# \(L_{1}\) Adaptive Control for a Vertical Rotor Orientation System
## 2 Problem Statement
\begin{array}{cccc}0&0&1&0\\ 0&0&0&1\\ \frac{Pc}{J_{xy0}}&0&-\frac{J_{z0}\omega_{z}}{J_{xy0}}\\ 0&\frac{Pc}{J_{xy0}}&\frac{J_{z0}\omega_{z}}{J_{xy0}}&0\end{array} \right]\]  
\[\mathbf{A}_{\theta}=\left[ \begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ -\frac{Pc\theta_{1}}{J_{xy0}(J_{xy0}+\theta_{1})}&0&-\frac{\omega_{z}(J_{xy0} \theta_{2}-J_{z0}\theta_{1})}{J_{xy0}(J_{xy0}+\theta_{1})}\\ 0&-\frac{Pc\theta_{1}}{J_{xy0}(J_{xy0}+\theta_{1})}&\frac{\omega_{z}(J_{xy0} \theta_{2}-J_{z0}\theta_{1})}{J_{xy0}(J_{xy0}+\theta_{1})}&0\end{array} \right]\]  
\[\mathbf{B}=\left[ \begin{array}{cccc}0&0\\ 0&0\\ \frac{1}{J_{xy0}}&0\\ 0&\frac{1}{J_{xy0}}\end{array} \right],\mathbf{B}_{\theta}=\left[ \begin{array}{ ccccc}0&0\\ 0&0\\ -\frac{\theta_{1}}{J_{xy0}(J_{xy0}+\theta_{1})}&0\\ 0&-\frac{\theta_{1}}{J_{xy0}(J_{xy0}+\theta_{1})}\end{array} \right]\]  
Next, we define:  
\[\mathbf{f}(t,\mathbf{x}(t))=\mathbf{f}_{1}(\mathbf{x})+\mathbf{A}_{\theta}\mathbf{x}(t)+\mathbf{B}_{\theta} \mathbf{u}(t) \tag{6}\]  
and  
\[\mathbf{u}=\mathbf{u}_{\mathrm{m}}+\mathbf{u}_{\mathrm{ad}} \tag{7}\]  
where \(\mathbf{u}_{\mathrm{m}}=-\mathbf{K}_{\mathrm{m}}\mathbf{x}(t)\) is a linear state feedback that renders \(\mathbf{A}_{\mathrm{m}}\triangleq\mathbf{A}-\mathbf{B}\mathbf{K}_{\mathrm{m}}\) Hurwitz. \(\mathbf{u}_{\mathrm{ad}}\) is the adaptive input vector used to compensate for the unknown parameter, system nonlinearity and other disturbances. Then, Equation (4) can be rewritten as:  
\[\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}(\mathbf{u}_{\mathrm{m}}+\mathbf{u}_{\mathrm{ad} })+\mathbf{f}(t,\mathbf{x}(t))\quad\mathbf{x}(0)=\mathbf{x}_{0} \tag{8}\]  
\(\mathbf{f}(t,\mathbf{x})\) should satisfies the following two assumptions:  
**Assumption 1**. : _(Uniform boundedness of \(\mathbf{f}(t,\mathbf{0})\))._  
_For any \(t\geq 0\), there exists \(F>0\), such that \(\mathbf{f}(t,\mathbf{0})\leq F\)._Assumption 2: (Boundedness of partial derivatives): For arbitrary \(\delta>0\), there exist \(d_{f_{x}}(\delta)\) and \(d_{f_{t}}(\delta)\), such that, for arbitrary \(\mathbf{x}\), as long as \(\left\|\mathbf{x}\right\|_{\infty}\leq\delta\), the partial derivatives of \(\mathbf{f}(t,\mathbf{x})\) are bounded:  
\[\left\|\frac{\partial\mathbf{f}}{\partial\mathbf{x}}\right\|_{\infty}\leq d_{f_{x}}( \delta),\ \left\|\frac{\partial\mathbf{f}}{\partial t}\right\|_{1}\leq d_{f_{t}}(\delta) \tag{9}\]  
In this paper, the system equilibrium \(\mathbf{x}=\mathbf{0}\) implies that the rotor is along the vertical direction, which is just the control aim of the system. In other words, the control input must ensure that the equilibrium of Equation (8) is \(\mathbf{x}=\mathbf{0}\). While the rotor lies vertically, the external control torques also need to be zero, i.e., \(\left(\mathbf{u}_{\text{m}}+\mathbf{u}_{\text{ad}}\right)\rvert_{\mathbf{x}=\mathbf{0}}=\mathbf{0}\). Therefore, \(\mathbf{f}(t,\mathbf{0})=\mathbf{0}\) for all \(t\geq 0\), and Assumption 1 is satisfied naturally. Assumption 2 requires that \(\mathbf{f}(t,\mathbf{x})\) is continuous about \(\mathbf{x}\) and slowly-varying about \(t\). \(\mathbf{f}(t,\mathbf{x})\) consists of three parts: \(f_{1}(\mathbf{x})\), \(A_{\theta}\mathbf{x}\) and \(B_{\theta}\mathbf{u}(t)\). It is obvious that \(f_{1}\) and \(A_{\theta}\mathbf{x}\) satisfy the requirement, and the key issue is \(\mathbf{u}(t)\). Equation (7) indicates that \(\mathbf{u}(t)\) contains \(\mathbf{u}_{\text{m}}\) and \(\mathbf{u}_{\text{ad}}\), while \(\mathbf{u}_{\text{m}}=-\mathbf{K}_{\mathbf{m}}\mathbf{x}\) satisfies the requirement, and \(\mathbf{u}_{\text{ad}}\) is an artificial adaptive control input, which can be designed to be continuous and slowly varying. Therefore, Assumption 2 can be satisfied. The control aim is to seek a proper adaptive control signal \(\mathbf{u}_{\text{ad}}\), such that the state in Equation (2) is stable about the origin.