## 2 Trajectory Tracking of Nonlinear Control Systems  
In this section, we introduce the basics of trajectory tracking control of nonlinear systems based on a Lyapunov-based approach. Let us consider the following input affine nonlinear system:  
\[\dot{x}=f(x)+g(x)u=f(x)+\sum_{i=1}^{m}g_{i}(x)u_{i}, \tag{1}\]  
where \(x\in\mathbb{R}^{n}\) is the state and \(u\in\mathbb{R}^{m}\) is the control input. The mappings \(f,g_{i}:\mathbb{R}^{n}\to\mathbb{R}^{n}\) are assumed to be locally Lipschitz continuous. **Assumption 1** (Desired trajectory). : _Let \(x_{d}:[0,\infty)\to\mathbb{R}^{n};t\mapsto x_{f}(t)\) be a \(C^{2}\) desired trajectory for system (1) such that_  
\[\dot{x}_{d}(t)=f(x_{d}(t))+g(x_{d}(t))u_{r}(t),\;\forall t\in[0,\infty), \tag{2}\]  
_where \(u_{r}:[0,\infty)\to\mathbb{R}^{m};t\mapsto u_{r}(t)\) is the corresponding \(C_{1}\) reference input._  
_Throughout the paper, we suppose that \(\dot{x}_{d}(t)\), \(\dot{x}_{d}(t)\), \(u_{r}(t)\), and \(\dot{u}_{r}(t)\) are bounded on \([0,\infty)\)._  
By using the tracking error \(e(t):=x(t)-x_{d}(t)\) and the new control input \(\tilde{u}:=u-u_{r}(t)\), system (1) can be transformed into the following time-varying error system:  
\[\dot{e}=\tilde{f}(t,e)+\tilde{g}(t,e)\tilde{u}, \tag{3}\]  
where  
\[\tilde{f}(t,e) :=f(e+x_{d}(t))-f(x_{d}(t))+[g(e+x_{d}(t))-g(x_{d}(t))]u_{r}(t), \tag{4}\] \[\tilde{g}(t,e) :=g(e+x_{d}(t)). \tag{5}\]  
Then, the trajectory tracking control problem of system (1) is to design a time-varying continuous state feedback controller \(\tilde{u}=\tilde{k}(t,e)\) such that the origin \(e=0\) of the error system (3) is asymptotically stable. **Remark 1**. : _Since \(e(t)=0\) is equivalent to \(x(t)=x_{d}(t)\), the asymptotic stability of \(e=0\) of the error system (3) implies that the following holds for the original system (1):_  
* _For every initial condition_ \(x_{0}\in\mathbb{R}^{n}\)_,_ \(x(t)\) _is bounded for all_ \(t\in[0,\infty)\)_;_
* \(x(t)\) _converges to_ \(x_{d}(t)\) _as_ \(t\to\infty\)_._  
_Moreover, a trajectory-tracking controller achieves the above conditions given by the following input transformation:_  
\[u=k(t,x)=\tilde{k}(t,x-x_{d})+u_{r}(t), \tag{6}\]  
_where \(\tilde{u}=\tilde{k}(t,e)\) is an asymptotic stabilizing controller for the error system (3)._  
To design an asymptotic stabilizing controller \(\tilde{u}=\tilde{k}(t,e)\), the tracking control Lyapunov function (TCLF), which is defined as follows, plays an important role:  
**Definition 1** (TCLF [30; 31]). : _Let \(D\subset\mathbb{R}^{n}\) be a neighborhood of \(e=0\). Then, a tracking control Lyapunov function (TCLF) for the error system (3) defined on \(D\) is a \(C^{1}\)-differentiable function \(V:[0,\infty)\times D\to\mathbb{R}\) satisfying the following conditions:_  
* _There exist positive definite proper functions_ \(\underline{V},\overline{V}:D\to\mathbb{R}\) _such that_ \[\underline{V}(e)\leq V(t,e)\leq\overline{V}(e),\;\forall t\geq 0,\;\forall e \in D;\] (7)  
**(B2)**: _There exists a positive definite function_ \(Q:D\to\mathbb{R}\) _such that_  
\[\inf_{\tilde{u}}\biggl{(}\frac{\partial V}{\partial t}+L_{\tilde{f}}V+L_{\tilde{g} }V\cdot\tilde{u}\biggr{)}<-Q(e),\ \forall e\
eq 0, \tag{8}\]  
_where_  
\[L_{\tilde{f}}V(t,e) =\frac{\partial V}{\partial e}\tilde{f}(t,e),\ L_{\tilde{g}}V(t,e )=\frac{\partial V}{\partial e}\tilde{g}(t,e)=[L_{\tilde{g}_{1}}V,\ldots,L_{ \tilde{g}_{n}}V], \tag{9}\] \[L_{\tilde{g}_{i}}V(t,e) =\frac{\partial V}{\partial e}\tilde{g}_{i}(t,e)\ (i=1,\cdots,m).\]  
_Moreover, if the conditions (B1)-(B2) holds on \(D=\mathbb{R}^{n}\), \(V(t,e)\) is said to be a global TCLF._  
By using a TCLF \(V(t,e)\), we can construct the following Sontag-type trajectory-tracking controller:  
**Theorem 1** (Theorem 5.5 of [30]). : _Let \(V(t,e)\) be a TCLF for the error system (3). Then, the state feedback controller:_  
\[\dot{x}_{d}(t)=f(x_{d}(t))+g(x_{d}(t))u_{r}(t),\;\forall t\in[0,\infty), \tag{2}\]0 \[p_{1}(t,e) :=