## 5 Simulations Results
### Hypothetical Model  
Consider the following uncertain nonlinear SISO system  
\[\left\{ \begin{array}{c}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=f(x_{1},x_{2})+w(t)+b(t)u\\ y=x_{1}\end{array} \right. \tag{50}\]  
where \(f(x_{1},x_{2})=a_{1}x_{1}+a_{2}sin(x_{2})\), \(a_{1}=0.2\), \(a_{2}=0.1\), \(b(t)=(1+\text{a}_{3}\sin(t))\), \(a_{3}=0.1\), and the exogenous disturbance \(w(t)\) is given as \(w(t)=\exp(-t)\cos(t)\). In this example, \(L(t)=f(x_{1},x_{2})+w(t)+b(t)u-b_{0}u\). This system is uncertain due to the time-varying parameter \(b(t)\) and time-varying periodic disturbance \(w(t)\) with varying amplitude and constant frequency. Firstly, the Conventional ADRC (C-ADRC), given in Figure 1 was first applied on Equation (50) to reject the generalized disturbance \(L(t)\) from Equation (50) with the following elements,  
(a) LESO:  
\[\left\{ \begin{array}{c}\dot{x}_{1}=\hat{x}_{2}+\beta_{1}(y-\hat{x}_{1})\\ \dot{x}_{2}=\hat{x}_{3}+\beta_{2}(y-\hat{x}_{1})\\ \dot{\hat{x}}_{3}=\beta_{3}(y-x_{1})\end{array} \right. \tag{51}\]where \(\hat{x}=\left( \begin{array}{cc}\hat{x}_{1}&\hat{x}_{2}&\hat{x}_{3}\end{array}  \right)^{T}\) is the observer state vector, and \(\beta=\left( \begin{array}{cc}\beta_{1}&\beta_{2}&\beta_{3}\end{array}  \right)^{T}=\left(a_{1}\omega_{0}\;a_{2}\omega_{0}^{2}\;a_{3}\omega_{0}^{3} \right)^{T}\) is the observer gain vector. The design parameters of the LESO were set to \(a_{1}=0.0255\), \(a_{2}=0.2400\), \(a_{3}=0.0717\), \(b_{0}=1\), and \(\omega_{0}=100\). 2. The NLSEF control law: \[u=fal(\overline{e}_{1}.\alpha_{1}.\delta_{1})+fal(\overline{e}_{2}.\alpha_{2}.\delta_{2})-\frac{\hat{x}_{3}}{b_{0}}\] (52) where \(fal(\cdot)\) is described as in Equation (7), and \(e=\left(\overline{e}_{1}\;\overline{e}_{2}\right)^{T}\) is the tracking error vector which can be defined as \(\overline{e}_{i}=r_{i}-\hat{x}_{i}\), \(i=1,\;2\). The design parameters of the control law were set to \(\alpha_{1}=0.0047\), \(\delta_{1}=0.0158\), \(\alpha_{2}=0.0498\), and \(\delta_{2}=0.3316\). 3. The TD is given as [11]: \[\left\{ \begin{array}{c}\dot{r}_{1}=r_{2}\\ \dot{r}_{2}=-R\;sign\Big{(}r_{1}-r(t)+\frac{r_{2}\;|r_{2}|}{2R}\Big{)}\end{array} \right.\] (53) where \(r_{1}\) is tracking signal of the input \(r\), and \(r_{2}\) tracking signal of the derivative of the input \(r\), and where \(R=31.6350\). The Novel-ADRC (N-ADRC) based on nested LESO was also implemented for the system Equation (50) with the following configuration,  
1. Inner-loop LESO The inner-loop LESO is the same as the conventional LESO of Equation (51) with the same set of parameter values. 2. Outer-loop LESO \[\left\{ \begin{array}{c}\dot{\hat{z}}_{1}=\hat{z}_{2}+l_{1}(y-\hat{z}_{1})\\ \dot{\hat{z}}_{2}=\hat{z}_{3}+l_{2}(y-\hat{z}_{1})\\ \dot{\hat{z}}_{3}=l_{3}(y-\hat{z}_{1})\end{array} \right.\] (54) where \(\hat{z}=\left( \begin{array}{cc}\hat{z}_{1}&\hat{z}_{2}&\hat{z}_{3}\end{array}  \right)^{T}\) is the observer state vector, and \(l=\left( \begin{array}{cc}l_{1}&l_{2}&l_{3}\end{array} \right)^{T}=\left(a_{1 }\omega_{0}^{\prime}\;a_{2}\Big{(}\omega_{0}^{\prime}\Big{)}^{2}\;a_{3}\Big{(} \omega_{0}^{\prime}\Big{)}^{3}\;\right)^{T}\) is the observer gain vector. The design parameters where selected as \(a_{1}=0.1305\), \(a_{2}=0.0922\), \(a_{3}=0.5119\), and \(b_{0}=1\), and \(\omega_{0}^{\prime}=22.83\). 3. The control law is selected as in Equation (52) with the same parameter values and tracking error vector defined as \(\overline{e}_{i}=r_{i}-\hat{z}_{i}\), \(i=1,\;2\) as illustrated in Figure 2.
4. The TD for the N-ADRC is identical to Equation (53) with the same parameter values. Both controllers and the suggested system were numerically simulated using MATLAB(r)/Simulink(r) ODE45 solver for models with continuous states. The reference input (\(r(t)\)) to the system was \(\cos(0.5t)\) applied at \(t=0\) sec. Two test conditions were considered for this work.