## 3 Results
### Nonlinear Disturbance Observer  
This section aims to employ a finite-time disturbance observer to estimate the time-varying mismatched disturbance existing in (1). Based on the research on globally convergent differentiators in [50], a nonlinear disturbance observer is designed as follows:  
\[\left\{\begin{array}{l}\dot{z}_{1}=v_{1}+x_{2}\\ v_{1}=-\lambda_{1}L^{1/3}|z_{1}-x_{1}|^{2/3}\text{sign}(z_{1}-x_{1})-\mu_{1}(z_ {1}-x_{1})+z_{2}\\ \dot{z}_{2}=v_{2}\\ v_{2}=-\lambda_{2}L^{1/2}|z_{2}-v_{1}|^{1/2}\text{sign}(z_{2}-v_{1})-\mu_{2}(z_ {2}-v_{1})+z_{3}\\ \dot{z}_{3}=-\lambda_{3}\text{Lsign}(z_{3}-v_{2})-\mu_{3}(z_{3}-v_{2})\end{array}\right. \tag{15}\]  
where \(z_{1}=\hat{x}_{1}\), \(z_{2}=\hat{d}_{1}\), \(z_{3}=\hat{\hat{d}}_{1}\), and \(L>0\), \(\lambda_{i},\mu_{i}>0\)\((i=1,2,3)\).  
**Assumption 2**.: _The time-varying mismatched disturbance \(d_{1}(t)\) is assumed to be second-order differentiable and has a Lipschitz constant \(L\)._  
**Theorem 1**.: _Given that the disturbance observer is designed as in (15), if the design parameters are properly selected, the following results can be achieved in finite time:_  
\[z_{1}=x_{1},\ \ z_{2}=d_{1},\ \ z_{3}=\dot{d}_{1} \tag{16}\]  
**Proof of Theorem 1.** For the disturbance observer (15), estimation errors are defined as follows:  
\[\varepsilon_{1}=\tfrac{z_{1}-x_{1}}{L},\ \ \ \varepsilon_{2}=\tfrac{z_{2}-d_{1}}{L},\ \ \ \varepsilon_{3}=\tfrac{z_{3}-\dot{d}_{1}}{L} \tag{17}\]  
By taking the derivative of \(\varepsilon_{1}\), \(\varepsilon_{2}\) and \(\varepsilon_{3}\), the observer error dynamics can be attained as:  
\[\left\{\begin{array}{l}\dot{\varepsilon}_{1}=-\lambda_{1}|\varepsilon_{1}|^{ 2/3}\text{sign}(\varepsilon_{1})-\mu_{1}\varepsilon_{1}+\varepsilon_{2}\\ \dot{\varepsilon}_{2}=-\lambda_{2}\big{|}\varepsilon_{2}-\dot{\varepsilon}_{1 }\big{|}^{1/2}\text{sign}(\varepsilon_{2}-\dot{\varepsilon}_{1})-\mu_{2} \big{(}\varepsilon_{2}-\dot{\varepsilon}_{1}\big{)}+\varepsilon_{3}\\ \dot{\varepsilon}_{3}=-\lambda_{3}\text{sign}(\varepsilon_{3}-\dot{ \varepsilon}_{2})-\mu_{3}(\varepsilon_{3}-\dot{\varepsilon}_{2})-\tfrac{1}{L} \dot{d}_{1}\end{array}\right. \tag{18}\]  
Then, according to Theorem 3.1, Lemma B.1, and their proofs in [51], the estimation errors (18) will converge to zero in finite time, indicating that the nonlinear disturbance observer can accurately estimate the system state \(x_{1}(t)\), the time-varying mismatched disturbance \(d_{1}(t)\), and its derivative \(\dot{d}_{1}(t)\). This completes the proof of Theorem 1.  
**Remark 1**.: _Readers are strongly encouraged to inspect the works [21; 50; 51] and the references therein for better understanding of the rigorous stability analysis of the disturbance observer, and the appropriate selection of its design parameters. It is noted that the observer design parameters, \(\{\lambda_{i},\mu_{i}\}(i=1,2,3)\), are chosen recursively in such a manner that \(\{\lambda_{i},\mu_{i}\}\) provide for the convergence of the observer with the Lipschitz constant \(L\) indicated in Assumption 2. It is also mentioned by the authors that these parameters can be easily changed because the design is not highly sensitive to their values. Nonetheless, the tradeoff is as follows: generally, the greater the parameters, the faster the convergence of the disturbance observer, but the higher the sensitivity to input noises and the sampling step, and the larger the peaking phenomenon in the control signal, if the initial values of the system states and those of the observer states are different. Thus, in the controller design, particularly in the case of different initial values, a tradeoff should be made to ensure fast convergence yet an acceptable peaking phenomenon._