Keywords

1 Introduction

Collaboration control of multi-agent systems is one of the hottest topics in the control field due to the broadly satisfactory applications in formation control [1, 2], containment control [3, 4], and target tracking [5]. Target-tracking is an active research area, which aims to activate multiple follower agents to track the target’s trajectory and accomplish the assigned tasks simultaneously.

Specifically, according to the quantity of the tracking target, target-tracking can be divided into single-target tracking and multi-target tracking. Under a leader-follower framework, several single-target tracking problems of MASs have been addressed successfully [2, 6]. The results focused on the single-target tracking can not handle with the multi-target tracking problem directly, which motivates researchers to achieve the control objective of multi-target tracking [7]. Different from the single-target tracking, multi-target tracking means that all the followers are divided into several groups and track the trajectories of the multiple targets respectively in complex operation environments. Besides, the difficulties of multi-target tracking lie in the complexity of the construct of the Lyapunov function. Therefore, a significative consideration of coordinated control for MASs lies in multi-target tracking.

It is worth mentioning that the existing research on multi-target tracking control can only be achieved in the asymptotic or finite-time manner. However, the convergence rate is also highly considered when evaluates the designed algorithms [7, 8]. Early years, the control methods mainly contain the asymptotic control, finite-time control and fixed-time control [9, 10]. None of them can obtain a precise settling time. Then, to make the settling time certainty, the prescribed-time control methods have been provided [11,12,13]. The prescribed-time stability has been highly considered on account of the performance of the user-defined settling time and the independent of initial conditions. Therefore, the combination of the prescribed-time stability and multi-target tracking becomes a significant but challenging problem.

Inspired by the aforementioned discussions, we propose a prescribed-time multi-target tracking algorithm for second-order MASs under a directed graph. The main contributions can be listed as follows.

  1. 1.

    Unlike the prescribed-time control of single-target tracking [14], the proposed algorithm is designed to achieve multi-target tracking under a leader-follower framework. The tracking targets are time-varying and unknown to the followers.

  2. 2.

    Unlike the existing references on achieving multi-target tracking in the asymptotic or finite-time manner, this is the first work on solving the prescribed-time multi-target tracking problem. The settling time can be prescribed and is unrelated to the initial condition.

2 Problem Formulation and Preliminaries

2.1 Graph Theory

A directed graph \(G = \{ V,E,A\}\) is introduced to depict the interaction of MASs, where \(V = \{ 1,2, \ldots ,N\}\), \(E \in V \times V\). The weighted adjacent matrix is defined as \(A = [{a_{ij}}] \in {{R}^{N \times N}}\), and \({a_{ij}} > 0\) if \(( i,j) \in E\), \({a_{ij}} = 0\) otherwise. The neighbor set of the ith agent is \({N_i} = \{ j \in V\left| {(i,j)} \right. \in E\}\). The Laplacian matrix is \(L\mathrm{{ = [}}{l_{ij}}\mathrm{{]}}\in {{R}^{N \times N}} \), in which \({l_{ii}} = \sum \nolimits _{j = 1}^N {{a_{ij}}}, i =j\) and \({l_{ij}}\mathrm{{ = - }}{a_{ij}}\), \(i \ne j\). Moreover, \(B = \mathrm{diag}({b_1}, \ldots ,{b_N})\) is the pining matrix, and \(b_i > 0\) if there is a directed path between the ith agent and its leader, \(b_i = 0\) otherwise.

Consider the graph can be divided into k subgraphs \(\{G_1,G_2, \ldots , G_k\}\), in which \(\{ {V_1},{V_2}, \ldots ,{V_k}\} \) are the corresponding sets of nodes. \(V_1=\{1,2, \dots , o_1\}\), \(V_2=\{o_1+1,o_1+2, \dots , o_2\}\),\(\dots \), \(V_k=\{o_{k-1}+1,o_{k-1}+2, \dots , o_k\}\), \(o_k=N\) and the number of the agents in each subgroup is defined as \(n_l\), \(\forall l \in \{ 1,2, \ldots ,k\}\).

Assumption 1

For each subgraph, there is a spanning tree rooted in a leader node.

Assumption 2

The sets \(\{V_1, V_2,\dots , V_k\}\) are acyclic partition in directed graph G.

Under Assumption 2, the Laplacian matrix can be redefined into the following form [16]

$$\begin{aligned} L = \left[ {\begin{array}{*{20}{c}} L_{11}&{}0&{} \ldots &{}0\\ {{L_{21}}}&{}L_{22}&{} \ldots &{}0\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ {{L_{k1}}}&{}{{L_{k2}}}&{} \ldots &{}L_{kk} \end{array}} \right] , \end{aligned}$$

where \({L_l}\) is the Laplacian matrix of \(G_l\), and \(L_{ml}\) denotes the interaction between \(G_m\) and \(G_l\), \(\forall l,m \in \{ 1,2, \ldots ,k\}\).

2.2 Problem Formulation

The considered system is molded as second-order integrator,

$$\begin{aligned} \left\{ \begin{array}{l} {{\dot{x}}_i} = {v_i},\\ {{\dot{v}}_i} = {u_i}, \end{array} \right. \end{aligned}$$
(1)

where \(i=1,2,\dots , N\), \({x_i} \in { R^r}\) is the position, \({v_i} \in { R^r}\) is the velocity, and \({u_i} \in { R^r}\) is the control input to be designed.

The sub-leaders can be described as \({{\dot{x}}_{l,0}} = {v_{l,0}}\) and \({{\dot{v}}_{l,0}} = {a_{l,0}}\), where \(x_{l,0}, v_{l,0}, a_{l,0}\in {R^r}\) is position, velocity and acceleration, \(\forall l \in \{ 1,2, \ldots ,k\} \).

The prescribed-time multi-target tracking for (1) will be achieved if there exists a \(u_i\) for followers such that the agents can track the sub-leaders’ trajectories in a prescribed time respectively.

2.3 Preliminaries

Lemma 1

[15] Under Assumption 1, there exists a positive-definite matrix \(P_l =\mathrm{diag}({\xi _i})= \mathrm{diag}({y_i}/{x_i})\) such that \(Q_l = P_lH_{ll} + {H_{ll}^T}P_l\), in which \(H_{ll} = L_l + B_l\), \(x = {[{x_1},{x_2}, \ldots {x_N}]^T} = {H_{ll}^{ - 1}}{1_N}\), \(y = {[{y_1},{y_2}, \ldots {y_N}]^T} = {H_{ll}^{ - T}}{1_N}\), \(\forall l \in \{ 1,2, \ldots ,k\}\).

Before moving on, a time-relevant function \(\eta (t)\) is proposed as

$$\begin{aligned} \eta (t) = {\left( {\frac{{{T_u}}}{{{t_0} + {T_u} - t}}} \right) ^\rho }, \end{aligned}$$

where \(\rho > 1\) is a positive constant, \({t_0}\) and \({{T_u}}\) are initial time and the prescribed time.

Lemma 2

[11] For system (1), if there exists a Lyapunov function V(y) such that \({{\dot{V}(y)}} \le - b{V(y)} - c\varphi (t)V(y)\), where \( b \ge 0, c > 0\), \(\varphi (t)\) is given as

$$\begin{aligned} \varphi (t) = \left\{ {\begin{array}{*{20}{l}} {\frac{{\dot{\eta }(t)}}{{\eta (t)}},{t_0} \le t < {t_0} + {T_u}},\\ {\frac{\rho }{{{T_u}}},t \ge {t_0} + {T_u}}. \end{array}} \right. \end{aligned}$$
(2)

Then, (1) is said to be prescribed-time stability in the prescribed time \(T_u\). Further, it has \({V}(y) \le {\eta ^{ - c}}(t){\exp ^{ - b\left( {t - {t_0}} \right) }}{V}\left( {{t_0}} \right) \) on \(t \in [{t_0},{t_0} + {T_u})\), and \({V(y) = 0}\) on \(t \in [{t_0} + {T_u},\infty )\).

Lemma 3

For any vectors x, y, there exists \(\sigma > 0\), then

$$\begin{aligned} \left\| x \right\| \left\| y \right\| \le \sigma {\left\| x \right\| ^2} + \frac{1}{{4\sigma }}{\left\| y \right\| ^2}. \end{aligned}$$
(3)

3 Main Results

In this section, we propose an algorithm to force followers to track their leaders’ trajectory in the prescribed time over the directed graph. Further, we demonstrate the prescribed-time stability of system (1) under the control of the designed algorithm.

3.1 Prescribed-Time Multi-target Tracking Control Algorithm

For the prescribed-time multi-target tracking problem, we propose a novel algorithm for (1), namely,

(4)

where \(\alpha _1, \alpha _2\) are positive parameters, \(\varphi (t)\) is defined in (2).

3.2 Analysis for Prescribed-Time Multi-target Tracking

Theorem 1

Under Assumptions 1-2, the prescribed-time multi-target tracking of (1) is achieved under the control algorithm (4) with the following limitation.

$$\begin{aligned} \begin{array}{lll} 2 + {\alpha _1}\rho - {\alpha _2} + b{T_u} + c\rho - {\alpha _2}\rho \frac{{{\lambda _{\max }}({Q_l})}}{{{\lambda _{\max }}({P_l})}} \le 0,\\ {\rho }{\alpha _1} - \alpha _2 \le 0,\\ \end{array} \end{aligned}$$
(5)

and the prescribed time is \(T=k(t_0+T_u)\), \(\forall l \in \{ 1,2, \ldots ,k\} \).

Proof

The tracking errors are \(\bar{x}_i=x_i-x_{l,0}\) and \({\bar{v}_i}=v_i-v_{l,0}\). The related compact form are \(\bar{x}=\mathrm{col}(\bar{x}_1,\bar{x}_2,\ldots ,\bar{x}_N)\) and \({\bar{v}}=\mathrm{col}({\bar{v}_1},{\bar{v}_2},\ldots ,{\bar{v}_N})\).

Define the following auxiliary variables

$$\begin{aligned} {\tilde{x}}= & {} (H \otimes {I_r})\varphi (t){\bar{x}}, \nonumber \\ {\tilde{v}}= & {} (H \otimes {I_r}){\bar{v}}, \end{aligned}$$
(6)

where

$$\begin{aligned} H = L + B = \left[ {\begin{array}{*{20}{c}} {{h_{11}}}&{}0&{} \ldots &{}0\\ {{h_{21}}}&{}{{h_{22}}}&{} \ldots &{}0\\ \vdots &{} \vdots &{} \ddots &{}0\\ {{h_{k1}}}&{}{{h_{k2}}}&{} \ldots &{}{{h_{kk}}} \end{array}} \right] , \end{aligned}$$

and \(h_{ll} = L_l + B_l\). Similarly, the compact forms are \(\tilde{x} = \mathrm{col}({{\tilde{x}}_1},{{\tilde{x}}_2}, \ldots ,{{\tilde{x}}_N})\) and \(\tilde{v} = \mathrm{col}({{\tilde{v}}_1},{{\tilde{v}}_2}, \ldots ,{{\tilde{v}}_N})\).

Combining (6) with (4), it obtains the following newly closed-loop system

$$\begin{aligned} \left\{ \begin{array}{l} \dot{\tilde{x}} = \bar{\varphi }(t)\tilde{x} + \varphi (t)\tilde{v},\\ \dot{\tilde{v}} = - \varphi (t)(H \otimes {I_r})({\alpha _1}\tilde{x} + {\alpha _2}\tilde{v}), \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} {\bar{\varphi }(t)} = \frac{{\dot{\varphi }(t)}}{{\varphi (t)}}=\left\{ \begin{array}{l} \frac{{\varphi (t)}}{\rho },{t_0} \le t \le {t_0} + {T_u},\\ 0,t \ge {t_0} + {T_u}. \end{array} \right. \end{aligned}$$

Let \(z = {\alpha _1}\tilde{x} + {\alpha _2}\tilde{v}\). Differentiating z yields that

$$\begin{aligned} \dot{z} = - {\alpha _2}\varphi (t)(H \otimes {I_r})z + {\alpha _1}\bar{\varphi }(t)\tilde{x} + {\alpha _1}\varphi (t)\tilde{v}. \end{aligned}$$

Specifically, it follows that

(7)

where \(z_l, {\tilde{x}}_l, {\tilde{v}_l}, \forall l \in \{ 1,2, \ldots ,k\}\) are associated with the lth subgroup.

The Lyapunov function candidate is given as

$$\begin{aligned} V_l = \frac{1}{2}{z_l^T}({P_l} \otimes {I_{{n_l}}})z_l, l \in \{1,2,\ldots ,k\}. \end{aligned}$$

The remaining proof is based on mathematical induction and the Lyapunov argument.

Step 1: Suppose that \(l=1\), it follows

$$\begin{aligned} {V_{1}} = \frac{1}{2}z_1^T({P_1} \otimes {I_{{n_1}}}){z_1}. \end{aligned}$$
(8)

For \(t \in [{t_0},{t_0} + {T_u})\), taking the derivative of \({V_{1}}\) yields that

(9)

According to Lemma 3, we have

$$\begin{aligned} {\lambda _{\min }}(Q_l){{\tilde{x}}^T}\tilde{x}\le & {} {{\tilde{x}}^T}(Q_l \otimes {I_{{n_1}}})\tilde{x} \le {\lambda _{\max }}(Q_l){{\tilde{x}}^T}\tilde{x},\nonumber \\ \tilde{z}_1^T({P_l} \otimes {I_{{n_1}}})\tilde{v}= & {} \sum \nolimits _{i = 1}^N {{\xi _i}\tilde{z}_{1i}^T{{\tilde{v}}_i}}\nonumber \\ ~~~~~~~~~~~~~~~~\le & {} {\lambda _{\max }}(P_l)(\sigma \tilde{z}_1^T{{\tilde{z}}_1} + \frac{1}{{4\sigma }}{{\tilde{v}}^T}\tilde{v}). \end{aligned}$$
(10)

Let \(R(t) = {{\dot{V}}_{1}} + (b + c\varphi (t)){V_{1}}\). Then we can obtain that

It can be concluded that \(R(t) \le 0\) if (5) holds.

Based on (2), it provides that

$$\begin{aligned} \begin{array}{l} {\left\| {\tilde{x}} \right\| ^2} + {\left\| {\tilde{v}} \right\| ^2} = {H^2}({\varphi ^2}(t){\left\| {\bar{x}} \right\| ^2} + {\left\| {\bar{v}} \right\| ^2})\\ ~~~~~~~~~~~~~~~\ge {\varepsilon _1}({\left\| {\bar{x}} \right\| ^2} + {\left\| {\bar{v}} \right\| ^2}), \end{array} \end{aligned}$$
(11)

where \({\varepsilon _1} = \min \left\{ {{\rho ^2}/{T_u}^2,1}\right\} \).

In addition, it follows that

$$\begin{aligned} {\left\| {\tilde{x}} \right\| ^2} + {\left\| {\tilde{v}} \right\| ^2} \le {\frac{1}{{{\varepsilon _2}}}}{\eta ^{ - c}}(t){\exp ^{ - b\left( {t - {t_0}} \right) }}{V_1}\left( {{t_0}} \right) , \end{aligned}$$
(12)

where \(\frac{1}{{{\varepsilon _2}}} = {\frac{1}{2}\lambda _{\max }}(P_1)\min ({\alpha _1^2,\alpha _2^2})\).

Combining (11) with (12), it follows

$$\begin{aligned} {\left\| {\bar{x}} \right\| ^2} + {\left\| {\bar{v}} \right\| ^2} \le \frac{1}{{{\varepsilon _1}{\varepsilon _2}}}{\eta ^{ - c}}(t){\exp ^{ - b\left( {t - {t_0}} \right) }}{V_1}\left( {{t_0}} \right) . \end{aligned}$$

Based on Lemma 2, it concludes that \({\lim _{t \rightarrow {t_0} + {T_u}}}{\eta ^{ - c}}(t) = 0\). Then, it follows that \({\lim _{t \rightarrow {t_0} + {T_u}}}\left\| {\bar{x}}\right\| = 0\) and \({\lim _{t \rightarrow {t_0} + {T_u}}}\left\| {\bar{v}} \right\| = 0\).

Similarly, for \({t \ge {t_0} + {T_u}}\), if (5) holds, we can easily obtain that

$$\begin{aligned} \begin{array}{lll} {{\dot{V}}_1} &{}\le &{} - b{V_1} - c\varphi (t){V_1} \\ &{}=&{} - \left( {b + c\rho /{T_u}} \right) {V_1}\\ &{}\le &{} 0. \end{array} \end{aligned}$$
(13)

Hence, it yields that \({\lim _{t \rightarrow {t_0} + {T_u}}}\left\| {\bar{x}}\right\| = 0\) and \({\lim _{t \rightarrow {t_0} + {T_u}}}\left\| {\bar{v}} \right\| = 0\) for \(t \ge {t_0} + T\), \(\forall i \in {V_1}\).

Step 2: Suppose that \(l = 2\). When \(t \ge {t_0} + {T_u}\), it follows that \(z_1=0\). Then \({\dot{z}_2}\) can be rewritten as

$$\begin{aligned} {{\dot{z}}_2} = - {\alpha _2}\varphi (t)({h_{22}\otimes {I_r}}){z_2} + {\alpha _1}\bar{\varphi }(t){{\tilde{x}}_2} + {\alpha _1}\varphi (t){{\tilde{v}}_2}. \end{aligned}$$

By employing the similar manipulation as presented in (8)-(13), it can be obtained that \(\bar{x}_i\) and \(\bar{v}_i\) converge to zero as \(t \ge 2({t_0}+{T_u})\), \(\forall i \in {V_2}\).

Step 3: For \(l = k\), when \(t \ge (k - 1)(t_0 + T_u)\), it can be concluded that

$$\begin{aligned} {{\dot{z}}_k} = - {\alpha _2}\varphi (t)({h_{kk}\otimes {I_r}}){z_k} + {\alpha _1}\bar{\varphi }(t){{\tilde{x}}_k} + {\alpha _1}\varphi (t){{\tilde{v}}_k}. \end{aligned}$$

Similarly, the prescribed-time convergence of \(\bar{x}_i ,\bar{v}_i\) will be achieved in \(k({t_0} + {T_u})\), \(\forall i \in V\).

Based on the mathematical induction, it obtains that \(\bar{x}_i ,\bar{v}_i\) approach zero within the prescribed time \(k({t_0} + {T_u})\). This ends the proof.

4 Simulation Results

In this section, the effectiveness of the proposed algorithm is proved through simulation experiments.

The studied MASs contain thirteen agents, including three sub-leaders and ten followers. The interaction network is shown in Fig ??, in which nodes L1, L2, etc are the sub-leaders, and nodes \(1-5, 6-9, 10-13\) are the corresponding followers. Specifically, the pinning matrix is \(B=\mathrm diag(0,1,1,0,0,0,0,5,0,6,0,4,0)\). The trajectories of sub-leaders are selected as

$$\begin{aligned} \begin{array}{l} \left\{ \begin{array}{l} x _{1,0} ={[2+\cos (0.2t),-1+\sin (0.2t)]^T},\\ v_{1,0} = {[-0.2\sin (0.2t),0.2\cos (0.2t)]^T},\\ a_{1,0} = {[ - 0.04\cos (0.2t),-0.04\sin (0.2t)]^T}, \end{array} \right. \\ \\ \left\{ \begin{array}{l} x _{2,0} = {[0.3 + \cos (0.3t),0.5+\sin (0.3t)]^T},\\ v_{2,0} = {[-0.3\sin (0.3t),0.3\cos (0.3t)]^T},\\ a_{2,0} = {[ - 0.09\cos (0.3t),-0.09\sin (0.3t)]^T}, \end{array} \right. \\ \\ \left\{ \begin{array}{l} x _{3,0} = {[ 2\sin (t),- 2\cos (t)]^T},\\ v_{3,0} = {[2\cos (t),2\sin (t)]^T},\\ a_{3,0} = {[ - 2\sin (t),2\cos (t)]^T}. \end{array} \right. \end{array} \end{aligned}$$

The control parameters of (4) are set as follows. To satisfy the conditions (5), let \( \alpha _1=\alpha _2=5, \rho =7\), \(t_0=0.1, T_u=4\), and then the prescribed time is \(T=12.3s\). The simulation results are shown in Figs. ??. For more details, it can be easily observed from Fig. ?? that all the followers are divided into three subgroups and track the corresponding trajectories of the sub-leaders in the prescribed time T.

5 Conclusion

In this paper, by employing a time-relevant function, the prescribed-time multi-target tracking problem of second-order MASs has been solved successfully under the directed graph. The proposed algorithm has been demonstrated that the error states between the follower agents and the corresponding leaders converge to zero within a prescribed time. Further, combining the Lyapunov stability theory with the mathematical induction, the necessary conditions for the achievement of the designed protocol are obtained. Moreover, the simulation results have been presented to verify the prescribed-time performance and multi-target tracking ability of the proposed method. Future works will be concentrated on solving the prescribed-time multi-target tracking for nonlinear physical models that agree with industrial machining practice.