# Robust Backstepping Control of Wing Rock Using Disturbance Observer
## 2 Problem Formulation and Preliminaries
### Review of the Extended State Observer  
An uncertain nonlinear single-input and single-output dynamic system is given by [43]  
\[y^{\prime\left(n\right)}\left(t\right)=f\left(y^{\prime\left(n-1\right)}\left( t\right),\cdots,y^{\prime}\left(t\right),d\left(t\right)\right)+bu\left(t\right) \tag{8}\]  
where \(d(t)\) is the external disturbance; \(b\
eq 0\) is the control gain; \(f\left(y^{\prime\left(n-1\right)}\left(t\right),\cdots,y^{\prime}\left(t \right),d\left(t\right)\right)\) is the unknown nonlinear function. Figure 4: Sideslip angle oscillation with disturbances. Treat \(f\) as an extended system state and define \(\chi_{n+1}=f\). Then, the system (8) can be written in the following augmented state space form [44]:  
\[\dot{\chi}_{1} = \chi_{2}\] \[\vdots\] \[\dot{\chi}_{n} = \chi_{n+1}+bu \tag{9}\] \[\dot{\chi}_{n+1} = \hbar\] \[y^{\prime} = \chi_{1}\]  
where \(\hbar\) represents the time derivative of \(f\), \(\chi=\left[\chi_{1},\chi_{2},\cdots,\chi_{n},\chi_{n+1}\right]^{T}=\left[y^{ \prime},\dot{y}^{\prime},\cdots,y^{\prime(n-1)},f\right]^{T}\in R^{n+1}\), \(u\in R\) and \(y^{\prime}\in R\) are the state, input and output of the system, respectively. Hence, an ESO can be constructed as [44]  
\[\dot{z}_{1} = z_{2}-l_{1}\phi_{1}\left(z_{1}-y^{\prime}\right)\] \[\vdots\] \[\dot{z}_{n-1} = z_{n}-l_{n-1}\phi_{n-1}\left(z_{1}-y^{\prime}\right) \tag{10}\] Figure 4:0 Figure 4:1  
where \(z=\left[z_{1},z_{2},\cdots,z_{n+1}\right]^{T}\in R^{n+1}\) is the state vector of the ESO; \(l_{i},i=1,2,\cdots,n+1\), are the observer gain parameters, and \(\phi_{i},i=1,2,\cdots,n+1\) are the linear functions or nonlinear functions of \(\left(z_{1}-y^{\prime}\right)\) to be chosen. If the functions \(\phi_{i},i=1,2,\cdots,n+1\) and the observer gain parameters \(l_{i},i=1,2,\cdots,n+1\) are chosen properly [45; 46], we can obtain  
Figure 4:2  
In this paper, the RBFNN and ESO technique are combined to improve the robustness of the wing rock controller. Based on the output of the RBFNN and ESO, a robust backstepping attitude control law will be developed. To promote the control law design, the following lemma and assumptions are required. **Lemma 1**. : _[_22_]_ _For bounded initial conditions, if there exists a \(C^{1}\) continuous and positive-definite Lyapunov function \(V(x)\) satisfying \(\pi_{1}\left(\left\|x\right\|\right)\leq V\left(x\right)\leq\pi_{2}\left(\left\| x\right\|\right)\), such that \(\dot{V}\left(x\right)\leq-\kappa V\left(x\right)+c\), where \(\pi_{1},\pi_{2}:R^{n}\to R\) are class \(K\) functions, \(\kappa\) and \(c\) are positive constants, then the solution \(x(t)\) is uniformly bounded._  
**Assumption 1**. : _All the states of the uncertain nonlinear MIMO lateral model (1) are measurable._  
**Assumption 2**. : _For the uncertain nonlinear MIMO lateral model (1), the time derivatives of \(d_{i},i=1,2\) are bounded._  
**Assumption 3**. : _For the uncertain nonlinear MIMO system (1), the desired system output \(y_{d}\) and the derivatives \(\dot{y}_{d}\), \(\dot{y}_{d}\) are bounded. Namely, the set \(\Pi_{0}=\left\{\left(y_{d},\dot{y}_{d},\dot{y}_{d}\right):\left\|y_{d}\right\| ^{2}+\left\|\dot{y}_{d}\right\|^{2}+\left\|\dot{y}_{d}\right\|^{2}\leq M_{0}\right\}\) exists, where \(M_{0}>0\) is an unknown positive constant._  
**Assumption 4**. : _For the uncertain nonlinear MIMO lateral model (1), there exist unknown positive constants \(\bar{\lambda}_{i}\) such that \(\lambda_{\max}(g_{i}g_{i}^{T})\leq\bar{\lambda}_{i}(i=1,2)\) where \(\lambda_{\max}(\bullet)\) denotes the maximum eigenvalue._  
**Remark 1**. : _Considering that the wing rock motion is a typical lateral attitude motion, the flight speed can be assumed to be constant. In practice, the wing rock motion is a limit-cycle oscillation, and the aerodynamic angle \(\beta\) is bounded in the wing rock motion. Hence, invoking the expression of the gain matrices \(g_{1}(x_{1})\), and \(g_{2}(x_{1},x_{2})\) in [40], Assumption 4 is reasonable._  
**Remark 2**.