1 Introduction

Over the past few years, distributed cooperative control of multi-agent systems has received compelling attention from various scientific communities with the advent of communication networks and powerful embedded systems. Research on this topic arms to understand how various group behaviors emerge as a result of local interactions among individuals. Distributed cooperative control has broad applications in a wide range of areas, such as consensus, formation control, distributed tracking, containment control and cooperative surveillance [1–4] and [5] with such advantages such as low cost, high robustness and easy maintenance [6].

As one of the interesting and important research issues arising from distributed cooperative control for multi-agent system, consensus problem has been extensively studied in the past two decades. The objective is to design distributed protocols based on local relative information to guarantee the states of the agents to reach an agreement. According to the number of leaders in the network, consensus can be classified into leaderless consensus, distributed tracking with only one leader and containment control with multiple leaders. Recently, a variety of consensus algorithms has been proposed to solve the leaderless consensus problem under different scenarios [2, 7–10] and [11]. Further, significant progress has been made in the leaderless consensus problem [5, 12] and [13]. However, in practice, there might exist leaders in a network for achieving different objectives. Thus, distributed tracking as a more challenging problem in distributed cooperative control of multi-agent systems is arising. Distributed tracking problem for multi-agent systems in the single-leader case is known as a consensus tracking problem, where the objective is to drive the states of the followers to approach the state of the leader. Consensus tracking problem and its extensions were investigated in [1, 14] and [15] with integrator type of dynamics. More recently, some effective algorithms were proposed in [16] for tracking a dynamic leader. Furthermore, a class of distributed tracking topologies was designed in [5] for multi-agent systems with general linear dynamics. In the multiple leaders case, the containment control problem arises, where the objective is to drive the followers to move into a convex hull spanned by the states of the leaders. Containment control problems were investigated in [4] and [17] for multi-agent systems with single-integrator dynamics under fixed or switching communication network topologies. In [18], containment control problem was studied under the assumption that the followers have double-integrator dynamics, but the leaders have single-integrator dynamics. Authors in [19] then addressed the containment control for both leaders and followers with double-integrator dynamics. Moreover, containment control for multiple Euler–Lagrange systems was studied in [20].

In reality, convergence rate as a significant performance index for evaluating the effectiveness of a controller design in the study of the distributed tracking problem is a focal research topic in the area of consensus problems. Numerous researchers endeavored to improve the convergence rate by enlarging the coupling strength, optimizing the system gain or designing better communication topology [21–24] and [25]. However, the above-mentioned methods may only guarantee the achievement of asymptotic consensus, which implies that the convergence rate is at best exponential with an infinite settling time. In practical applications, it is often desirable to achieve consensus in a finite time. Therefore, it is essential to investigate finite-time consensus algorithms. Compared with the asymptotic control approach, finite-time control is an effective approach with high performance and good robustness to uncertainty and disturbance rejection. Finite-time consensus problem in the leaderless case was first studied in [26], where finite-time consensus for single-integrator systems was solved by using some discontinuous algorithms. Then, finite-time consensus problem with double-integrator dynamics was investigated in [27]. Next, the authors in [28] and [29] studied nonlinear finite-time consensus problems. Furthermore, distributed finite-time tracking control and containment control were addressed in [30] and [31]. In [32] and [33], finite-time formation control problems were studied with state feedback. It is worthy of noting that most of the above-mentioned works focused on distributed finite-time tracking control by using both position and velocity measurements. However, it is usually difficult to obtain velocity and acceleration measurements rather than position measurements in realistic circumstances. Besides, the dynamics of both the followers and the leaders are assumed the same in most previous works. This assumption is always too strict to accomplish in some complex tracking tasks, for instance, to avoid hazardous obstacles or to reach a desirable tracking target. Thus, how to solve the distributed finite-time tracking problem under a leader or multiple leaders with acceleration inputs for a multi-agent system by only using relative position measurements becomes an important and challenging issue.

Motivated by the above observations, this paper investigates the distributed finite-time containment control problem under several dynamic leaders with unknown acceleration inputs. The contribution of this paper is that a distributed finite-time containment protocol based only on the relative position measurements is designed such that followers with double-integrator dynamics can track a dynamic convex hull formed by different leaders with unknown inputs in finite time. A distributed containment protocol based on the finite-time observer-type algorithm is designed by using only relative position measurements. Compared with [31], where the distributed algorithms can guarantee finite-time convergence only when the acceleration of the leader was zero, in this paper the new protocols can ensure the followers to move into the convex hull shaped by multiple dynamic leaders with nonzero acceleration inputs. Also, compared with the work in [16], where the distributed finite-time tracking problem with a dynamic leader was studied by utilizing both relative position and velocity measurements, the new algorithm here can solve the same problem without using velocity measurements. Further, it is theoretically significant and practically important to provide an estimation on the settling time of the second-order distributed control problems, which is another achievement of the present paper.

The rest of this paper is organized as follows. Section 2 introduces some useful preliminaries and the model description. Section 3 shows the main results of this paper. Section 4 gives a numerical example to verify the theoretical analysis. Finally, Sect. 5 presents some concluding remarks.

2 Preliminaries and model description

2.1 Notations

Let \(R^{n\times n}\) and \(C^{n\times n}\) be the sets of \(n\times n\) real and complex matrices, respectively, \(R^{+}\) the set of positive real numbers and \(I_p\) the \(p\)-dimensional identity matrix. \(P\,{>}\,0\; (P\,{<}\,0)\) means that the matrix \(P\) is positive (negative) definite. \(\mathbf {1}\) represents the vector with all entries being one. Given a vector \(\xi =[\xi _1, \xi _2, \ldots , \xi _p]^T\) and constant \(\kappa >0\), define \(\mathrm{sig}(\xi )^{\kappa }=[\mathrm{sgn}(\xi _1)|\xi _1|^{\kappa }, \mathrm{sgn}(\xi _2)|\xi _2|^{\kappa }, \ldots , \mathrm{sgn}(\xi _p)|\xi _p|^{\kappa }]^T\) and \(\mathrm{sgn}(\xi )=[\mathrm{sgn}(\xi _1), \mathrm{sgn}(\xi _2), \ldots , \mathrm{sgn}(\xi _p)]^T\), where \(\mathrm{sgn}(\cdot )\) is the signum function. Let \(\mathrm{diag}(\xi _1, \ldots , \xi _p)\) represent a diagonal matrix with diagonal elements \(\xi _1, \xi _2, \ldots , \xi _p\). \(\Vert T\Vert _\infty \) denotes the \(\infty \)-norm of matrix \(T\). The Kronecker product of matrices \(A\in R^{m\times n}\) and \(B\in R^{p\times q}\) is defined as

$$\begin{aligned} A\otimes B=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} a_{11}B&{} \cdots &{}a_{1n}B\\ \vdots &{}\ddots &{}\vdots \\ a_{m1}B&{}\cdots &{}a_{mn}B\\ \end{array}\right) . \end{aligned}$$

2.2 Preliminaries

Consider a group of \(N+M\) agents. Denote a graph \(\mathcal {G}=(\mathcal {V},\mathcal {E})\) to be a communication topology among \(N+M\) agents (nodes), where \(\mathcal {V}=\{1,2,\ldots ,N+M\}\) and \(\mathcal {E}\subseteq \mathcal {V}\times \mathcal {V}\) represent the set of nodes and the set of edges, respectively. A directed edge \((i, j)\in \mathcal {E}\) in graph \(\mathcal {G}\) means that agent \(j\) can obtain information from agent \(i\), but not conversely. The neighbors of agent \(i\) are denoted as \(\mathcal {N}_i=\{j\in \mathcal {V}|(i, j)\in \mathcal {E}\}\). An undirected edge \((i, j)\in \mathcal {E}\) in graph \(\mathcal {G}\) if agent \(i\) and \(j\) can access information from each other. In this paper, it is assumed that \(\mathcal {G}\) is a simple graph, which means \((i, i)\notin \mathcal {V}\). Furthermore, a graph is undirected if \((i, j)\in \mathcal {E}\) implies \((j, i)\in \mathcal {E}\). A path from agent \(i_1\) to agent \(i_s\) is a sequence of ordered edges in the form of \((i_k, i_{k+1})\in \mathcal {E}, \; k=1, 2,\ldots ,s-1\). A graph \(\mathcal {G}\) is said to be connected if there exists a path among each pair of distinct nodes.

The adjacency matrix of a graph \(\mathcal {G}\) is denoted by \(A=(a_{ij})\in R^{(N+M)\times (N+M)}\), where \(a_{ij}=1\) if \((j, i)\in \mathcal {E}\) and \(a_{ij}=0\) otherwise. The Laplacian matrix of the graph \(\mathcal {G}\) associated with adjacency matrix \(A\) is designed as \(\mathcal {L}=(l_{ij})\), where \(l_{ii}=\sum _{j=1}^{N+M}a_{ij}\) and \(l_{ij}=-a_{ij}, \; i\ne j,\; i, j=1,2,\ldots ,N+M\).

In this paper, agents labeled \(1, 2, \ldots , N, \) are followers, while agents labeled \(N+1,\ldots , N+M\) are the leaders. Suppose that all the followers have at least one neighbor, but the leaders have no neighbors. Throughout, the network communication topology satisfies the following assumption.

Assumption 1

The communication topology among the followers is undirected. For each follower, there exists at least one directed path from the leader to the follower.

Then, \(\mathcal {L}\) can be rewritten as

$$\begin{aligned} \mathcal {L}= & {} \left[ \begin{array}{l@{\quad }l} L_1&{} L_2\\ 0_{M\times N} &{}0\\ \end{array}\right] , \end{aligned}$$
(1)

where \(L_2\in R^{N\times M}\) has at least one negative entry, \(L_1\) is symmetric and positive definite and \(-L^{-1}_1L_2\mathbf {1}_{M\times 1}=\mathbf {1}_{N\times 1}\) [20] under Assumption 1.

Lemma 1

(Young’s inequality) For every real numbers \(a>0, b>0, c>0, p>1, q>1\), with \(\frac{1}{p}+\frac{1}{q}=1\), the following inequality is satisfied:

$$\begin{aligned} ab\le c^p\frac{a^p}{p}+c^{-q}\frac{b^q}{q}. \end{aligned}$$

Lemma 2

(Jensen’s inequality) For every real numbers \(a\ge 0, b\ge 0\) and \(0<p<q\), the following inequality is satisfied:

$$\begin{aligned} (a^q+b^q)^\frac{1}{q}\le (a^p+b^p)^\frac{1}{p}. \end{aligned}$$

Lemma 3

[34] Consider the system

$$\begin{aligned} \dot{p}= & {} q-l_1\mathrm{sig}(p)^{\frac{1}{2}},\nonumber \\ \dot{q}= & {} -l_2\mathrm{sgn}(p)+f(t), \end{aligned}$$
(2)

where \(l_1, l_2\) are positive constants and the bounded perturbation satisfies \(\Vert f(t)\Vert _{\infty }\le \varepsilon \). Then, \(p\) and \(q\) will converge to zero in finite time if there exists a symmetric and positive definite matrix \(P\) such that

$$\begin{aligned} A^TP+PA+\varepsilon ^2C^TC+PBB^TP=-Q<0, \end{aligned}$$
(3)

where \( A=\left[ \begin{array}{c@{\quad }c} -1/2l_1&{} 1/2\\ -l_2 &{}0\\ \end{array}\right] , \;B=\left[ \begin{array}{c} 0\\ 1\\ \end{array}\right] , \;C=\left[ \begin{array}{c@{\quad }c} 1&{} 0\\ \end{array}\right] .\) Moreover, let \(V_1=\vartheta ^TP\vartheta ,\) with \(\vartheta =[\mathrm{sig}(p)^\frac{1}{2}; q]\). Then, the trajectory of (2) reaches the origin in a time smaller than

$$\begin{aligned} T_1=\frac{2\lambda _{\max }(P)}{\lambda _{\min }(Q) \lambda _{\min }^{\frac{1}{2}}(P)}V_1^{\frac{1}{2}}(\vartheta _0), \end{aligned}$$
(4)

where \(\vartheta _0\) is the initial state of \(\vartheta \) .

The following lemma shows a way to select \(l_1,\; l_2\) such that (3) is feasible.

Lemma 4

For a given constant \(\varepsilon >0\), if

$$\begin{aligned} l_2+ & {} 2-\sqrt{l_2^2-\varepsilon ^2}< l_1< l_2+2+\sqrt{l_2^2-\varepsilon ^2}, \end{aligned}$$
(5)
$$\begin{aligned} l_2> & {} \varepsilon , \end{aligned}$$
(6)

then there exists a positive definite matrix \(P\) such that (3) is feasible.

Proof

See the section of “Appendix”.\(\square \)

2.3 Model description

Suppose that the followers are governed by double-integrator dynamics given by

$$\begin{aligned} \dot{x}_i=v_i, \quad \dot{v}_i=u_i+w_i(t),\; i=1, 2, \ldots , N, \end{aligned}$$
(7)

where \(x_i\) is the position, \(v_i\) is the velocity, \(u_i\) is the control input and \(w_i(t)\) is the external noises of the \(i\)th agent satisfying \(\Vert w_i(t)\Vert _\infty <\varpi <\infty \). The dynamics of the multiple leaders, labeled by \(N+1,\ldots , N+M\), is described by

$$\begin{aligned} \dot{x}_{N+i}= & {} v_{N+i},\nonumber \\ \dot{v}_{N+i}= & {} a_{N+i}(t), \; i=1,\ldots ,M, \end{aligned}$$
(8)

where \(a_{N+i}(t)\) are the leaders’ time-varying accelerations. It is assumed that \(\Vert a_{N+i}(t)\Vert _\infty \le c <\infty \). It is worth mentioning that, compare with existing work in [35], the dynamics of the leader with \(a_{N+i}(t), i=1,\ldots ,M,\) can describe more complex tracking tasks, therefore is more general.

Definition 1

For systems (7) and (8), the finite-time containment control problem is said to be solved if and only if there exist a distributed control protocol \(u_i\) and \(T>0\), such that the positions and velocities of the followers converge into the convex hulls \(co(X_h)\) and \(co(V_h)\), respectively, formed by those of the multiple leaders in finite time, where

$$\begin{aligned} co(X_h)= & {} \left\{ \sum _{i=N+1}^{N+M}\theta _i x_i |\, \theta _i\ge 0,\, \sum _{i=N+1}^{N+M}\theta _i=1\right\} ,\\ co(V_h)= & {} \left\{ \sum _{i=N+1}^{N+M}\theta _i v_i |\, \theta _i\ge 0,\, \sum _{i=N+1}^{N+M}\theta _i=1\right\} . \end{aligned}$$

The main objective of this paper is to design distributed algorithm without velocity and acceleration measurements of the agents such that the states of the followers converge to a convex hull spanned by those of the multiple dynamic leaders in finite time.

3 Main results

Consider the multi-agent systems described by (7) and (8). The protocol based only on relative position measurements is proposed as

$$\begin{aligned} u_i= & {} {-}k_1 \mathrm{sgn}(\xi _i)-k_2 \mathrm{sgn}(\eta _i), \nonumber \\ \dot{\xi }_i= & {} \eta _i-l_1\mathrm{sig}\bigg (\xi _i-\sum _{j=1}^{N+M}a_{ij}(x_i{-}x_j)\bigg )^{\frac{1}{2}},\nonumber \\ \dot{\eta }_i= & {} -l_2\mathrm{sgn}\bigg (\xi _i-\sum _{j=1}^{N+M}a_{ij}(x_i{-}x_j)\bigg ),\nonumber \\&\quad i=1,2, \ldots , N, \end{aligned}$$
(9)

where \( k_1, k_2, l_1, l_2\) are parameters to be determined and \(a_{ij}\) is the \((i, j)\)th entry of the adjacency matrix \(A\). Note that the controller (9) is distributed, since it is based only on local information. Throughout this paper, the solutions of the error systems are subject to the sense of Filippov [36].

Remark 1

Compared with the existing observer-based protocols designed in [5] and [35], which require the relative input information, \(\sum _{i=1}^{N+1}(u_i-u_j), \; i=1,2,\ldots ,N\), the protocol proposed in (9) does not require relative input information. This is more favorable and cost savings in real applications.

The following result provides a sufficient condition for the designed consensus tracking protocol (9).

Theorem 1

Suppose that Assumption 1 holds. Then, the positions and velocities of the followers under protocol (9) will converge into the convex hulls \(co(X_h)\) and \(co(V_h)\) formed by those of the multiple leaders in finite settling time, respectively, if the following inequalities are fulfilled:

$$\begin{aligned}&l_2+2-\sqrt{l_2^2-\varepsilon ^2}<l_1< l_2+2+\sqrt{l_2^2-\varepsilon ^2}, \end{aligned}$$
(10)
$$\begin{aligned}&l_2>\varepsilon ,\end{aligned}$$
(11)
$$\begin{aligned}&k_1>\frac{\kappa \theta }{3}({4Nn})^{\frac{1}{3}}\sqrt{\frac{2Nn}{\lambda _{\max }(L_1)}},\end{aligned}$$
(12)
$$\begin{aligned}&k_1>k_2+c+\varpi ,\end{aligned}$$
(13)
$$\begin{aligned}&k_2>\frac{2\theta Nn\lambda _{\min }(L_1)}{3}\sqrt{\frac{2Nn}{\lambda _{\max }(L_1)}}{+}c+\varpi , \end{aligned}$$
(14)

where \(\varepsilon =2DNn(c+k_1+k_2+\varpi )\), \(\theta >0\) and \(\kappa =\Vert L_1^{-1}\Vert _{\infty }\). Moreover, provide the Lyapunov function

$$\begin{aligned} V_2= & {} \bigg [k_1\Vert e_x\Vert _1{+}\frac{1}{2}e_v^T(L_1^{-1}{\otimes } I)e_v\bigg ]^{\frac{3}{2}}\nonumber \\&+\, \theta e_x^T(L_1^{-1}{\otimes } I)e_v. \end{aligned}$$
(15)

where \(e_x=(L_1\otimes I)\widetilde{X}\), \(e_v=(L_1\otimes I)\widetilde{V}\), \(\widetilde{X}={X}+(L_1^{-1}L_2{\otimes } I) {X_l}\), \(\widetilde{V}={V}+(L_1^{-1}L_2{\otimes } I) {V_l}\), \(X=(x_1^T, \ldots , x_{N}^T)^T\), \(X_l=(x_{N+1}^T, \ldots , x_{N+M}^T)^T\), \(V=(v_1^T, \ldots , v_{N}^T)^T\) and \(V_l=(v_{N+1}^T, \ldots , v_{N+M}^T)^T\). The settling-time estimation can be computed by

$$\begin{aligned} T=T_1+T_2, \end{aligned}$$
(16)

where \(T_1\)is given by (4), \(T_2=\frac{3}{\gamma }V_2^{\frac{1}{3}}(e_x(0), e_v(0))\) with \(\gamma \) satisfying \(\alpha _1^{\frac{3}{2}}-\gamma ^{\frac{3}{2}}\beta _1>0\) and \(\alpha _2^{\frac{3}{2}}-\gamma ^{\frac{3}{2}}\beta _2>0\), where \(\alpha _1= \theta (k_1-k_2- c ) \), \(\alpha _2=\frac{3}{2}\bigg (\frac{\lambda _{\max }(L_1)}{2Nn}\bigg )^{\frac{1}{2}}(k_2-c)-\theta Nn\lambda _{\min }(L_1)\), \(\beta _1=\sqrt{2}k_1^{\frac{3}{2}}+ \theta \frac{2h^\frac{3}{2}}{3}\) and \(\beta _2=\theta \frac{Nn \kappa ^3 }{3h^3}\).

Proof

Firstly, we will show the global finite-time stability of system (9). Let \(\delta _i=\xi _i-\sum _{j=1}^{N+M}a_{ij}(x_i{-}x_j)\) and \(\zeta _i=\eta _i-\sum _{j=1}^{N+M}a_{ij}(v_i{-}v_j)\). It follows from (7), (8) and (9) that

$$\begin{aligned} \dot{\delta }_i= & {} \zeta _i-l_1\mathrm{sig}({\delta }_i)^{\frac{1}{2}} \nonumber \\ \dot{\zeta }_i= & {} {-}l_2 \mathrm{sgn}({\delta }_i)-\sum _{j=1}^{N}l_{ij}u_j \nonumber \\&-\sum _{j=1}^{N}l_{ij}w_j(t)-\sum _{j=N+1}^{N+M}l_{ij}a_{j}(t)\nonumber \\ u_i= & {} {-}k_1 \mathrm{sgn}(\xi _i)-k_2 \mathrm{sgn}(\eta _i), \end{aligned}$$
(17)

Denote \({\delta }=(\delta _1^T, \ldots , \delta _{N}^T)^T\), \(\zeta =(\zeta _1^T, \ldots , \zeta _{N}^T)^T\), \(U=(u_1^{T}, u_2^{T}, \ldots , u_N^{T})^T\), \(W(t)=(w_1^T(t), \ldots , w_{N}^T(t))^T\), \(F(t)=(a_{N+1}^T(t), \ldots , a_{N+M}^T(t))^T\). One has

$$\begin{aligned} \dot{\delta }= & {} \zeta -l_1\mathrm{sig}({\delta })^{\frac{1}{2}} \nonumber \\ \dot{\zeta }= & {} {-}l_2 \mathrm{sgn}({\delta })+(L_1\otimes I)[k_1 \mathrm{sgn}(\xi )+k_2 \mathrm{sgn}(\eta )\nonumber \\&-W(t)-(L_1^{-1}L_2\otimes I)F(t)]. \end{aligned}$$
(18)

Since \(\Vert (L_1\otimes I)\Vert _1\le 2DNn, \; \Vert [k_1 \mathrm{sgn}(\xi )\Vert _\infty \le k_1,\;\Vert k_2 \mathrm{sgn}(\eta )\Vert _\infty \le k_2,\; \Vert W(t)\Vert _\infty \le \varpi ,\; \Vert (L_1^{-1} L_2\otimes I)\Vert _1\le 1\) and \(\Vert F(t)]\Vert _\infty \le c\), where \(D\) is the maximal in-degree of network. Then, one has

$$\begin{aligned}&\Vert (L_1\otimes I)[k_1 \mathrm{sgn}(\xi )+k_2 \mathrm{sgn}(\eta )-W(t)\\&\qquad -\,(L_1^{-1} L_2\otimes I)F(t)]\Vert _\infty \\&\quad \le \Vert (L_1\otimes I)\Vert _1\Vert [k_1 \mathrm{sgn}(\xi )+k_2 \mathrm{sgn}(\eta )-W(t)\\&\qquad -\,(L_1^{-1} L_2\otimes I)F(t)]\Vert _\infty \\&\quad \le \Vert (L_1\otimes I)\Vert _1(\Vert [k_1 \mathrm{sgn}(\xi )\Vert _\infty +\Vert k_2 \mathrm{sgn}(\eta )\Vert _\infty \\&\qquad +\,\Vert W(t)\Vert _\infty +\Vert (L_1^{-1} L_2\otimes I)\Vert _1\Vert F(t)]\Vert _\infty )\\&\quad \le 2DNn(k_1+k_2+\varpi +c), \end{aligned}$$

It follows from Lemma 3 that there exists a finite time \(T_1>0\), such that \(\delta =0\) and \(\zeta =0\) when \(t>T_1\), if there exists a positive definite matrix \(P\) such that (3) holds with \(\varepsilon =2DNn(c+k_1+k_2+\varpi )\). Furthermore, for a given bound \(\varepsilon =2DNn(c+k_1+k_2+\varpi )\), it follows from Lemma 4 that (3) is feasible if \(l_2>\varepsilon \) and \(l_2+2-\sqrt{l_2^2-\varepsilon ^2}<l_1< l_2+2+\sqrt{l_2^2-\varepsilon ^2}\). In this case, \(\delta _i\) and \(\zeta _i\) will converge in finite time to the origin, which means \(\xi _i=\sum _{j=1}^{N+M}a_{ij}(x_i-x_j)\) and \(\eta _i=\sum _{j=1}^{N+M}a_{ij}(v_i-v_j)\), respectively, when \(t>T_1\). Therefore,

$$\begin{aligned} u_i= & {} {-}k_1 \mathrm{sgn}\bigg [\sum _{j=1}^{N+M}a_{ij}(x_i{-}x_j)\bigg ]\nonumber \\&-\,k_2 \mathrm{sgn}\bigg [\sum _{j=1}^{N+M}a_{ij}(v_i{-}v_j)\bigg ], \nonumber \\&i=1,2,\ldots , N, \end{aligned}$$
(19)

when \(t>T_1\). Thus, the closed-loop system of (7) and (9) is transformed into

$$\begin{aligned} \dot{x}_i= & {} v_i \nonumber \\ \dot{v}_i= & {} {-}k_1 \mathrm{sgn}\bigg [\sum _{j=1}^{N+M}a_{ij}(x_i{-}x_j)\bigg ]\nonumber \\&-\,k_2 \mathrm{sgn}\bigg [\sum _{j=1}^{N+M}a_{ij}(v_i{-}v_j)\bigg ]+w_i(t). \end{aligned}$$
(20)

Let \(X=(x_1^T, \ldots , x_{N}^T)^T\), \(X_l=(x_{N+1}^T, \ldots , x_{N+M}^T)^T\), \(V=(v_1^T, \ldots , v_{N}^T)^T\), \(V_l=(v_{N+1}^T, \ldots , v_{N+M}^T)^T\). Then, (20) can be rewritten in a matrix form as

$$\begin{aligned} \dot{ {X}}= & {} {V} \nonumber \\ \dot{ {V}}= & {} {-}k_1 \mathrm{sgn} \{(L_1{\otimes } I) [{X}+(L_1^{-1}L_2{\otimes } I) {X_l}]\} \nonumber \\&-\,k_2 \mathrm{sgn} \{(L_1{\otimes } I)[{V}+(L_1^{-1}L_2{\otimes } I) {V_l}]\} \nonumber \\&+\, W(t), \end{aligned}$$
(21)

where \(L_1>0\). Notice that systems (8) can be rewritten as

$$\begin{aligned}&\dot{ {X_l}} = {V_l} \nonumber \\&\dot{ {V_l}} = F(t). \end{aligned}$$
(22)

Denote \(\widetilde{X}={X}+(L_1^{-1}L_2{\otimes } I) {X_l}\) and \(\widetilde{V}={V}+(L_1^{-1}L_2{\otimes } I) {V_l}\). From (21) and (22), one obtains the error systems as follows,

$$\begin{aligned} \dot{\widetilde{X}}= & {} \widetilde{V} \nonumber \\ \dot{\widetilde{V}}= & {} {-}k_1 \mathrm{sgn} [(L_1{\otimes } I)\widetilde{X}] {-}k_2 \mathrm{sgn} [(L_1{\otimes } I)\widetilde{V}]\nonumber \\&+\,(L_1^{-1}L_2{\otimes } I)F( t)+W(t), \end{aligned}$$
(23)

where \(L_1>0\). From \(-L^{-1}_1L_2\mathbf {1}_{M\times 1}=\mathbf {1}_{N\times 1}\) in [20], one has that finite-time containment control problems are solved if and only in system (23) can converge to zero in finite time.

Next, taking linear transformation \(e_x=(L_1\otimes I)\widetilde{X}\) and \(e_v=(L_1\otimes I)\widetilde{V}\), one obtains that

$$\begin{aligned} \dot{e}_x= & {} e_v \nonumber \\ \dot{e}_v= & {} -(L_1\otimes I)[k_1 \mathrm{sgn} (e_x)+k_2 \mathrm{sgn} (e_v)\nonumber \\&+\,(L_1^{-1}L_2{\otimes } I)F(t)+W(t)]. \end{aligned}$$
(24)

Then, consider the Lyapunov function (15). First of all, let us show that \(V_2\) is positive definite. Denote \(e_{x}=(e_{x1}^T,\ldots ,e_{xN}^T)^T\), where \(e_{xi}=(e_{xi1},\ldots ,e_{xin})^T\), \( i=1,\ldots , N\). According to Lemma 1 and 2, one has

$$\begin{aligned} V_2= & {} \bigg [k_1\Vert e_x\Vert _1+\frac{1}{2}e_v^T(L_1^{-1}\otimes I)e_v\bigg ]^{\frac{3}{2}}\nonumber \\&+\,\theta e_x^T(L_1^{-1}\otimes I)e_v \nonumber \\\ge & {} (k_1\Vert e_x\Vert _1)^{\frac{3}{2}}+\bigg [\frac{1}{2}e_v^T(L_1^{-1}\otimes I)e_v\bigg ]^{\frac{3}{2}}\nonumber \\&-\,\theta \bigg (\frac{2h^\frac{3}{2}}{3}\sum _{i=1}^{N}\sum _{m=1}^n|e_{xim}|^{\frac{3}{2}}\nonumber \\&+ \frac{1}{3h^3}\Vert (L_1^{-1}\otimes I) e_v\Vert _{3}^3\bigg )\nonumber \\\ge & {} \bigg (k_1^{\frac{3}{2}}-\frac{2\theta h^\frac{3}{2}}{3}\bigg )\bigg (\sum _{i=1}^{N}\sum _{m=1}^n|e_{xim}|^{\frac{3}{2}}\bigg ) \nonumber \\&+\bigg [\bigg (\frac{\lambda _{\max }(L_1)}{2Nn}\bigg )^{\frac{3}{2}} -\theta \frac{Nn \kappa ^3 }{3h^3}\bigg ] \Vert e_v\Vert _1^{3}. \end{aligned}$$
(25)

Both coefficients, i.e., \( k_1^{\frac{3}{2}}-\frac{2\theta h^\frac{3}{2}}{3} \) and \( \bigg (\frac{\lambda _{\max }(L_1)}{2Nn}\bigg )^{\frac{3}{2}} -\theta \frac{Nn \kappa ^3 }{3h^3} \) in the above expression are positive if \(h\) is selected such that \(\kappa \Big (\frac{\theta Nn}{3}\Big )^{\frac{1}{3}}\sqrt{\frac{2Nn}{\lambda _{\max }(L_1)}}<h<\Big (\frac{3}{2\theta }\Big )^{\frac{2}{3}}k_1\). Such an \(h\) exists if

$$\begin{aligned} k_1>\frac{\kappa \theta }{3}({4Nn})^{\frac{1}{3}}\sqrt{\frac{2Nn}{\lambda _{\max }(L_1)}}. \end{aligned}$$
(26)

Select \(h=\chi \kappa \Big (\frac{\theta Nn}{3}\Big )^{\frac{1}{3}}\sqrt{\frac{2Nn}{\lambda _{\max }(L_1)}}+(1-\chi )\Big (\frac{3}{2\theta }\Big )^{\frac{2}{3}}k_1\), \(0<\chi <1\), which guarantees to the positive definiteness of \(V_2\). Then, differentiating the Lyapunov function \(V_2\) along the trajectory of (24) gives

$$\begin{aligned} \dot{V}_2= & {} \frac{3}{2}\bigg [k_1\Vert e_x\Vert _1+\frac{1}{2}e_v^T(L_1^{-1}\otimes I)e_v\bigg ]^{\frac{1}{2}} \bigg [k_1 e_v^T \mathrm{sgn}(e_x)\\&-\, e_v^T\bigg (k_1 \mathrm{sgn} (e_x)+k_2 \mathrm{sgn} (e_v)\\&+\, (L_1^{-1}L_2{\otimes } I)F(t)+W(t)\bigg )\bigg ]+\theta e_v^T(L_1^{-1}{\otimes } I)e_v\\&-\, \theta e_x^T\bigg (k_1 \mathrm{sgn} (e_x)+k_2 \mathrm{sgn} (e_v)\\&+\,(L_1^{-1}L_2{\otimes } I)F(t)+W(t)\bigg )\\= & {} -\frac{3}{2}\bigg [k_1\Vert e_x\Vert _1{+}\frac{1}{2}e_v^T(L_1^{-1}{\otimes } I)e_v\bigg ]^{\frac{1}{2}}\\&\times \,\bigg (k_2\Vert e_{v}\Vert _1 {-} e_{v}^T[(L_1^{-1}L_2{\otimes } I)F(t)+W(t)]\bigg ) \\&+\,\theta e_v^T(L_1^{-1}{\otimes } I)e_v {-}\theta \{k_1 \Vert e_{x}\Vert _1\\&+\,k_2 e_{x}^T\mathrm{sgn} (e_{v}){+} e_{x}^T[(L_1^{-1}L_2{\otimes } I)F(t)+W(t)]\}. \end{aligned}$$

Note that the following inequalities hold for all \(t, e_x, e_v\),

$$\begin{aligned}&k_2\Vert e_{v}\Vert _1 {-} e_{v}^T[(L_1^{-1}L_2{\otimes } I)F(t)+W(t)]{>}0,\\&k_1 \Vert e_{x}\Vert _1+ k_2 e_{x}^T\mathrm{sgn} (e_{v}) \\&\quad +\, e_{x}^T[(L_1^{-1}L_2{\otimes } I)F(t)+W(t)]>0, \end{aligned}$$

if

$$\begin{aligned} k_2 >c+\varpi ,\quad k_1 >k_2 +c+\varpi . \end{aligned}$$
(27)

In this case, since

$$\begin{aligned}&[k_1\Vert e_x\Vert _1+\frac{1}{2}e_v^T(L_1^{-1}\otimes I) e_v]^{\frac{1}{2}}\\&\ge \bigg (\frac{\lambda _{\max }(L_1)}{2}\bigg )^{\frac{1}{2}}\Vert e_v\Vert \ge \bigg (\frac{\lambda _{\max }(L_1)}{2Nn}\bigg )^{\frac{1}{2}}\Vert e_v\Vert _1, \end{aligned}$$

and

$$\begin{aligned} e_v^T(L_1^{-1}\otimes I)e_v\le & {} \lambda _{\min }(L_1)\Vert e_v\Vert ^2\\\le & {} Nn\lambda _{\min }(L_1) \Vert e_v\Vert _1^2, \end{aligned}$$

it follows that

$$\begin{aligned} \dot{V}_2\le & {} -\bigg [\frac{3}{2}\bigg (\frac{\lambda _{\max }(L_1)}{2Nn}\bigg )^{\frac{1}{2}} (k_2-c-\varpi )-\theta Nn\lambda _{\min }(L_1)\bigg ]\nonumber \\&\times \,\Vert e_{v}\Vert _1^2 - \theta \bigg (k_1-k_2- c-\varpi \bigg )\Vert e_{x}\Vert _1. \end{aligned}$$
(28)

Then, it is clear that \(\dot{V}_2\) is negative definite if

$$\begin{aligned}&k_2>\frac{2\theta Nn\lambda _{\min }(L_1)}{3} \sqrt{\frac{2Nn}{\lambda _{\max }(L_1)}}{+}c+\varpi , \nonumber \\&k_1>k_2{+}c+\varpi . \end{aligned}$$

Note that

$$\begin{aligned}&\bigg [k_1\Vert e_x\Vert _1+\frac{1}{2}e_v^T(L_1^{-1}\otimes I)e_v\bigg ]^{\frac{3}{2}}\nonumber \\&\quad \le \sqrt{2}\bigg [(k_1\Vert e_x\Vert _1)^{\frac{3}{2}}+\Big (\frac{1}{2}e_v^T(L_1^{-1}\otimes I)e_v\Big )^{\frac{3}{2}}\bigg ]\nonumber \\&\quad \le \sqrt{2}\bigg [(k_1\Vert e_x\Vert _1)^{\frac{3}{2}}+ \bigg (\frac{\lambda _{\min }(L_1)}{2Nn}\bigg )^{\frac{3}{2}}\Vert e_x\Vert _1^3\bigg ]. \end{aligned}$$
(29)

One has

$$\begin{aligned} V_2= & {} \bigg [k_1\Vert e_x\Vert _1+\frac{1}{2}e_v^T(L_1^{-1}\otimes I)e_v\bigg ]^{\frac{3}{2}}\nonumber \\&+\,\theta e_x^T(L_1^{-1}\otimes I)e_v \nonumber \\\le & {} \sqrt{2}\bigg [(k_1\Vert e_x\Vert _1)^{\frac{3}{2}}+\Big (\frac{1}{2}e_v^T(L_1^{-1}\otimes I)e_v\Big )^{\frac{3}{2}}\bigg ]\nonumber \\&+\,\theta \bigg (\frac{2h^\frac{3}{2}}{3}\sum _{i=1}^{N}\sum _{m=1}^n|e_{xim}|^{\frac{3}{2}}\nonumber \\&+ \frac{1}{3h^3}\Vert (L_1^{-1}\otimes I) e_v\Vert _{3}^3\bigg )\nonumber \\\le & {} \bigg (\sqrt{2}k_1^{\frac{3}{2}}+ \theta \frac{2h^\frac{3}{2}}{3}\bigg )\Vert e_x\Vert _1^{\frac{3}{2}}\nonumber \\&+\, \bigg [\bigg (\frac{\lambda _{\min }(L_1)}{2Nn}\bigg )^{\frac{3}{2}} +\theta \frac{Nn \kappa ^3 }{3h^3}\bigg ]\Vert e_v\Vert _1^3\nonumber \\= & {} \beta _1\Vert e_x\Vert _1^{\frac{3}{2}}+\beta _2\Vert e_v\Vert _1^3, \end{aligned}$$
(30)

where \(\beta _1=\sqrt{2}k_1^{\frac{3}{2}}+ \theta \frac{2h^\frac{3}{2}}{3}\) and \(\beta _2=\theta \frac{Nn \kappa ^3 }{3h^3}\). Let \(\alpha _1= \theta \bigg (k_1-k_2- c-\varpi \bigg ) \) and \(\alpha _2=\frac{3}{2}\bigg (\frac{\lambda _{\max }(L_1)}{2Nn}\bigg )^{\frac{1}{2}}(k_2-c-\varpi )-\theta Nn\lambda _{\min }(L_1)\). Recalling (28), it follows from Lemma 2 that

$$\begin{aligned} \dot{V}_2\le & {} -\alpha _1\Vert e_{x}\Vert _1 - \alpha _2\Vert e_{v}\Vert _1^2\nonumber \\\le & {} -(\alpha _1^{\frac{3}{2}}\Vert e_{x}\Vert _1^{\frac{3}{2}} + \alpha _2^{\frac{3}{2}}\Vert e_{v}\Vert _1^3)^{\frac{2}{3}}\nonumber \\\le & {} -\gamma {V}_2^{\frac{2}{3}}, \end{aligned}$$
(31)

where \(\gamma \) satisfies \(\alpha _1^{\frac{3}{2}}-\gamma ^{\frac{3}{2}}\beta _1>0\) and \(\alpha _2^{\frac{3}{2}}-\gamma ^{\frac{3}{2}}\beta _2>0\). Thus, \(e_x\) and \(e_v\) will converge to the origin in finite time, which means that \(\widetilde{X}\) and \(\widetilde{V}\) would be zero after a finite time. Therefore, finite-time containment control problems with multiple dynamic leaders can be solved without velocity and acceleration measurements. Finally, from the differential equation (31), one has \(V_2(t)=\bigg (V_2(0)^{\frac{1}{3}}-\frac{\gamma }{3}t\bigg )^3\), which implies that the trajectory starting at the initial point \((e_x(0), e_y(0))\) will converge to the origin in finite time less than \(T=T_1+T_2\) computed by (16). This completes the proof.\(\square \)

Remark 2

Note that the protocol (9) is distributed and based only on relative position measurements among the neighboring agents. It is worth mentioning that Lemma 3 plays an important role in the derivation of this result. Furthermore, for the case of one leader, the protocol (9) can also solve the finite-time tracking control problem using only relative position measurements.

Remark 3

From the proof of Theorem 1, the settling time is less than \(T=T_1+T_2\), where \(T_1\) and \(T_2\) are computed by (4) and (16), respectively.

Remark 4

It follows from the proof of Theorem 1 that the dynamic protocol (9) can obtain the relative velocity measurements. In fact, the multi-agent systems (7) and (8) and the observer systems (9) can be decoupled in finite time since the error systems (18) will be zero for \(t>T_1\). Therefore, one can obtain the information of the relative positions and relative velocities after a finite time \(T_1\). Then, the observer-based controller (9) is equivalent to the state feedback controller shown in Theorem 1.

4 Simulation results

Consider a network system of nine agents with a undirected topology among the six followers as shown in Fig. 1. Assume that there are three leaders described by the following dynamics

$$\begin{aligned}&\dot{x}_i=v_{xi},\nonumber \\&\dot{y}_i=v_{y i},\nonumber \\&\dot{v}_{xi}=f_i(t),\nonumber \\&\dot{v}_{yi}=0,\quad i=7,8,9, \end{aligned}$$
(32)

where \(f_7(t)=-\frac{1}{4}x_7+\frac{1}{4}\), \(f_8(t)=-\frac{1}{4}x_8+\frac{3}{8}\) and \(f_9(t)=-\frac{1}{4}x_9+\frac{1}{2}\) with initial states \(x_7(0)=3,\;y_7(0)=1\), \(v_{x7}(0)=0.5,\;v_{y7}(0)=0.5\), \(x_8(0)=4,\;y_8(0)=1.5\), \(v_{x8}(0)=0.5,\;v_{y8}(0)=0.5\), \(x_9(0)=3,\;y_9(0)=2\), and \(v_{x9}(0)=0.5,\;v_{y9}(0)=0.5\). By choosing the controller parameters \(l_1=3,\; l_2=1,\; k_1=25\) and \(k_2=20\), the conditions (10–14) in Theorem 1 are satisfied. Figures 2 and 3 show the positions and velocities of agents 1–9 using (9), respectively. It can be seen that the followers can track the dynamic convex hull formed by the leaders at about \(t=4.5 s\), which verifies the effectiveness of the theoretical results.

Fig. 1
figure 1

Communication topology

Full size image
Fig. 2
figure 2

Trajectories of six agents with three dynamic leaders in two-dimensional space

Full size image
Fig. 3
figure 3

Velocities of agents

Full size image

5 Conclusion

This paper considers the distributed finite-time containment control problem, which has been investigated for a group of agents modeled by double-integrator dynamics under a time-invariant network topology among the followers using only relative state measurements. A class of distributed finite-time protocols is proposed for achieving containment tracking without velocity and acceleration measurements. Future work will focus on finite-time control problem under a directed communication topology.