## 3 Control System Design
### Nonlinear ESO Design  
Suppose that the nonlinear system dynamic is given by  
\[\dot{x}=f(x)+\varphi\cdot u+d, \tag{6}\]  
where \(x\) is the state, \(u\) denotes control input and \(d\) represents the total disturbance. \(f(x)\) is a known nonlinear function, while \(\varphi\) used in this paper is a constant expression. Accordingly, Equation (6) has the same form compared with the pitch angular rate and flight path angle loop equations in system (2). **Assumption 2:**_The disturbance item \(d\) is bounded, and its derivative exists and is also bounded, namely \(\text{max}\Big{\{}|d|,\Big{|}\dot{d}|\Big{\}}\) is bounded._  
_Test d as the augmented state of the nonlinear system; the nonlinear ESO can be constructed as follows:_  
\[\left\{ \begin{array}{l}\dot{\stackrel{{\wedge}}{{x}}}=f(x)+ \varphi\cdot u+\stackrel{{\wedge}}{{d}}+w_{1}\Big{[}\frac{x- \stackrel{{\wedge}}{{x}}}{\varepsilon_{1}}+w_{2}\frac{ \varepsilon_{1}-\varepsilon_{2}}{\varepsilon_{2}}\cdot\text{sat}\Big{(}\frac{x -\stackrel{{\wedge}}{{x}}}{\varepsilon_{1}\omega_{2}}\Big{)}\Big{]} \end{array} \right.,\\ \dot{\stackrel{{\wedge}}{{d}}}=\frac{w_{2}}{\varepsilon_{1}}\Big{[} \frac{x-\stackrel{{\wedge}}{{x}}}{\varepsilon_{1}}+w_{2}\frac{ \varepsilon_{1}^{2}-\varepsilon_{2}^{2}}{\varepsilon_{2}^{2}}\cdot\text{sat} \Big{(}\frac{x-\stackrel{{\wedge}}{{x}}}{\varepsilon_{1}\omega_{2}} \Big{)}\Big{]},\end{array}\right. \tag{7}\]_where \(\overset{\wedge}{x}\) and \(\overset{\wedge}{d}\) are the observer estimated states; \(w_{i}>0\) and \(\varsigma_{j}>0\) are positive observer-designing parameters; \(i=1,2,3\); and \(j=1,2\), satisfying that \(\varsigma_{2}<\varsigma_{1}\ll 1\). The symbol \(sat(\cdot)\) represents the standard unity saturation function, which is defined as \(sat(x)=sign(x)\cdot min\{1,|x|\}\)._  
**Remark 3:**_As can be seen from the expressions of \(d_{\mu}\) and \(d_{q}\) in Equation (2), the uncertain items (including parametric uncertainties, un-modelled additional force coefficient and coupling torque coefficient) are considered within the composite disturbance. To deal with time-varying disturbances and uncertain or un-modelled dynamics, several potential solutions have been proposed for unmanned vehicles such as the intelligent control method based on recurrent neural networks (NN) [39] and fuzzy logic system (FLS) [40]. The core of these controllers is to adaptively estimate or compensate for the model uncertainties and unknown nonlinear functions approximately based on the universal approximation ability of NN or FLS. [PERSON] et al. [41] designed two iterative learning control schemes for suppressing the uncertainties and distributed disturbances online. Nevertheless, the above methods suffer from the heavy computational load and slow convergence rate as multiple iterations are needed. During the flight process for the UAV with multiple control surfaces, uncertainties derived from additional force and coupling torque may repeatedly change. To obtain better real-time performance, the proposed observer is adopted to tackle the uncertain dynamics._  
**Remark 4:**_Essentially speaking, in most of the existing literatures, the design process of disturbance observers is a trade-off between the transient performance and the steady-state error. The high-gain observer has advantages in accelerating the convergence speed and in improving the robust performance against large uncertainties, but meanwhile, high gain leads to the notorious peaking value problem in the initial stage caused by different initial values of the system and the ESO. It can be seen in the proposed observer that saturation function is invoked to weaken the peaking value problem.