## 2 Problem Formulation and Preliminaries
### Problem Statement
\(\beta\) _is locally integrable over_ \(t\)_,_ \(\forall\chi(t)\in\Pi_{\chi}\)A unique maximal solution can be obtained that \(\chi\): [0,tmax] \(\rightarrow\Pi_{\chi}\) of \(\hat{\chi}(t)=\beta(t,\chi(t))\) on the time domain \([0,t_{max})\) with \(t_{max}\in\{\mathbb{R}_{+},\infty\}\), and consequently, \(\chi(t)\in\Pi_{\chi}\), \(t\in[0,t_{max})\). Furthermore, if \(t_{max}<\infty\) and \(\Pi_{\chi}^{*}\subset\Pi_{\chi}\), there exists a time moment \(t^{*}\in[0,t_{max})\) such that \(\chi(t^{*})\
otin\Pi_{\chi}^{*}\). **3. Main Results**  
Define \(e_{x}=[e_{1x},\cdots,e_{Nx}]^{T}\), \(e_{v}=[e_{1v},\cdots,e_{Nv}]^{T}\). The dynamic model of the cumulative difference in vector form is written as follows:  
\[ \begin{cases}\dot{e}_{x}=e_{v}\\ \dot{e}_{v}=(L+B)(u+d-1\dot{v}_{0})\end{cases}  \tag{8}\]  
where \(d=[d_{1},\cdots,d_{N}]^{T}\) and \(\mathbf{1}=[1,\cdots,1]^{T}\in\mathbb{R}^{N}\). Let us define the normalized error \(\xi_{i}\) as \(\xi_{i}=\frac{e_{1x}}{\zeta_{i}},i=1,\cdots,N\). Then, let  
\[\xi=[\xi_{1},\cdots,\xi_{N}]^{T}=[\frac{e_{1x}}{\zeta_{1}},\cdots,\frac{e_{Nx }}{\zeta_{N}}]^{T}=C(t)e_{x} \tag{9}\]  
where \(C(t)=\text{diag}(\frac{1}{\zeta_{1}(t)},\cdots,\frac{1}{\zeta_{N}(t)})\), with \(\zeta_{i}(t),i=1,\cdots,N\) being performance functions to be determined later. Define the transformation error  
\[\varepsilon_{i}=\ln(\frac{1+\zeta_{i}}{1-\xi_{i}}),\ i=1,\cdots,N. \tag{10}\]  
Let \(\varepsilon=[\varepsilon_{1},\cdots,\varepsilon_{N}]^{T}\). Differentiating \(\varepsilon\) with respect to \(t\) obtains  
\[\dot{\varepsilon}=\Phi(t)(e_{v}-\dot{S}(t)\xi) \tag{11}\]  
where \(\Phi(t)=\text{diag}(\frac{2}{(1-\xi_{1}^{2})\zeta_{1}},\cdots,\frac{2}{(1-\xi_ {N}^{2})\zeta_{N}})\), \(S(t)=\text{diag}(\zeta_{1}(t),\ldots,\zeta_{N}(t))\). In order to simplify the description of \(\Phi(t)\) and \(S(t)\), \(\Phi\) and \(S\) are used to denote \(\Phi(t)\) and \(S(t)\), respectively, in the following content. Next, partly inspired by [39], a novel smooth performance function is defined as  
\[\zeta_{i}(t)= \begin{cases}(\zeta_{i}(0)-\frac{t}{t_{f}})e^{(1-\frac{t_{f}}{t_ {f}-1})}+\zeta_{i}(t_{f}),&t\in[0,t_{f})\\ \dot{\zeta}_{i}(t_{f}),&t\in[t_{f},+\infty)\end{cases}  \tag{12}\]  
where \(i=1,\cdots,N\), \(\zeta_{i}(0)>\frac{5}{4}\), and \(\zeta_{i}(t_{f})>0\). With the given performance function (12), Lemma 4 is introduced. The proof of Lemma 4 can be found in Appendix A. Remark 1: _In the present research results on PPC, a conventional performance function [13] is shown in (13). Compared with \(\xi_{i}(t)\), our proposed performance function in (12) has the property that it guarantees appointed-time convergence, which does not hold for \(\zeta_{i}(t)\) due to \(\lim_{t\rightarrow\infty}\zeta_{i}(t)=\xi_{i}(\infty)\)_  
\[\xi_{i}(t)=(\xi_{i}(0)-\xi_{i}(\infty))e^{-l_{i}t}+\zeta_{i}(\infty) \tag{13}\]  
_where \(\xi_{i}(\infty)>0\), \(\xi_{i}(0)>0\) and \(l_{i}>0\)._The distributed control protocol is given as follows:  
\[u_{i}=-k_{1}\varepsilon_{i}(t)-k_{2}e_{iv}(t),\ \theta_{i}\in\mathcal{V} \tag{14}\]  
with \(k_{1}\), \(k_{2}\) being positive constants, and \(e_{iv}(t)\), \(\varepsilon_{i}(t)\) have been defined in (6) and (10), respectively. With control input (14), system (8) can be rewritten as follows:  
\[ \begin{cases}\dot{\varepsilon}_{x}=\varepsilon_{v}\\ \dot{\varepsilon}_{v}=(L+B)(-k_{1}\varepsilon(t)-k_{2}\varepsilon_{v}(t)+d(t )-1\dot{v}_{0}).\end{cases}  \tag{15}\]  
The following theorem presents our main results. Theorem 4.1: _Under Assumptions 1 and 2, consider MASs (3) and (4) with a directed graph.