## 2 Problem Formulation  
This paper studies the control problem of an FJM with one flexible joint and one rigid link, whose physical model is shown in Figure 1.  
The parameters and variables in Figure 1 are given as follows: \(q\) and \(\theta\) represent the angles of the link and the motor shaft, respectively; \(u\) is the control torque generated by the motor; \(g\) is the acceleration of gravity; \(M\), \(K\) and \(L\) are the mass of the link, the spring stiffness of the flexible joint, the distance between the flexible joint, and the mass centre of the link, respectively; \(I\) and \(J\) are the rotational inertia of the rigid link and the motor, respectively. According to the Euler-Lagrangian equation, the mathematical model of the single-link FJM is obtained as [16; 48]:  
\[\left\{ \begin{array}{l}I\bar{q}+K(q-\theta)+MgL\sin q=w_{1}\\ J\bar{\theta}-K(q-\theta)=u+w_{2}\end{array} \right. \tag{1}\]  
where \(w_{1}\) and \(w_{2}\) are the lumped disturbances including system friction/natural damping terms and unknown external disturbances. Define \(x=[x_{1},x_{2},x_{3},x_{4}]^{\top}=[q,\theta,\dot{q},\dot{\theta}]^{\top}\) as the state variable, and \(y\) as the output of the system, then the state-space equation of model (1) is written as:  
\[\left\{ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=x_{3}+g_{1}(x)+d_{1}\\ \dot{x}_{3}=x_{4}\\ \dot{x}_{4}=\frac{1}{J}u+g_{2}(x)+d_{2}\\ y=x_{1}\end{array} \right. \tag{2}\]  
Figure 1: Physical model of a single-link FJM. where  
\[ \begin{split} g_{1}(x)&=-x_{3}-\frac{K}{I}(x_{1}-x_{3})- \frac{MgL}{I}\sin x_{1}\\ g_{2}(x)&=\frac{K}{J}(x_{1}-x_{3})\\ d_{1}&=\frac{1}{I}w_{1}\\ d_{2}&=\frac{1}{J}w_{2}\end{split}  \tag{3}\]  
Note that the terms \(g_{1}(x)\) and \(g_{2}(x)\) are straightforwardly derived from (1), and they are treated as nominal functions of the FJM model, which will be used for the control design. The main feature of the FJM system expressed as (2) is that it contains two terms \(d_{1}\) and \(d_{2}\), which are the mismatched and matched lumped disturbances, respectively. Generally speaking, the mismatched lumped disturbance \(d_{1}\) is determined by the friction term and external disturbance, while \(d_{2}\) represents the lumped disturbance caused by the natural damping and the disturbance generated by the control torque. In this paper, the mismatched and matched disturbances are simultaneously estimated by employing two RESOs, and the estimation values are fed back to the controller to compensate for their effects. Considering the physical limitations on the FJM in practical applications, some assumptions are given as follows. **Assumption 1** ([30]). : _The desired reference trajectory \(y_{\rm d}\) and its first-order time derivative \(\dot{y}_{\rm d}\) are available._  
**Assumption 2** ([43]). : _The lumped disturbances \(d_{1}\), \(d_{2}\), and their derivatives are all bounded, i.e., there exist positive constants \(\bar{d}_{1}\), \(\bar{d}_{1d}\), \(\bar{d}_{2}\) and \(\bar{d}_{2d}\) that satisfy \(|d_{1}|\leq\bar{d}_{1}\), \(|\dot{d}_{1}|\leq\bar{d}_{1d}\), \(|d_{2}|\leq\bar{d}_{2}\), \(\left|d_{2}\right|\leq d_{2d}\), \(\forall t\in[0,+\infty)\)._  
To facilitate the control design and stability analysis, the following lemmas are needed in the subsequent context. **Lemma 1** ([25]). : _Consider the second-order command filters (SCFs) defined as_  
\[\left\{ \begin{array}{l}\dot{z}_{i}=z_{\rm id}\\ \dot{z}_{\rm id}=-2\zeta\omega_{\rm n}z_{\rm id}-\omega_{\rm n}^{2}(z_{i}- \alpha_{\rm i})\end{array} \right. (i=1,2,3) \tag{4}\]  
_where \(\alpha_{i}\) are the inputs and \(\alpha_{i}^{c}=z_{i}\), \(\dot{\alpha}_{i}^{c}=z_{\rm id}\) are the outputs of the SCFs; \(\zeta\) and \(\omega_{\rm n}\) are the damping ratio and bandwidth, respectively. Set the initial conditions as \(z_{i}(0)=\alpha_{i}(0)\) and \(z_{\rm id}(0)=0\).