## 3 Problem Formulation  
In this article, we focus on trajectory tracking of the following input-constrained nonlinear system:  
\[\dot{x}=f(x)+g(x)u,\ u\in U(x)\subset\mathbb{R}^{m}, \tag{13}\]  
where \(U(x)\) is a _convex input constraint_ defined as follows:  
**Definition 2** (Convex input constraint). : _A continuous-set-valued map \(U:\mathbb{R}^{n}\to 2^{\mathbb{R}^{m}}\) is said to be a convex input constraint if the following conditions hold:_  
**(H1)**: \(U(x)\) _is defined by_ \(U(x):=\{u\in\mathbb{R}^{m}\mid G(x,u)\leq 0\}\)_, where the mapping_ \(G:\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}\) _is_ \(C^{1}\) _on_ \(\mathbb{R}^{n}\times\mathbb{R}^{m}\) _and strictly convex with respect to_ \(u\)_;_
**(H2)**: \(u=0\in\dot{U}(x),\ \forall x\in\mathbb{R}^{n}\)_;_
**(H3)**: _There exists a compact set_ \(\overline{U}\subset\mathbb{R}^{m}\) _such that_ \(U(x)\subset\overline{U},\ \forall x\in\mathbb{R}^{n}\)_._  
Since level sets of strictly convex functions are strictly convex sets, \(U(x)\) is a compact strictly convex set with the \  
\[G_{1}(x,u)=\|u\|-\alpha. \tag{15}\]  
_Note that the expression of \(U(x)\) is not unique. In fact, by using_  
\[G_{2}(x,u)=\|u\|^{2}-\alpha^{2}, \tag{16}\]  
_the constraint (14) is also represented by_  
\[U=\{u\in\mathbb{R}^{m}\mid G_{2}(x,u)\leq 0\}. \tag{17}\]  
**Example 2**. : _Let us consider the following state-dependent constraint for \(u\in\mathbb{R}\):_  
\[-\underline{a}(x)\leq u\leq\overline{a}(x), \tag{18}\]  
_where \(\underline{a}(x)\) and \(\overline{a}(x)\) are continuous non-negative functions._  
_If the constraint is symmetric, i.e., there exists \(a(x)\) such that \(\underline{a}(x)=\overline{a}(x)=a(x)\), the constraint (18) reduces to the following (state-dependent) norm constraint:_  
\[\|u\|\leq a(x). \tag{19}\]  
_In the same manner as Example 1, this constraint is also represented by the form of (H1) with \(G(x,u)=\|u\|-a(x)\). Note, however, that asymmetric constraints cannot be represented as the form of a simple norm constraint. Note, however, that asymmetric constraints are not represented as the form of a simple norm constraint._  
_In contrast, we can represent asymmetric constraints as convex constraints. Let \(b(x)\) be the function defined by_  
\[b(x)=(\overline{a}(x)-\underline{a}(x))/2. \tag{20}\]  
_Then, the constraint (18) becomes_  
\[-\frac{\overline{a}(x)+\underline{a}(x)}{2}\leq u+b(x)\leq\frac{\overline{a}( x)+\underline{a}(x)}{2}. \tag{21}\]  
_This is equivalent to_  
\[\|u+b(x)\|\leq\frac{\overline{a}(x)+\underline{a}(x)}{2}, \tag{22}\]  
_and hence, the corresponding function \(G(x,u)\) of (H1) is given by_  
\[G(x,u)=\|u+b(x)\|-\frac{\overline{a}(x)+\underline{a}(x)}{2}. \tag{23}\]  
_Note that (23) is not a simple norm constraint, but is a convex constraint for \(u\). This example shows that convex constraints can deal with asymmetric input constraints. Note also that this type of asymmetric state-dependent constraint appears in some practical problems, such as vehicle tracking control subject to tire slip angle constraints (see, e.g., [36])._  
We introduce the following additional assumption to consider the input-constrained trajectory tracking problem:  
**Assumption 2**. : _Let \(x_{d}(t)\) and \(u_{r}(t)\) be a reference trajectory and the corresponding reference input satisfying Assumption 1, respectively. Then, we assume that the following holds:_  
\[G_{1}(x,u)=\|u\|-\alpha. \tag{15}\]0According to the discussion in Section 2, system (13) can be transformed into the following constrained error system:  
\[G_{1}(x,u)=\|u\|-\alpha. \tag{15}\]1  
where  
\[G_{1}(x,u)=\|u\|-\alpha. \tag{15}\]2 \[G_{1}(x,u)=\|u\|-\alpha. \tag{15}\]3  
It should be noted that the original convex input constraint \(u\in U(x)\) is transformed into the time-varying input constraint \(\tilde{u}\in\tilde{U}(t,e)\) due to the input transformation \(\tilde{u}=u-u_{r}(t)\). In the following, we refer to this constraint as the _time-varying convex input constraint_. **Remark 3**.