## 2 Problem Formulation and Preliminaries
### Problem Statement  
We consider the MASs with \(N\) followers and a leader over directed graphs. The double-integrator dynamics of follower \(i\) are described as follows:  
\[ \begin{cases}\dot{x}_{i}(t)=v_{i}(t)\\ \dot{v}_{i}(t)=u_{i}(t)+d_{i}(t)\end{cases}  \tag{3}\]  
where \(x_{i}(t),v_{i}(t),u_{i}(t),d_{i}(t)\in\mathbb{R}\) are the position, velocity, control input, and external disturbance of the \(i\)th follower, respectively, for \(i=1,\cdots,N\). Let \(x=[x_{1},x_{2},\cdots,x_{N}]^{T}\) and \(v=[v_{1},v_{2},\cdots,v_{N}]^{T}\) be the stack vector of MASs. The leader's dynamics is described as  
\[\dot{x}_{0}(t)=v_{0}(t) \tag{4}\]  
where \(x_{0}\in\mathbb{R}\) is leader's position and \(v_{0}\in\mathbb{R}\) is leader's velocity. In order to simplify the subsequent controller design of each follower, the following assumptions are required. **Assumption 1**. : _The graph \(\bar{\mathcal{G}}\) contains a spanning tree with the root node being the leader, and there exists no self-loop for each vertex._  
**Assumption 2**. : \(v_{0}(t)\) _is piecewise continuous and \(v_{0}\) is bounded._  
Figure 1: Error transformation: (**a**) case of \(e(0)\geq 0\), and (**b**) case of \(e(0)<0\). **Assumption 3**. : _For the external disturbance \(d_{i}(t)\), there exists a positive constant \(\bar{d}\) such that \(|d_{i}|\leq\bar{d}\)._  
We use \(e_{ix}\) and \(e_{iv}\) to represent the cumulative position difference and the cumulative velocity difference of the \(i\)th follower agent, respectively, which are defined as follows:  
\[e_{ix}=\sum_{j=1}^{N}a_{ij}(x_{i}-x_{j})+b_{i}(x_{i}-x_{0}) \tag{5}\]  
\[e_{iv}=\sum_{j=1}^{N}a_{ij}(v_{i}-v_{j})+b_{i}(v_{i}-v_{0}). \tag{6}\]  
Then, we define the position tracking error \(\delta_{ix}\) and the velocity tracking error \(\delta_{iv}\) for the \(i\)th follower as \(\delta_{ix}=x_{i}-x_{0}\), \(\delta_{iv}=v_{i}-v_{0}\). **Definition 1** ([35]). : _(Practical appointed-time consensus for leader-following networks): For the leader-follower networks described by MASs (3) and (4), the practical appointed-time consensus can be achieved if the following conditions hold:_  
\[\mid\delta_{ix}(t)\mid\leq k_{x},\forall t\geq t_{f}\]  
_and_  
\[\mid\delta_{iv}(t)\mid\leq k_{v},\forall t\geq t_{f}\]  
_where \(t_{f}\) is the appointed setting time, \(k_{x}>0\) is the maximum allowable steady-state error, and \(k_{v}>0\)._  
Our goal is to design a distributed consensus control protocol for linear MASs expressed by (3) and (4) such that the following apply:  
(1) The practical appointed-time consensus as defined by Definition 1 is achieved. (2) The cumulative position difference \(e_{ix}\) evolves within a prescribed region, which is quantified as  
\[-\varsigma_{i}(t)<e_{ix}(t)<\varsigma_{i}(t),\;i=1,\cdots,N \tag{7}\]  
where \(\varsigma_{i}(t)>0\) is the performance function. (3) The control input \(u_{i}\) and the velocity of the \(i\)th follower, i.e., \(v_{i}\), are bounded. Before moving on, three lemmas are introduced as follows, which will be employed in the proof of the main results. **Lemma 1** ([36]). : _Let \(H\in R^{N\times N}\) be a nonsingular M-matrix, then there exists a diagonal positive definite matrix \(Q=(\text{diag}(q))^{-1}\) such that \(QH+H^{\mathsf{T}}Q\) is positive definite, where \(q=H^{-1}\mathbf{1}\) and \(\mathbf{1}=[1,\cdots,1]^{T}\in\mathbb{R}^{N}\)._  
**Lemma 2** (Holder's Inequality [37]). : _For \(p_{i}\in R\) and \(q_{i}\in R\), the following inequality is satisfied:_  
\[\sum_{i=1}^{n}|p_{i}q_{i}|\leq\{\sum_{i=1}^{n}|p_{i}|^{a}\}^{1/a}\{\sum_{i=1} ^{n}|q_{i}|^{b}\}^{1/b}\]  
_with \(a>1\) and \(b>1\) being real numbers and \(1/a+1/b=1\)._  
**Lemma 3** ([38]). : _Consider a class of differential equation such that \(\dot{\chi}=\beta(t,\chi(t))\), \(\chi(0)=\chi^{0}\in\Pi_{\chi}\) with \(\beta\): \(\mathbb{R}_{+}\times\Pi_{\chi}\to\mathbb{R}^{N}\) and \(\Pi_{\chi}\subset\mathbb{R}^{N}\) being a non-empty open set. If the conditions hold:_  
1. \(\beta\) _is locally Lipschitz over_ \(\chi\)_;_
2. \(\beta\) _is continuous over_ \(t\)_;_
3.