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www.ietdl.org Published in IET Control Theory and Applications Received on 7th January 2013 Revised on 28th May 2013 Accepted on 2nd June 2013 doi: 10.1049/iet-cta.2013.0013

ISSN 1751-8644

Brief Paper Finite-time tracking for double-integrator multi-agent systems with bounded control input Xiaoqing Lu1,2 , Shihua Chen2 , Jinhu Lü3 1 College

of Electrical and Information Engineering, Hunan University, Changsha 410082, People’s Republic of China of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China 3 Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China E-mail: [email protected] 2 College

Abstract: This study investigates the finite-time tracking for double-integrator multi-agent systems with bounded control input under the conditions of fixed and switching jointly reachable digraphes. In detail, two continuous distributed tracking protocols are designed to track the virtual leader in finite time with the same velocity and converging position. In particular, one introduces a special continuous distributed tracking protocol with bounded control input to track the virtual leader in finite time and reduce the chatter together. An example is also given to validate the proposed approaches.

1

Introduction

The distributed coordination problems of dynamic agent networks have received increasing attention in various disciplines of sciences and engineering recently. Typical examples include the distributed tracking control [1, 2], consensus of multi-agent systems [3–6], formation control [7], swarming and aggregation of agent-based systems [8]. This paper focuses on the finite-time distributed tracking control problems, the main objective of which is to design various tracking protocols to guide all agents or followers to track the given time-varying dynamic trajectories or virtual leader via local information exchange. The main difficulty lies in how to design the effective distributed control protocols as well as increasing the speed of tracking consensus. Generally, the design of double-integrator consensus algorithms [9] is essential important in practical engineering systems. Besides, various approaches have been developed to increase the speed of tracking consensus [10, 11]. It should be especially pointed out that Lipschitz condition is often essential necessary for many methods in literature [12, 13]. In 2006, based on the method of non-smooth gradient flows, Cortes initially discussed the finite-time consensus problems for multi-agent systems [3]. Afterwards, various efforts have been devoted to increase the convergence rate and improve the performance against various disturbances and uncertainties [14–23]. Very recently, many works on finite-time consensus have been proposed. In 2009, Xiao et al. investigated the finitetime formation control with non-Lipschitz protocols [15]. In 2011, Sayyaadi and Doostmohammadian studied the 1562 © The Institution of Engineering and Technology 2013

finite-time consensus problems with switching topology and communication delay [17]. For double-integrator multi-agent systems, Wang and Hong [19] addressed the finite-time consensus problem under the condition of time-variant undirected connected topology. Khoo et al. [20] introduced a robust finite-time consensus tracking algorithm with timeinvariant topology by using sliding-mode control technique. Following this line, Li and his colleagues further explored the finite-time consensus problem for leaderless and leader– follower double-integrator multi-agent systems with external disturbances [21]. Of late, Du et al., investigated the global finite-time stabilisation for a class of non-linear systems by bounded feedback method [24], and then they further extended it partially to the case of time-varying non-linear systems using the homogeneous domination approach [25]. Note that finite-time convergence implies the nonLipschitz continuous of multi-agent systems [24–26]. Moreover, the necessary condition of finite-time convergence may generate some undesired characteristics, such as the chattering phenomenon [27–29], which leads to the essential difficulty for the analysis and synthesis of practical finite-time control systems. Unfortunately, there are few results in the literature to resolve this important issue. Besides, most existing results [19–21] of finite-time double-integrator consensus protocols did not consider the switching topology which is directed. In general, it is very intractable when the discussed multi-agent system is non-linear with the asymmetric Laplacian matrix while the variable communication topology is directed. In this paper, one aims to solve this problem by introducing a detail-balanced digraph [13]. In addition, since all actuators of physical devices are subject to amplitudes IET Control Theory Appl., 2013, Vol. 7, Iss. 11, pp. 1562–1573 doi: 10.1049/iet-cta.2013.0013

www.ietdl.org saturation, the actuators are unable to implement control commands out of their ranges. It simulates us to design some specific finite-time consensus protocols with bounded control input to reduce the chatter to some extent. Furthermore, different from the terminal sliding-mode control method that used in [20], a new approach is presented in this paper to ensure the finite-time tracking consensus with bounded control input under switching jointly reachable digraphes. This paper aims at exploring the finite-time tracking problem for double-integrator dynamics with bounded control input by merging the traditional finite-time consensus idea [15, 16] and the classical distributed tracking control approach [1–3]. In detail, with the assumption that the weighted and directed switching topologies during each switching dwell time interval is not always connected, this paper generalises the results in [26] to the case of double-integrator multi-agent systems with switching jointly reachable digraphs by using the algebraic graph theory and Lyapunov analysis. Especially, in order to meet the need of bounded input and chatter reduction, a saturated function is introduced to smooth out the sign function in a thin boundary layer of the neighbourhood of stability state as well as adjust the control speed. The rest of this paper is organised as follows. In Section 2, some preliminaries and the problem formulation are briefly outlined. The main finite-time tracking protocols are proposed in Section 3. Section 4 introduces a typical example to illustrate the effectiveness of the proposed protocols. Some conclusions are drawn in Section 5.

2

Problem statements and preliminaries

To begin with, some necessary notations are listed. Let IN be the N × N identity matrix, A−1 be the inverse matrix of a reversible matrix A, and ⊗ be the Kronecher product. For any vectors a = (a1 , . . . , aN )T and b = (b1 , . . . , bN )T , denote a  b = (a1 b1 , . . . , aN bN )T . Let sig(x)α = sign(x)|x|α , where α > 0, x ∈ R, and sign(·) is the sign function. If x = (x1 , . . . , xN )T ∈ RN , then sig(x)α = sign(x)  |ˆx|α , where sign(x) = (sign(x1 ), . . . , sign(xN ))T , and |ˆx|α = (|x1 |α , . . . , |xN |α )T . If w = (w1 , . . . , wN ) ∈ RN , then diag(w) denotes the diagonal matrix with wi being as the (i, i) entry. Consider the finite-time tracking problem for the doubleintegrator multi-agent systems  x˙ i (t) = vi (t), i = 1, 2, . . . , N (1) v˙ i (t) = ui (t), and the dynamics of the virtual leader described by  x˙ 0 (t) = v0 (t) v˙ 0 (t) = a0 (t) where xi (t), vi (t), ui (t) ∈ R are the positions, velocities and control inputs of N followers, respectively. Hereafter, x0 (t), v0 (t), a0 (t) ∈ R are the position, velocity, and the acceleration of the virtual leader. Denote x = (x1 , . . . , xN )T and v = (v1 , . . . , vN )T . Since most of the existing protocols are still asymptotical convergence, that is lim |xi (t) − x0 (t)| = 0, lim |vi (t) − v0 (t)| = 0,

t→+∞

t→+∞

i = 1, 2, . . . , N then the above convergence time is infinite. Next, one introduces the following definition. IET Control Theory Appl., 2013, Vol. 7, Iss. 11, pp. 1562–1573 doi: 10.1049/iet-cta.2013.0013

Definition 1 (finite-time tracking): The tracking protocol ui (t) in system (1) can guide system (1) to reach the finite-time tracking control if and only if, for any initial state, there exists some finite time t ∗ , called settling time, which may be associated with the initial state, such that 

lim |xi (t) − x0 (t)| = 0, lim∗ − |vi (t) − v0 (t)| = 0

t→(t ∗ )−

t→(t )

xi (t) = x0 (t),

vi (t) = v0 (t),

t ≥ t ∗ , i = 1, 2, . . . , N

Let G = (V , ε, A) be a weighted digraph of order N with nodes V = {1, 2, . . . , N }, arcs ε ⊆ V × V , and a weighted adjacency A = (aij ) ∈ RN ×N , where aii = 0 and aij ≥ 0 for all i = j. aij > 0 if and only if there is an edge from node j to node i. A path between two distinct nodes i and j means that there is a sequence of distinct edges in the form (vi , vk1 ), (vk1 , vk2 ), . . . , (vkl , vj ). Moreover, a diagonal matrix D = diag{d1 , d2 , . . . , dN } ∈ RN ×N with di = j∈Ni aij , i = 1, 2, . . . , N is called a degree matrix of G, where Ni = {j ∈ V : (i, j) ∈ ε} is the set of all neighbours of node i. The matrix L = D − A ∈ RN ×N is called the Laplacian of the weighted digraph G. Weighted directed graph G(A) is said to satisfy the detail-balanced condition [13] in weights if there exist some scalars wi > 0 with i = 1, 2, . . . , N satisfying wi aij = wj aji for all i, j = 1, 2, . . . , N . In the following, the digraph G¯ is used to describe the interconnection topology of a multi-agent system consisting of one active leader-agent, denoted by 0, and N follower-agents, denoted by 1, . . . , N . Diagonal matrix B = diag{a10 , . . . , aN 0 } ∈ RN ×N is called the leader adjacency matrix, where ai0 > 0 if follower i is connected to the leader across the communication link (0, i), otherwise ai0 = 0. If there is a path in G¯ from every other node i to node 0, then node 0 is said to be globally reachable in ¯ The union graph of digraph families G¯1 , G¯2 , . . . , G¯m , with G. the same node set V for some m ≥ 1 is defined as graph G¯1−m ([30]), whose node set is V¯ = {V , 0}, and the edge set is the union of the edge sets of all graphs in the collections, and the connected weight between agents i and j is the sum weight of G¯1 , G¯2 , . . . , G¯m . Lemma 1 [9]: All eigenvalues of matrix L + B associated with digraph G¯ have positive real parts if and only if node ¯ 0 is globally reachable in digraph G. Lemma 2 [31]: Consider a family of systems x˙ = fσ (t) (x),

fσ (t) (0) = 0,

x ∈ Rn

(2)

Let  denote the finite switching index set, σ (t) : [0, ∞) →  be a piecewise constant continuous function of time, fk be continuous with respect to x for any fixed k ∈ , and τ be the dwell time. If the switched system (2) is asymptotically stable and x˙ = fk (x) is finite-time stable for any fixed k ∈ , then system (2) is finite-time stable.

3 3.1

Main results Finite-time tracking control

3.1.1 Double-integrator multi-agent systems with fixed topology: To begin with, the finite-time distributed 1563 © The Institution of Engineering and Technology 2013

www.ietdl.org tracking protocol is described by

ui (t) = a0 (t) + sig + sig

 

 

Construct Lyapunov candidate for system (5) as follows α  2−α

V (y, z) =

aij (xj − xi )

j∈Ni



aij (vj − vi )

where i = 1, 2, . . . , N , 0 < α < 1, and a0 (t) is a given control input. System (1) with protocol (3) is non-linear for parameter α ∈ (0, 1). For α = 1, it becomes a general double-integrator linear tracking control system with infinite converging time [4–11].

Clearly, V is non-negative and continuously differentiable for α ∈ (0, 1). Differentiate V (y, z) along system (5), one has 1 (L + B)−1 z˙ w 1 1 α−1 + (|ˆy| 2−α )T diag (sig(y) 2−α  y˙ ) w 1 1 α = −z T diag sig(y) 2−α − z T diag sig(z)α w w 1 α T 2−α z + (sig(y) ) diag w 1 1+α 1+α = −(|ˆz | 2 )T diag (|ˆz | 2 ) ≤ 0 w

V˙ (y, z) = z T diag

Theorem 1: If digraph G is detail-balanced and the leader node 0 of the corresponding digraph G¯ is globally reachable, then the tracking protocol (3) can guide the controlled system (1) to realise the finite-time tracking control. Proof: Let L = D − A, then L is the Laplacian matrix of digraph G. Since digraph G is detail-balanced, there exist scalars wi > 0 satisfying wi aij = wj aji for all i, j = 1, 2, . . . , N . Thus, diag(w)A is symmetric and then diag(w)(L + B) is also symmetric. As node 0 of digraph G¯ is globally reachable, diag(w)(L + B) is positive definite by lemma 1.  Letx¯ i = xi − x0 , v¯ i = vi − v0 , yi = j∈Ni aij (xj − xi ) and zi = j∈Ni aij (vj − vi ). Denote x¯ = (¯x1 , x¯ 2 , . . . , x¯ N )T , v¯ = (¯v1 , v¯ 2 , . . . , v¯ N )T , y = (y1 , y2 , . . . , yN )T and z = (z1 , z2 , . . . , zN )T . For agent i, its neighbours Ni can be divided into the f follower–neighbour set Ni  and the virtual leader–neighbour  set Ni , then one has yi = j∈N f aij (¯xj − x¯ i ) − ai0 x¯ i . Hence, i y = −(L + B)¯x. Similarly, one deduces z = −(L + B)¯v. Since L + B is positive stable, in order to obtain (¯xT , v¯ T )T → (0T , 0T )T in finite time, it suffices to prove (yT , z T )T → (0T , 0T )T in finite time. By (3), taking the above variable transformations on (1) yields ⎧ y˙ i (t) = zi (t) ⎪ ⎪ ⎪  ⎪ α α ⎪ ⎪ z ˙ (t) = aij [sig(yj ) 2−α − sig(yi ) 2−α ] ⎪ i ⎪ ⎪ ⎨ f j∈Ni  ⎪ aij [sig(zj )α − sig(zi )α ] ⎪ + ⎪ ⎪ ⎪ f ⎪ j∈Ni ⎪ ⎪ ⎪ α ⎩ −ai0 [sig(yi ) 2−α + sig(zi )α ], i = 1, 2, . . . , N

V (k 2−α y, kz) = k 2 V (y, z)

(7)

V˙ (k 2−α y, kz) = k 1+α V˙ (y, z)

(8)

and Since (7) and (8) are homogeneous, V (y, z) is radially 1 unbounded. Let k = [V (y, z)]− 2 in (8), one obtains V˙ (y, z) [V (y, z)]

1+α 2

= V˙ ([V (y, z)]−

2−α 2

y, [V (y, z)]− 2 z) 1

≤ sup V˙ (y, z) (y,z)∈ 2−α

Rewrite (4) in a compact form (5)

The right-hand side of system (5) is continuous but nonLipschitz continuous on some special points. Then, there exists a unique solution in forward time for any initial value. Thus, the solutions of system (5) are also continuous for the initial conditions, which indicates us that the settling time t ∗ in Definition 1 is well defined. 1564 © The Institution of Engineering and Technology 2013

where the derivative of V at 0 is the right derivative. For any k > 0, one obtains

where  = {([V (y, z)]− 2 y, [V (y, z)]− 2 z) : (yT , z T )T ∈ R2N \ {(0T , 0T )T }}. Since (7) is homogeneous, one has V ([V (y, 2−α 1 1 z)]− 2 y, [V (y, z)]− 2 z) = ([V (y, z)]− 2 )2 V (y, z) = 1. Therefore,  = {(y, z) : V (y, z) = 1}. Since V (y, z) is radially unbounded, it is a compact set. Moreover, V˙ (y, z) is continuous and non-positive on set . Thus, one obtains (4)

y˙ = z, α z˙ = −(L + B)sig(y) 2−α − (L + B)sig(z)α

(6)

(3)

j∈Ni



1 T z [(L + B)diag(w)]−1 z 2 2−α 1 1 + (|ˆy| 2−α )T [diag(w)]−1 (|ˆy| 2−α ) 2

V˙ (y, z) [V (y, z)]

1+α 2

1

≤ sup V˙ (y, z) = max V˙ (y, z) = −c (y,z)∈

(y,z)∈

with c ≥ 0. Furthermore, since {(yT , z T )T : V˙ (y, z) = 0} = {(0T , 0T )T }, one obtains c > 0. Then one has V˙ (y, z) ≤ 1+α −c[V (y, z)] 2 with α ∈ (0, 1). Thus, one deduces 1+α ∈ (0, 1). 2 1−α By Comparison Lemma, one obtains (V (y, z)) 2 ≤ 1−α 1−α (V (y(0), z(0))) 2 − 2 ct. 2 For (V (y(0), z(0))) 2 − 1−α ct = 0, it yields t ∗ = c(1−α) 2 1−α (V (y(0), z(0))) 2 . When t ≥ t ∗ , the origin of system (5) is a finite-time stable equilibrium. It follows that (¯xT , v¯ T )T will reach (0T , 0T )T in finite time t ∗ . This completes the proof of Theorem 1.  1−α

IET Control Theory Appl., 2013, Vol. 7, Iss. 11, pp. 1562–1573 doi: 10.1049/iet-cta.2013.0013

www.ietdl.org 3.1.2 Double-integrator multi-agent systems with switching topology: Consider the distributed tracking protocol with switching topology ⎡ ui (t) = a0 (t) + sig ⎣ ⎡ + sig ⎣

 j∈Ni (σ (t))



α ⎤ 2−α

with (9) V˙ = V˙ 1 + V˙ 2 + V˙ 3

aij (σ )(xj − xi )⎦

j=1

i=1

⎤α

aij (σ )(vj − vi )⎦

N N   wi aij [sig(zj )α − sig(zi )α ] zi

=− −

where i = 1, 2, . . . , N and σ (t) : [0, +∞) → ρ = {1, 2, . . . , m} is a piecewise constant switching signal defined on digraphs  = {G¯1 , G¯2 , . . . , G¯m }. Assume that the communication topology G¯σ switches at ts between the switching topologies {G¯1 , . . . , G¯m , G¯1 , . . . , G¯m , . . .} periodically in the order, where the time interval [ts , ts+1 ) with s = 0, 1, 2, . . . is an infinite sequence of non-empty, bounded and contiguous. Theorem 2: If the leader of system (1) with digraph G¯1−m is switching jointly reachable, it switches at time ts , s = 0, 1, 2, . . ., and in each time interval [ts , ts+1 ) with 0 < ts+1 − ts ≤ T1 for some constant T1 > 0, digraphs G1 , . . . , Gm are always detail-balanced with a time-invariant positive column vector w, then the tracking protocol (9) can guide the controlled system (1) to realise finite-time tracking control.

wi zi ai0 sig(zi )α

i=1

(9)

j∈Ni (σ )

N 

1  wi aij (zj − zi )(sig(zj )α − sig(zi )α ) 2 i=1 j=1 N

=− −

N 

N

wi zi ai0 sig(zi )α

i=1

≤ 0,

t ∈ [tsk , tsk+1 ), k = 0, 1, . . . , ms − 1, s = 0, 1, . . .

The last inequality comes from the fact that node 0 of G¯1−m is jointly reachable. In fact, since node 0 is jointly reachable in each time interval [ts , ts+1 ), there is at least one r ∈ {1, . . . , m} such that ai0 > 0 or one j ∈ {1, 2, . . . , N } such that aij (t) > 0 for any i ∈ {1, . . . , N }. Then V˙ = 0 if and only if zi = zj = 0 for any i = j. Hence z˙i (t) =

N 

α

α

α

aij (sig(yj ) 2−α − sig(yi ) 2−α ) − ai0 sig(yi ) 2−α = 0

j=1

(11) Proof: For t ∈ [ts , ts+1 ), denote yi , zi , y and z as those in the proof of Theorem 1. By (9), taking these variable transformations on system (1) yields (see (10)) Since in each time interval [ts , ts+1 ), digraphs G1 , . . . , Gm are always detail-balanced with a time-invariant positive column vector w, then diag(w)A(t) = [wi aij (t)]N ×N is always symmetric as t ∈ [ts , ts+1 ). Suppose that in each time interval [ts , ts+1 ) with ts+1 − ts ≤ T1 , t0 = 0, there is a sequence of non-overlapping subintervals [ts0 , ts1 ), [ts1 , ts2 ), . . . , [tsms −1 , tsms ) with ts0 = ts , tsms = ts+1 , satisfying Tsk+1 − tsk ≥ T2 , 0 ≤ k ≤ ms − 1 for some integer ms ≥ 0 and constant T2 > 0, such that the digraph G¯σ switches at tsk and it is invariant during each interval [tsk , tsk+1 ). Obviously, there are at most N˜ = T1 /T2 subintervals in [ts , ts+1 ). For t ∈ [tsk , tsk+1 ) with s = 0, 1, . . . and k = 0, 1, . . . , ms − 1, take the Lyapunov function candidate V = V1 + V2 + V3 with ⎧  yi N N  ⎪ 1 α ⎪ 2 ⎪ ⎪ V = w z , V = w ai0 (t)sig(s) 2−α ds, i 2 i 1 i ⎪ ⎨ 2 i=1 0 i=1  N N yj ⎪  ⎪ α α ⎪ ⎪ V3 = wi aij (t)[sig(yj ) 2−α − sig(s) 2−α ] ds. ⎪ ⎩ yi i=1 j=1

Since the elements aij (t) and ai0 (t) are invariant in each time interval [tsk , tsk+1 ), along the trajectories of system (1)

which implies that  N  N   α α α yi w i aij (sig(yj ) 2−α − sig(yi ) 2−α ) − ai0 sig(yi ) 2−α i=1

j=1

1  α α wi aij (yi − yj )(sig(yj ) 2−α − sig(yi ) 2−α ) 2 i=1 j=1 N

=− −

N 

N

α

wi yi ai0 sig(yi ) 2−α = 0

(12)

i=1

Since node 0 of digraph G¯1−m is jointly reachable, then (12) implies yi = yj = 0 for all i = j. Thus, for t ∈ [tsk , tsk+1 ), the origin of system (10) is globally asymptotically stable. Next, one proves that the origin of system (10) with t ∈ [tsk , tsk+1 ) is also finite-time stable. For any k¯ > 0, α α one obtains V (k¯ 2−α y, k¯ 2 z) = k¯ α V (y, z) and V˙ (k¯ 2−α y, k¯ 2 z) = α(1+α) 1 k¯ 2 V˙ (y, z). Let k¯ = [V (y, z)]− α , one obtains V˙ (y, z) [V (y, z)]

1+α 2

= V˙ ([V (y, z)]−

2−α α

1

≤ sup V˙ (y, z), ¯ (y,z)∈

¯ = {([V (y, z)]− α y, [V (y, z)]− 2 z) : (yT , z T )T ∈ R2N \ where  T T T {(0 , 0 ) }}. Similar to the proof of Theorem 1, one can 2−α

⎧ ⎪ zi (t), y˙ (t) =  ⎪  ⎪ i α α ⎨ aij (t)[sig(yj ) 2−α − sig(yi ) 2−α ] + aij (t)[sig(zj )α − sig(zi )α ] z˙i (t) = f f ⎪ j∈Ni j∈Ni ⎪ ⎪ α ⎩ −ai0 (t)[sig(yi ) 2−α + sig(zi )α ], t ∈ [ts , ts+1 ), s = 0, 1, . . . , i = 1, 2, . . . , N . IET Control Theory Appl., 2013, Vol. 7, Iss. 11, pp. 1562–1573 doi: 10.1049/iet-cta.2013.0013

y, [V (y, z)]− 2 z)

1

(10)

1565 © The Institution of Engineering and Technology 2013

www.ietdl.org obtain c > 0 such that 1+α V˙ (y, z) ≤ −c[V (y, z)] 2 ,

Therefore next one gives a novel approach to resolve this problem. t ∈ [tsk , tsk+1 ) ∗

By Comparison Lemma again, one obtains t = 1−α (V (y(tsk ), z(tsk ))) 2 . Therefore for t ∈ [tsk , tsk+1 ), when ∗ t ≥ t (> tsk ), the origin of system (10) is a finite-time stable equilibrium. As the origin of system (10) is also globally asymptotically stable and there are at most N˜ < +∞ subsystems in each time interval [tsk , tsk+1 ), this together with Lemma 2 gives that the origin of system (10) is globally finite time stable. Combining the assumption of the jointly reachable leader in each [ts , ts+1 ) with the aforementioned variable transformation yi and zi , one obtains that (¯xi , v¯ i )T will reach (0, 0)T in finite time, that is, system (1) with (9) will converge to (x0 , v0 )T in finite time globally. Theorem 2 holds.  2 c(1−α)

3.2 Finite-time tracking control with bounded control input Although the sign function with fractional power used in (3) and (9) is a continuous approximation to the discontinuous control function, the large gain can lead to high frequencies in the control signal in real-world applications. Thus, the influence of undesirable chatter is still very serious [17]. Until now, many scholars have done research to reduce the chatter [27–29] by using an ideal continuous approximate function or by adjusting the arrival rate. Since a saturation protection method is an effective solution to mitigate the high frequencies, then next one combines the two methods through adding a saturation function in (3) and (9). By adjusting the arrival rate appropriately, the undesirable chatter can be reduced to a certain extent. Moreover, since all actuators of physical devices are unable to deliver control commands out of their ranges, then the designed tracking protocols are also with bounded control input. 3.2.1 Double-integrator multi-agent systems with fixed topology: The bounded controllers are described by ⎡  α ⎤  2−α  ⎦ ui (t) = a0 (t) + sat 2−α ⎣sig aij (xj − xi ) ε

 + satε α1

sig

Theorem 3: If digraph G is detail-balanced and the leader node 0 of the corresponding digraph G¯ is globally reachable, then the bounded tracking protocol (13) can guide the controlled system (1) to realise the finite-time tracking control. Proof: Similarly, recast system (1) with protocol (13) in the following compact form  y˙ (t) = z(t) α (sig(y) 2−α ) − (L + B)satε α1 (sig(z)α ) z˙ (t) = −(L + B)satε 2−α α (14) where y = −(L + B)¯x and z = −(L + B)¯v. Since L + B is positive stable, then it suffices to prove (yT , z T )T → (0T , 0T )T in finite time. Denote

j∈Ni

2−α α

⊗ 1, z ≥ ε α ⊗ 1},

B = {(yT , z T )T : y > −ε

2−α α

⊗ 1, z ≥ ε α ⊗ 1}

C = {(yT , z T )T : y ≥ ε

2−α α

1

1

1

⊗ 1, |ˆz | ≤ ε α ⊗ 1},   2−α 1 D = {(yT , z T )T : |ˆy| ≤ ε α ⊗ 1, zˆ  ≤ ε α ⊗ 1} Then the set D is a positively invariant set for system (14) since all trajectories are bounded by its boundaries ∂D = ∪4i=1 Di , where D1 = {(yT , z T )T : y = −ε D2 = {(yT , z T )T : |ˆy| ≤ ε D3 = {(yT , z T )T : y = ε

2−α α

2−α α

2−α α

D4 = {(yT , z T )T : |ˆy| ≤ ε

1

⊗ 1, |ˆz | ≤ ε α ⊗ 1}, 1

⊗ 1, z = ε α ⊗ 1} 1

⊗ 1, |ˆz | ≤ ε α ⊗ 1},

2−α α

1

⊗ 1, z = −ε α ⊗ 1}

Fig. 1 shows the above sets: A, B, C, D and Di for i = 1, . . . , 4. Since z˙ |D1 = (L + B)(ε ⊗ 1 ± |ˆz |α )|D1 ≥ 0,

α

 

A = {(yT , z T )T : y ≤ −ε

α

z˙ |D2 = (L + B)(±|ˆy| 2−α − ε ⊗ 1)|D2 ≤ 0

α 

aij (vj − vi )

(13)

j∈Ni

where i = 1, 2, . . . , N , a0 (t) is a known bounded control input, ε ≥ 1 is a given positive constant, and for any vector x = (x1 , x2 , . . . , xN )T , the saturation function satε (·) is given by  x, |ˆx| ≤ ε satε (x) = ε sign(x), |ˆx| > ε Note that the protocol (13) has several advantages. First, the variable communication topology is directed and unconnected. Second, although there exist some control protocols including sat(·), the accordingly religious theoretical proof in the case of switching directed graph had not been given. Nevertheless, it is generally very unmanageable to investigate the finite-time consensus problems for non-linear double-integrator dynamics when the variable communication topology is directed and unconnected. Third, most of the existing works did not consider the undesirable chatter. 1566 © The Institution of Engineering and Technology 2013

Fig. 1

Regions in the phase plane for y, z ∈ RN with N = 1

IET Control Theory Appl., 2013, Vol. 7, Iss. 11, pp. 1562–1573 doi: 10.1049/iet-cta.2013.0013

www.ietdl.org z˙ |D3 = (L + B)(−ε ⊗ 1 ± |ˆz |α )|D3 ≤ 0, z˙ |D4 = (L + B)(±|ˆy|

α 2−α

switching topology are described by ⎡

+ ε ⊗ 1)|D4 ≥ 0

then no trajectories can leave D through ∂D. Hence, D is a positively invariant set for (14). Next, one proves that all trajectories starting in A, B and C enter D in a finite time interval. Clearly, z˙ |A = (L + 2−α 1 B)(ε α − ε α ) ⊗ 1 ≥ 0 and y˙ |A = z|A > 0. If (yT , z T )T ∈ A, 2−α 1 then after a finite time, y > −ε α ⊗ 1 and z ≥ ε α ⊗ 1. It follows that every trajectory originating in A enters B. 1 Moreover, (yT , z T )T ∈ B, z˙ |B ≤ (L + B)(ε ⊗ 1 − ε α ⊗ 1) < 0 and y˙ |B = z|B > 0. Since all trajectories starting in B cannot leave B through the set ∂A ∩ ∂B, then 1 after a finite time, |ˆz | ≤ ε α ⊗ 1 and the trajectories T T T enter C ∪ D. Since (y , z ) ∈ C, one obtains z˙ |C ≤ (L + 2−α B)(−ε α ⊗ 1 + ε ⊗ 1) < 0. Hence, after a finite time, 1 z(t) < −β ⊗ 1 with 0 < β ⊗ 1 < ε α ⊗ 1. Note that all trajectories can leave the region {(yT , z T )T : y > ε

2−α α

1

⊗ 1, −ε α ⊗ 1 < z < −β ⊗ 1} 2−α

1

only through the set {(yT , z T )T : y = ε α ⊗ 1, −ε α ⊗ 1 < z < −β ⊗ 1}. In this region, y˙ = z < 0. Therefore after a finite time, all trajectories enter D. Similarly, all trajectories starting in −A, −B and −C will certainly enter into D after a finite time. Moreover, the right-hand sides of (13) and (3) are identical in D. Since D is positive invariant, Theorem 3 holds.  3.2.2 Double-integrator multi-agent systems with switching topology: The bounded controllers with

Fig. 2



⎢ ⎝ sig ui (t) = a0 (t) + satε 2−α α ⎣



α ⎤ ⎞ 2−α ⎥ aij (σ )(xj − xi )⎠ ⎦

j∈Ni (σ )





+ satε α1 ⎣sig ⎝



⎞α ⎤ aij (σ )(vj − vi )⎠ ⎦

(15)

j∈Ni (σ )

Theorem 4: If the leader of system (1) with digraph G¯1−m is switching jointly reachable, it switches at time ts , s = 0, 1, 2, . . ., and in each time interval [ts , ts+1 ) with 0 < ts+1 − ts ≤ T1 for some constant T1 > 0, digraphs G1 , . . . , Gm are always detail-balanced with a time-invariant positive column vector w, then the tracking protocol (15) can guide the controlled system (1) to realise finite-time tracking control.

4

Typical example

Consider a multi-agent system consisting of a virtual leaderagent and nine follower-agents, denoted by 0, 1, 2, . . . , 9, respectively. In all simulations, the dynamics of the virtual leader is described by x˙ 0 (t) = v0 (t) and v˙ 0 (t) = sin t for t ≥ 0 with (x0 (0), v0 (0)) = (3, 1). The initial values of the followers are given by x(0) = (−1/2, 3, 5, 7, 9, 3, −7, 1, 2)T , v(0) = (2, 3, 1, −3, 1, 2, −2, 3, 3)T . Fig. 2 shows six different weighted digraphs with ten agents. Node 0 of G¯ is globally reachable. The digraphes ¯ thus the G¯i with i = 1, 2, . . . , 5 are the subgraphs of G, number of digraphes in  = {G¯1 , G¯2 , G¯3 , . . .} is finite. Moreover, the leader agent 0 in the union digraph G¯1−5 (i.e. ¯ is jointly reachable. All weights of the adjacency G) matrixes are labelled in Fig. 2. Obviously, the weighted

Five examples of directed topologies: G¯1 , G¯2 , . . . , G¯5 , which are the subgraphs of G¯

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6 5 4 Velocity evolutions

250 225 200 175 150 125 100 75 50 25 0 −25 −50

0

1 0 −1

−4

10 20 30 40 50 60 70 80 90 100 Time

10 8 6 4 2 0 −2 −4 −6 −8 −10 −12 −14

0

10 20 30 40 50 60 70 80 90 100 Time

0

10 20 30 40 50 60 70 80 90 100 Time

3 2 1 0 −1 −2 −3 −4

10 20 30 40 50 60 70 80 90 100 Time

Velocity evolutions

250 225 200 175 150 125 100 75 50 25 0 −25 −50

0

10 20 30 40 50 60 70 80 90 100 Time

10 8 6 4 2 0 −2 −4 −6 −8 −10 −12 −14

6 5 4 3 2 1 0 −1 −2 −3 −4

0

10 20 30 40 50 60 70 80 90 100 Time

0

10 20 30 40 50 60 70 80 90 100 Time

3 2 Velocity errors

Position errors

Position evolutions

Finite-time tracking consensus of system (1) under protocol (3) with α = 0.2 and fixed topology G¯

1 0 −1 −2 −3

0

Fig. 4

2

−3

0

Fig. 3

3

−2

Velocity errors

Position errors

Position evolutions

www.ietdl.org

10 20 30 40 50 60 70 80 90 100 Time

−4

Tracking evolutions of system (1) under protocol (3) with α = 1 and fixed topology G¯

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IET Control Theory Appl., 2013, Vol. 7, Iss. 11, pp. 1562–1573 doi: 10.1049/iet-cta.2013.0013

www.ietdl.org digraphes G¯ and G¯i (without virtual leader 0) are always detail-balanced with a time-invariant positive column vector w = (0.3, 2.8, 1.2, 4.2, 2.8, 1.8, 1.2, 0.2, 0.3)T . Take ε = 1 in all simulations. ¯ Figs. 3 and 4 show (1) Case 1: Consider the fixed digraph G. the tracking curves of system (1) under (3) with α = 0.2 and α = 1 respectively. Clearly, the convergence time is finite for 0 < α < 1 but infinite for α = 1. Fig. 5 shows the inner relationship between parameter α and convergence time t ∗ . It indicates us that the shortest convergence time occurs when 80

The convergence time

70

60

50

40

In the previous works on double-integrator multi-agent systems[19, 21], the proposed control inputs are given below

30

20 0

0.2

0.4 0.6 The parameter α

0.8

1

ui (t) = a0 (t) +

Relationship between convergence time t ∗ and parame-

250 225 200 175 150 125 100 75 50 25 0 −25 −50

aij [sig(xj − xi )α1 + sig(vj − vi )α2 ]

j∈Ni (t)

(16)

6 5 4 3 2 1 0 −1 −2 −3 −4 0

10

20

30

40

50 60 Time

70

80

90 100

10 8 6 4 2 0 −2 −4 −6 −8 −10 −12 −14

0

10

20

30

40

50 60 Time

0

10

20

30

40 50 Time

70

80

90 100

70

80

90 100

3 2 Velocity errors

Position errors



− bi [sig(xi − x0 )α1 + sig(vi − v0 )α2 ]

Velocity evolutions

Position evolutions

Fig. 5 ter α

α is approaching to 0.2. For the chatter reduction in tracking consensus, one simulates the tracking curves of system (1) with (13) in Fig. 6. Compared with that in Fig. 3, one obtains that the convergence time in Fig. 6 is much longer since the restrictions on the range of control inputs postpone the convergence time. Moreover, Fig. 6 indicates that the chatter induced in system (1) with (3) (as shown in Fig. 3) has been largely reduced under protocol (13). (2) Case 2: Consider the switching digraph  = {G¯1 , . . . , G¯5 }. The system (1) with (9) and (15) begins at the state G¯1 and switches at t = 0.05 to the next state with the order: {G¯1 , G¯2 , . . . , G¯5 , G¯1 , G¯2 , . . .}. Fig. 7 shows the tracking curves of system (1) under (9). For the chatter reduction of tracking consensus in system (1) under (15), the similar numerical results are given in Fig. 8. Compared with that in Figs. 7, 8 illustrates that the chatter induced in system (1) under (9) (as shown in Fig. 7) has been largely reduced under (15). When ε = 1 and a0 (t) = sin(t), the control inputs ui (t) in protocols (13) and (15) are bounded by [−3, 3] as shown in Fig. 9. However, for another switching digraphes 1 = {G¯3 , G¯4 , G¯5 , G¯3 , G¯4 , G¯5 , G¯3 , . . .}, since leader agent 0 in G¯3−5 is not jointly reachable, then different from the numerical results in Fig. 8, the states of the followers cannot reach the leader, as shown in Fig. 10.

1 0 −1 −2 −3 −4

0

Fig. 6

10

20

30

40

50 60 Time

70

80

90 100

60

Finite-time tracking consensus of system (1) under protocol (13) with α = 0.2 and fixed topology G¯

IET Control Theory Appl., 2013, Vol. 7, Iss. 11, pp. 1562–1573 doi: 10.1049/iet-cta.2013.0013

1569 © The Institution of Engineering and Technology 2013

www.ietdl.org 250

6

225

5

200

4 Velocity evolutions

Position evolutions

175 150 125 100 75 50

3 2 1 0 −1

25 −2

0

−3

−25 −50

−4 20

30

40

50 60 Time

70

80

90 100

10 8 6 4 2 0 −2 −4 −6 −8 −10 −12 −14

0

10

20

30

40

50 60 Time

70

80

90 100

0

10

20

30

40

50 60 Time

70

80

90 100

3 2 1 0 −1 −2 −3 −4 0

Fig. 7

10

Velocity errors

Position errors

0

10

20

30

40

50 60 Time

70

80

90 100

Finite-time tracking consensus of system (1) under protocol (9) with α = 0.2 and switching topology 

250 225

5 4

175

Velocity evolutions

Position evolutions

200 150 125 100 75 50

3 2 1 0 −1

25 −2

0

−3

−25 −50 0

10

20

30

40

50

60

70

80

90 100

0

10

20

30

40

Time

50

60

70

80

90 100

60

70

80

90 100

Time

10

2

8 6

1

2

0

0

Velocity errors

Position errors

4

−2 −4 −6 −8 −10 −12

−2 −3

−14

−4 0

10

20

30

40

50

60

70

Time

Fig. 8

−1

80

90 100

0

10

20

30

40

50

Time

Finite-time tracking consensus of system (1) under protocol (15) with α = 0.2 and switching topology 

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www.ietdl.org 4

Unbounded control input (9)

Unbounded control input (3)

4

2

0

−2

2

0

−2

−4

−4 0

10

20

30

0

10

3

3

2

2

1 0 −1 −2

20

30

0 −1 −2 −3

0

10 Time

20

30

0

10 Time

300

6

200

4 Velocity evolutions

Position evolutions

Evolutions of control inputs ui (t) with protocols (3), (9), (13) and (15)

100 0 −100

2 0 −2

−200

−4 0

20

40

60

80

100

0

20

40

60

80

100

60

80

100

Time

100

4

0

2 Velocity errors

Position errors

Time

−100 −200

0

−2

−300

−4

−400

−6 0

20

40

60

80

Time

Fig. 10

30

1

−3

Fig. 9

20 Time

Bounded control input (15)

Bounded control input (13)

Time

100

0

20

40 Time

Tracking evolutions and errors of system (1) under protocol (15) with α = 0.2 and switching topology 1

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250 225 200 175 150 125 100 75 50 25 0 −25 −50

6 5 4

Velocity evolutions

Position evolutions

www.ietdl.org

1 0 −1

−3 −4 10

20

30

40

50 60 Time

70

80

90 100

10 8 6 4 2 0 −2 −4 −6 −8 −10 −12 −14

0

10

20

30

40

50 60 Time

0

10

20

30

40 50 Time

70

80

90 100

70

80

90 100

3 2 1

Velocity errors

Position errors

2

−2

0

0

−1 −2 −3 −4

0

Fig. 11

3

10

20

30

40

50 60 Time

70

80

90 100

60

Finite-time tracking consensus of system (1) under protocol (16) with α1 = 1/9 (α = 0.2) and G¯ 250

20

225 15

Velocity evolutions

Position evolutions

200 175 150 125 100 75 50 25 0

10 5 0 −5 −10

−25 −50

0

10

20

30

40

50 Time

60

70

80

90

−15

100

80

20

60

15

Velocity errors

Position errors

40 20 0 −20

20

30

40

50 Time

60

70

80

90

100

0

10

20

30

40

50 Time

60

70

80

90

100

10 5 0

−10

−60

Fig. 12

10

−5

−40

−80

0

0

10

20

30

40

50 Time

60

70

80

90

100

−15

Tracking evolutions and errors of system (1) under protocol (17) with α1 = 1/9 (α = 0.2) and 

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IET Control Theory Appl., 2013, Vol. 7, Iss. 11, pp. 1562–1573 doi: 10.1049/iet-cta.2013.0013

www.ietdl.org ui (t) = a0 (t) +



5

aij (t){sat[sig(xj − xi )α1 ]

j∈Ni (t)

+ sat[sig(vj − vi )α2 ]} − bi (t){sat[sig(xi − x0 )α1 ] + sat[sig(vi − v0 )α2 ]}

(17)

where 0 < α1 < 1, α2 = and a0 (t) = 0. However, most of the results are established with the assumptions that the underlying digraph G¯ is fixed or the switching graph  (without leader) is undirected. Take a0 (t) = sin(t), using the same parameters as above, one gives the following simulation results. Fig. 11 describes the tracking curves of system (1) with ¯ the same evolving curves of system (1) with (16) under G, (17) under  is shown in Fig. 12. Obviously, for the fixed ¯ protocol (16) can also guide the controlled sysdigraph G, tem (1) to realise the finite-time tracking consensus, but compared with Fig. 6, the convergence process of Fig. 11 near the stable state is full of chatter. For the switching digraph , compared with protocol (15) shown in Fig. 8, the controlled system (1) with (17) could not achieve tracking consensus, as shown in Fig. 12. In fact, protocols (16) and (17) are proposed based on the fixed directed graphs or switching undirected graphs, it may not hold in the case of switching digraphs. Therefore our proposed finite-time distributed tracking protocols are feasible and effective. 2α1 , α1 +1

6 7 8 9 10 11 12 13 14 15 16 17 18

5

Conclusions

This paper has investigated the finite-time distributed tracking control problems for double-integrator multi-agent systems with bounded control input under fixed and switching jointly-reachable digraphes, respectively. Especially, a continuous bounded tracking protocol has been proposed to reduce the chatter and track the virtual leader in finite time. It sheds some light on the potential applications in realworld engineering systems [32], such as the multi-machine synchronisation in power grid.

19 20 21 22 23

6

Acknowledgment

This work was supported by the National Natural Science Foundation of China (grant numbers 61273215, 61025017, 11072254, 61203148 and 61175075), by the Young Teachers Growth Plan of Hunan University (grant no. 531107040651) and by the Postdoctoral Special Foundation of Hunan Province (grant no. 2013RS4042).

24 25 26 27

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