## 3 Main Results  
In this section, the data driven containment control law is first designed under the stationary leaders and fixed topology case. Let \(\xi_{i}(k)\) denote the local containment error:  
\[\xi_{i}(k)=\sum_{j=1}^{N}a_{ij}(y_{j}(k)-y_{i}(k))+\sum_{l=N+1}^{N+M}a_{il}(w_{ l}(k)-y_{i}(k)) \tag{4}\]  
where \(a_{ij}\) (\(i,j\in\mathscr{V}\)) is the entry in the adjacency matrix, and \(y_{i}(k)\) and \(w_{l}(k)\) denote the outputs of the follower agent \(i\) and leader agent \(l\), respectively. From this definition, \(\xi_{j}(k)\) can be regarded as the tracking error of the \(i\)th agent with all its neighboring agents at time instant \(k\). The distributed model free adaptive containment control (MFACC) is designed as follows:  
**Distributed PPD updating algorithm:**  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(k-1)+\frac{\eta\Delta u_{i}(k-1)}{ \mu+|\Delta u_{i}(k-1)|^{2}}\times(\Delta y_{i}(k)-\widehat{\phi}_{i}(k-1) \Delta u_{i}(k-1)) \tag{5}\]  
**PPD reset mechanism:**  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(1),if\ |\widehat{\phi}_{i}(k)|\leq\epsilon\] \[\text{or}\ |\Delta u_{i}(k-1)|\leq\epsilon\] \[\text{or}\ \text{sign}(\widehat{\phi}_{i}(k))\
eq\text{sign}( \widehat{\phi}_{i}(1)) \tag{6}\]  
**Distributed containment input law:**  
\[u_{i}(k)=u_{i}(k-1)+\frac{\rho_{i}\widehat{\phi}_{i}(k)}{\lambda+|\widehat{ \phi}_{i}(k)|^{2}}\xi_{i}(k) \tag{7}\]  
where \(\rho_{i}\in(0,1]\), \(\eta\in(0,2]\), \(\mu>0\) and \(\lambda>0\) are adjustable parameters, \(\widehat{\phi}_{i}(k)\) is the estimated value of \(\phi_{i}(k)\), \(\widehat{\phi}_{i}(1)\) is the initial value of \(\widehat{\phi}_{i}(k)\), and \(\epsilon>0\) is a preset small constant. **Remark 3**. : _The PPD is estimated by the distributed updating algorithm (5) and (6), which is the well-known projection algorithm [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33] and derived by minimizing the following performance index \(J(\widehat{\phi}_{i}(k))=|\Delta y_{i}(k)-\widehat{\phi}_{i}(k)\Delta u_{i}(k- 1)|^{2}-\mu|\widehat{\phi}_{i}(k)-\widehat{\phi}_{i}(k-1)|^{2}\). With this distributedalgorithm, the PPDs of follower agents are estimated by using their own I/O data. Theoretical results show these estimated values will be bounded if some parameters are appropriately chosen, which is presented in the next section. Furthermore, due to the heterogeneity of the agent dynamics, the trajectories of the estimated PPD \(\widehat{\phi}_{i}(k)\) may vary from different follower agents._  
The distributed containment control law (7) is derived by minimizing the following index \(J(u_{i}(k))=|\sum_{j=1}^{N}a_{ij}(y_{j}(k)-y_{i}(k+1))\ +\sum_{l=N+1}^{N+M}a_{il} \big{(}w_{l}(k)-y_{i}(k+1)\big{)}|^{2}+\lambda|\Delta u_{i}(k)|^{2}\). As we can see, this performance index tries to decrease the errors between the \(i\)th agent and its neighbors. It will be proved in the following part that the MASs will finally achieve the containment control goal. Observing the control scheme (5)-(7), one will find it only uses the local I/O data to estimate the PPD and calculate the input for each follower agent, i.e., no model information of \(f_{i}(\cdot)\) is involved; that is why it is called a data-driven method. The control gain in (7) is also adaptively tuned by \(\widehat{\phi}_{i}(k)\), which is continuously updated by the new received I/O data. In addition, the protocols (5)-(7) will regress to the consensus control law in [34] when there is only one leader. The following assumptions and lemma are prepared before the main result. For all follower agents, the sign of the PPD \(\phi_{i}(k)\) (\(i=1,...,N\)) is unchanged, i.e., \(\phi_{i}(k)>\bar{\epsilon}>0\) (or \(\phi_{i}(k)<-\bar{\epsilon}<0\)), where \(\bar{\epsilon}\) is an arbitrarily small positive constant. Without loss of generality, it is assumed that \(\phi_{i}(k)>\bar{\epsilon}>0\) in the discussion. Assumption 3 indicates that the follower agent's output incremental direction caused by its input do not change during the containment control process.