## 3 Main Results
Similar assumptions can also be found in model-based control theory. The communication graph among followers is strongly connected, and each follower can receive the information through a direct path from at least one leader. Assumption 4 guarantees that there are no isolated followers and the containment control problem is well-posed. For a dynamic system \(x(k+1)=f(x(k))\), \(V(k)=g(x(k))\) is a positive definite and radially unbounded Lyapunov function. If \(\Delta V(k)\leq 0\) and \(\Delta V(k)\
ot\equiv 0\) for any \(x(k)\
eq 0\), then the equilibrium \(x_{e}=0\) is global asymptotic stable. In the stationary leader case, the trajectory of the \(l\)th leader is written as \(w_{l}(k)=w_{l}\ (l\in\mathcal{W})\), where \(w_{l}\) is a constant. The main result is summarized as Theorem 3.1. Assume that the MASs dynamics satisfies Assumptions 3-3 and the communication topology satisfies Assumption 4. In the stationary leaders case, that is, \(w_{l}(k)=w_{l}\ (l\in\mathcal{W})\), let the distributed MFACC (5)-(7) be applied. If the parameter \(\rho_{i}\ (i=1,...,N)\) is selected to satisfy the following convergence condition  
\[0<\rho_{i}<\frac{1}{\overline{d}_{i}}\]  
then there exists a \(\lambda_{min}>b^{2}/4\), and the MASs will achieve the containment control objective defined in Definition 2 when \(\lambda>\lambda_{min}\) and \(k\rightarrow\infty\). The proof consists of three steps. Step one is to prove that the PPD estimation is bounded. Step two is to derive the convergence condition. Step three is to prove the convergence of the global containment error. \(Step\ 1\) The boundedness of \(\widehat{\phi}_{i}(k)\) is straightforward if \(\|\widehat{\phi}_{i}(k)\|\leq\varepsilon\) or \(\|\Delta u_{i}(k)\|\leq\varepsilon\) or \(\operatorname{sign}(\widehat{\phi}_{i}(k))\
eq\operatorname{sign}(\widehat{ \phi}_{i}(1))\). In the other case, the PPD estimation error is defined as \(\tilde{\phi}_{i}(k)=\widehat{\phi}_{i}(k)-\phi_{i}(k)\). Subtracting \(\phi_{i}(k)\) from both sides of the PPD estimation algorithm (5) yields  
\[\tilde{\phi}_{i}(k) =\tilde{\phi}_{i}(k-1)-\left(\phi_{i}(k)-\phi_{i}(k-1)\right)\] \[+\frac{\eta\Delta u_{i}(k-1)}{\mu+|\Delta u_{i}(k-1)|^{2}}\times \left(\Delta y_{j}(k-1)\right.\] \[\left.-\widehat{\phi}_{i}(k-1))\Delta u_{i}(k-1)\right) \tag{8}\]  
Define \(\Delta\phi_{i}(k)=\phi_{i}(k)-\phi_{i}(k-1)\). Then, substituting CFDL data description (3) into (8) gives  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(k-1)+\frac{\eta\Delta u_{i}(k-1)}{ \mu+|\Delta u_{i}(k-1)|^{2}}\times(\Delta y_{i}(k)-\widehat{\phi}_{i}(k-1) \Delta u_{i}(k-1)) \tag{5}\]0 \[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(k-1)+\frac{\eta\Delta u_{i}(k-1)}{ \mu+|\Delta u_{i}(k-1)|^{2}}\times(\Delta y_{i}(k)-\widehat{\phi}_{i}(k-1) \Delta u_{i}(k-1)) \tag{5}\]1  
From Lemma 1, it can be known that \(\phi_{i}(k)\) is bounded, and assume the bound of it as \(\tilde{b}\). Taking the norm on both sides of (9) leads to  
\[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(k-1)+\frac{\eta\Delta u_{i}(k-1)}{ \mu+|\Delta u_{i}(k-1)|^{2}}\times(\Delta y_{i}(k)-\widehat{\phi}_{i}(k-1) \Delta u_{i}(k-1)) \tag{5}\]2 \[\widehat{\phi}_{i}(k)=\widehat{\phi}_{i}(k-1)+\frac{\eta\Delta u_{i}(k-1)}{ \mu+|\Delta u_{i}(k-1)|^{2}}\times(\Delta y_{i}(k)-\widehat{\phi}_{i}(k-1) \Delta u_{i}(k-1)) \tag{5}\]3  
Note the fact that the term \(\eta|\Delta u_{i}(k-1)|^{2}/(\mu+|\Delta u_{i}(k-1)|^{2})\) is monotonically increasing with respect to \(|\Delta u(k-1)|\).