# Trajectory Tracking Control Method Based on Adaptive Higher Order Sliding Mode
## 2 Design of Adaptive High-Order Sliding Mode Trajectory Tracking Controller
### Adaptive Higher-Order Sliding Mode Algorithms  
The nonlinear system is defined as  
\[\dot{z} =f(z)+g(z)v \tag{10}\] \[y =h(z,t)\]  
where \(z\in Z\subset R^{n}\) is the state variable, \(v\in U\subset R\) is the input variable (it should be noted that the input variable is not the actual control variable), functions \(f(z)\) and \(g(z)\) are unknown but are bounded nonlinear functions, and \(y\) is the output observation of the system. To counteract problems related to strong nonlinearity, external disturbance, and the presence of deviation in the initial tension state, this paper introduces a robust, relatively strong sliding mode control. The main idea behind the sliding mode control is the design of the input variable \(v\) to cause the sliding mode variable \(s\) to converge to 0. If \(s=0\), the system trajectory will be attracted to the plane and the desired state will tend to be 0 [22]. To simplify the design of the controller, the following two reasonable assumptions are made for the system nonlinear model (10). Assumption 1: The relative degree of the system (10) is \(\rho\), where \(\rho>1\) and is defined as  
\[s^{(\rho)}=a(z,t)+b(z,t)v \tag{11}\]  
Assumption 2: \(a(z,t)\) and \(b(z,t)\) are unknown but bounded functions. Moreover, \(|a|\leq a_{M}\), \(0\leq b_{m}\leq b\leq b_{M}\) in regards to \(z\in Z\) and \(t>0\); \(a_{M}\),\(b_{m}\) and \(b_{M}\) are positive constants. Figure 1: Optimal trajectory in the two-dimensional plane. Figure 2: Time curve of missile overload. Based on the above assumptions, this paper seeks to enable sliding mode variable \(s\) to approach 0 during a finite time by designing input \(v\) under the conditions of model uncertainty and external disturbances. [PERSON] et al. [23] proposed a class of higher-order sliding mode control algorithms based on the Lyapunov function stability proof. The controller is expressed as  
\[ \begin{split}& v=-k\text{sig}^{0}(s)\\ & s_{i}=\text{sig}^{\frac{\gamma_{1}}{\gamma_{i}}}\left(s^{(i-1)} \right)+k_{i-1}^{\frac{\gamma_{1}}{\gamma_{1}}}s_{i-1}\,i=2,\cdots,\rho\end{split}  \tag{12}\]  
where \(\text{sig}^{k}(s)=|s|^{k}\text{sign}(s)\), relative degree \(\rho\geq 2\), \(r=\left(r_{1},\cdots,r_{\rho}\right)=\left(\rho,\rho-1,\cdots,1\right)\), and \(s_{1}=s\), \(\left(k_{1},\cdots,k_{\rho}\right)\) is the controller gain. To establish the \(\rho\)-order sliding mode control, assuming that gain \(k_{1}\) has been defined, controller gain \(k_{2}\) should satisfy the following conditions:  
1. When \(\rho=2\) \[b_{m}k_{2}-a_{M}\geq\gamma_{1}k_{1}^{2}\] (13)
2. When \(\rho>2\) \[ \begin{split}& k_{i}=\gamma_{i-1}k_{1}^{\frac{\rho}{\gamma_{i} -(i-1)}},\forall i=2,\cdots,\rho-1\\ & b_{m}k_{\rho}-a_{M}\geq\gamma_{1}k_{1}^{\rho}\end{split} \] (14) where parameter \(\gamma_{i-1}\) can be obtained from Table 1.  
To simplify the controller model, using \(\rho=2\) in this paper, we obtain  
\[ \begin{cases}s_{1}=s,\ s_{2}=\text{sig}^{2}(\dot{s})+k_{1}^{2}s,\gamma_{1}=1.2 6\\ \ v=-k_{2}\text{sign}(s_{2})\end{cases}  \tag{15}\]  
To address the problem of different control gains required under different control precisions, this paper introduces an adaptive algorithm [24]. Based on Equation (12), parameter index \(\overline{\alpha}\) is introduced as follows:  
\[v=-k_{\rho}\text{sig}^{\overline{\alpha}}(s_{\rho}) \tag{16}\]  
here, the parameters \(k_{\rho}\) and \(s_{\rho}\) are adjusted as expressed in Equations (12)-(14), and \(\overline{\alpha}\in[0,1]\) satisfies  
\[\overline{\alpha}=\text{max}\left(-\overline{\beta}\sum_{i=1}^{\rho}\frac{|s^ {i-1}|}{|s^{i-1}|+\varepsilon_{s_{i}}}+1,0\right) \tag{17}\]  
where \(\varepsilon_{s_{i}}\) is a constant and \(\overline{\beta}>1\). The stability proof of the adaptive method is given in [24; 25], and the core concepts are as follows:  
1. If the value of the time derivative of \(|s|\) and \(s\) is sufficiently small, a higher-order sliding mode surface is established; that is, \(s=\dot{s}=\cdots=s^{(\rho-1)}=0\), and the exponential term \(\overline{\alpha}\) will approach 1.