2. 北京交通大学数学系 北京 100044
2. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China
In the past decades, the distributed coordination control problems of multiagent networks have received a great deal of attention. This is mainly because they have important potential applications in practical systems, such as power grids, wireless sensor networks, transportation networks and so on [1]. To achieve coordination control, we require that every agent is forced to reach consensus firstly. Therefore, consensus problem in multiagent networks is the basic issue of the coordination control. Based on the local information, which each agent can receive from its neighbors, a mass of consensus protocols have been designed for multiagent systems with firstorder, secondorder, highorder, fractionalorder or mixedorder dynamics [2]
Moreover, in many real systems, it is often required that the consensus can be reached in finitetime. Compared with asymptotic convergence, finitetime convergence has higher convergence rate. Therefore, finitetime consensus problem is more appealing. In [21], finitetime consensus problems were studied and two valid distributed protocols were proposed for making the closedloop systems achieve finitetime consensus. In [22], the finitetime consensus problem was solved by nonsmooth stability theory. In [23], exponential finitetime protocols were proposed for both containment control and consensus control. For other finitetime consensus results, one can refer to [24]
In modern military applications, it is often needed to make all agents encircle some targets to attack and reconnoiter the situations of the enemy in many cases, such as groundattack missions [30]. Therefore, it is meaningful to investigate the encirclement motion of agents in a distributed manner. However, to the best of our knowledge, few results exist on this issue to date. And the results in literatures cannot be extended directly to encirclement control, as advocated by Lin and Jia [31]. In [32], a protocol was proposed to make agents surround and track targets, and the protocol therein can make agents surround and track targets with the same formation as that of targets asymptotically. In [33]
In response to the lack of research dealing specifically with the encirclement control problems, in this paper, we investigate the finitetime distributed rotating encirclement control problem of multiagent systems, where each agent tracks one target and every target can be tracked by one agent. First, we propose a finitetime encirclement tracking protocol, which contains the estimators for the geometrical center of the targets and the estimators for the maximum distance between the target and their geometrical center. Then, we give some sufficient conditions for making the multiagent systems achieve finitetime rotating encirclement tracking in the manner of circular formation. Finally, simulations are provided to demonstrate the effectiveness of presented results.
2 Preliminaries 2.1 Graph TheoryTo solve the coordination control problems, let us introduce some basic concepts about graph theory (see [36] for details). Consider a multiagent system with
Consider the multiagent system with
$ \begin{equation} \dot{x}_{i}( t) =u_{i}( t), \ i\in \mathcal{I} \label{eq1} \end{equation} $  (1) 
where
$ \begin{equation} \left\{ \begin{array}{c} {{{\dot{l}}}}_{{{i}}}{\left( t\right) =v}_{{i}}\left( t\right) \\ {{{\dot{\theta}}_{i}}\left( t\right) ={\omega _{i}}}\left( t\right) \end{array}% \right., \, \, i\in \mathcal{I} \label{eq2} \end{equation} $  (2) 
where
Combining the definitions of encirclement control [33]
Definition 1: The protocol
$ \begin{eqnarray} &&\lim\limits_{t\rightarrow T}\Bigg[\ x_{i}(t)\frac{1}{n}% \sum\limits_{k=1}^{n}r_{k}(t)\ \nonumber\\ &&~~~~~~k \times \max\limits_{j\in I}\bigg\{ \ r_{j}(t)\frac{1}{n}\sum\limits_{k=1}^{n}r_{k}(t)\\bigg \} \Bigg] =0 \nonumber\\ &&\lim\limits_{t\rightarrow T }\left[\theta _{i}(t)\theta _{j}(t)\frac{% 2\pi (ij)}{n}\right] =0\nonumber\\ &&\lim\limits_{t\rightarrow T }\left[\dot{\theta} _{i}(t)\Omega(t)\right] =0 \end{eqnarray} $  (3) 
where
Remark 1: In the above definition,
In practice, the exact position information of geometry center of targets cannot always be obtained by all agents, so we need to design an observer to calculate the geometry center of targets for agent
$ \begin{equation} \left\{ \begin{array}{lll} \dot{\varphi}_{i}\left( t\right)=\alpha \sum\limits_{j\in N_{i}}a_{ij}\dfrac{p_{j}\left( t\right) p_{i}\left( t\right) }{\left\Vert p_{j}\left( t\right) p_{i}\left( t\right) \right\Vert } \\ {p}_{i}\left( t\right)=\varphi _{i}\left( t\right) +r_{i}\left( t\right) \end{array}% \right. \label{eq6} \end{equation} $  (4) 
where
Let
$ \begin{equation}\label{eq7} \left\{ \begin{array}{lll} \dot{\rho}^1_{i}(t)=k^1_{1}{\rm sign}[\rho^1_{i}(t)\max\limits_{j\in N_i\cup\{i\}}(d_{j}(t))]\\ \dot{\rho}^2_{i}(t)=k^2_{1}{\rm sign}[\rho^2_{i}(t)\max\limits_{j\in N_i\cup\{i\}}(\rho^1_{j}(t))\}]\\ ~~~~~~~~~~~~~~~~~~~~~~~~~\vdots\\ \dot{\rho}^{N1}_{i}(t)=k^{N1}_{1}{\rm sign}[\rho^{N1}_{i}(t)\max\limits_{j\in N_i\cup\{i\}}(\rho^{N2}_{j}(t))]\\ \dot{\rho}_{i}(t)=k_{1}{\rm sign}[\rho_{i}(t)\max\limits_{j\in N_i\cup\{i\}}(\rho^{N1}_{j}(t))] \end{array} \right. \end{equation} $  (5) 
where
Remark 2: In [33] and [34], the maximum distance between all targets and the geometry center of targets by agent
Remark 3: In [33]
In polar coordinate system, we propose the following protocol for system (2):
$ \begin{equation} \left\{ \begin{array}{lll} v_{i}\left( t\right) &\!\!\!\!\!\! = &\!\!\!\!\!\!\!\! k_{2}{\rm sign}\left( l_{i}\left( t\right) k\rho _{i}\left( t\right) \right), \quad i\in \mathcal{I} \\ {\omega }_{i}\left( t\right) &\!\!\!\!\!\! =&\!\!\!\!\!\!\!\! k_3 \sum\limits_{j\in N_{i} \ }a_{ij} {\rm sign} \left( \theta _{i}\left( t\right) \theta _{j}\left( t\right) \dfrac{2\pi (ji)}{n}\right)\\ & &\times \theta _{i}\left( t\right) \theta _{j}\left( t\right) \dfrac{2\pi (ji)}{n}^\kappa +\Omega(t) \end{array}% \right. \label{eq5} \end{equation} $  (6) 
where
Assumption 1:
Assumption 2:
Remark 4: In [33] and [34], it is assumed that
Let
$ \begin{equation} \left\{ \begin{array}{lll} \dot{l}_{i}(t)\!\!\!\! &=&\!\!\!\! k_{2}{\rm sign}(l_{i}(t)k\rho _{i}(t)), \quad i\in \mathcal{I} \\ \dot{\hat{\theta}}_{i}(t)\!\!\!\! &=&\!\!\!\! k_3 \sum\limits_{j\in N_{i}\left( t\right) \ }a_{ij} {\rm sign} \left(\hat{\theta}_{i}\left( t\right) \hat{\theta} _{j}\left( t\right) \right)\\ &&\!\!\times\hat{\theta} _{i}\left( t\right) \hat{\theta} _{j}\left( t\right)^\kappa. \end{array}% \right. \label{eq8} \end{equation} $  (7) 
To solve the finitetime encirclement control problem of system (1), we introduce the following protocol:
$ \begin{equation} u_i=\gamma {\rm sign} (x_i\widehat{x}_i)\label{eq22} \end{equation} $  (8) 
where
Remark 5: The protocols proposed in [33]
To analyze the finitetime rotating encirclement control problem of multiagent systems, we first introduce the following lemmas.
Lemma 1 [37]: Consider system
Lemma 2: Consider system (4). Take
The proof of this lemma is provided in Appendix A.
Remark 6: This lemma shows that the estimated value of each agent for the geometrical center of the targets can converge to the precise value. In addition, it is easy to see that
Lemma 3: Consider system (5). Suppose Assumption 1 holds. If
The proof of this lemma is provided in Appendix B.
Remark 7: In practical applications, instead of (5) we can represent maximum estimated velocity by the equation
$ \begin{eqnarray} \ \ \ \ \ \ &&\!\!\!\!\!\!\!\!\!\!\!\!\dot{\rho}_i(t)=k_1 {\rm sign}[\rho_{i}(t)\nonumber\\ &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sum\limits_{k=1}^{N1}\!\!\max\limits_{j\in N_i}(\rho^{k}_{i}(t))1_{\{\rho^k_i(t)\neq \max\limits_{j\in N_i}(\rho^{k1}_j(t)); ~\rho^{k1}_i(t)= \max\limits_{j\in N_i}(\rho^{k2}_j(t))\} }]\nonumber \end{eqnarray} $ 
where
In fact, this equation is equivalent to the first equation when
Lemma 4: Consider system (2) with protocol (6). Suppose Assumption 1 holds. If
The proof of this lemma is provided in Appendix C.
Next, let us state the main result of this paper.
Theorem 1: Consider system (1). Suppose graph
Proof: Choose the following Lyapunov function:
$ \begin{equation} V_1=x_i\widehat{x}_i.\label{eq23} \end{equation} $  (9) 
Compute the derivative of
$ \begin{array}{lll} \dot{V}_1\!\!\!\!\!\!\!\!\!\!\!&&={\rm sign}(x_i\widehat{x}_i)(\dot{x}_i\dot{\widehat{x}}_i)\\ \!\!\!\!\!\!\!\!\!\!\!&&=\gamma{\rm sign}(x_i\widehat{x}_i)\dot{\widehat{x}}_i\leq \gamma + \\dot{\widehat{x}}_i\. \end{array} $ 
According to Lemma 2, there exists
Remark 8: Theorem 1 shows that the protocols proposed here can force agents not only achieve encirclement tracking in the manner of distribute formation, but also rotate around targets.
Remark 9: In contrast to the asymptotic rotating control [31] and encirclement control [33]
Remark 10: Although the discontinuous control has strong antijamming capability, while, the chattering phenomenon is inevitable in practice, how to eliminate or weaken the chattering phenomenon is the common problem for discontinuous dynamical systems (such as [33]
$ u_i=\gamma {\rm sign} (x_i\widehat{x}_i)x_i\widehat{x}_i^{\sigma_1} + \dot{\widehat{x}}_i $ 
where
$ \left\{ \begin{array}{lll} \dot{\varphi}_{i}\left( t\right)=\alpha \sum\limits_{j\in N_{i}}a_{ij}\dfrac{p_{j}\left( t\right) p_{i}\left( t\right) }{\left\Vert p_{j}\left( t\right) p_{i}\left( t\right) \right\Vert + \sigma_2 } \\ {p}_{i}\left( t\right)=\varphi _{i}\left( t\right) +r_{i}\left( t\right) \end{array}% \right. $ 
where
In this section, numerical simulations are given to illustrate the theoretical results obtained in the previous sections. Fig. 1 shows a graph with
Suppose the dynamics of targets are
$ \begin{eqnarray} \dot{r}_{11}(t)&=&(r_{12}(t)1)(r_{11}(t)1)\nonumber\\ && \times[(r_{11}(t)1)^2+(r_{12}(t)1)^20.01]\nonumber\\ \dot{r}_{12}(t)&=&(r_{11}(t)1)(r_{12}(t)1)\nonumber\\ && \times[(r_{11}(t)1)^2+(r_{12}(t)1)^20.01]\nonumber\\ \dot{r}_{21}(t)&=&(r_{22}(t)1)r_{21}(t)\nonumber\\ &&\times[(r_{21}(t))^2+(r_{22}(t)1)^20.01]\nonumber\\ \dot{r}_{22}(t)&=&r_{21}(t)(r_{22}(t)1)\nonumber\\ &&\times[(r_{21}(t))^2+(r_{22}(t)1)^20.01]\\ \end{eqnarray} $  (10) 
$ \begin{eqnarray} \dot{r}_{31}(t)&=&(r_{32}(t)1)(r_{31}(t)2)\nonumber\\ &&\times[(r_{31}(t)2)^2+(r_{32}(t)1)^20.01]\nonumber\\ \dot{r}_{32}(t)&=&(r_{31}(t)2)(r_{32}(t)1)\nonumber\\ &&\times[(r_{31}(t)2)^2+(r_{32}(t)1)^20.01]\\ \end{eqnarray} $  (11) 
$ \begin{eqnarray} \dot{r}_{41}(t)&=&(r_{42}(t)2)(r_{41}(t)1)\nonumber\\ &&\times[(r_{41}(t)1)^2+(r_{42}(t)2)^20.01]\nonumber\\ \dot{r}_{42}(t)&=&(r_{41}(t)1)(r_{42}(t)2)\nonumber\\ &&\times[(r_{41}(t)1)^2+(r_{42}(t)2)^20.01]\\ \end{eqnarray} $  (12) 
$ \begin{eqnarray} \dot{r}_{51}(t)&=& r_{52}(t)(r_{51}(t)1)\nonumber\\ &&\times[(r_{51}(t)1)^2+(r_{52}(t))^20.01]\nonumber\\ \dot{r}_{52}(t)&=&(r_{51}(t)1)r_{52}(t)\nonumber\\ &&\times[(r_{51}(t)1)^2+(r_{52}(t))^20.01]. \end{eqnarray} $  (13) 
And the initial conditions of system (10)
Fig. 2 shows the movement paths of targets. Fig. 3 shows the estimates of targets' geometry center and the real path of targets' geometry center, which is marked by black line. Fig. 4 shows the estimates of polar radius. From Figs. 3 and 4, it is easy to see that these estimators are finitetimeconvergent. Fig. 5 shows the paths of agents and targets of multiagent systems, which implies that the closedloop multiagent system can achieve circle formation in finitetime, and the task of distributed rotating encirclement tracking moving targets is completed.
In this paper, the focus was mainly on the problem of how to implement encircling and tracking of moving targets for agents. We designed some protocols, under which, the task of rotating encirclement and tracking moving targets for agents was completed. Based on the neighbor information of each agent, some estimators were proposed to estimate geometry center of targets, the maximum distance between all targets and their geometry center. Sufficient conditions were obtained for making agents encircle and track targets in circle formation, and the radius of formation can be changed with targets simultaneously. By the theory of finitetime Lyapunov stability, it is proved that geometry center position of targets can be gotten in finitetime by agents and circle formation can be formed in finitetime. Finally, simulation results are provided to demonstrate the effectiveness of presented results. The future work will focus on the encirclement control of the timedelay systems with external disturbances.
AppendixA. Proof of Lemma 2
Let
$ \begin{eqnarray} V_{1}(t) &=&\frac{1}{2}\left[ {\overline{p}(t)}\frac{1}{n}\mathbf{1}_{n}\otimes \left[ \begin{array}{c} \mathbf{1}_{n}^{T}\otimes E_2 {\overline{p}(t)} \nonumber\\ \end{array}% \right] \right] ^{T}\\%[0.8cm] &&\times\left[{\overline{p}(t)}\frac{1}{n}\mathbf{1}_{n}\otimes \left[ \begin{array}{c} \mathbf{1}_{n}^{T}\otimes E_2 {\overline{p}(t)} \\ \end{array}% \right] \right] \end{eqnarray} $  (14) 
where
Computing the derivative of
$ \begin{array}{lll} \dot{V_{1}}(t) &=&\sum\limits_{i=1}^{n}\Bigg( \left[p_{i}(t)\dfrac{1}{n}% \sum\limits_{k=1}^{n}p_{k}(t)\right] ^{T}\\[0.4cm] &&\times\left[\dot{p}_{i}(t)\dfrac{1}{n}% \sum\limits_{k=1}^{n}\dot{p}_{k}(t)\right] \Bigg) \\[0.4cm] &=&\sum\limits_{i=1}^{n}\Bigg\{ \left[p_{i}(t)\dfrac{1}{n}\sum\limits_{k=1}^{n}p_{k}(t)% \right] ^{T}\\[0.4cm] &&\times \big[\alpha \sum\limits_{j\in N_{i}}a_{ij} \dfrac{p_{j}\left( t\right)p_{i}\left( t\right) }{\left\Vert p_{j}\left( t\right) p_{i}\left( t\right) \right\Vert } \\[0.4cm] & &+\dot{r}_{i}(t)\dfrac{1}{n}\sum\limits_{k=1}^{n} \dot{p}_{k}(t) \big] \Bigg\}. \end{array} $ 
Since
$ \begin{array}{lll} &&\alpha \sum\limits_{i=1}^{n}\Big\{\big[p_{i}(t)\dfrac{1}{n}\sum\limits_{k=1}^{n}p_{k}(t)\big]^{T} \sum\limits_{j\in N_{i}} a_{ij}\\ &&\times\dfrac{p_{j}\left( t\right) p_{i}\left( t\right) }{\left\Vert p_{j}\left( t\right) p_{i}\left( t\right) \right\Vert }\Big\} \\ &=&\dfrac{\alpha }{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n} \Big\{ a_{ij}\big[p_{i}(t)\dfrac{1}{n} \sum\limits_{k=1}^{n}p_{k}(t)\big]^{T}\\ &&\times\dfrac{% p_{j}\left( t\right) p_{i}\left( t\right) }{\left\Vert p_{j}\left( t\right) p_{i}\left( t\right) \right\Vert }\Big\} \dfrac{\alpha }{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n} \Big\{ a_{ij}[p_{j}(t) \\ &&\dfrac{1}{n} \sum\limits_{k=1}^{n}p_{k}(t)]^{T}\dfrac{% p_{j}\left( t\right) p_{i}\left( t\right) }{\left\Vert p_{j}\left( t\right) p_{i}\left( t\right) \right\Vert }\Big\}\\ &=&\dfrac{\alpha }{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\Big\{a_{ij}[p_{i}(t)p_{j}(t)]^{T}\dfrac{p_{j}\left( t\right) p_{i}\left( t\right) }{% \left\Vert p_{j}\left( t\right) p_{i}\left( t\right) \right\Vert }\Big\}\\ &\leq &\dfrac{\alpha }{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}a_{ij}\Vert p_{i}(t)p_{j}(t)\Vert. \end{array} $ 
From Assumption 1, we can get
$ \begin{array}{lll} &&\sum\limits_{i=1}^{n}\Big\{(p_{i}(t)\dfrac{1}{n}\sum\limits_{k=1}^{n}p_{k}(t))^{T}\dot{r}% _{i}(t)\Big\} \label{eq12} \\[0.3cm] &\leq &\beta \sum\limits_{i=1}^{n}\Vert (p_{i}(t)\dfrac{1}{n}% \sum\limits_{k=1}^{n}p_{k}(t))\Vert \\[0.3cm] &=&\dfrac{\beta }{n}\sum\limits_{i=1}^{n}\Vert n{p_{i}(t)\sum\limits_{k=1}^{n}p_{k}(t)}% \Vert \\[0.3cm] &\leq &\beta \sum\limits_{j=1, j\neq i}^{n}\max\limits_{i=1, 2\ldots , n}(\Vert p_{i}(t)p_{j}(t)\Vert ) \\[0.3cm] &\leq &(n1)\beta \max\limits_{i, j=1, 2\ldots, n}\Vert p_{i}(t)p_{j}(t)\Vert \\[0.3cm] &\leq &\dfrac{\beta (n1)}{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}a_{ij}\Vert p_{i}(t))p_{j}(t)\Vert \end{array} $ 
and
$ \begin{array}{lll} &&\sum\limits_{i=1}^{n}\left\{ \left[{p_{i}(t)\dfrac{1}{n}\sum\limits_{k=1}^{n}p_{k}(t)}% \right] ^{T}\bigg[\dfrac{1}{n}\sum\limits_{k=1}^{n}\dot{p}_{k}(t)\bigg]\right\}\\[0.3cm] &=&\left[\sum\limits_{i=1}^{n}{p_{i}(t)\sum\limits_{k=1}^{n}p_{k}(t)}\right] \left[% \dfrac{1}{n}\sum\limits_{k=1}^{n}\dot{p}_{k}(t)\right] =0. \end{array} $ 
So we have
$ \begin{array}{lll} \dot{V_{1}}(t)&\leq &\dfrac{\alpha }{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}a_{ij}\Vert p_{i}(t)p_{j}(t)\Vert \\[0.2cm] &&+\dfrac{\beta (n1)}{2}% \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}a_{ij}\Vert p_{i}(t)p_{j}(t)\Vert \\[0.2cm] &\leq&\big(\dfrac{\beta (n1)}{2}\dfrac{\alpha }{2}\big)n^{2}s\left( t\right) \end{array} $ 
where
$ \begin{array}{lll} &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\dot{V}_{1}\left( t\right) +\sqrt{\dfrac{2}{n}}\left( \dfrac{\alpha }{2}% \dfrac{\beta (n1)}{2}\right) n^{2}V_{1}\left( t\right) ^{\frac{1}{2}} \label{eq15} \\[0.2cm] &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\leq \left( \dfrac{\beta (n1)}{2}\dfrac{\alpha }{2}\right) n^{2}s\left( t\right) +\left( \dfrac{\alpha }{2}\dfrac{\beta (n1)}{2}\right) n^{2}s\left( t\right)=0. \end{array} $ 
by the lyapunov finitetime stability theorem [38], there exists
B. Proof of Lemma 3
Define the following candidate Lyapunov function:
$ \begin{equation} V_{2}(t)=\left\vert \varepsilon _{i}\right\vert \end{equation} $  (15) 
where
$ \begin{array}{lll} \dot{V_{2}}(t)&=&{\rm sign}(\varepsilon _{i})\dot{\varepsilon}_{i}\\[0.2cm] &=&{\rm sign}(\varepsilon _{i})[k_{1}{\rm sign}(\rho _{i}(t)\max\limits_{j\in N_i\cup\{i\}}(d_{j}(t))\\[0.2cm] &&\max\limits_{j\in N_i\cup\{i\}}(\dot{d}_{j}(t))% ]. \end{array} $ 
Let
$ \begin{equation} \begin{array}{lll} d\left( t\right)& =\max\limits_{j\in N_i\cup\{i\}}\left\{ \left\Vert r_{j}\left( t\right) \dfrac{1}{n}\sum\limits_{k=1}^{n}r_{k}\left( t\right) \right\Vert \right\} \\[5mm] &=\max\limits_{j\in N_i\cup\{i\}}\left\{ \left\Vert \dfrac{% n1}{n}r_{j}\left( t\right) \dfrac{1}{n}\sum\limits_{k=1, k\neq j}^{n}r_{k}\left( t\right) \right\Vert \right\}. \end{array} \end{equation} $  (16) 
Having noted Assumption 1, it is easy to see that
$ \begin{eqnarray*} \dot{d}\left( t\right) &=&\left\vert \max\limits_{j\in N_i\cup\{i\}}(\dot{d}% _{j}(t))\right\vert \\ &=&\left\vert \max\limits_{j\in N_i\cup\{i\}}\left\{ \left\Vert \dot{r}% _{j}\left( t\right) \frac{1}{n}\sum\limits_{k=1}^{n}\dot{r}_{k}\left( t\right) \right\Vert \right\} \right\vert \\ &=&\left\vert \max\limits_{j\in N_i\cup\{i\}}\left\{ \left\Vert \frac{n1}{n}% \dot{r}_{j}\left( t\right) \frac{1}{n}\sum\limits_{k=1, k\neq j}^{n}\dot{r}_{k}\left( t\right) \right\Vert \right\} \right\vert \\ &\leq &\max\limits_{j\in N_i\cup\{i\}}\left\{ \left\Vert \frac{n1}{n}% \dot{r}_{j}\left( t\right) \frac{1}{n}\sum\limits_{k=1, k\neq j}^{n}\dot{r}_{k}\left( t\right) \right\Vert \right\} \\ &\leq &2\beta. \end{eqnarray*} $ 
Therefore,
$ \begin{equation} \dot{V_{2}}(t)\leq {\rm sign}(\varepsilon _{i})\left[k_{1} {\rm sign}% (\varepsilon _{i})\right] +2\beta =k_{1}\left\vert {\rm sign}\left( \varepsilon _{i}\right) \right\vert +2\beta. \end{equation} $  (17) 
From
Similarly, we can prove that the other equations in system (5) are also finitetime stable, i.e., there exist
C. Proof of Lemma 4
Choose the candidate Lyapunov function as follows:
$ \begin{equation} V_{3}(t)=\left\vert e_{i}\right\vert \end{equation} $  (18) 
where
Computing the derivative of
$ \begin{eqnarray*} \dot{V_{3}}(t) &=&{\rm sign}(e_{i}\left( t\right) )\dot{e}_{i} \\ &=&{\rm sign}(e_{i}\left( t\right) )\left[k_{2}{\rm sign}% (l_{i}(t)k\rho _{i}\left( t\right) )k\dot{\rho}_{i}\left( t\right) \right] \\ &\leq &k_{2}\left\vert {\rm sign}(e_{i}(t))\right\vert {\rm sign}% (e_{i}(t)k\dot{\rho}_{i}\left( t\right). \end{eqnarray*} $ 
By Lemma 3, there exists
$ \begin{equation} \dot{V_{3}}(t)\leq k_{2}\left\vert {\rm sign}(e_{i}(t))\right\vert +2k\beta. \end{equation} $  (19) 
It is easy to see that the closedloop system is finitetime stable when
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