## 3 Main Results  
### Preliminaries and Notations  
Consider the following nonlinear system:  
\[\dot{x}(t)=f(t,x),x(0)=0,f(x(0))=0, \tag{7}\]  
where \(x(t)\in R^{n},f(t,x)\) denotes the smooth nonlinear function and it is assumed that the origin is the equilibrium point of system (7). **Definition 1** ([25]). : _Let us assume that system (7) is globally asymptotically stable. If there exists a finite convergence time \(T\) for all \(t\geqslant T\) satisfying \(x(t)\equiv 0\), then system (7) is globally finite-time-stable._  
**Definition 2** ([34]). : _System (7) is globally finite-time-stable. For \(\forall x_{0}\in R^{n}\), there exists \(T\leqslant T_{s}\), \(T_{s}\in R\), denoting a bounded positive value; then, system (7) is globally fixed-time-stable._  
**Lemma 1** ([35]). : _If a continuous radial bounded function \(V(x):\)\(R^{n}\to R_{+}\cup\{0\}\) satisfies \(\dot{V}(x)\leqslant-\big{(}j_{1}V_{1}^{a}(x)-j_{2}V_{2}^{a}(x)\big{)}^{a_{3}}+\theta\), with \(V(x)=0\Leftrightarrow x=0\), where \(\theta\in(0,+\infty)\), \(\{a_{1},a_{2},a_{3},j_{1},j_{2}\}>  
positive constants. The FTESO (13) gains are assigned to ensure that the matrices \(A\), \(A_{1}\) are Hurwitz, with \(A=\left[ \begin{array}{ccc}-\mu_{1}&1&0\\ -\mu_{2}&0&1\\ -\mu_{3}&0&0\end{array} \right]\), \(A_{1}=\left[ \begin{array}{ccc}-\varepsilon_{1}&1&0\\ -\varepsilon_{2}&0&1\\ -\varepsilon_{3}&0&0\end{array} \right]\). Theorem 1: _The states \(\theta\), \(\omega\) and the disturbance \(\rho_{4}\) can be estimated using FTESO (13) in fixed time \(T_{1}\), with \(T_{1}\) being denoted as_  
\[T_{1}\leqslant\frac{\lambda_{\max}^{\rho}(P)}{r_{2}\rho}+\frac{1}{r_{2}\sigma \varpi^{\sigma}}, \tag{14}\]  
_where \(\rho=1-\bar{\alpha},\sigma=\bar{\beta}-1\), \(r_{1}=\frac{\lambda_{\min}(Q)}{\lambda_{\max}(P)},r_{2}=\frac{\lambda_{\min}( Q_{1})}{\lambda_{\max}(P_{1})}\), \(\varpi\leqslant\lambda_{\min}(P_{1})\) is a positive constant. The symmetric positive matrices \(P,Q,P_{1},Q_{1}\) satisfy the function_  
\[ \begin{cases}PA+A^{T}P=-Q,\\ P_{1}A_{1}+A_{1}^{T}P_{1}=-Q_{1}.\end{cases}  \tag{15}\]  
Proof: The estimation error of the observer is defined as  
\[ \begin{cases}\bar{\theta}{=}\theta-\bar{\theta},\\ \bar{\omega}{=}\omega-\bar{\omega},\\ \bar{\rho}_{4}{=}\rho_{4}-\hat{\rho}_{4}.\end{cases}  \tag{16}\]  
The derivative of (16) is given as  
\[ \begin{cases}\dot{\bar{\theta}}=\bar{\omega}-\mu_{1}sig^{\alpha_{1}}(\theta- \hat{\theta})-\varepsilon_{1}sig^{\beta_{1}}(\theta-\hat{\theta}),\\ \dot{\omega}=\frac{1}{\bar{\gamma}}\hat{\rho}_{4}-\mu_{2}(sig^{\alpha_{2}}( \theta-\hat{\theta})-\varepsilon_{2}sig^{\beta_{2}}(\theta-\hat{\theta}),\\ \hat{\rho}_{4}=-\mu_{3}sig^{\alpha_{3}}(\theta-\hat{\theta})-\varepsilon_{3} sig^{\beta_{3}}(\theta-\hat{\theta})-\Upsilon\tanh(\theta-\hat{\theta}).\end{cases}  \tag{17}\]  
The remaining proof is similar to that of Theorem 1 in [36] and is thus omitted here due to space constraints. If \(t\geqslant T_{1}\), \(\bar{\theta},\bar{\omega},\bar{\rho}_{4}\) can converge to zero in fixed time. The proof is completed. Remark 1: _The smooth function \(\hat{\rho}_{4}\) is used to approximate \(\rho_{4}\). It should be noted that \(\hat{\rho}_{4}=\rho_{4}\) cannot be obtained due to problems such as sampling noise and sampling delay._  
According to (11), an auxiliary variable is defined as \(\psi=x\cos(\theta)+y\sin(\theta)\), with its derivative being given as \(\dot{\psi}=v+\omega(-x\sin(\theta)+y\cos(\theta))\). Then, the FTESO for estimating the linear velocity \(v\) and the lumped disturbance signal \(\rho_{3}\) is shown as follows:  
\[ \begin{cases}\dot{\bar{\psi}}=\bar{\sigma}+\bar{\omega}(-x\sin(\theta)+y\cos( \theta))+\mu_{1}sig^{\alpha_{1}}(\psi-\bar{\psi})+\varepsilon_{1}sig^{\beta_{ 1}}(\psi-\bar{\psi}),\\ \dot{\psi}=\frac{1}{mr}\tau_{v}+\frac{1}{m}\hat{\rho}_{3}+\mu_{2}sig^{\alpha_{2} }(\psi-\hat{\psi})+\varepsilon_{2}sig^{\beta_{2}}(\psi-\hat{\psi}),\\ \dot{\rho}_{3}=\mu_{3}sig^{\alpha_{3}}(\psi-\hat{\psi})+\varepsilon_{3}sig^{ \alpha_{3}}(\psi-\hat{\psi})+\Upsilon\tanh(x).\end{cases}