t)\|\infty+\boldsymbol{u}_{\text{id}}(t )\right)\quad\boldsymbol{x}_{\text{id}}(0)=\boldsymbol{x}_{0} \tag{31}\]  
The ideal control signal is:  
\[\boldsymbol{u}_{\text{id}}(s)=-C(s)\boldsymbol{\eta}(s) \tag{32}\]  
where \(\boldsymbol{\eta}(s)\) is the Laplace transform of \(\boldsymbol{\eta}(t)=\boldsymbol{K}_{\boldsymbol{\theta}}(t)\|\boldsymbol{x }_{\text{id}}(t)\|_{\infty}\).  
We define \(G(s)\triangleq(s\mathbb{I}_{4}-\boldsymbol{A}_{\text{m}})^{-1}\boldsymbol{B }(s\mathbb{I}_{2}-C(s))\) and \(L\triangleq\max_{\theta\in\Theta}\|\theta\|_{1}\), where \(\mathbb{I}_{4}\) and \(\mathbb{I}_{2}\) are, respectively, fourth-order and second-order identity matrices. According to the definition of \(\boldsymbol{K}_{\boldsymbol{\theta}}\) in Equation (10), \(\|\boldsymbol{K}_{\boldsymbol{\theta}}\|_{L_{\infty}}\leq L\). It can be proven that \(\boldsymbol{x}_{\text{id}}\) in Equation (31) is banded input banded state (BIBS) stable when \(\|G(s)\|_{L_{1}}L<1\).  
Proof.: According to the definition of \(G(s)\), Equation (31) can be transformed to:  
\[\boldsymbol{s}\boldsymbol{x}_{\text{id}}(s)-\boldsymbol{x}(0)= \boldsymbol{A}_{\text{m}}\boldsymbol{x}_{\text{id}}(s)+\boldsymbol{B}(s \mathbb{I}_{2}-C(s))\boldsymbol{\eta}(s) \tag{33}\]  
After performing some algebraic operations, we get:  
\[\boldsymbol{x}_{\text{id}}(s)=G(s)\boldsymbol{\eta}(s)+\boldsymbol{x}_{\text{ in}}(s) \tag{34}\]  
where \(\boldsymbol{x}_{\text{in}}(s)=(s\mathbb{I}_{4}-\boldsymbol{A}_{\text{m}})^{-1 }\boldsymbol{x}_{0}\). For \(\forall\tau\in(0,\,\infty)\) and \(t\in[0,\,\tau]\), the following bound exists [12]:  
\[\|\boldsymbol{x}_{\text{id}\,\tau}\|_{L_{\infty}}\leq\|G(s)\boldsymbol{K}_{ \boldsymbol{\theta}}\|_{L_{1}}\cdot\|\|\boldsymbol{x}_{\text{id}}(t)\|_{ \infty\,\tau}\|_{L_{\infty}}+\|\boldsymbol{x}_{\text{in}\,\tau}\|_{L_{\infty}} \tag{35}\]  
According to the definition of the \(\infty\)-norm and the \(L_{1}\)-norm, the following bound exists:  
\[\|\boldsymbol{x}_{\text{id}\,\tau}\|_{L_{\infty}}\leq\|\,\| \boldsymbol{x}_{\text{id}\,\tau}(t)\|_{\infty}\|_{L_{\infty}}\] (36) Meanwhile, since \(\|G(s)\boldsymbol{K}_{\boldsymbol{\theta}}\|_{L_{1}}\leq\|G(s)\|_{L_{1}}\| \boldsymbol{K}_{\boldsymbol{\theta}}\|_{\infty}\leq\|G(s)\|_{L_{1}}L<1\), therefore, \[\|\boldsymbol{x}_{\text{id}\,\tau}\|_{L_{\infty}}\leq\frac{\| \boldsymbol{x}_{\text{in}}\|_{L_{\infty}}}{1-\|G(s)\boldsymbol{K}_{\boldsymbol {\theta}}\|_{L_{1}}} \tag{37}\]  
According to the definition of \(\boldsymbol{x}_{\text{in}}\), if it is bounded, \(\boldsymbol{x}_{\text{id}}\) will be bounded. Therefore, System (31) is BIBS stable.  
The system in Equation (31) is equivalent to the system predictor in Equation (12) while \(\boldsymbol{\hat{K}}_{\boldsymbol{\theta}}=\boldsymbol{K}_{\boldsymbol{\theta}}\). Because \(\boldsymbol{\theta}_{e}\) and \(\boldsymbol{x}_{\text{in}}\) are both bounded, \(\boldsymbol{\hat{x}}\) is bounded. Meanwhile, since \(\boldsymbol{e}=\boldsymbol{\hat{x}}-\boldsymbol{x}\) and \(\boldsymbol{e}\) is also bounded, therefore, \(\boldsymbol{x}\) is bounded, i.e., System (2) is BIBS stable under the \(L_{1}\)-controller.