# Rotating Lorentz Force Magnetic Bearings' Dynamics Modeling and Adaptive Controller Design
## 4 RLFMB Online Controller Design
### Design of Adaptive Control Method Based on RBF Network  
The tracking error can be employed as an input index and quickly converges to zero in a finite amount of time if RLFMB is required to have a modest static difference and overshoot of the input signal. As a single-input, single-output system, think of RLFMB:  
\[ \begin{array}{l}\dot{\mathbf{x_{1}}}=\mathbf{x_{2}}\\ \dot{\mathbf{x_{2}}}=b\mathbf{u}+f(\mathbf{x_{1}},\mathbf{x_{2}})\end{array} . \tag{14}\]  
In the above formula, \(u\) is the control input, while \(x_{1},x_{2}\) are the angular displacement signal and angular velocity signal, respectively. Because the controller input is not constant at zero, take \(u=\sigma u_{c}\), where \(0<\sigma<1\). If the angle position input is \(x_{d}\) and the tracking error is \(e=x_{1}-x_{d}\), then \(\dot{e}=x_{2}-\dot{x}_{d}\).Take position instruction \(x_{d}=\frac{A\sin\omega_{e}t}{\omega_{e}k}\) and design the control law when \(t\rightarrow\infty\), \(e\to 0\), \(\dot{e}\to 0\). Equation (5) shows that there is an induced electromotive force disturbance and that the input voltage of the RLFMB is not linear with the coil current. The spacecraft is also affected by environmental torque, and the stable platform is affected by other attitude actuators, such as the flywheel and thruster, which will result in an increase in pointing error or even the target being lost during the attitude maneuver. Let \(f(x_{1},x_{2})\) be a function of angular displacement and angular velocity; the specific parameters and form are unknown, and we simulate the disturbance during the stable pointing of RLFMB. The approximation of the unknown perturbation function \(f(x_{1},x_{2})\) can be implemented using the RBF network. The network algorithm is  
\[ \begin{array}{l}h_{i}=g(\left\|\mathbf{x}-\mathbf{c}_{ij}\right\|^{2}/b_{j} ^{\,2})\\ f=\mathbf{W}^{*T}\mathbf{h}(\mathbf{x})+\varepsilon\end{array} . \tag{15}\]  
In the above formula, \(x\) is the network input, \(i\) is the number of inputs to the network, \(j\) is the jth node of the implied layer, \(\mathbf{h}=[h_{1},h_{2},\ldots h_{n}]^{T}\) is the output of a Gaussian function, \(\mathbf{W}^{*}\) is the ideal weight for the network, \(\varepsilon\) is the error approximation of the network, and \(|\varepsilon|\leq\varepsilon_{N}\). Using RBF to approximate the unknown function \(f(x_{1},x_{2})\), the input of the network takes \(\mathbf{x}=[x_{1}\;\;x_{2}]^{T}\), and then the output of the RBF network is  
\[\overset{\wedge}{f}(\mathbf{x})=\overset{\wedge}{\mathbf{W}}^{T}\mathbf{h}(x), \tag{16}\]  
Further expansion yields  
\[\overset{\sim}{f}(\mathbf{x})=f(\mathbf{x})-\overset{\wedge}{f}(\mathbf{x})= \mathbf{W}^{*T}\mathbf{h}(x)+\varepsilon-\overset{\wedge}{\mathbf{W}}^{T} \mathbf{h}(x)=\overset{\sim}{\mathbf{W}}^{T}\mathbf{h}(x)+\varepsilon, \tag{17}\]  
and we define \(\overset{\sim}{\mathbf{W}}=\mathbf{W}^{*}-\overset{\wedge}{\mathbf{W}}\). The designed slipmode function is  
\[s=ce+\dot{e}, \tag{18}\]  
In the above formula, \(c>0\), and  
\[\dot{s}=\dot{c}\dot{e}+\dddot{e}=\dot{c}\dot{e}+b\sigma u_{c}+f(x)-\dddot{x}_{d} =\dot{c}\dot{e}+\beta u_{c}+f(x)-\dddot{x}_{d}, \tag{19}\]  
In the above formula, \(\beta=b\sigma\). Taking \(\lambda=\frac{1}{\beta}\), we designed the Lyapunov function as  
\[V=\frac{1}{2}s^{2}+\frac{|\beta|}{2\gamma}\dddot{\lambda}+\frac{1}{2\gamma}W^{ \widetilde{\gamma}}\dddot{W}. \tag{20}\]  
In the above formula, \(\dddot{\lambda}=\dddot{\lambda}-\lambda,\gamma>0\). The following is a derivation of Equation (20):  
\[\dot{V}=s\dot{s}+\frac{|\beta|}{\gamma}\dddot{\lambda}-\frac{1}{\gamma}W^{ \widetilde{\gamma}}\dddot{W}=s(\dot{c}\dot{e}+\beta u_{c}+f(x)-\dddot{x}_{d})+ \frac{|\beta|}{\gamma}\dddot{\lambda}-\frac{1}{\gamma}\dddot{W}^{\widetilde{ \gamma}}\dddot{W}, \tag{21}\]  
Take  
\[ \begin{array}{c}\alpha=ks+\dot{c}\dot{e}+\dddot{f}(x)-\dddot{x}_{d}+\eta \mathrm{sgns}\\ k>0,\eta\geq\varepsilon_{N}\end{array} , \tag{22}\]  
Equation (20) can be further expanded to  
\[