## 3 Controller Design  
### Preliminaries  
Since the HMW has strong nonlinearity and uncertainty and is easily affected by external disturbances, a small bounded convergence time is necessary for the controller. Nevertheless, traditional methods based on finite-time stability are unable to guarantee the system states to converge within a bounded settling time, which has led to limitations in the application of such methods. Afterward, the introduction of the fixed-time stability concept provides new techniques. Before starting controller design, it is necessary to introduce relevant concepts and lemmas. Consider the following system:  
\[\dot{x}=g(t,x),\,x(0)=x_{0} \tag{21}\]  
where \(x\in\mathbb{R}^{n}\) is the system state vector, and \(g(t,x):\mathbb{R}_{+}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is a nonlinear function. The solutions of Equation (21) are defined in the sense of Filippov, and it is assumed that the system has a zero equilibrium point. **Definition 1** ([26]). : _The equilibrium point \(x=0\) of the system (21) is said to be a fixed-time stable equilibrium point if it is finite-time stable with a bounded settling-time function \(T(x_{0})\) whose upper bound is a positive number \(T_{max}>0\); thus, \(T(x_{0})\leq T_{max},\forall x_{0}\in\mathbb{R}^{n}\)._  
**Definition 2** ([26]). : _The set \(M\) is said to be a fixed-time attractive for the system (21) if it is globally finite-time attractive with a bounded settling-time function \(T(x_{0})\) whose upper bound is a positive number \(T_{max}>0\); thus, \(T(x_{0})\leq T_{max},\forall x_{0}\in\mathbb{R}^{n}\)._  
**Lemma 1** ([26; 31]). : _Suppose that there exists a Lyapunov function \(V(x):\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}\cup\{0\}\) satisfying_  
\[\dot{V}(x)\leq-\left(l_{1}V^{m_{1}}(x)+l_{2}V^{m_{2}}(x)\right)^{k} \tag{22}\]  
_where \(l_{1}\), \(l_{2}\), \(m_{1}\), \(m_{2}\) and \(k\) are some positive constants satisfying \(m_{1}k>1\), \(0<m_{2}k<1\); then, the equilibrium point \(x=0\) is fixed-time stable, and the settling time satisfying \(T(x_{0})\leq T_{max}=\frac{1}{l_{1}(m_{1}k-1)}+\frac{1}{l_{2}^{k}(1-m_{2}k)}, \forall x_{0}\in\mathbb{R}^{n}\)._  
In Lemma 1, set \(k=1\), and the following corollary can be obtained. **Corollary 1**. : _Suppose that there exists a Lyapunov function \(V(x):\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}\cup\{0\}\) satisfying_  
\[\dot{V}(x)\leq-l_{1}V^{m_{1}}(x)-l_{2}V^{m_{2}}(x) \tag{23}\]  
_where \(l_{1}\), \(l_{2}\), \(m_{1}\), \(m_{2}\) are some positive constants satisfying \(m_{1}>1\), \(0<m_{2}<1\); then, the equilibrium point \(x=0\) is fixed-time stable, and the settling time \(T(x_{0})\leq T_{max}=\frac{1}{l_{1}(m_{1}-1)}+\frac{1}{l_{2}(1-m_{2})}, \forall x_{0}\in\mathbb{R}^{n}\)._  
**Lemma 2** ([33]). : _Suppose that there exists a Lyapunov function \(V(x):\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}\cup\{0\}\) satisfying_  
\[\dot{V}(x)\leq-l_{1}V^{m_{1}}(x)-l_{2}V^{m_{2}}(x)+\varsigma \tag{24}\]  
_where \(l_{1}\), \(l_{2}\), \(m_{1}\), \(m_{2}\) and \(k\) are some positive constants satisfying \(m_{1}>1\), \(0<m_{2}<1\), and \(\varsigma\) is a small positive number; then, the system is fixed-time stable and the system states will converge to an arbitrarily small neighborhood of the origin, i.e.,\(V(x)\leq 2\gamma\) with \(l_{1}\gamma^{m_{1}}+l_{2}\gamma^{m_{2}}=\varsigma\) in fixed time bounded by_  
\[T<T_{max}=\frac{1}{l_{1}}\frac{1}{m_{1}-1}+\frac{1}{l_{2}(2^{m_{2}}-1)}\frac{ 1}{1-m_{2}} \tag{25}\]  
**Lemma 3** ([34]). : _Consider the following system:_  
\[\left\{ \begin{array}{l}\dot{\eta}_{1}=-k_{1}\phi_{1}\big{(}\frac{\eta_{1}}{ \varepsilon}\big{)}+\eta_{2}\\ \dot{\eta}_{2}=-\frac{k_{2}}{\varepsilon}\phi_{2}\big{(}\frac{\eta_{1}}{ \varepsilon}\big{)}+\xi(t)\end{array} \right.