## 2 Preliminaries and Problem Formulation
### Problem Formulation  
The MASs consist of \(N\) follower agents and the dynamics of the \(i\)th agent is described as:  
\[y_{i}(k+1)=f_{i}\big{(}y_{i}(k),u_{i}(k)\big{)} \tag{2}\]  
where \(y_{i}(k)\in\mathbb{R}\), \(u_{i}(k)\in\mathbb{R}\) are the agent output and control input of the \(i\)th agent at time instant \(k\), respectively, and \(f_{i}(\cdot)\) is an unknown nonlinear function. Denote the output of the leader as \(w_{l}(k)\in\mathbb{R},l=N+1,...,N+M\), which can only be received by a subset of follower agents. **Definition 1** (Convex Set and Convex Hull [43]). : _A set \(\mathcal{C}\subseteq\mathbb{R}^{n}\) is said to be convex if \((1-\alpha)x+\alpha y\in\mathcal{C}\), for any \(\alpha\in[0,1]\) and \(x,y\in\mathcal{C}\). For a finite set of points \(\mathcal{W}=\{w_{1},w_{2},...,w_{q}\}\in\mathbb{R}^{n}\), the convex hull \(\mathcal{C}(\mathcal{W})\) is the minimal convex set containing all points in \(\mathcal{W}\), that is, \(\mathcal{C}(\mathcal{W})=\{\sum_{i=1}^{q}a_{i}w_{i}|w_{i}\in\mathcal{W},a_{i} \geq 0,\sum_{i=1}^{q}a_{i}=1\}\). In particular, when \(\mathcal{W}\subseteq\mathbb{R}\), \(\mathcal{C}(\mathcal{W})=\{w|w\in[\min_{i}w_{i},\max_{i}w_{i}]\}\)._  
**Definition 2** (Containment Control [10]). : _The MASs achieve containment if all followers are driven by a distributed control law and move into the convex hull spanned by the leader set \(\mathcal{W}\), that is, \(\forall i\in 1,...,N\), there exist an element \(\bar{w}_{i}\in\mathcal{C}(\mathcal{W})\), such that_  
\[\lim_{k\rightarrow\infty}\|y_{i}(k)-\bar{w}_{i}\|=0.\]  
_where \(\bar{w}_{i}\) is the convergence value for the \(i\)th follower agent._  
The following assumptions are given before further discussions. **Assumption 1**. : _The partial derivative of \(f_{i}(\cdot)\) with respect to \(u_{i}(k)\) is continuous._  
**Assumption 2**. : _For all \(k\) with finite exceptions, if \(\|\Delta u_{i}(k)\|\
eq 0\), the follower agent dynamics (2) satisfies the following generalized Lipschitz condition, that is_  
\[|\Delta y_{i}(k+1)|\leq b|\Delta u_{i}(k)|\]  
_where \(\Delta y_{i}(k+1)=y_{i}(k+1)-y_{i}(k)\), \(\Delta u_{i}(k)=u_{i}(k)-u_{i}(k-1)\) and \(b\) is a positive constant._  
**Remark 1**. : _Assumption 1 is a general condition for controller design. Assumption 2 gives a bound of the change rate of the agent's output, i.e., the agent's output change should be finite if the control input change is bounded. A wide range of real agent systems satisfy the assumptions. For instance, the velocity change of an autonomous car must be finite with a bounded change of the accelerator. Similar assumptions can also be found in [22; 23; 24; 25; 26; 27; 28; 29; 32; 33]._  
Under Assumptions 1 and 2, the unknown agent dynamics (2) can be transformed into the following dynamic linearization model and then the distributed control law will be designed based on it. **Lemma 1** ([44]). : _Consider follower agents with dynamics (2) satisfying Assumptions 1 and 2. If \(\|\Delta u_{i}(k)\|\
eq 0\) holds, then agent (2) can be transformed into a compact form dynamic linearization (CFDL) data model:_  
\[\Delta y_{i}(k+1)=\phi_{i}(k)\Delta u_{i}(k) \tag{3}\]  
_where the time-varying variable \(\phi_{i}(k)\in\mathbb{R}\) is named as the pseudo partial derivative (PPD), which is bounded for any time instant \(k\)._  
**Remark 2**. : _By virtual of CFDL technique, the unknown nonlinear agent dynamics (2) are now transformed into an incremental form data description (3) in every operation point with a bounded PPD. In implementation, with proper selected algorithms, PPD can be estimated by using the I/O data of the agents and the estimated value can also be proved to be bounded, which is shown in Theorem 1. Here, the CFDL data model is derived under the condition \(\|\Delta u_{i}(k)\|\
eq 0\). In fact, if the case \(\|\Delta u_{i}(k)\|=0\) happens at some sampling time, a new CFDL data model can also be established after shifting \(\gamma_{i}\in\mathbb{Z}^{+}\) time steps until \(u_{i}(k+\gamma_{i})\
eq u_{i}(k)\) holds.