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Robust consensus of fractional multi-agent systems with external disturbances
Author’s Accepted Manuscript Robust Consensus of Fractional Multi-agent Systems with External Disturbances Guojian Ren, Yongguang Yu
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PII: DOI: Reference:
S0925-2312(16)30984-5 http://dx.doi.org/10.1016/j.neucom.2016.08.088 NEUCOM17522
To appear in: Neurocomputing Received date: 23 November 2015 Revised date: 12 July 2016 Accepted date: 28 August 2016 Cite this article as: Guojian Ren and Yongguang Yu, Robust Consensus of Fractional Multi-agent Systems with External Disturbances, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.08.088 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Robust Consensus of Fractional Multi-agent Systems with External Disturbances Yongguang Yu∗
Guojian Ren
Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, P.R.China
Abstract: In this paper, the problem of robust consensus for fractional multi-agent systems with external disturbances is investigated over a directed fixed interaction graph. Based on Mittag-Leffler stability theory and the inequality techniques, both linear and nonlinear systems are considered. Firstly, for fractional linear multi-agent systems, it is shown that consensus can be achieved asymptotically in the absence of disturbances. In the presence of disturbances, the steady-state errors of any two agens can reach a small region determined by the bound of disturbances. Secondly, for fractional nonlinear multi-agent systems, a pinning control input is proposed such that robust consensus can be realized. Finally, the numerical simulations are given to verify the correctness of the presented theories. Keywords: Robust consensus; Fractional; Multi-agent; External disturbances; Pinning control
1
Introduction In recent years, the research of multi-agent systems has become a hot issue in the engineering com-
munity. The multi-agent system which is composed of multiple interacting agents can be used to solve problems that are difficult or impossible for an individual agent to solve. As a mian research direction of multi-agent systems, the distributed coordination has received considerable attention of many researchers. The objective of the distributed coordination is to achieve collective group behavior through local interaction. And the distributed coordination has a large number of applications, such as rendezvous [1], flocking [2], formation [3,4] and consensus [5-11], and so on. Particularly, consensus plays an important role in the distributed coordination. The basic idea of consensus is that each agent updates its state based on the states of its neighbors and its own such that ∗ Corresponding
author E-mail: [email protected]
1
the states of all agents will converge to a common value. In the control community, the consensus problem has been widely studied in the systems with single-integrator or double-integrator dynamics [5-11]. At present, the research of the distributed coordination becomes more and more challenging and interesting due to the introduction of fractional calculus. Moreover, many phenomena can not be modelled by integer-order dynamics, including the food searching of microbes and ground vehicles moving on the road covered in mud or grass [12, 13]. Similarly, many financial and engineering systems often demonstrate fractional dynamics either. Therefore, more and more researchers mainly pay attention to the distributed coordination of multi-agent systems with fractional dynamics. In Ref. [14], Cao et al. studied the coordination algorithms for networked fractional systems and this literature is the first paper that studied the distributed coordination of networked fractional systems. In Ref. [15], Ren et al. investigated the leader-following consensus for both single-integrator and double-integrator dynamics over an undirected fixed interaction graph. In Ref. [16], Bai et al. considered consensus of fractionalorder multi-agent systems with a constant or time-varying reference state. Yang et al. [17] investigated the distributed coordination of fractional multi-agent systems with communication delays, and obtained a critical value of time delay to reach consensus. Yu et al. [18] considered the leader-following consensus problem of fractional multi-agent systems via adaptive pinning control. Chen et al. [19] studied the multiconsensus problem of fractional uncertain multi-agent systems when the fractional order α satisfies 0 < α < 2. And in Ref. [20], Yu et al. investigated the leader-following consensus problem of fractional multi-agent systems with nonlinear dynamics. In practical applications, dynamical systems are often subjected to various perturbations, such as stochastic noises, external interferences, communication delays and model uncertainty. The existence of disturbances may impact the system performance and even cause networks divergent if those imprecise information is used. Hence, it is meaningful to discuss their effects on the behaviour of multi-agent systems. These issues recently have been extensively investigated. In Ref. [21], it was proved that the steady-state errors of any two agents can reach a small region for first-order multi-agent systems under bounded unknown external disturbances. In Ref. [22], continuous finite-time control laws were proposed to analyze the disturbance rejection property for second-order multi-agent systems under bounded unknown external disturbances. And Du et al. [23] investigated the problem of robust consensus for second-order multi-agent systems under external disturbances with/without communication delays. However, many existing tools developed for the study of the robust consensus problem can only be applied to integerorder dynamics. This is partly due to the limited theories for fractional systems. In Ref. [24], Wong et al. investigated robust synchronization of fractional-order complex dynamical networks with parametric uncertainties. In Ref. [25], Song et al. addressed the robust consensus problem of fractional-order linear multi-agent systems with positive real uncertainty. The results in Ref. [24, 25] are all concerned with the influences of parameter uncertainty. However, to the best of our knowledge, only a few authors consider the effects of external disturbances on fractional systems [26]. In Ref. [26], under the assumption
2
that the external disturbance generated by a fractional linear system, Yang et al. proposed a disturbance observers-based consensus protocol to study the distributed coordination of fractional multi-agent systems with external disturbancesNevertheless, when the external disturbance model is unknown in advance, the above mentioned mothods all don’t work. In this technical note, we consider the fractional multi-agent system under bounded unknown external disturbances and analyze the disturbance rejection properties for both linear and nonlinear systems. On the other hand, in reality, mobile agents may be governed by more complicated intrinsic dynamics. Indeed, nonlinear dynamics are commonly considered in integer-order multi-agent systems. In Ref. [10], Yu et al. studied the consensus problem for cooperative agents with nonlinear dynamics in a directed network. In Ref. [27, 28], Li et al. considered the consensus problem of both first-order and secondorder multi-agent systems with nonlinear dynamic and directed topologies, where each agent can only communicate with its neighbors on some disconnected timeintervals. And in Ref. [29], Song et al. investigated the second-order leader-following consensus problem of nonlinear multi-agent systems. The consensus protocol proposed in Ref. [29] is based on a pinning control, which means that only a small fraction agents can make use of leader’s information, because it is practically impossible to add controllers to all the agents in multi-agent aystems. Motivated by the above discussion and the significance of robust control, the problem of robust consensus for fractional multi-agent systems with external disturbances is investigated in this paper. Comparing with the results in [21-23], the systems with fractional dynamics are considered. Different from Ref. [26], the external disturbances are unknown in advance in this paper. A combination of the tools of Mittag-Leffler stability theory, the inequality techniques and Laplace transform is utilized to analyze the disturbance rejection properties for both linear and nonlinear systems. Firstly, it is proved that consensus can be achieved asymptotically in the absence of diturbances. In the presence of disturbances, the steady-state errors of any two agents can reach a small region. Secondly, the obtained results are extended to the leader-follower case with nonlinear dynamics. Finally, some simulation results are presented as a proof of concept. The rest of this paper is arranged as follows. The graph theory notions and Mittag-Leffler stability theory are introduced as a basis in Section 2. The main results on robust consensus for fractional multiagent systems are presented in Section 3 and Section 4. Then, some simulation results are given in Section 5. Finally, a short conclusion is shown to close the paper in Section 6.
2
Preliminaries In this section, some basic notions about the graph thoery and some definitions of fractional calculus
are introduced as the preliminaries of this paper. And some necessary conclusions are presented for the use of next several sections.
3
2.1
Graph Theory Notions
A directed graph G = (V, W) is used to model the interaction topology in the multi-agent system consisting of n agents, with the agent set V = {v1 , v2 , . . . , vn } and the edge set W ⊆ V 2 . A directed edge denoted by (vi , vj ) means that agent j can receive the state information of agent i. Consequently, agent i is a neighbor of agent j. All neighbors of agent i are denoted by Ni . Then, two types of matrices are introduced to represent the interaction graph: 1) the adjacency matrix A = [aij ] ∈ Rn×n with aij > 0 if (vi , vj ) ∈ W and aij = 0 otherwise, 2) the (nonsymmetric) Laplacian matrix L = [lij ] ∈ Rn×n with lii = j∈Ni aij and lij = −aij , i = j. The in-degree and out-degree of node i can be defined as follows N N aij , degout (i) = aji . [31], degin (i) = j=1,j=i
j=1,j=i
Lemma 1 [30]: Assume that the directed communication topology G has a directed spanning tree. Then, the matrix L has a simple zero eigenvalue and all the other eigenvalues have positive real parts. Let B = diag {b1 , b2 , · · · , bn } , i = 1, 2, . . . , n, where bi is a nonnegative real number that does not always equal 0, then all eigenvalues of L + B have positive real parts.
2.2
Caputo Fractional Operator and Mittag-Leffler stability
Definition 1 [33]: The fractional integral of order α for a function f is defined as t 1 α−1 α (t − τ ) f (τ ) dτ , t0 It f (t) = Γ (α) t0 where t ≥ t0 and α > 0. Definition 2 [33]: Caputo’s fractional derivative of order α for a function f ∈ C n ([t0 , +∞] , R) is defined by α t0 Dt f (t) =
1 Γ (n − α)
t t0
f (n) (τ )
α−n+1 dτ ,
(t − τ )
where t ≥ t0 and n is a positive integer such that n − 1 < α < n. Particularly, when 0 < α < 1, t f (τ ) 1 α D f (t) = dτ . t0 t Γ (1 − α) t0 (t − τ )α The Laplace transform of the Caputo fractional derivative is L {t0 Dtα f (t) ; s} = sα F (s) −
n−1
sα−k−1 f (k) (t0 ), n − 1 < α ≤ n,
k=0
where s is the variable in Laplace domain. Some properties of Caputo fractional derivative are listed as follows: Property 1:
α t0 Dt C
= 0 holds, where C is any constant.
Property 2: For constants μ and ν, the linearity of Caputo fractional derivative is described by α t0 Dt
(μf (t) + νg (t)) = μt0 Dtα f (t) + ν t0 Dtα g (t) .
4
In the following, we will introduce the Mittag-Leffler function [32] and Mittag-Leffler stability [34]. For α, β ∈ C, the Mittag-Leffler function in two parameters is defined as Eα,β (z) =
∞ k=0
zk . Γ (kα + β)
(1)
When β = 1, and α > 0, (1) can be written in a special case as Eα (z) = Consider the following n-dimensional Caputo fractional-order system ⎧ ⎨ t Dα x(t) = f (t, x(t)) 0
∞ k=0
zk Γ(kα+1) .
t
⎩ x(t ) = x 0 t0
(2)
where α ∈ (0, 1), x = (x1 , x2 , · · · , xn )T ∈ Rn , t0 ≥ 0, f is piecewise continuous on t and satisfies locally Lipschitz condition on x. Lemma 2 [34]: There exists a unique solution of system (2) for any initial value, if f (t, x) satisfies locally Lipschitz condition on x. Definition 3 [34]: The constant x ¯ is an equilibrium point of Caputo fractional dynamic system (2) if and only if f (t, x ¯) = 0. Definition 4 [34]: If x ¯ = 0 is an equilibrium point of system (2), the solution of (2) is said to be Mittag-Leffler stable if x(t) ≤ [m(xt0 )Eα (−λ(t − t0 )α )]b ,
(3)
where λ > 0, b > 0, m(0) = 0, · denotes an arbitrary norm and m(x) ≥ 0 satisfies locally Lipschitz condition on x ∈ Rn with Lipschitz constant m0 . Remark 1: Mittag-Leffler stability for system (2) implies asymptotic stability for any initial value, i.e., x → 0 with t → +∞. Lemma 3 [34]: The equilibrium point x ¯ = 0 of fractional-order system (2) is Mittag-Leffler stable if there exist positive α1 , α2 , α3 , a, b and a continuously differentiable function V (t, x (t)) satisfying a
ab
α1 x ≤ V (t, x (t)) ≤ α2 x , β t0 Dt V
(t, x (t)) ≤ −α3 xab ,
(4) (5)
where t ≥ 0, β ∈ (0, 1) , V (t, x (t)) : [ t0 , ∞) × D → R satisfies locally Lipschitz condition on x; D ∈ Rn is a domin containing the origin. If the assumptions hold globally on Rn , x¯ = 0 is globally Mittag-Leffler stable. Lemma 4 [35]: Let x (t) ∈ R be a continuous and derivable function. Then, for any time instant t ≥ t0 , 1 β 2 β t D x (t) ≤ x (t) t0 Dt x (t) , ∀β ∈ (0, 1) . 2 0 t Obviously, when x (t) ∈ Rn is continuous and derivable, it implies 1 β T β T t D x (t) x (t) ≤ x (t) t0 Dt x (t) , ∀β ∈ (0, 1) . 2 0 t 5
(6)
3
Robust Consensus of Fractional Linear Multi-agent Systems In this section, the problem of robust consensus for fractional linear multi-agent systems with external
disturbances is discussed. Different from most of the existing work [24-26], here we assume that the external disturbances are unknown and bounded. Consider the multi-agent system consisting of n agents, labeled as agents 1 to n. The dynamics of each agent with external disturbances is given by α 0 Dt xi
(t) = ui (t) + wi (t) , i ∈ {1, 2, . . . , n} ,
(7)
where α ∈ (0, 1), xi (t) is the state for the ith agent, ui (t) is the control input for the ith agent, and wi (t) is the external disturbance. Definition 5 [36]: The multi-agent system (7) is said to achieve robust consensus if its solution satisfies lim xi (t) − xj (t) ≤ h (l) , ∀i, j ∈ {1, 2, . . . , n} ,
t→∞
where the function h(l) satisfies h(0) = 0 and is monotonous increasing with respect to l. In the following, assume that all agents are in a one-dimensional space for the simplicity of presentation. However, all results hereafter are still valid for the m-dimensional (m > 1) space by the introduction of the Kronecker product. As in [21-23], the disturbance signal wi (t) satisfies the following assumption. Assumption 1: The external disturbances wi (t) satisfy |wi (t)| ≤ l < +∞ for all i ∈ {1, 2, . . . , n}. Consider the following control protocol to achieve robust consensus of system (7): ui (t) = −β
n
aij (xi (t) − xj (t)),
(8)
i=1
where aij (i, j = 1, 2, . . . , n) is the (i, j)th entry of the adjacency matrix A ∈ Rn×n associated with the directed graph G; β is a nonnegative constant. The control protocol is independent of the disturbance signal, because the external disturbance model is unknown in advance. With the contol input (8), system (7) can be rewritten as α 0 Dt xi
(t) = −β
n
aij (xi (t) − xj (t)) + wi (t) ,
(9)
i=1
which can be written in a vector form as α 0 Dt X
(t) = −βLX (t) + W (t) ,
T
(10) T
where X (t) = [x1 (t) , x2 (t) , . . . , xn (t)] , W (t) = [w1 (t) , w2 (t) , . . . , wn (t)] . Before presenting the main result, we need the following changes of variables: y12 (t) = x1 (t) − x2 (t) , y13 (t) = x1 (t) − x3 (t) , . . . , y1n (t) = x1 (t) − xn (t) . 6
(11)
⎛ Let Y (t) = [y12 (t) , y13 (t) , . . . , y1n (t)]T , E = [1n−1 , −In−1 ] ∈ R(n−1)×n , and F = ⎝
0Tn−1 −In−1
⎞ ⎠ ∈
Rn×(n−1) . Then the coordinate changes (11) can be written in a vector form as Y (t) = EX (t) , X (t) = x1 1n + F Y (t) .
(12)
As a result, it follows from (12) and (10) that α 0 Dt Y
(t) = E 0 Dtα X (t) = −βx1 EL1n − βELF Y (t) + EW (t)
(13)
= −βELF Y (t) + EW (t) = βCY (t) + EW (t) , where C = −ELF .
Lemma 5 [23, 37]: The matrix C = −ELF is Hurwitz if and only if the communication topology G has a directed spanning tree, where L is the Laplacian matrix of graph G. Proof: From the definitions of matrixes E, L, F , we have C = −ELF ⎛
1 −1 ⎜ .. ⎜ .. = −⎜ . . ⎝ 1 0 ⎛ ⎜ ⎜ = −⎜ ⎝
··· .. .
0 .. .
...
−1
⎛
⎞ ⎟ ⎟ ⎟ ⎠ (n−1)×n
l12 − l22 .. .
··· .. .
l1n − l2n .. .
l12 − ln2
···
l1n − lnn
l11 ⎜ ⎜ .. ⎜ . ⎝ ln1
··· .. .
l1n .. .
···
lnn
⎛
⎞
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎟ ⎟ ⎟ ⎠ n×n
⎞
···
0
−1 · · · .. .. . .
0 .. .
0
0
···
−1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ n×(n−1)
⎟ ⎟ ⎟ ⎠ (n−1)×(n−1)
Then, it follows from Lemma 1 in Ref. [33] that C is Hurwitz. Thus, the proof is completed. Theorem 1: Consider the fractional linear multi-agent system (7) under the control input (8). If the fixed directed graph G has a directed spanning tree, then, (i) consensus can be achieved asymptotically in system (7) when wi (t) = 0. (ii) when wi (t) = 0 and Assumption 1 is satisfied, the steady-state errors of any two agents will converge to the region M1 asymptotically, where 2nλmax (P )/λmin (P ) P E l , M1 = |xi (t) − xj (t)| ≤ β
(14)
P > 0 is the solution of the Lyapunov equation C T P + P C = −2In−1 . Proof: Since matrix C is Hurwitz matrix from Lemma 5, there is a positive-definite P = P T > 0 such that C T P + P C = −2In−1 . For the system (13), construct Lyapunov functions candidate in the form of V (t) = Y T (t) P Y (t) . Obviously, the Lyapunov function satisfies inequality (4). 7
Then, by Lemma 4 and Eq. (13), it is easy to obtain α 0 Dt V
(t) = 0 Dtα Y T (t) P Y (t) ≤ 2Y T (t) P 0 Dtα Y (t) = 2Y T (t) P [βCY (t) + EW (t)]
Then, noting equality ξ T ζ = ζ T ξ and inequality ξ T ς + ς T ξ ≤ κξ T ξ + κ1 ς T ς for any ξ, ζ ∈ Rm , m ∈ N+ , one has α 0 Dt V
(t) ≤ β Y T (t) P CY (t) + Y T (t) C T P Y (t) + Y T (t) P EW (t) + W T (t) E T P Y (t) 1 ≤ βY T (t) C T P + P C Y (t) + βY (t)2 + P E2 W (t)2 β ≤ −βY (t)2 +
2
nP E l2 . β
(15)
2
Noticing that V (t) ≤ λmax (P ) Y (t) , we have α 0 Dt V (t) ≤ −
Let u (t) = V (t) −
nλmax (P )P E2 l2 , β2
β λmax (P )
V (t) +
nP E2 l2 . β
this together with (15) implies that α 0 Dt u (t)
≤−
β u (t) . λmax (P )
Then, there exists a nonnegative function m (t) satisfying α 0 Dt u (t)
+ m (t) = −
β u (t) . λmax (P )
(16)
Taking the Laplace transform of (16) yields sα U (s) − u (0) sα−1 + M (s) = −
β U (s) , λmax (P )
where U (s) = L {u (t)} and M (s) = L {m (t)}. Thus, we have U (s) =
u (0) sα−1 − M (s) sα +
β λmax (P )
.
It follows from Lemma 2 that there exists a unique solution of (16). With the inverse Laplace transform, it is easy to obtain u (t) = u (0) Eα −
βtα λmax (P )
α−1
− m (t) ∗ t
Eα,α −
βtα λmax (P )
.
βtα Because tα−1 and Eα,α − λmax are nonnegative functions, then the following is satisfied (P ) V (t) −
2 βtα nλmax (P ) P E l2 → 0, − = u (t) ≤ u (0) E α β2 λmax (P ) 8
2
when t → +∞. Noticing that V (t) ≥ λmin (P ) Y (t) and letting t tend to indinity, one has nλmax (P )/λmin (P ) P E l 2 . lim Y (t) ≤ t→+∞ β
(17)
From Eq. (11), we obtain for any i, j ∈ {1, 2, . . . , n}, |xi (t) − xj (t)| ≤ |x1 (t) − xj (t)| + |x1 (t) − xj (t)| = |y1i (t)| + |y1j (t)| √ √ 2 2 ≤ 2 |y1i (t)| + |y1j (t)| ≤ 2 Y (t) .
(18)
Combining (17) and (18), one obtains 2nλmax (P )/λmin (P ) P E l lim |xi (t) − xj (t)| ≤ , t→+∞ β for any i, j ∈ {1, 2, . . . , n}. When wi (t) = 0, from Eq. (15), one has α 0 Dt V
(t) ≤ −βY (t)2 .
(19)
From Eq. (19), inequality (5) can be satisfied. According to Lemma 3, system (13) is Mittag-Leffler stable, which implies that system (13) is asympototically stable. From the definition of Y (t), we have xi (t) − xj (t) → 0 as t → +∞ for any i, j ∈ {1, 2, . . . , n}. Therefore, consensus can be achieved in system (7) asymptotically under the control law (8) in the absence of disturbances. This completes the proof.
4
Robust Consensus of Fractional Nonlinear Multi-agent Systems In this section, robust consensus of fractional nonlinear multi-agent systems with external disturbances
is discussed, and a controller based on a pinning control is designed to implement robust consensus. The fractional nonlinear multi-agent system with external disturbances is described by α 0 Dt xi
(t) = f (t, xi (t)) + ui (t) + wi (t) , i ∈ {1, 2, . . . , n} ,
(20)
where α ∈ (0, 1), xi (t) is the state for the ith agent, ui (t) is the control input for the ith agent, f (t, xi (t)) is the intrinsic nonlinear dynamics for the ith agent, and wi (t) is the external disturbance. The virtual leader for multi-agent system (20) is an isolated agent described by α 0 Dt xr
(t) = f (t, xr (t)) ,
where xr (t) is the state for the virtual leader.
9
(21)
Assumption 2: The nonlinear function f (t, x) (x, y ∈ R) is continuous and satisfies Lipschitz condition on x with Lipschitz constant θ ≥ 0, i.e., |f (t, x) − f (t, u)| ≤ θ|x − u|, for any t ≥ 0. The following pinning control input is considered to implement robust consensus of multi-agent system (20): ⎡ ui (t) = −β ⎣
n
⎤ aij (xi (t) − xj (t)) + bi (xi (t) − xr (t))⎦ ,
(22)
j=1
where aij (i, j = 1, 2, . . . , n) is the (i, j)th entry of the adjacency matrix A ∈ Rn×n associated with the directed graph G; bi (i = 1, 2, . . . , n) is a nonnegative real number that does not always equal 0; β is a nonnegative constant. When bi > 0, the agent i is pinned. Consequently, we assume that the agent should be pinned if it receives little information from its neighbour but outputs more, i.e., the agent i should be pinned if degin (i) < degout (i) [18]. The agents in Fig. 1 which should be pinned are agent 1 and agent 2. Let H = L + B, where L is the Laplacian matrix of G. It follows from Lemma 1 that −H is Hurwitz. With the contol input (21), system (19) can be rewritten as ⎡ ⎤ n α aij (xi (t) − xj (t)) + bi (xi (t) − xr (t))⎦ + wi (t) . 0 Dt xi (t) = f (t, xi (t)) − β ⎣
(23)
j=1
Noting that zi (t) = xi (t) − xr (t) , i = 1, 2, . . . , n, we have ⎡ ⎤ n α ⎣ aij (zi (t) − zj (t)) + bi zi (t)⎦ + wi (t) , 0 Dt zi (t) = f (t, zi (t) + xr (t)) − f (t, xr (t)) − β
(24)
j=1
which can be written in a vector form as α 0 Dt Z
(t) = −βHZ (t) + F (t, Z (t)) + W (t) ,
(25)
where Z (t) = [z1 (t) , z2 (t) , . . . , zn (t)]T , F (t, Z (t)) = [f (t, z1 (t) + xr (t)) − f (t, xr (t)) , f (t, z2 (t) T
+xr (t)) − f (t, xr (t)) , . . . , f (t, zn (t) + xr (t)) − f (t, xr (t))] . Theorem 2: Suppose that the fixed directed graph G has a directed spanning tree and the nonlinear function f in system (20) and (21) satisfies Assumption 2. If β ≥ Q , θ
(26)
where Q > 0 is the solution of the Lyapunov equation H T Q + QH = 3In , then, (i) all agents in system (20) can achieve leader-following consensus asympototically when wi (t) = 0.
10
(ii) when wi (t) = 0 and Assumption 1 is satisfied, the state of ith agent can reach the region M2 asympototically, 2nλmax (P ) Q l , μβλmin (P )
|xi (t) − xr (t)| ≤
M2 = where μ = β −
(27)
Q2 θ 2 . β
Proof: It follows from Lemma 1 that -H is Hurwitz matrix. Then there is a positive-definite Q = QT > 0 such that −H T Q − QH = −3In . For the system (25), construct Lyapunov functions candidate in the form of V (t) = Y T (t) QY (t) . Obviously, the Lyapunov function satisfies inequality (4). Then, by Lemma 4, Eq. (25) and Assumptions 1 and 2, we have α 0 Dt V
(t) ≤ 2Z T (t) Q [−βHZ (t) + F (t, Z (t)) + W (t)] = βZ T (t) −QH − H T Q Z (t) + 2Z T (t) QF (t, Z (t)) + 2Z T (t) QW (t) ≤ −3βZ (t)2 + βZ (t)2 + β1 Q2 W (t)2 + βZ (t)2 + β1 Q2 F (t, Z (t))2 n 2 2 l Q2 ≤ −βZ (t)2 + nQ + (f (t, zi (t) + xr (t)) − f (t, xr (t)))2 β β ≤ −βZ (t)2 +
nQ2 l2 β
!
Q2 θ2 =− β− β where μ = β −
Q2 θ 2 β
+
i=1 Q2 θ 2 Z β
" 2
Z (t) +
(t)2
nQ2 l2 nQ2 l2 2 = −μZ (t) + , β β
(28)
> 0 by condition (26). Applying the proof procedure in Theorem 1, we can
immediately obtain lim |xi (t) − xr (t)| ≤
t→+∞
2nλmax (P ) Q l, μβλmin (P )
for any i ∈ {1, 2, . . . , n}. When wi (t) = 0, from Eq. (28), one has α 0 Dt V
(t) ≤ −μZ (t)2 .
(29)
From Eq. (29), inequality (5) can be satisfied. According to Lemma 4, when condition (26) is satisfied, system (25) is Mittag-Leffler stable, which implies that system (25) is asympototically stable. Therefore, zi (t) = xi (t) − xr (t) → 0, i ∈ {1, 2, . . . , n}, as t → +∞, and all agents in system (20) can achieve leader-following consensus asympototically. The proof is completed.
11
Figure 1: Interaction graph for 7 agents.
5
Numerical simulations Several numerical simulations are presented to verify the results of Theorem 1 proposed in Section
3 and Theorem 2 proposed in Section 4. We consider a group of 7 agents with an interaction graph given by Fig. 1. Note that the interaction graph has a directed spanning tree with node 1 being the root. The corresponding Laplacian matrix is chosen aij = 1 if (vj , vi ) ∈ W, and aij = 0 otherwise. The T
initial conditions are selected as x (0) = [5, −4, 3, −2, −6, 1, 8] , and the external disturbances are given as: w1 (t) = 1.1 sin (8t − 1), w2 (t) = − cos (2t), w3 (t) = 0.8 sin (t) + 0.7 cos (t), w4 (t) = 0.2 cos (11t − 4), w5 (t) = −1.5 sin (−7t + 1), w6 (t) = cos (−3t), w7 (t) = −0.6 sin (t) − 0.9 cos (t). Let the fractional order α = 0.9. Firstly, the simulations results are showed to verify Theorem 1 in Section 3. Let β = 1. i) When wi (t) = 0, i = 1, 2, . . . , 7, according to Theorem 1, the system (7) can achieve cosnensus asympototically, which is presented in Fig. 2. ii) When wi (t) = 0, i = 1, 2, . . . , 7, the trajectories of xi (t) in system (7) are showed in Fig. 3, and it can be observed that the steady-state errors of any two agents can reach a small region, which confirms the effectiveness of Theorem 1. We next present the simulation results to demonstrate Theorem 2 in Section 4. Suppose that the nonlinear intrinsic dynamics of each agent in system (20) and system (21) is described by f (t, x) = sin x + cos t, which satisfies Assumption 2 with θ = 1. It can be computed that Q = 4.3853 and we choose β = 5 to satisfy condition (26). i) When wi (t) = 0, i = 1, 2, . . . , 7, according to Theorem 2, the system (20) can realize leader-following consensus asympototically, which is presented in Fig. 4. ii) When wi (t) = 0, i = 1, 2, . . . , 7, the trajectories of xi (t) in system (20) and xr (t) in system (21) are showed in Fig. 5, and it can be shown that all agents can reach the region around the leader asympototically, which verifies the correctness of Theorem 2.
12
8
6
4
x i (t)
2
0
-2
-4
-6 0
5
10
15
20
25
30
t
Figure 2: Trajectories of xi (t) in system (7) with wi (t) = 0.
8
6
4
x i (t)
2
0
-2
-4
-6
-8 0
5
10
15
20
25
t
Figure 3: Trajectories of xi (t) in system (7) with wi (t) = 0.
13
30
8
6
4
2
xr(t)
0
x (t) 1
x (t) 2
-2
x3 (t) x (t) 4
x5 (t)
-4
x (t) 6
x7 (t)
-6 0
0.5
1
1.5
2
2.5
3
3.5
4
t
Figure 4: Trajectories of xr (t) and xi (t) in system (20) with wi (t) = 0.
8
6
4
2
xr(t)
0
x1 (t) x2 (t)
-2
x3 (t) x (t) 4
x5 (t)
-4
x6 (t) x7 (t)
-6 0
0.5
1
1.5
2
2.5
3
3.5
4
t
Figure 5: Trajectories of xr (t) and xi (t) in system (20) with wi (t) = 0.
14
6
Conclusion In this paper, the problem of robust consensus for fractional multi-agent systems is investigated via
a directed fixed interaction graph. The effects of external disturbances on the behaviour of both linear and nonlinear systems are considered over a combination of the tools of Lyapunov method, matrix theory and fractional calculus theory. Firstly, for fractional linear multi-agent systems under bounded unknown external disturbances, it is proved that the steady-state errors of any two agents can reach a small region. Then, a pinning control input is proposed to achieve robust consensus for fractional nonlinear multi-agent systems with external disturbances. Future works include the study of robust consensus for fractional time-delayed multi-agent systems and extending the results in this article to the switching topology case.
Acknowledgment This work is supported by the National Nature Science Foundation of China (No. 11371049).
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17
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PII: DOI: Reference:
S0925-2312(16)30984-5 http://dx.doi.org/10.1016/j.neucom.2016.08.088 NEUCOM17522
To appear in: Neurocomputing Received date: 23 November 2015 Revised date: 12 July 2016 Accepted date: 28 August 2016 Cite this article as: Guojian Ren and Yongguang Yu, Robust Consensus of Fractional Multi-agent Systems with External Disturbances, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.08.088 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Robust Consensus of Fractional Multi-agent Systems with External Disturbances Yongguang Yu∗
Guojian Ren
Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, P.R.China
Abstract: In this paper, the problem of robust consensus for fractional multi-agent systems with external disturbances is investigated over a directed fixed interaction graph. Based on Mittag-Leffler stability theory and the inequality techniques, both linear and nonlinear systems are considered. Firstly, for fractional linear multi-agent systems, it is shown that consensus can be achieved asymptotically in the absence of disturbances. In the presence of disturbances, the steady-state errors of any two agens can reach a small region determined by the bound of disturbances. Secondly, for fractional nonlinear multi-agent systems, a pinning control input is proposed such that robust consensus can be realized. Finally, the numerical simulations are given to verify the correctness of the presented theories. Keywords: Robust consensus; Fractional; Multi-agent; External disturbances; Pinning control
1
Introduction In recent years, the research of multi-agent systems has become a hot issue in the engineering com-
munity. The multi-agent system which is composed of multiple interacting agents can be used to solve problems that are difficult or impossible for an individual agent to solve. As a mian research direction of multi-agent systems, the distributed coordination has received considerable attention of many researchers. The objective of the distributed coordination is to achieve collective group behavior through local interaction. And the distributed coordination has a large number of applications, such as rendezvous [1], flocking [2], formation [3,4] and consensus [5-11], and so on. Particularly, consensus plays an important role in the distributed coordination. The basic idea of consensus is that each agent updates its state based on the states of its neighbors and its own such that ∗ Corresponding
author E-mail: [email protected]
1
the states of all agents will converge to a common value. In the control community, the consensus problem has been widely studied in the systems with single-integrator or double-integrator dynamics [5-11]. At present, the research of the distributed coordination becomes more and more challenging and interesting due to the introduction of fractional calculus. Moreover, many phenomena can not be modelled by integer-order dynamics, including the food searching of microbes and ground vehicles moving on the road covered in mud or grass [12, 13]. Similarly, many financial and engineering systems often demonstrate fractional dynamics either. Therefore, more and more researchers mainly pay attention to the distributed coordination of multi-agent systems with fractional dynamics. In Ref. [14], Cao et al. studied the coordination algorithms for networked fractional systems and this literature is the first paper that studied the distributed coordination of networked fractional systems. In Ref. [15], Ren et al. investigated the leader-following consensus for both single-integrator and double-integrator dynamics over an undirected fixed interaction graph. In Ref. [16], Bai et al. considered consensus of fractionalorder multi-agent systems with a constant or time-varying reference state. Yang et al. [17] investigated the distributed coordination of fractional multi-agent systems with communication delays, and obtained a critical value of time delay to reach consensus. Yu et al. [18] considered the leader-following consensus problem of fractional multi-agent systems via adaptive pinning control. Chen et al. [19] studied the multiconsensus problem of fractional uncertain multi-agent systems when the fractional order α satisfies 0 < α < 2. And in Ref. [20], Yu et al. investigated the leader-following consensus problem of fractional multi-agent systems with nonlinear dynamics. In practical applications, dynamical systems are often subjected to various perturbations, such as stochastic noises, external interferences, communication delays and model uncertainty. The existence of disturbances may impact the system performance and even cause networks divergent if those imprecise information is used. Hence, it is meaningful to discuss their effects on the behaviour of multi-agent systems. These issues recently have been extensively investigated. In Ref. [21], it was proved that the steady-state errors of any two agents can reach a small region for first-order multi-agent systems under bounded unknown external disturbances. In Ref. [22], continuous finite-time control laws were proposed to analyze the disturbance rejection property for second-order multi-agent systems under bounded unknown external disturbances. And Du et al. [23] investigated the problem of robust consensus for second-order multi-agent systems under external disturbances with/without communication delays. However, many existing tools developed for the study of the robust consensus problem can only be applied to integerorder dynamics. This is partly due to the limited theories for fractional systems. In Ref. [24], Wong et al. investigated robust synchronization of fractional-order complex dynamical networks with parametric uncertainties. In Ref. [25], Song et al. addressed the robust consensus problem of fractional-order linear multi-agent systems with positive real uncertainty. The results in Ref. [24, 25] are all concerned with the influences of parameter uncertainty. However, to the best of our knowledge, only a few authors consider the effects of external disturbances on fractional systems [26]. In Ref. [26], under the assumption
2
that the external disturbance generated by a fractional linear system, Yang et al. proposed a disturbance observers-based consensus protocol to study the distributed coordination of fractional multi-agent systems with external disturbancesNevertheless, when the external disturbance model is unknown in advance, the above mentioned mothods all don’t work. In this technical note, we consider the fractional multi-agent system under bounded unknown external disturbances and analyze the disturbance rejection properties for both linear and nonlinear systems. On the other hand, in reality, mobile agents may be governed by more complicated intrinsic dynamics. Indeed, nonlinear dynamics are commonly considered in integer-order multi-agent systems. In Ref. [10], Yu et al. studied the consensus problem for cooperative agents with nonlinear dynamics in a directed network. In Ref. [27, 28], Li et al. considered the consensus problem of both first-order and secondorder multi-agent systems with nonlinear dynamic and directed topologies, where each agent can only communicate with its neighbors on some disconnected timeintervals. And in Ref. [29], Song et al. investigated the second-order leader-following consensus problem of nonlinear multi-agent systems. The consensus protocol proposed in Ref. [29] is based on a pinning control, which means that only a small fraction agents can make use of leader’s information, because it is practically impossible to add controllers to all the agents in multi-agent aystems. Motivated by the above discussion and the significance of robust control, the problem of robust consensus for fractional multi-agent systems with external disturbances is investigated in this paper. Comparing with the results in [21-23], the systems with fractional dynamics are considered. Different from Ref. [26], the external disturbances are unknown in advance in this paper. A combination of the tools of Mittag-Leffler stability theory, the inequality techniques and Laplace transform is utilized to analyze the disturbance rejection properties for both linear and nonlinear systems. Firstly, it is proved that consensus can be achieved asymptotically in the absence of diturbances. In the presence of disturbances, the steady-state errors of any two agents can reach a small region. Secondly, the obtained results are extended to the leader-follower case with nonlinear dynamics. Finally, some simulation results are presented as a proof of concept. The rest of this paper is arranged as follows. The graph theory notions and Mittag-Leffler stability theory are introduced as a basis in Section 2. The main results on robust consensus for fractional multiagent systems are presented in Section 3 and Section 4. Then, some simulation results are given in Section 5. Finally, a short conclusion is shown to close the paper in Section 6.
2
Preliminaries In this section, some basic notions about the graph thoery and some definitions of fractional calculus
are introduced as the preliminaries of this paper. And some necessary conclusions are presented for the use of next several sections.
3
2.1
Graph Theory Notions
A directed graph G = (V, W) is used to model the interaction topology in the multi-agent system consisting of n agents, with the agent set V = {v1 , v2 , . . . , vn } and the edge set W ⊆ V 2 . A directed edge denoted by (vi , vj ) means that agent j can receive the state information of agent i. Consequently, agent i is a neighbor of agent j. All neighbors of agent i are denoted by Ni . Then, two types of matrices are introduced to represent the interaction graph: 1) the adjacency matrix A = [aij ] ∈ Rn×n with aij > 0 if (vi , vj ) ∈ W and aij = 0 otherwise, 2) the (nonsymmetric) Laplacian matrix L = [lij ] ∈ Rn×n with lii = j∈Ni aij and lij = −aij , i = j. The in-degree and out-degree of node i can be defined as follows N N aij , degout (i) = aji . [31], degin (i) = j=1,j=i
j=1,j=i
Lemma 1 [30]: Assume that the directed communication topology G has a directed spanning tree. Then, the matrix L has a simple zero eigenvalue and all the other eigenvalues have positive real parts. Let B = diag {b1 , b2 , · · · , bn } , i = 1, 2, . . . , n, where bi is a nonnegative real number that does not always equal 0, then all eigenvalues of L + B have positive real parts.
2.2
Caputo Fractional Operator and Mittag-Leffler stability
Definition 1 [33]: The fractional integral of order α for a function f is defined as t 1 α−1 α (t − τ ) f (τ ) dτ , t0 It f (t) = Γ (α) t0 where t ≥ t0 and α > 0. Definition 2 [33]: Caputo’s fractional derivative of order α for a function f ∈ C n ([t0 , +∞] , R) is defined by α t0 Dt f (t) =
1 Γ (n − α)
t t0
f (n) (τ )
α−n+1 dτ ,
(t − τ )
where t ≥ t0 and n is a positive integer such that n − 1 < α < n. Particularly, when 0 < α < 1, t f (τ ) 1 α D f (t) = dτ . t0 t Γ (1 − α) t0 (t − τ )α The Laplace transform of the Caputo fractional derivative is L {t0 Dtα f (t) ; s} = sα F (s) −
n−1
sα−k−1 f (k) (t0 ), n − 1 < α ≤ n,
k=0
where s is the variable in Laplace domain. Some properties of Caputo fractional derivative are listed as follows: Property 1:
α t0 Dt C
= 0 holds, where C is any constant.
Property 2: For constants μ and ν, the linearity of Caputo fractional derivative is described by α t0 Dt
(μf (t) + νg (t)) = μt0 Dtα f (t) + ν t0 Dtα g (t) .
4
In the following, we will introduce the Mittag-Leffler function [32] and Mittag-Leffler stability [34]. For α, β ∈ C, the Mittag-Leffler function in two parameters is defined as Eα,β (z) =
∞ k=0
zk . Γ (kα + β)
(1)
When β = 1, and α > 0, (1) can be written in a special case as Eα (z) = Consider the following n-dimensional Caputo fractional-order system ⎧ ⎨ t Dα x(t) = f (t, x(t)) 0
∞ k=0
zk Γ(kα+1) .
t
⎩ x(t ) = x 0 t0
(2)
where α ∈ (0, 1), x = (x1 , x2 , · · · , xn )T ∈ Rn , t0 ≥ 0, f is piecewise continuous on t and satisfies locally Lipschitz condition on x. Lemma 2 [34]: There exists a unique solution of system (2) for any initial value, if f (t, x) satisfies locally Lipschitz condition on x. Definition 3 [34]: The constant x ¯ is an equilibrium point of Caputo fractional dynamic system (2) if and only if f (t, x ¯) = 0. Definition 4 [34]: If x ¯ = 0 is an equilibrium point of system (2), the solution of (2) is said to be Mittag-Leffler stable if x(t) ≤ [m(xt0 )Eα (−λ(t − t0 )α )]b ,
(3)
where λ > 0, b > 0, m(0) = 0, · denotes an arbitrary norm and m(x) ≥ 0 satisfies locally Lipschitz condition on x ∈ Rn with Lipschitz constant m0 . Remark 1: Mittag-Leffler stability for system (2) implies asymptotic stability for any initial value, i.e., x → 0 with t → +∞. Lemma 3 [34]: The equilibrium point x ¯ = 0 of fractional-order system (2) is Mittag-Leffler stable if there exist positive α1 , α2 , α3 , a, b and a continuously differentiable function V (t, x (t)) satisfying a
ab
α1 x ≤ V (t, x (t)) ≤ α2 x , β t0 Dt V
(t, x (t)) ≤ −α3 xab ,
(4) (5)
where t ≥ 0, β ∈ (0, 1) , V (t, x (t)) : [ t0 , ∞) × D → R satisfies locally Lipschitz condition on x; D ∈ Rn is a domin containing the origin. If the assumptions hold globally on Rn , x¯ = 0 is globally Mittag-Leffler stable. Lemma 4 [35]: Let x (t) ∈ R be a continuous and derivable function. Then, for any time instant t ≥ t0 , 1 β 2 β t D x (t) ≤ x (t) t0 Dt x (t) , ∀β ∈ (0, 1) . 2 0 t Obviously, when x (t) ∈ Rn is continuous and derivable, it implies 1 β T β T t D x (t) x (t) ≤ x (t) t0 Dt x (t) , ∀β ∈ (0, 1) . 2 0 t 5
(6)
3
Robust Consensus of Fractional Linear Multi-agent Systems In this section, the problem of robust consensus for fractional linear multi-agent systems with external
disturbances is discussed. Different from most of the existing work [24-26], here we assume that the external disturbances are unknown and bounded. Consider the multi-agent system consisting of n agents, labeled as agents 1 to n. The dynamics of each agent with external disturbances is given by α 0 Dt xi
(t) = ui (t) + wi (t) , i ∈ {1, 2, . . . , n} ,
(7)
where α ∈ (0, 1), xi (t) is the state for the ith agent, ui (t) is the control input for the ith agent, and wi (t) is the external disturbance. Definition 5 [36]: The multi-agent system (7) is said to achieve robust consensus if its solution satisfies lim xi (t) − xj (t) ≤ h (l) , ∀i, j ∈ {1, 2, . . . , n} ,
t→∞
where the function h(l) satisfies h(0) = 0 and is monotonous increasing with respect to l. In the following, assume that all agents are in a one-dimensional space for the simplicity of presentation. However, all results hereafter are still valid for the m-dimensional (m > 1) space by the introduction of the Kronecker product. As in [21-23], the disturbance signal wi (t) satisfies the following assumption. Assumption 1: The external disturbances wi (t) satisfy |wi (t)| ≤ l < +∞ for all i ∈ {1, 2, . . . , n}. Consider the following control protocol to achieve robust consensus of system (7): ui (t) = −β
n
aij (xi (t) − xj (t)),
(8)
i=1
where aij (i, j = 1, 2, . . . , n) is the (i, j)th entry of the adjacency matrix A ∈ Rn×n associated with the directed graph G; β is a nonnegative constant. The control protocol is independent of the disturbance signal, because the external disturbance model is unknown in advance. With the contol input (8), system (7) can be rewritten as α 0 Dt xi
(t) = −β
n
aij (xi (t) − xj (t)) + wi (t) ,
(9)
i=1
which can be written in a vector form as α 0 Dt X
(t) = −βLX (t) + W (t) ,
T
(10) T
where X (t) = [x1 (t) , x2 (t) , . . . , xn (t)] , W (t) = [w1 (t) , w2 (t) , . . . , wn (t)] . Before presenting the main result, we need the following changes of variables: y12 (t) = x1 (t) − x2 (t) , y13 (t) = x1 (t) − x3 (t) , . . . , y1n (t) = x1 (t) − xn (t) . 6
(11)
⎛ Let Y (t) = [y12 (t) , y13 (t) , . . . , y1n (t)]T , E = [1n−1 , −In−1 ] ∈ R(n−1)×n , and F = ⎝
0Tn−1 −In−1
⎞ ⎠ ∈
Rn×(n−1) . Then the coordinate changes (11) can be written in a vector form as Y (t) = EX (t) , X (t) = x1 1n + F Y (t) .
(12)
As a result, it follows from (12) and (10) that α 0 Dt Y
(t) = E 0 Dtα X (t) = −βx1 EL1n − βELF Y (t) + EW (t)
(13)
= −βELF Y (t) + EW (t) = βCY (t) + EW (t) , where C = −ELF .
Lemma 5 [23, 37]: The matrix C = −ELF is Hurwitz if and only if the communication topology G has a directed spanning tree, where L is the Laplacian matrix of graph G. Proof: From the definitions of matrixes E, L, F , we have C = −ELF ⎛
1 −1 ⎜ .. ⎜ .. = −⎜ . . ⎝ 1 0 ⎛ ⎜ ⎜ = −⎜ ⎝
··· .. .
0 .. .
...
−1
⎛
⎞ ⎟ ⎟ ⎟ ⎠ (n−1)×n
l12 − l22 .. .
··· .. .
l1n − l2n .. .
l12 − ln2
···
l1n − lnn
l11 ⎜ ⎜ .. ⎜ . ⎝ ln1
··· .. .
l1n .. .
···
lnn
⎛
⎞
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
⎟ ⎟ ⎟ ⎠ n×n
⎞
···
0
−1 · · · .. .. . .
0 .. .
0
0
···
−1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ n×(n−1)
⎟ ⎟ ⎟ ⎠ (n−1)×(n−1)
Then, it follows from Lemma 1 in Ref. [33] that C is Hurwitz. Thus, the proof is completed. Theorem 1: Consider the fractional linear multi-agent system (7) under the control input (8). If the fixed directed graph G has a directed spanning tree, then, (i) consensus can be achieved asymptotically in system (7) when wi (t) = 0. (ii) when wi (t) = 0 and Assumption 1 is satisfied, the steady-state errors of any two agents will converge to the region M1 asymptotically, where 2nλmax (P )/λmin (P ) P E l , M1 = |xi (t) − xj (t)| ≤ β
(14)
P > 0 is the solution of the Lyapunov equation C T P + P C = −2In−1 . Proof: Since matrix C is Hurwitz matrix from Lemma 5, there is a positive-definite P = P T > 0 such that C T P + P C = −2In−1 . For the system (13), construct Lyapunov functions candidate in the form of V (t) = Y T (t) P Y (t) . Obviously, the Lyapunov function satisfies inequality (4). 7
Then, by Lemma 4 and Eq. (13), it is easy to obtain α 0 Dt V
(t) = 0 Dtα Y T (t) P Y (t) ≤ 2Y T (t) P 0 Dtα Y (t) = 2Y T (t) P [βCY (t) + EW (t)]
Then, noting equality ξ T ζ = ζ T ξ and inequality ξ T ς + ς T ξ ≤ κξ T ξ + κ1 ς T ς for any ξ, ζ ∈ Rm , m ∈ N+ , one has α 0 Dt V
(t) ≤ β Y T (t) P CY (t) + Y T (t) C T P Y (t) + Y T (t) P EW (t) + W T (t) E T P Y (t) 1 ≤ βY T (t) C T P + P C Y (t) + βY (t)2 + P E2 W (t)2 β ≤ −βY (t)2 +
2
nP E l2 . β
(15)
2
Noticing that V (t) ≤ λmax (P ) Y (t) , we have α 0 Dt V (t) ≤ −
Let u (t) = V (t) −
nλmax (P )P E2 l2 , β2
β λmax (P )
V (t) +
nP E2 l2 . β
this together with (15) implies that α 0 Dt u (t)
≤−
β u (t) . λmax (P )
Then, there exists a nonnegative function m (t) satisfying α 0 Dt u (t)
+ m (t) = −
β u (t) . λmax (P )
(16)
Taking the Laplace transform of (16) yields sα U (s) − u (0) sα−1 + M (s) = −
β U (s) , λmax (P )
where U (s) = L {u (t)} and M (s) = L {m (t)}. Thus, we have U (s) =
u (0) sα−1 − M (s) sα +
β λmax (P )
.
It follows from Lemma 2 that there exists a unique solution of (16). With the inverse Laplace transform, it is easy to obtain u (t) = u (0) Eα −
βtα λmax (P )
α−1
− m (t) ∗ t
Eα,α −
βtα λmax (P )
.
βtα Because tα−1 and Eα,α − λmax are nonnegative functions, then the following is satisfied (P ) V (t) −
2 βtα nλmax (P ) P E l2 → 0, − = u (t) ≤ u (0) E α β2 λmax (P ) 8
2
when t → +∞. Noticing that V (t) ≥ λmin (P ) Y (t) and letting t tend to indinity, one has nλmax (P )/λmin (P ) P E l 2 . lim Y (t) ≤ t→+∞ β
(17)
From Eq. (11), we obtain for any i, j ∈ {1, 2, . . . , n}, |xi (t) − xj (t)| ≤ |x1 (t) − xj (t)| + |x1 (t) − xj (t)| = |y1i (t)| + |y1j (t)| √ √ 2 2 ≤ 2 |y1i (t)| + |y1j (t)| ≤ 2 Y (t) .
(18)
Combining (17) and (18), one obtains 2nλmax (P )/λmin (P ) P E l lim |xi (t) − xj (t)| ≤ , t→+∞ β for any i, j ∈ {1, 2, . . . , n}. When wi (t) = 0, from Eq. (15), one has α 0 Dt V
(t) ≤ −βY (t)2 .
(19)
From Eq. (19), inequality (5) can be satisfied. According to Lemma 3, system (13) is Mittag-Leffler stable, which implies that system (13) is asympototically stable. From the definition of Y (t), we have xi (t) − xj (t) → 0 as t → +∞ for any i, j ∈ {1, 2, . . . , n}. Therefore, consensus can be achieved in system (7) asymptotically under the control law (8) in the absence of disturbances. This completes the proof.
4
Robust Consensus of Fractional Nonlinear Multi-agent Systems In this section, robust consensus of fractional nonlinear multi-agent systems with external disturbances
is discussed, and a controller based on a pinning control is designed to implement robust consensus. The fractional nonlinear multi-agent system with external disturbances is described by α 0 Dt xi
(t) = f (t, xi (t)) + ui (t) + wi (t) , i ∈ {1, 2, . . . , n} ,
(20)
where α ∈ (0, 1), xi (t) is the state for the ith agent, ui (t) is the control input for the ith agent, f (t, xi (t)) is the intrinsic nonlinear dynamics for the ith agent, and wi (t) is the external disturbance. The virtual leader for multi-agent system (20) is an isolated agent described by α 0 Dt xr
(t) = f (t, xr (t)) ,
where xr (t) is the state for the virtual leader.
9
(21)
Assumption 2: The nonlinear function f (t, x) (x, y ∈ R) is continuous and satisfies Lipschitz condition on x with Lipschitz constant θ ≥ 0, i.e., |f (t, x) − f (t, u)| ≤ θ|x − u|, for any t ≥ 0. The following pinning control input is considered to implement robust consensus of multi-agent system (20): ⎡ ui (t) = −β ⎣
n
⎤ aij (xi (t) − xj (t)) + bi (xi (t) − xr (t))⎦ ,
(22)
j=1
where aij (i, j = 1, 2, . . . , n) is the (i, j)th entry of the adjacency matrix A ∈ Rn×n associated with the directed graph G; bi (i = 1, 2, . . . , n) is a nonnegative real number that does not always equal 0; β is a nonnegative constant. When bi > 0, the agent i is pinned. Consequently, we assume that the agent should be pinned if it receives little information from its neighbour but outputs more, i.e., the agent i should be pinned if degin (i) < degout (i) [18]. The agents in Fig. 1 which should be pinned are agent 1 and agent 2. Let H = L + B, where L is the Laplacian matrix of G. It follows from Lemma 1 that −H is Hurwitz. With the contol input (21), system (19) can be rewritten as ⎡ ⎤ n α aij (xi (t) − xj (t)) + bi (xi (t) − xr (t))⎦ + wi (t) . 0 Dt xi (t) = f (t, xi (t)) − β ⎣
(23)
j=1
Noting that zi (t) = xi (t) − xr (t) , i = 1, 2, . . . , n, we have ⎡ ⎤ n α ⎣ aij (zi (t) − zj (t)) + bi zi (t)⎦ + wi (t) , 0 Dt zi (t) = f (t, zi (t) + xr (t)) − f (t, xr (t)) − β
(24)
j=1
which can be written in a vector form as α 0 Dt Z
(t) = −βHZ (t) + F (t, Z (t)) + W (t) ,
(25)
where Z (t) = [z1 (t) , z2 (t) , . . . , zn (t)]T , F (t, Z (t)) = [f (t, z1 (t) + xr (t)) − f (t, xr (t)) , f (t, z2 (t) T
+xr (t)) − f (t, xr (t)) , . . . , f (t, zn (t) + xr (t)) − f (t, xr (t))] . Theorem 2: Suppose that the fixed directed graph G has a directed spanning tree and the nonlinear function f in system (20) and (21) satisfies Assumption 2. If β ≥ Q , θ
(26)
where Q > 0 is the solution of the Lyapunov equation H T Q + QH = 3In , then, (i) all agents in system (20) can achieve leader-following consensus asympototically when wi (t) = 0.
10
(ii) when wi (t) = 0 and Assumption 1 is satisfied, the state of ith agent can reach the region M2 asympototically, 2nλmax (P ) Q l , μβλmin (P )
|xi (t) − xr (t)| ≤
M2 = where μ = β −
(27)
Q2 θ 2 . β
Proof: It follows from Lemma 1 that -H is Hurwitz matrix. Then there is a positive-definite Q = QT > 0 such that −H T Q − QH = −3In . For the system (25), construct Lyapunov functions candidate in the form of V (t) = Y T (t) QY (t) . Obviously, the Lyapunov function satisfies inequality (4). Then, by Lemma 4, Eq. (25) and Assumptions 1 and 2, we have α 0 Dt V
(t) ≤ 2Z T (t) Q [−βHZ (t) + F (t, Z (t)) + W (t)] = βZ T (t) −QH − H T Q Z (t) + 2Z T (t) QF (t, Z (t)) + 2Z T (t) QW (t) ≤ −3βZ (t)2 + βZ (t)2 + β1 Q2 W (t)2 + βZ (t)2 + β1 Q2 F (t, Z (t))2 n 2 2 l Q2 ≤ −βZ (t)2 + nQ + (f (t, zi (t) + xr (t)) − f (t, xr (t)))2 β β ≤ −βZ (t)2 +
nQ2 l2 β
!
Q2 θ2 =− β− β where μ = β −
Q2 θ 2 β
+
i=1 Q2 θ 2 Z β
" 2
Z (t) +
(t)2
nQ2 l2 nQ2 l2 2 = −μZ (t) + , β β
(28)
> 0 by condition (26). Applying the proof procedure in Theorem 1, we can
immediately obtain lim |xi (t) − xr (t)| ≤
t→+∞
2nλmax (P ) Q l, μβλmin (P )
for any i ∈ {1, 2, . . . , n}. When wi (t) = 0, from Eq. (28), one has α 0 Dt V
(t) ≤ −μZ (t)2 .
(29)
From Eq. (29), inequality (5) can be satisfied. According to Lemma 4, when condition (26) is satisfied, system (25) is Mittag-Leffler stable, which implies that system (25) is asympototically stable. Therefore, zi (t) = xi (t) − xr (t) → 0, i ∈ {1, 2, . . . , n}, as t → +∞, and all agents in system (20) can achieve leader-following consensus asympototically. The proof is completed.
11
Figure 1: Interaction graph for 7 agents.
5
Numerical simulations Several numerical simulations are presented to verify the results of Theorem 1 proposed in Section
3 and Theorem 2 proposed in Section 4. We consider a group of 7 agents with an interaction graph given by Fig. 1. Note that the interaction graph has a directed spanning tree with node 1 being the root. The corresponding Laplacian matrix is chosen aij = 1 if (vj , vi ) ∈ W, and aij = 0 otherwise. The T
initial conditions are selected as x (0) = [5, −4, 3, −2, −6, 1, 8] , and the external disturbances are given as: w1 (t) = 1.1 sin (8t − 1), w2 (t) = − cos (2t), w3 (t) = 0.8 sin (t) + 0.7 cos (t), w4 (t) = 0.2 cos (11t − 4), w5 (t) = −1.5 sin (−7t + 1), w6 (t) = cos (−3t), w7 (t) = −0.6 sin (t) − 0.9 cos (t). Let the fractional order α = 0.9. Firstly, the simulations results are showed to verify Theorem 1 in Section 3. Let β = 1. i) When wi (t) = 0, i = 1, 2, . . . , 7, according to Theorem 1, the system (7) can achieve cosnensus asympototically, which is presented in Fig. 2. ii) When wi (t) = 0, i = 1, 2, . . . , 7, the trajectories of xi (t) in system (7) are showed in Fig. 3, and it can be observed that the steady-state errors of any two agents can reach a small region, which confirms the effectiveness of Theorem 1. We next present the simulation results to demonstrate Theorem 2 in Section 4. Suppose that the nonlinear intrinsic dynamics of each agent in system (20) and system (21) is described by f (t, x) = sin x + cos t, which satisfies Assumption 2 with θ = 1. It can be computed that Q = 4.3853 and we choose β = 5 to satisfy condition (26). i) When wi (t) = 0, i = 1, 2, . . . , 7, according to Theorem 2, the system (20) can realize leader-following consensus asympototically, which is presented in Fig. 4. ii) When wi (t) = 0, i = 1, 2, . . . , 7, the trajectories of xi (t) in system (20) and xr (t) in system (21) are showed in Fig. 5, and it can be shown that all agents can reach the region around the leader asympototically, which verifies the correctness of Theorem 2.
12
8
6
4
x i (t)
2
0
-2
-4
-6 0
5
10
15
20
25
30
t
Figure 2: Trajectories of xi (t) in system (7) with wi (t) = 0.
8
6
4
x i (t)
2
0
-2
-4
-6
-8 0
5
10
15
20
25
t
Figure 3: Trajectories of xi (t) in system (7) with wi (t) = 0.
13
30
8
6
4
2
xr(t)
0
x (t) 1
x (t) 2
-2
x3 (t) x (t) 4
x5 (t)
-4
x (t) 6
x7 (t)
-6 0
0.5
1
1.5
2
2.5
3
3.5
4
t
Figure 4: Trajectories of xr (t) and xi (t) in system (20) with wi (t) = 0.
8
6
4
2
xr(t)
0
x1 (t) x2 (t)
-2
x3 (t) x (t) 4
x5 (t)
-4
x6 (t) x7 (t)
-6 0
0.5
1
1.5
2
2.5
3
3.5
4
t
Figure 5: Trajectories of xr (t) and xi (t) in system (20) with wi (t) = 0.
14
6
Conclusion In this paper, the problem of robust consensus for fractional multi-agent systems is investigated via
a directed fixed interaction graph. The effects of external disturbances on the behaviour of both linear and nonlinear systems are considered over a combination of the tools of Lyapunov method, matrix theory and fractional calculus theory. Firstly, for fractional linear multi-agent systems under bounded unknown external disturbances, it is proved that the steady-state errors of any two agents can reach a small region. Then, a pinning control input is proposed to achieve robust consensus for fractional nonlinear multi-agent systems with external disturbances. Future works include the study of robust consensus for fractional time-delayed multi-agent systems and extending the results in this article to the switching topology case.
Acknowledgment This work is supported by the National Nature Science Foundation of China (No. 11371049).
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