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Scaled consensus seeking in multiple non-identical linear autonomous agents
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Research article
Scaled consensus seeking in multiple non-identical linear autonomous agents Cheng-Lin Liu Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Wuxi 214122, China
art ic l e i nf o
a b s t r a c t
Article history: Received 12 November 2016 Received in revised form 21 June 2017 Accepted 22 June 2017
Scaled consensus problem is studied for a heterogeneous multi-agent system composed of non-identical stable linear agents, and a leader-following scaled consensus algorithm is designed. By using frequencydomain analysis, consensus conditions are obtained for the agents without communication delay under undirected and directed topologies, respectively. Moreover, consensus criteria are also gained for the agents suffering from communication delay under directed topology. Simulation examples show the correctness of theoretical results. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Heterogeneous linear multi-agent systems Scaled consensus problem Leader-following algorithm Communication delay
1. Introduction As a fundamental phenomenon in flocking, swarming, and coupled synchronization, consensus behavior has attracted more and more attention of researchers in the control theory field in recent years. Consensus problem requires several autonomous agents to reach a common agreement on their states, and has broad engineering applications, e.g., coordination control of unmanned aerial and ground vehicles, smart grid, wireless sensor networks, etc. Current research results on consensus problem have been mainly focused on homogeneous multi-agent systems, of which the agent's dynamics are identical [1–10]. With the help of algebraic graph theory [1], matrix theory [2–6], frequency-domain analysis [7–9], and Lyapunov functions [10], various consensus algorithms have been designed and analyzed for the agents with fixed topology or switching topologies. Over the past few years, furthermore, a lot of research effort has also been put into the consensus analysis and synthesis of heterogeneous multi-agent system composed of the agents with distinct dynamics. As a simple heterogeneous multi-agent system, mixed system with multi-class agents has attracted many researchers’ interests, such as the system of first-order agents and second-order agents [11,12] and the system of Euler-Lagrange agents and doubleintegrator agents [13,14]. For the heterogeneous linear multi-agent systems, state-feedback and output-feedback consensus algorithms were proposed by using the internal model principle [16,17] and the harmonic control [18]. To solve the consensus problem of heterogeneous linear multi-agent system with diverse communication delays, Lee and Spong [15] used the spectral radius theorem to obtain E-mail address: [email protected]
the delay-independent consensus conditions. Meanwhile, Muenz [19] and Tian and Zhang [20] adopted the consensus algorithms with devisable self delays distinct from the corresponding communication delays, and provided the consensus criteria based on frequency-domain analysis. In addition, Tian and Zhang [20] designed an adaptive adjustment algorithm in order to adjust the self delays on-line if the communication delays were unknown. Liu and Liu [21] proposed three adaptive consensus algorithms for the heterogeneous first-order time-invariant multi-agent systems, and obtained the consensus conditions for the agents under fixed topology without and with time delays respectively. Specially, non-identical disturbances also lead to the heterogeneity of multi-agent systems [22]. Based on Lyapunov functions, Kim et al. [22] adopted a general consensus algorithm and analyzed the robust consensus convergence condition For the heterogeneous first-order time-varying multi-agent systems with distinct disturbances. Treated as the generalized consensus problem, scaled consensus problem defined in [23] means that the network components’ scalar states reach the assigned proportions rather than a common value, and has attracted some researchers’ interests for its engineering applications, such as compartmental mass-action systems, closed queueing networks, and water distribution systems [23]. In addition to usual consensus problem, group consensus problem [24], wherein the agents form several sub-groups reach corresponding agreement values respectively, is also a special case of scaled consensus problem. With a connected topology, Roy [23] proved that the first-order multi-agent systems not only achieved a stationary scaled consensus asymptotically, but also could track on the stable manifold. For the first-order multi-agent systems, Meng and Jia [25] studied the scaled consensus problem with time-varying scaled ratios, and demonstrated that the agents converged to the scaled consensus asymptotically under jointly-connected switching
http://dx.doi.org/10.1016/j.isatra.2017.06.022 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Liu C-L. Scaled consensus seeking in multiple non-identical linear autonomous agents. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.022i
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2
topologies. By constructing Lyapunov-Krasovskii functionals, Aghbolagh et al. [26] considered the scaled consensus of first-order multi-agent systems with time-varying delay, and gained the consensus conditions in the form of linear matrix inequalities, which were used to calculate the allowable delay value. In this paper, we consider the scaled consensus problem of a class of heterogeneous multi-agent systems with the agents modeled by single-input and single output linear stable dynamics, and an adaptive scaled consensus algorithm is constructed in the leader-following structure. Based on frequency-domain analysis, firstly, we obtain the consensus conditions for the multi-agent systems without communication delay under undirected and directed topologies, respectively. For the multi-agent systems with identical communication delay, besides, the delay-independent and delay-dependent consensus conditions are obtained, respectively, for the system under directed topology.
2. Problem description
lim (ri yi (t ) − r j yj (t )) = 0, i, j = 1, 2, …, N ,
t →∞
(4)
where the ratios ri ∈ R, i = 1, … , N are assumed to be non-zero. If ri = r j, ∀ i, j ∈ {1, … , N}, the output scaled consensus in Definition 1 becomes an output synchronization. Hence, the output scaled consensus problem (4) is a generalized output synchronization problem. Usually, the interconnection topology of multi-agent system (1) is described as an N-order weighted digraph G = (V , E, ), which is composed of a set of vertices V = {1, … , N}, a set of edges E ⊆ V × V and a weighted adjacency matrix = [aij ] ∈ R N × N with aij ≥ 0. In the diagraph G, a directed edge from the node i to the node j is denoted by eij = (i, j ) ∈ E , and it is assumed that aij > 0 ⇔ eij ∈ E . Moreover, we assume aii = 0 for all i ∈ V . Besides, the digraph G is undirected graph or bidirectional if eij ⇔ eji , and the digraph is symmetric if aij = aji . The set of node i's neighbors is denoted by Ni = {j ∈ V : (i, j ) ∈ E}. The Laplacian matrix of the weighted digraph G is defined as L = D − = [lij ] ∈ Rn × n , where N
2.1. Non-identical agents and topology Consider a class of heterogeneous multi-agent system composed of N non-identical single-input single-output linear agents given by
ẋi (t ) = Ai xi (t ) + Bi(ui (t ) + ni ), yi (t ) = cxi(t ), i = 1, …, N ,
(1)
D = diag{ ∑ j = 1 aij , i ∈ V} is the degree matrix. In the digraph, if there is a path from one node i to another node j, then j is said to be reachable from i, otherwise, j is said to be not reachable from i. If a node is reachable from every other node in the digraph, then we say it globally reachable. A globally reachable node is precisely the degree of connectedness required of the digraph. An undirected graph is connected if it has a globally reachable node. 2.2. Design of scaled consensus algorithm
⎡ 0 1 0 ⎢ 0 1 ⎢ 0 Ai = ⎢ ⋮ ⋮ ⋮ ⎢ 0 0 0 ⎢⎣ −α −α −α i0 i1 i2
⋯ ⋯ ⋯ ⋯ ⋯
⎡ 0⎤ 0 ⎤ ⎢ ⎥ ⎥ 0 ⎥ ⎢ 0⎥ ⋮ ⎥, Bi = ⎢ ⋮ ⎥, ⎢ 0⎥ 1 ⎥ ⎢ ⎥ −αin − 1⎥⎦ ⎣ bi ⎦
In this paper, we focus on the leader-following coordination control, which has been extensively studied for multi-agent systems [27–31]. For the agents (1), a usual scaled consensus algorithm is given by
c = ⎡⎣ 1 0 0 ⋯ 0⎤⎦.
ξi(t ) =
where x i(t ) ∈ Rn , yi (t ) ∈ R and ui(t ) ∈ R are the state, the output and the input of agent i, respectively, ni ∈ R is the constant disturbance of each agent i, Ai ∈ Rn × n , Bi ∈ Rn , c ∈ R1 × n , bi ∈ R , and αik ∈ R, k = 0, … , n − 1. Correspondingly, the frequency-domain description of (1) is formulated as
Yi(s ) = gi (s )(Ui(s ) + Ni(s )), i = 1, …, N ,
(2)
where Yi(s), Ui(s) and Ni(s) are the Laplace transforms of yi(t), ui(t) and ni respectively, and gi(s) is the transfer function. Since the agents’ dynamics (1) are formulated as a controllable canonical form, we get
gi (s ) = =
bi di(s )
⎛ 1⎜ ri ⎜⎝
⎞ aij(r j yj (t ) − ri yi (t )) + pi ( y0 − ri yi (t ))⎟⎟, i ∈ V , ⎠ j ∈ Ni
∑
where the non-zero constants ri, i ∈ V denotes the ratios of agents’ states, Ni denotes the neighbors of agent i, aij > 0 is the adjacency element of in the digraph G = (V , E, ), y0 is the static leader's state, and pi denotes the linking weight from agent i to the leader. Assume that pi > 0 if there is a directed edge from agent i to the leader; otherwise, pi ¼0. In the rest of this paper, the notation P = diag{pi , i ∈ V} is used. Apparently, the algorithm (1) is similar to that in [25]. Taking ui(t ) = κξi(t ) with κ > 0, we get the closed-loop form of multi-agent system (1) as follows
ẋi (t ) = Ai xi (t ) + Bi(κξi(t ) + ni ), i ∈ V .
bi s n + αin − 1s n − 1 + ⋯ + αi1s + αi0
. (3)
Assumption 1. The system matrix Ai , i = 1, … , N is Hurwitz, i.e., di(s ) , i = 1, … , N is Hurwitz. Evidently, Ai is non-singular with Assumption 1 guaranteeing that all the eigenvalues of Ai are non-zero. Assumption 2. Without loss of generality, bi is assumed to be positive, i.e., bi > 0. Definition 1. The multi-agent network (1) achieves Scaled Consensus asymptotically, if
(5)
(6)
Remark 1. With the interconnection topology that has a globally reachable node, it is easily proved that the multi-agent system (6) converges to an asymptotic scaled consensus, only if −1 rcA i i Bini = y0 , ∀ i ∈ V .
(7)
Condition (7) can be obtained by assuming that the agents converge to the scaled consensus asymptotically, i.e., limt →∞ri yi (t ) = y0 . Evidently, the condition (7) is dependent on the parameters strictly, i.e., the agents cannot achieve the asymptotic scaled consensus when the condition (7) is not satisfied. Moreover, (7) also implies that the agents (1) without disturbances cannot reach the asymptotic scaled consensus with the algorithm (5).
Please cite this article as: Liu C-L. Scaled consensus seeking in multiple non-identical linear autonomous agents. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.022i
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To achieve the asymptotic scaled consensus and cancel the impact of the diversity of the disturbances ni, we design the following adaptive algorithm for multi-agent system (1),
ui (t ) = zi(t ) + κξi(t ), zi̇ (t ) = γξi(t ),
3
̇ x^ i (t ) = A i x^ i (t ) + Bi(z^i(t ) + κ (
a ij(y^j (t ) − y^i (t )) − pi y^i (t )) ,
∑ j ∈ Ni
̇ z^i(t ) = γ (
a ij(y^j (t ) − y^i (t )) − pi y^i (t )) ,
∑ j ∈ Ni
(8)
where κ > 0, γ > 0, and zi(t) is an additional variable of consensus controller ui(t). Furthermore, the algorithm (8) is a proportionalintegral form of algorithm (5).
y^i (t ) = cx^ i (t ) .
(11)
Moreover, the scaled consensus problem of system (9) turns to be an asymptotic stability problem of system (11). Taking the Laplace transform of system (11) yields ^ ^ ^ sX i (s) = A i X i (s) + Bi(Z i(s) + κ (
2.3. Useful lemmas
^ ^ ^ a ij(Yj(s) − Yi (s)) − pi Yi (s)),
∑ j ∈ Ni
For the leader-following control structure, we come to the following property from Lemma 2 in [32], Lemma 2 in [28], Lemma 3 in [27] and Lemma 1 in [21]. Lemma 1. uppose that the interconnection topology of N agents (3) and the static leader y0 has the leader as a globally reachable node. The eigenvalues of L + P are all none-zero, and D + P > μI , where μ > 0 and L is the Laplacian matrix of the topology of N agents (3) without leader. In addition, the following two lemmas will play important roles in the proof of main results. Lemma 2. [33] Gr (s ) ∈ R(s )N × N is proper, where N is a positive integrator. The zeros of Δ(s ) = det(I + Gr (s )) lie on the open left half complex plane if and only if the number of counterclockwise encirclements of ( − 1, j0) by the eigenloci of Gr (jω) as ω goes from −∞ to +∞ equals the total number of right-half plane poles of Gr(s).
^ sZ i(s) = γ (
∑
^ ^ ^ a ij(Yj(s) − Yi (s)) − pi Yi (s)),
j ∈ Ni
^ ^ Yi (s) = cX i (s),
(12)
^ ^ ^ where Xi(s ), Zi(s ) and Yi (s ) are the Laplace transforms of x^i(t ), z^i(t ) and y^ (t ) respectively. Thus, the characteristic equation of system (11) about y^ (t ) = [y^ (t ) , … , y^ (t )]T is given by 1
N
^ det(diag{sdi(s ), i ∈ V} + (κs + γ ) L ) = 0,
(13)
where = diag{bi , i ∈ V} and L^ = L + P . 3.1. Undirected topology At first, we provide the scaled consensus condition for multiagent system (9) with an undirected topology.
Lemma 3. [34] Let Q ∈ C n × n , Q = Q⋆ ≥ 0 and T = diag{ti, ti ∈ C}. Then
Theorem 1. Consider multi-agent system (9) with an undirected and symmetric topology that has the leader as a globally reachable node. Under Assumptions 1 and 2, then, the agents in system (9) converge to a scaled consensus asymptotically, if
λ(QT ) ∈ ρ(Q )Co(0 ∪ {ti}),
2bi (max( i∈V
where λ(·) denotes matrix eigenvalue, ρ(·) denotes the matrix spectral radius, and Co(·) denotes the convex hull.
∑
aij + pi ))Γ < 1,
(14)
j ∈ Ni
where Γ > 0 is the positive minimum value that satisfies jκω + γ ( − Γ , j0) ∉ Co(0 ∪ gi(jω) , i ∈ V ) with ω ∈ R , and gi(jω) = jωd (jω) . i
^ Proof. Let f (s ) = det(diag{sdi(s ) , i ∈ V} + (κs + γ ) L ), and ^ f (0) = det(diag{0(0n + αin − 10n − 1 + ⋯ + αi0) , i ∈ V} + (κ 0 + γ ) L ) ^ =det(γ L ) ≠ 0 holds from Lemma 1.
3. Scaled consensus seeking without time delay With (8), the dynamics of agents (1) are rewritten as
When s ≠ 0, the equation (13) is equivalent to
ẋi(t ) = A i x i(t ) + Bi(n i + z i(t ) + κξi(t )), z i̇ (t ) = γξi(t ), 1 ( ri
ξi(t ) =
∑
det(I + diag{ a ij (r j yj (t ) − ri yi (t )) + pi ( y0 − ri yi (t )), i ∈ V ,
j ∈ Ni
yi (t ) = cx i(t ).
(9)
Let y¯i (t ) = ri yi (t ) and x¯ i(t ) = rx i i(t ), and we get
aij( y¯j (t ) − y¯i (t )) + pi ( y0 − y¯i (t ), i ∈ V , (10)
Hence, the scaled consensus problem of system (9) becomes a consensus problem of system (10). Define y^i (t ) = y¯i (t ) − y0 , x^i(t ) = x¯ i(t ) − x0 with x0 = [y0 , 0, … , 0]T α and z^ (t ) = − i0 y + rn + rz (t ), and we obtain the dynamics of x^ i
as follows
bi 0
i i
i i
i
tion 1 and Lemma 2, the zeros of h(s) are on the open left half jκω + γ complex plane, if λ(diag{ jωd ( jω) , i ∈ V} L^ ) does not enclose the
λ(diag{
j ∈ Ni
y¯i (t ) = cx¯ i (t ).
κs + γ Denote h(s ) = det(I + diag{ sd (s) , i ∈ V} L^ ). According to Assump-
point ( − 1, j0) for ω ∈ R . The definition of Laplacian matrix and ^ ^T symmetric weights make L = L > 0 hold. Based on Assumption 2 and Lemma 3, we get
1 zi̇ (t ) = γ ξ¯i(t ), ri
∑
(15)
i
¯ x¯ i̇ (t ) = Ai x¯ i (t ) + Bi(rn i i + rz i i(t ) + κξi(t )),
ξ¯i(t ) =
κs + γ ^ , i ∈ V} L ) = 0. sdi(s )
i
jκω + γ ^ , i ∈ V } L ) jωdi(jω)
^ = λ(diag{gi (jω), i ∈ V}diag{ bi , i ∈ V}L diag{ bi , i ∈ V}) ^ ∈ ρ(diag{ bi , i ∈ V}L diag{ bi , i ∈ V})Co(0 ∪ gi (jω), i ∈ V ) ⊂ 2bi (max( i∈V
∑
aij + pi ))ΓΓ −1Co(0 ∪ gi (jω), i ∈ V ).
j ∈ Ni
Thus, gi(jω) , i ∈ V crosses the negative real axis at some certain
Please cite this article as: Liu C-L. Scaled consensus seeking in multiple non-identical linear autonomous agents. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.022i
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Γ must exist from the
frequencies under Assumption 1, and
definition. For the condition (14), hence, ρ(diag{ bi , i ∈ V} ^ L diag{ bi , i ∈ V})Co(0 ∪ gi(jω) , i ∈ V ) does not enclose the point ( − 1, j0), and the zeros of h(s) are on the open left half plane. Now, we have proved that the zeros of f(s) all have negative real parts, i.e., lim x^ (t ) = 0, i ∈ V . Therefore, the agents in (9) converge to a t →∞ i
□
scaled consensus asymptotically.
3.2. General directed topology Now, we analyze the scaled consensus seeking of system (9) with general directed topology. n 1 Define Ai = A + ΔAi , where A = n ∑i = 1 Ai , and the agents’ dyx i̇ (t ) = (A + ΔA i )x i(t ) + Bi(n i + z i(t ) + κξi(t )),
∑
a ij(r j yj (t ) − ri yi (t )) + pi ( y0 − ri yi (t )), i ∈ V ,
(16)
In (16), the characteristic polynomial of A is d(s ) = sn + αn − 1sn − 1 + ⋯ + α1s + α0 . ^ In the same way, let y^i (t ) = ry i i(t ) − x 0 with i i (t ) − y0 , x i(t ) = rx α x = [ y , 0, … , 0]T and z^ (t ) = − i0 y + rn + rz (t ), and we get i
0
bi 0
̇ x^i(t ) = (A + ΔA i )x^i(t ) + Bi(z^i(t ) + κ (
∑
i i
i i
a ij(y^j (t ) − y^i (t )) − pi y^i (t )),
j ∈ Ni
̇ z^i(t ) = γ (
∑
Therefore, the roots of (21) all lie on the open left half complex plane, i.e., the agents in (16) converge to a scaled consensus problem asymptotically. □ It should be highlighted that the roots’ distribution of equation (19) with complex coefficients can be determined by the Routhlike criterion [35].
u i (t ) = z i(t ) + κξiτ(t ),
j ∈ Ni
yi (t ) = cx i(t ).
0
i∈V
< 1.
With non-negligible communication delay, the asynchronously-coupled form of algorithm (8) is
z i̇ (t ) = γξi(t ), 1 ( ri
^ = {max|Δdi(jω)|}σ¯ (jω(jωd(jω)I + (jκω + γ ) L )−1)
4. Scaled consensus with communication delay
namics (9) are reformulated as
ξi(t ) =
^ σ¯ (jωΛ(jω)(jωd(jω)I + (jκω + γ ) L )−1) ^ ≤ σ¯ (Λ(jω))σ¯ (jω(jωd(jω)I + (jκω + γ ) L )−1)
a ij(y^j (t ) − y^i (t )) − pi y^i (t )),
j ∈ Ni
y^i (t ) = cx^i(t ).
(17)
Then, the characteristic equation y^ (t ) = [y^1(t ) , … , y^n (t )]T is given by
of
system
(17)
z i̇ (t ) = γξiτ(t ), ξiτ(t ) =
1 ri
∑
a ij (r j yj (t − τ ) − ri yi (t )) + pi ( y0 − ri yi (t )),
(22)
j ∈ Ni
where τ > 0 is the communication delay. Since the leader's state tracked by the agents (1) is stationary, the asynchronously-coupled algorithm (22) does not change the final consensus behavior [9]. With (22), the closed-loop dynamics of agents (1) are formulated as
xi̇ (t ) = Ai xi + Bi(ni + zi(t ) + κξiτ (t )), zi̇ (t ) = γξiτ (t ).
(23)
about 4.1. Delay-independent case
^ det(diag{s(d(s ) + Δdi(s )), i ∈ V} + (κs + γ ) L ) = 0,
(18)
where Δdi(s ) = di(s ) − d(s ).
Theorem 3. Consider multi-agent system (23), of which the directed topology has the leader as a globally reachable node. Assume that the roots of following equations
Theorem 2. Consider multi-agent system (16) with a directed topology that has the leader as a globally reachable node. Suppose that the roots of following equations
^ −1 sdi(s )bi−1di + (κs + γ ) = 0
sd(s ) + λ^i(κs + γ ) = 0, i = 1, …, N
(19)
lie on the open left half complex plane, where λ^i , i ∈ V is the eigenvalue of L^ . Then, the agents in system (16) converge to the scaled consensus asymptotically, if
|Δdi(jω)| <
1 ^ σ¯ (jω(jωd(jω)I + (jκω + γ ) L )−1)
^ all lie on the open left half complex plane, where di = ∑j ∈ N aij + pi . i
Then, all the agents in system (23) asymptotically converge to the scaled consensus asymptotically, if
|jωκ + γ |σ¯ ( −1(D + P )−1 ) < 1, i ∈ V ^ −1 |jωdi(jω)bi−1di + (κ jω + γ )|
(25)
holds for ∀ ω ∈ R .
,∀i∈V (20)
hold for ω ∈ R , where σ¯(·) denotes the largest singular value of matrix. Proof. From the assumption in Theorem 2, the equation (18) is rewritten as
^ det(I + sΛ(s )(sd(s )I + (κs + γ ) L )−1) = 0,
(24)
(21)
where Λ(s ) = diag{Δdi(s ) , i ∈ V}. It follows from condition (20) that
Proof. Similar to the proof of Theorem 1, we take variable transα formations as z^ (t ) = − i0 y + rn + rz (t ), y^ (t ) = ry (t ) − y , and i
bi 0
i i
i i
i
i i
0
T x^i(t ) = rx i i(t ) − x 0 with x 0 = [ y0 , 0, … , 0] , and get
̇ x^ i(t ) = A i x^ i(t ) + Bi(z^i(t ) + κ (
∑
a ij( y^j (t − τ ) − y^i (t )) − pi y^i (t )) ,
j ∈ Ni
̇ z^i(t ) = γ (
∑
a ij( y^j (t ) − y^i (t − τ )) − pi y^i (t )) ,
j ∈ Ni
y^i (t ) = cx^ i(t ) .
(26)
Please cite this article as: Liu C-L. Scaled consensus seeking in multiple non-identical linear autonomous agents. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.022i
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Taking the Laplace transform of system (26), we get the characteristic equation on y^ (t ) = [y^1(t ) , … , y^N (t )]T as
det(diag{sdi(s ), i ∈ V} + (κs + γ ) (D + P − e−sτ )) = 0.
(27)
Under the assumption in Theorem 3, (27) is equivalent to
det(I − M (s )) = 0,
(28)
where M(s) has no poles on the open right half complex plane. It follows from Lemma 1 and condition (25) that
ρ(M (jω)) jωκ + γ , i ∈ V} ^ −1 jωdi(jω)bi−1di + (κ jω + γ ) jωκ + γ ^ jωdi(jω)bi−1di
+ (κ jω + γ )
σ¯ ( −1(D + P )−1 ) < 1.
(29)
Intuitively, if the coupling weights aij between agents themselves are smaller and the coupling weights pi between the agents and the leader are higher, σ¯( −1(D + P )−1 ) turns to be smaller, i.e., the condition (25) in Theorem 3 hold more easily. Remark 2. Take into account a simplest case that the interconnection topology of N agents and a leader in system (23) has the leader as a globally reachable node, and each agent just has one direct path to reach the leader. Besides, we assume that the direct edge from agent i to j satisfies i > j , and we get
⋯ ⋯ ⋯ ⋯
0⎤ ⎥ 0⎥ . ⋮⎥ ⎥ 0⎦
Since the eigenvalues of adjacent matrix are all equivalent to zero, ρ(M (jω)) = ρ(diag{
jωκ + γ , ^ −1 jωdi( jω)b −1di + (κ jω + γ ) i
2|sin(
ωτ ^ )|σ¯ (M (jω)) < 1 2
(31)
Proof. With the supposition that the agents without communication delay converge to the scaled consensus asymptotically, the roots of following equation
^ det(diag{sdi(s ), i ∈ V} + (κs + γ ) L ) = 0
(32)
(33)
which equals
Therefore, the roots of Eq. (27) lie on the open left half complex plane, i.e., the agents in system (23) converge to the scaled consensus asymptotically. □
⎡ 0 0 ⎢ a 0 = ⎢ 21 ⎢ ⋮ ⋮ ⎢ ⎣ a n1 a n2
Then, all the agents in system (23) achieve an asymptotic scaled consensus, if
^ det(diag{sdi(s ), i ∈ V} + (κs + γ ) (L + (1 − e−sτ ))) = 0,
, i ∈ V})
−1
(30)
^ all lie on the open left half complex plane, i.e., M (s ) has its poles on ^ the open left half complex plane, so rank(L ) = N . In the same way as Theorem 3, the characteristic equation (27) is also rewritten as
−1(D + P )−1 e−jωτ ) ≤ σ¯ (diag{
(κs + γ ) . ^ diag{sdi(s ), i ∈ V} + (κs + γ ) L
holds with ω ∈ R .
(κs + γ ) M (s ) = diag{ , i ∈ V} −1(D + P )−1 e−sτ , ^ −1 sdi(s )bi−1di + (κs + γ )
≤ σ¯ (diag{
^ M (s ) =
5
i ∈ V} −1(D + P )−1 e−jωτ ) = 0 < 1
^ det(I + M (s )(1 − e−sτ )) = 0.
(34)
Then, we obtain from the condition (31) that
^ ρ(M (jω)(1 − e−jωτ )) ^ ≤ σ¯ (M (jω)(1 − e−jωτ )) ^ = |1 − e−jωτ |σ¯ (M (jω)) ωτ ^ < 2|sin( )|σ¯ (M (jω)) 2 < 1. Hence, the equation (34) has its roots on the open left half complex plane, i.e., the agents in system (23) converge to the scaled consensus asymptotically. □ Furthermore, we present a concrete delay bound for the achievement of asymptotic scaled consensus seeking. Corollary 1. It is assumed that the agents in system (23) without communication delay converges to the scaled consensus asymptotically. Then, all the agents in system (23) reach an asymptotic scaled consensus, if
τ<
1 ^ σ¯ ( jωM (jω))
holds with arbitrary communication delay if the assumption in Theorem 3 holds. Thus, asymptotic scaled consensus convergence in this case is delay-independent.
^ holds with ω ∈ R , where M (jω) is defined in (30).
It should be pointed out that the delay-independent scaled consensus condition in Theorem 3 is relatively conservative, and does not hold for arbitrary topology, coupling weights and control parameters according to Remark 3, Remark 5 and Remark 7 in [29].
5. Numerical simulation
(35)
5.1. Scaled consensus convergence without time delay
4.2. Delay-dependent case
Investigate a heterogeneous second-order multi-agent network given by
In this section, we will present a delay-dependent condition that is suitable for an arbitrary connected leader-following topology.
⎡ 0⎤ ⎡ 0 1 ⎤ ẋi (t ) = ⎢ x (t ) + ⎢ ⎥ (ui (t ) + ni ), ⎣ −αi0 −αi1⎦⎥ i ⎣ 1⎦ y (t ) = ⎣⎡ 1 0⎦⎤ xi (t ),
Theorem 4. Suppose that the agents in system (23) without communication delay converges to the scaled consensus asymptotically. Let
and we consider the agents (36) driven by the algorithm (8) under an undirected topology shown in Fig. 1 firstly. The undirected topology is connected and the leader is the globally reachable node.
i
(36)
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Fig. 1. Undirected topology of five agents and a leader.
The agents’ parameters are α10 = 1.5, α11 = 0.5, α20 = 1.6, α21 = 0.6, α30 = 1.7, α31 = 0.7, α40 = 1.8, α41 = 0.8, α50 = 1.9, α51 = 0.9 , b1= 0.8, b2 = 1.2, b3 = 0.5, b4 = 1, b5 = 1.5 , and the constant disturbances are n1 = 0.5, n2 = 0.3, n3 = 0.7, n4 = 1.0, n5 = − 0.2. The weights with respect to the edges are a12 = a21 = 0.5, a14 = a41 = 0.5, a23 =a32 = 0.4, a34 = a43 = 0.2, a45 = a54 = 0.3, p1 = 0.5, p5 = 0.2. . The scaled ratios for each agent are chosen as r1 = 0.3, r2 = 0.4, r3 = − 0.2, r4 = 0.5, r5 = − 0.4 , and the control parameters are set as κ = 3, γ = 1.2. Besides, the static leader's output is y0 = 1. In y addition, we define ei = ⎣⎡ 1 0⎦⎤x i − 0 as the errors of agents’ states. ri
Firstly, we take into account the usual scaled consensus algorithm (5). With the above coupling weights and control parameters of agents (36), the condition (7) is not satisfied, so the algorithm (5) cannot drive the agents (36) to achieve the assigned scaled consensus seeking (see Fig. 2). Then, we study the agents (36) driven by our proposed algorithm (8). Now, we analyze the Nyquist curves of
gi( jω) =
jκω + γ jω(( jω)2 + αi1( jω) + αi0)
γ κ
κ (jω + )
=
jω( jω + si )( jω + 1)
with the above assigned
parameters, and s1 = 0.5, s2 = 0.6, s3 = 0.7, s4 = 0.8, s5 = 0.9. Since γ = 0.4 < si , i = 1, 2, … , 5, we get arg(gi(jω)) ∈ ( − 2π , − π ), i.e., the κ curves of gi(jω) just lie in the third and fourth quadrants (see Fig. 3). In Theorem 1, Γ is an infinitesimal positive number, and it is concluded from Theorem 1 that the agents (36) with (8) converge to the scaled consensus asymptotically (see Fig. 4). Next, we study the scaled consensus seeking of second-order agents with a directed and asymmetric topology shown in Fig. 5. The adjacent weights are chosen as: a13 = 0.4, a21 = 0.5, a32 = 0.4, a34 = 0.2, a45 = 0.3, a51 = 0.5, p1 = 0.2 , and the eigenva^ lues of λ1 = 1.232, λ2 = 0.7334 + j0.0991, λ3 = 0.7334 L are
Fig. 3. Nyquist curves of gi(jω) . 0 1 −j0.0991, λ4 = 0.1519, λ5 = 0.3792 . Then, we set A = ⎡⎣ −1 −2 ⎤⎦, i.e.,
d(s ) = s2 + 2 s + 1, and other parameters are same as the above, and we will design ΔAi for each agent. For simplicity, we just in⎡ 0 0⎤ vestigate the case that ΔAi = ⎣⎢ −Δa 0 ⎦⎥, i.e., Δdi(s ) = Δαi0 . i0 According to the Routh-like criterion, the roots of Eq. (19) are proved to be lie on the open left half complex plane, and we obtain σ¯(jω(jωd(jω)I from the numerical simulation that ^ +(jκω + γ ) L )−1) < 1.0136 hold for ω ∈ R . Based on the condition 1
(20), we get |Δdi(jω)| = |Δαi0| < 1.0136 ≃ 0.9866, i = 1, 2, 3, 4, 5, so we choose Δα10 = − 0.2, Δα20 = 0.3, Δα30 = 0.2, Δα40 = − 0.3, Δα50 = 0.8 . Therefore, the agents (36) with (8) converge to the scaled consensus asymptotically (see Fig. 6). 5.2. Scaled consensus convergence with time delay Remark 2 has demonstrated a delay-independent scaled consensus seeking case based on Theorem 3, so this section just cares about the delay-dependent scaled consensus convergence in Theorem 4 and Corollary 1. Study second-order multi-agent system (36) with delayed scaled consensus algorithm (22) under the same directed and asymmetric topology shown in Fig. 5, and we choose the same
Fig. 2. Convergence under usual scaled consensus algorithm with undirected topology.
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Fig. 4. Scaled consensus convergence under our proposed algorithm with undirected topology.
Fig. 5. Directed topology.
coupling weights, agents’ parameters αi0, αi1, bi , scaled gains ri, and control parameters κ , γ as those of the directed topology case in above subsection. By numerical computing, then, we get ^ σ¯( jωM (jω)) < 2.044 for ω ∈ R , so the time delay bound is τ < 0.4892(s) from condition (35), i.e., the agents in system (36) with (22) converge to the scaled consensus asymptotically if τ < 0.4892(s) (see Fig. 7 with τ = 0.3(s)).
6. Conclusion In this paper, we propose an adaptive scaled consensus algorithm, which is in the leader-following form, to solve the scaled
consensus problem of a class of heterogeneous linear stable multiple agents with distinct constant disturbances. Based on frequency-domain analysis, we obtain the scaled consensus criteria for the multi-agent systems under the undirected and directed topologies, respectively. Moreover, the scaled consensus seeking is also analyzed for the multi-agent systems with communication delay under the general directed topology. With proper topologies and parameters, the asymptotic scaled consensus convergence can be delay-independent. Furthermore, the delay-dependent consensus condition, which is less conservative, is also provided for the agents converging to the scaled consensus asymptotically under general directed topology. Compared with current results on scaled consensus problem [23–26], we focus on a heterogeneous linear multi-agent system, which is relatively complicated, and scaled consensus algorithm is designed and analyzed for the agents with diverse disturbances and communication delay. However, the results herein are just for the agents under fixed topology and time-invariant identical communication delay. In our future work, thus, we will propose some proper analysis methods to gain the convergence conditions for the heterogeneous multiagent systems with time-varying disturbances, switching topologies and time-varying communication delays.
Fig. 6. Scaled consensus convergence under our proposed algorithm with directed topology.
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Fig. 7. Delay-dependent scaled consensus seeking under our proposed algorithm.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 61473138 and 61104092), Natural Science Foundation of Jiangsu Province (Grant No. BK20151130), Six Talent Peaks Project in Jiangsu Province (Grant no. 2015-DZXX011), and Fundamental Research Funds for the Central Universities (Grant No. JUSRP51407B).
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Research article
Scaled consensus seeking in multiple non-identical linear autonomous agents Cheng-Lin Liu Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, Wuxi 214122, China
art ic l e i nf o
a b s t r a c t
Article history: Received 12 November 2016 Received in revised form 21 June 2017 Accepted 22 June 2017
Scaled consensus problem is studied for a heterogeneous multi-agent system composed of non-identical stable linear agents, and a leader-following scaled consensus algorithm is designed. By using frequencydomain analysis, consensus conditions are obtained for the agents without communication delay under undirected and directed topologies, respectively. Moreover, consensus criteria are also gained for the agents suffering from communication delay under directed topology. Simulation examples show the correctness of theoretical results. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Heterogeneous linear multi-agent systems Scaled consensus problem Leader-following algorithm Communication delay
1. Introduction As a fundamental phenomenon in flocking, swarming, and coupled synchronization, consensus behavior has attracted more and more attention of researchers in the control theory field in recent years. Consensus problem requires several autonomous agents to reach a common agreement on their states, and has broad engineering applications, e.g., coordination control of unmanned aerial and ground vehicles, smart grid, wireless sensor networks, etc. Current research results on consensus problem have been mainly focused on homogeneous multi-agent systems, of which the agent's dynamics are identical [1–10]. With the help of algebraic graph theory [1], matrix theory [2–6], frequency-domain analysis [7–9], and Lyapunov functions [10], various consensus algorithms have been designed and analyzed for the agents with fixed topology or switching topologies. Over the past few years, furthermore, a lot of research effort has also been put into the consensus analysis and synthesis of heterogeneous multi-agent system composed of the agents with distinct dynamics. As a simple heterogeneous multi-agent system, mixed system with multi-class agents has attracted many researchers’ interests, such as the system of first-order agents and second-order agents [11,12] and the system of Euler-Lagrange agents and doubleintegrator agents [13,14]. For the heterogeneous linear multi-agent systems, state-feedback and output-feedback consensus algorithms were proposed by using the internal model principle [16,17] and the harmonic control [18]. To solve the consensus problem of heterogeneous linear multi-agent system with diverse communication delays, Lee and Spong [15] used the spectral radius theorem to obtain E-mail address: [email protected]
the delay-independent consensus conditions. Meanwhile, Muenz [19] and Tian and Zhang [20] adopted the consensus algorithms with devisable self delays distinct from the corresponding communication delays, and provided the consensus criteria based on frequency-domain analysis. In addition, Tian and Zhang [20] designed an adaptive adjustment algorithm in order to adjust the self delays on-line if the communication delays were unknown. Liu and Liu [21] proposed three adaptive consensus algorithms for the heterogeneous first-order time-invariant multi-agent systems, and obtained the consensus conditions for the agents under fixed topology without and with time delays respectively. Specially, non-identical disturbances also lead to the heterogeneity of multi-agent systems [22]. Based on Lyapunov functions, Kim et al. [22] adopted a general consensus algorithm and analyzed the robust consensus convergence condition For the heterogeneous first-order time-varying multi-agent systems with distinct disturbances. Treated as the generalized consensus problem, scaled consensus problem defined in [23] means that the network components’ scalar states reach the assigned proportions rather than a common value, and has attracted some researchers’ interests for its engineering applications, such as compartmental mass-action systems, closed queueing networks, and water distribution systems [23]. In addition to usual consensus problem, group consensus problem [24], wherein the agents form several sub-groups reach corresponding agreement values respectively, is also a special case of scaled consensus problem. With a connected topology, Roy [23] proved that the first-order multi-agent systems not only achieved a stationary scaled consensus asymptotically, but also could track on the stable manifold. For the first-order multi-agent systems, Meng and Jia [25] studied the scaled consensus problem with time-varying scaled ratios, and demonstrated that the agents converged to the scaled consensus asymptotically under jointly-connected switching
http://dx.doi.org/10.1016/j.isatra.2017.06.022 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Liu C-L. Scaled consensus seeking in multiple non-identical linear autonomous agents. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.022i
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topologies. By constructing Lyapunov-Krasovskii functionals, Aghbolagh et al. [26] considered the scaled consensus of first-order multi-agent systems with time-varying delay, and gained the consensus conditions in the form of linear matrix inequalities, which were used to calculate the allowable delay value. In this paper, we consider the scaled consensus problem of a class of heterogeneous multi-agent systems with the agents modeled by single-input and single output linear stable dynamics, and an adaptive scaled consensus algorithm is constructed in the leader-following structure. Based on frequency-domain analysis, firstly, we obtain the consensus conditions for the multi-agent systems without communication delay under undirected and directed topologies, respectively. For the multi-agent systems with identical communication delay, besides, the delay-independent and delay-dependent consensus conditions are obtained, respectively, for the system under directed topology.
2. Problem description
lim (ri yi (t ) − r j yj (t )) = 0, i, j = 1, 2, …, N ,
t →∞
(4)
where the ratios ri ∈ R, i = 1, … , N are assumed to be non-zero. If ri = r j, ∀ i, j ∈ {1, … , N}, the output scaled consensus in Definition 1 becomes an output synchronization. Hence, the output scaled consensus problem (4) is a generalized output synchronization problem. Usually, the interconnection topology of multi-agent system (1) is described as an N-order weighted digraph G = (V , E, ), which is composed of a set of vertices V = {1, … , N}, a set of edges E ⊆ V × V and a weighted adjacency matrix = [aij ] ∈ R N × N with aij ≥ 0. In the diagraph G, a directed edge from the node i to the node j is denoted by eij = (i, j ) ∈ E , and it is assumed that aij > 0 ⇔ eij ∈ E . Moreover, we assume aii = 0 for all i ∈ V . Besides, the digraph G is undirected graph or bidirectional if eij ⇔ eji , and the digraph is symmetric if aij = aji . The set of node i's neighbors is denoted by Ni = {j ∈ V : (i, j ) ∈ E}. The Laplacian matrix of the weighted digraph G is defined as L = D − = [lij ] ∈ Rn × n , where N
2.1. Non-identical agents and topology Consider a class of heterogeneous multi-agent system composed of N non-identical single-input single-output linear agents given by
ẋi (t ) = Ai xi (t ) + Bi(ui (t ) + ni ), yi (t ) = cxi(t ), i = 1, …, N ,
(1)
D = diag{ ∑ j = 1 aij , i ∈ V} is the degree matrix. In the digraph, if there is a path from one node i to another node j, then j is said to be reachable from i, otherwise, j is said to be not reachable from i. If a node is reachable from every other node in the digraph, then we say it globally reachable. A globally reachable node is precisely the degree of connectedness required of the digraph. An undirected graph is connected if it has a globally reachable node. 2.2. Design of scaled consensus algorithm
⎡ 0 1 0 ⎢ 0 1 ⎢ 0 Ai = ⎢ ⋮ ⋮ ⋮ ⎢ 0 0 0 ⎢⎣ −α −α −α i0 i1 i2
⋯ ⋯ ⋯ ⋯ ⋯
⎡ 0⎤ 0 ⎤ ⎢ ⎥ ⎥ 0 ⎥ ⎢ 0⎥ ⋮ ⎥, Bi = ⎢ ⋮ ⎥, ⎢ 0⎥ 1 ⎥ ⎢ ⎥ −αin − 1⎥⎦ ⎣ bi ⎦
In this paper, we focus on the leader-following coordination control, which has been extensively studied for multi-agent systems [27–31]. For the agents (1), a usual scaled consensus algorithm is given by
c = ⎡⎣ 1 0 0 ⋯ 0⎤⎦.
ξi(t ) =
where x i(t ) ∈ Rn , yi (t ) ∈ R and ui(t ) ∈ R are the state, the output and the input of agent i, respectively, ni ∈ R is the constant disturbance of each agent i, Ai ∈ Rn × n , Bi ∈ Rn , c ∈ R1 × n , bi ∈ R , and αik ∈ R, k = 0, … , n − 1. Correspondingly, the frequency-domain description of (1) is formulated as
Yi(s ) = gi (s )(Ui(s ) + Ni(s )), i = 1, …, N ,
(2)
where Yi(s), Ui(s) and Ni(s) are the Laplace transforms of yi(t), ui(t) and ni respectively, and gi(s) is the transfer function. Since the agents’ dynamics (1) are formulated as a controllable canonical form, we get
gi (s ) = =
bi di(s )
⎛ 1⎜ ri ⎜⎝
⎞ aij(r j yj (t ) − ri yi (t )) + pi ( y0 − ri yi (t ))⎟⎟, i ∈ V , ⎠ j ∈ Ni
∑
where the non-zero constants ri, i ∈ V denotes the ratios of agents’ states, Ni denotes the neighbors of agent i, aij > 0 is the adjacency element of in the digraph G = (V , E, ), y0 is the static leader's state, and pi denotes the linking weight from agent i to the leader. Assume that pi > 0 if there is a directed edge from agent i to the leader; otherwise, pi ¼0. In the rest of this paper, the notation P = diag{pi , i ∈ V} is used. Apparently, the algorithm (1) is similar to that in [25]. Taking ui(t ) = κξi(t ) with κ > 0, we get the closed-loop form of multi-agent system (1) as follows
ẋi (t ) = Ai xi (t ) + Bi(κξi(t ) + ni ), i ∈ V .
bi s n + αin − 1s n − 1 + ⋯ + αi1s + αi0
. (3)
Assumption 1. The system matrix Ai , i = 1, … , N is Hurwitz, i.e., di(s ) , i = 1, … , N is Hurwitz. Evidently, Ai is non-singular with Assumption 1 guaranteeing that all the eigenvalues of Ai are non-zero. Assumption 2. Without loss of generality, bi is assumed to be positive, i.e., bi > 0. Definition 1. The multi-agent network (1) achieves Scaled Consensus asymptotically, if
(5)
(6)
Remark 1. With the interconnection topology that has a globally reachable node, it is easily proved that the multi-agent system (6) converges to an asymptotic scaled consensus, only if −1 rcA i i Bini = y0 , ∀ i ∈ V .
(7)
Condition (7) can be obtained by assuming that the agents converge to the scaled consensus asymptotically, i.e., limt →∞ri yi (t ) = y0 . Evidently, the condition (7) is dependent on the parameters strictly, i.e., the agents cannot achieve the asymptotic scaled consensus when the condition (7) is not satisfied. Moreover, (7) also implies that the agents (1) without disturbances cannot reach the asymptotic scaled consensus with the algorithm (5).
Please cite this article as: Liu C-L. Scaled consensus seeking in multiple non-identical linear autonomous agents. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.022i
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To achieve the asymptotic scaled consensus and cancel the impact of the diversity of the disturbances ni, we design the following adaptive algorithm for multi-agent system (1),
ui (t ) = zi(t ) + κξi(t ), zi̇ (t ) = γξi(t ),
3
̇ x^ i (t ) = A i x^ i (t ) + Bi(z^i(t ) + κ (
a ij(y^j (t ) − y^i (t )) − pi y^i (t )) ,
∑ j ∈ Ni
̇ z^i(t ) = γ (
a ij(y^j (t ) − y^i (t )) − pi y^i (t )) ,
∑ j ∈ Ni
(8)
where κ > 0, γ > 0, and zi(t) is an additional variable of consensus controller ui(t). Furthermore, the algorithm (8) is a proportionalintegral form of algorithm (5).
y^i (t ) = cx^ i (t ) .
(11)
Moreover, the scaled consensus problem of system (9) turns to be an asymptotic stability problem of system (11). Taking the Laplace transform of system (11) yields ^ ^ ^ sX i (s) = A i X i (s) + Bi(Z i(s) + κ (
2.3. Useful lemmas
^ ^ ^ a ij(Yj(s) − Yi (s)) − pi Yi (s)),
∑ j ∈ Ni
For the leader-following control structure, we come to the following property from Lemma 2 in [32], Lemma 2 in [28], Lemma 3 in [27] and Lemma 1 in [21]. Lemma 1. uppose that the interconnection topology of N agents (3) and the static leader y0 has the leader as a globally reachable node. The eigenvalues of L + P are all none-zero, and D + P > μI , where μ > 0 and L is the Laplacian matrix of the topology of N agents (3) without leader. In addition, the following two lemmas will play important roles in the proof of main results. Lemma 2. [33] Gr (s ) ∈ R(s )N × N is proper, where N is a positive integrator. The zeros of Δ(s ) = det(I + Gr (s )) lie on the open left half complex plane if and only if the number of counterclockwise encirclements of ( − 1, j0) by the eigenloci of Gr (jω) as ω goes from −∞ to +∞ equals the total number of right-half plane poles of Gr(s).
^ sZ i(s) = γ (
∑
^ ^ ^ a ij(Yj(s) − Yi (s)) − pi Yi (s)),
j ∈ Ni
^ ^ Yi (s) = cX i (s),
(12)
^ ^ ^ where Xi(s ), Zi(s ) and Yi (s ) are the Laplace transforms of x^i(t ), z^i(t ) and y^ (t ) respectively. Thus, the characteristic equation of system (11) about y^ (t ) = [y^ (t ) , … , y^ (t )]T is given by 1
N
^ det(diag{sdi(s ), i ∈ V} + (κs + γ ) L ) = 0,
(13)
where = diag{bi , i ∈ V} and L^ = L + P . 3.1. Undirected topology At first, we provide the scaled consensus condition for multiagent system (9) with an undirected topology.
Lemma 3. [34] Let Q ∈ C n × n , Q = Q⋆ ≥ 0 and T = diag{ti, ti ∈ C}. Then
Theorem 1. Consider multi-agent system (9) with an undirected and symmetric topology that has the leader as a globally reachable node. Under Assumptions 1 and 2, then, the agents in system (9) converge to a scaled consensus asymptotically, if
λ(QT ) ∈ ρ(Q )Co(0 ∪ {ti}),
2bi (max( i∈V
where λ(·) denotes matrix eigenvalue, ρ(·) denotes the matrix spectral radius, and Co(·) denotes the convex hull.
∑
aij + pi ))Γ < 1,
(14)
j ∈ Ni
where Γ > 0 is the positive minimum value that satisfies jκω + γ ( − Γ , j0) ∉ Co(0 ∪ gi(jω) , i ∈ V ) with ω ∈ R , and gi(jω) = jωd (jω) . i
^ Proof. Let f (s ) = det(diag{sdi(s ) , i ∈ V} + (κs + γ ) L ), and ^ f (0) = det(diag{0(0n + αin − 10n − 1 + ⋯ + αi0) , i ∈ V} + (κ 0 + γ ) L ) ^ =det(γ L ) ≠ 0 holds from Lemma 1.
3. Scaled consensus seeking without time delay With (8), the dynamics of agents (1) are rewritten as
When s ≠ 0, the equation (13) is equivalent to
ẋi(t ) = A i x i(t ) + Bi(n i + z i(t ) + κξi(t )), z i̇ (t ) = γξi(t ), 1 ( ri
ξi(t ) =
∑
det(I + diag{ a ij (r j yj (t ) − ri yi (t )) + pi ( y0 − ri yi (t )), i ∈ V ,
j ∈ Ni
yi (t ) = cx i(t ).
(9)
Let y¯i (t ) = ri yi (t ) and x¯ i(t ) = rx i i(t ), and we get
aij( y¯j (t ) − y¯i (t )) + pi ( y0 − y¯i (t ), i ∈ V , (10)
Hence, the scaled consensus problem of system (9) becomes a consensus problem of system (10). Define y^i (t ) = y¯i (t ) − y0 , x^i(t ) = x¯ i(t ) − x0 with x0 = [y0 , 0, … , 0]T α and z^ (t ) = − i0 y + rn + rz (t ), and we obtain the dynamics of x^ i
as follows
bi 0
i i
i i
i
tion 1 and Lemma 2, the zeros of h(s) are on the open left half jκω + γ complex plane, if λ(diag{ jωd ( jω) , i ∈ V} L^ ) does not enclose the
λ(diag{
j ∈ Ni
y¯i (t ) = cx¯ i (t ).
κs + γ Denote h(s ) = det(I + diag{ sd (s) , i ∈ V} L^ ). According to Assump-
point ( − 1, j0) for ω ∈ R . The definition of Laplacian matrix and ^ ^T symmetric weights make L = L > 0 hold. Based on Assumption 2 and Lemma 3, we get
1 zi̇ (t ) = γ ξ¯i(t ), ri
∑
(15)
i
¯ x¯ i̇ (t ) = Ai x¯ i (t ) + Bi(rn i i + rz i i(t ) + κξi(t )),
ξ¯i(t ) =
κs + γ ^ , i ∈ V} L ) = 0. sdi(s )
i
jκω + γ ^ , i ∈ V } L ) jωdi(jω)
^ = λ(diag{gi (jω), i ∈ V}diag{ bi , i ∈ V}L diag{ bi , i ∈ V}) ^ ∈ ρ(diag{ bi , i ∈ V}L diag{ bi , i ∈ V})Co(0 ∪ gi (jω), i ∈ V ) ⊂ 2bi (max( i∈V
∑
aij + pi ))ΓΓ −1Co(0 ∪ gi (jω), i ∈ V ).
j ∈ Ni
Thus, gi(jω) , i ∈ V crosses the negative real axis at some certain
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Γ must exist from the
frequencies under Assumption 1, and
definition. For the condition (14), hence, ρ(diag{ bi , i ∈ V} ^ L diag{ bi , i ∈ V})Co(0 ∪ gi(jω) , i ∈ V ) does not enclose the point ( − 1, j0), and the zeros of h(s) are on the open left half plane. Now, we have proved that the zeros of f(s) all have negative real parts, i.e., lim x^ (t ) = 0, i ∈ V . Therefore, the agents in (9) converge to a t →∞ i
□
scaled consensus asymptotically.
3.2. General directed topology Now, we analyze the scaled consensus seeking of system (9) with general directed topology. n 1 Define Ai = A + ΔAi , where A = n ∑i = 1 Ai , and the agents’ dyx i̇ (t ) = (A + ΔA i )x i(t ) + Bi(n i + z i(t ) + κξi(t )),
∑
a ij(r j yj (t ) − ri yi (t )) + pi ( y0 − ri yi (t )), i ∈ V ,
(16)
In (16), the characteristic polynomial of A is d(s ) = sn + αn − 1sn − 1 + ⋯ + α1s + α0 . ^ In the same way, let y^i (t ) = ry i i(t ) − x 0 with i i (t ) − y0 , x i(t ) = rx α x = [ y , 0, … , 0]T and z^ (t ) = − i0 y + rn + rz (t ), and we get i
0
bi 0
̇ x^i(t ) = (A + ΔA i )x^i(t ) + Bi(z^i(t ) + κ (
∑
i i
i i
a ij(y^j (t ) − y^i (t )) − pi y^i (t )),
j ∈ Ni
̇ z^i(t ) = γ (
∑
Therefore, the roots of (21) all lie on the open left half complex plane, i.e., the agents in (16) converge to a scaled consensus problem asymptotically. □ It should be highlighted that the roots’ distribution of equation (19) with complex coefficients can be determined by the Routhlike criterion [35].
u i (t ) = z i(t ) + κξiτ(t ),
j ∈ Ni
yi (t ) = cx i(t ).
0
i∈V
< 1.
With non-negligible communication delay, the asynchronously-coupled form of algorithm (8) is
z i̇ (t ) = γξi(t ), 1 ( ri
^ = {max|Δdi(jω)|}σ¯ (jω(jωd(jω)I + (jκω + γ ) L )−1)
4. Scaled consensus with communication delay
namics (9) are reformulated as
ξi(t ) =
^ σ¯ (jωΛ(jω)(jωd(jω)I + (jκω + γ ) L )−1) ^ ≤ σ¯ (Λ(jω))σ¯ (jω(jωd(jω)I + (jκω + γ ) L )−1)
a ij(y^j (t ) − y^i (t )) − pi y^i (t )),
j ∈ Ni
y^i (t ) = cx^i(t ).
(17)
Then, the characteristic equation y^ (t ) = [y^1(t ) , … , y^n (t )]T is given by
of
system
(17)
z i̇ (t ) = γξiτ(t ), ξiτ(t ) =
1 ri
∑
a ij (r j yj (t − τ ) − ri yi (t )) + pi ( y0 − ri yi (t )),
(22)
j ∈ Ni
where τ > 0 is the communication delay. Since the leader's state tracked by the agents (1) is stationary, the asynchronously-coupled algorithm (22) does not change the final consensus behavior [9]. With (22), the closed-loop dynamics of agents (1) are formulated as
xi̇ (t ) = Ai xi + Bi(ni + zi(t ) + κξiτ (t )), zi̇ (t ) = γξiτ (t ).
(23)
about 4.1. Delay-independent case
^ det(diag{s(d(s ) + Δdi(s )), i ∈ V} + (κs + γ ) L ) = 0,
(18)
where Δdi(s ) = di(s ) − d(s ).
Theorem 3. Consider multi-agent system (23), of which the directed topology has the leader as a globally reachable node. Assume that the roots of following equations
Theorem 2. Consider multi-agent system (16) with a directed topology that has the leader as a globally reachable node. Suppose that the roots of following equations
^ −1 sdi(s )bi−1di + (κs + γ ) = 0
sd(s ) + λ^i(κs + γ ) = 0, i = 1, …, N
(19)
lie on the open left half complex plane, where λ^i , i ∈ V is the eigenvalue of L^ . Then, the agents in system (16) converge to the scaled consensus asymptotically, if
|Δdi(jω)| <
1 ^ σ¯ (jω(jωd(jω)I + (jκω + γ ) L )−1)
^ all lie on the open left half complex plane, where di = ∑j ∈ N aij + pi . i
Then, all the agents in system (23) asymptotically converge to the scaled consensus asymptotically, if
|jωκ + γ |σ¯ ( −1(D + P )−1 ) < 1, i ∈ V ^ −1 |jωdi(jω)bi−1di + (κ jω + γ )|
(25)
holds for ∀ ω ∈ R .
,∀i∈V (20)
hold for ω ∈ R , where σ¯(·) denotes the largest singular value of matrix. Proof. From the assumption in Theorem 2, the equation (18) is rewritten as
^ det(I + sΛ(s )(sd(s )I + (κs + γ ) L )−1) = 0,
(24)
(21)
where Λ(s ) = diag{Δdi(s ) , i ∈ V}. It follows from condition (20) that
Proof. Similar to the proof of Theorem 1, we take variable transα formations as z^ (t ) = − i0 y + rn + rz (t ), y^ (t ) = ry (t ) − y , and i
bi 0
i i
i i
i
i i
0
T x^i(t ) = rx i i(t ) − x 0 with x 0 = [ y0 , 0, … , 0] , and get
̇ x^ i(t ) = A i x^ i(t ) + Bi(z^i(t ) + κ (
∑
a ij( y^j (t − τ ) − y^i (t )) − pi y^i (t )) ,
j ∈ Ni
̇ z^i(t ) = γ (
∑
a ij( y^j (t ) − y^i (t − τ )) − pi y^i (t )) ,
j ∈ Ni
y^i (t ) = cx^ i(t ) .
(26)
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Taking the Laplace transform of system (26), we get the characteristic equation on y^ (t ) = [y^1(t ) , … , y^N (t )]T as
det(diag{sdi(s ), i ∈ V} + (κs + γ ) (D + P − e−sτ )) = 0.
(27)
Under the assumption in Theorem 3, (27) is equivalent to
det(I − M (s )) = 0,
(28)
where M(s) has no poles on the open right half complex plane. It follows from Lemma 1 and condition (25) that
ρ(M (jω)) jωκ + γ , i ∈ V} ^ −1 jωdi(jω)bi−1di + (κ jω + γ ) jωκ + γ ^ jωdi(jω)bi−1di
+ (κ jω + γ )
σ¯ ( −1(D + P )−1 ) < 1.
(29)
Intuitively, if the coupling weights aij between agents themselves are smaller and the coupling weights pi between the agents and the leader are higher, σ¯( −1(D + P )−1 ) turns to be smaller, i.e., the condition (25) in Theorem 3 hold more easily. Remark 2. Take into account a simplest case that the interconnection topology of N agents and a leader in system (23) has the leader as a globally reachable node, and each agent just has one direct path to reach the leader. Besides, we assume that the direct edge from agent i to j satisfies i > j , and we get
⋯ ⋯ ⋯ ⋯
0⎤ ⎥ 0⎥ . ⋮⎥ ⎥ 0⎦
Since the eigenvalues of adjacent matrix are all equivalent to zero, ρ(M (jω)) = ρ(diag{
jωκ + γ , ^ −1 jωdi( jω)b −1di + (κ jω + γ ) i
2|sin(
ωτ ^ )|σ¯ (M (jω)) < 1 2
(31)
Proof. With the supposition that the agents without communication delay converge to the scaled consensus asymptotically, the roots of following equation
^ det(diag{sdi(s ), i ∈ V} + (κs + γ ) L ) = 0
(32)
(33)
which equals
Therefore, the roots of Eq. (27) lie on the open left half complex plane, i.e., the agents in system (23) converge to the scaled consensus asymptotically. □
⎡ 0 0 ⎢ a 0 = ⎢ 21 ⎢ ⋮ ⋮ ⎢ ⎣ a n1 a n2
Then, all the agents in system (23) achieve an asymptotic scaled consensus, if
^ det(diag{sdi(s ), i ∈ V} + (κs + γ ) (L + (1 − e−sτ ))) = 0,
, i ∈ V})
−1
(30)
^ all lie on the open left half complex plane, i.e., M (s ) has its poles on ^ the open left half complex plane, so rank(L ) = N . In the same way as Theorem 3, the characteristic equation (27) is also rewritten as
−1(D + P )−1 e−jωτ ) ≤ σ¯ (diag{
(κs + γ ) . ^ diag{sdi(s ), i ∈ V} + (κs + γ ) L
holds with ω ∈ R .
(κs + γ ) M (s ) = diag{ , i ∈ V} −1(D + P )−1 e−sτ , ^ −1 sdi(s )bi−1di + (κs + γ )
≤ σ¯ (diag{
^ M (s ) =
5
i ∈ V} −1(D + P )−1 e−jωτ ) = 0 < 1
^ det(I + M (s )(1 − e−sτ )) = 0.
(34)
Then, we obtain from the condition (31) that
^ ρ(M (jω)(1 − e−jωτ )) ^ ≤ σ¯ (M (jω)(1 − e−jωτ )) ^ = |1 − e−jωτ |σ¯ (M (jω)) ωτ ^ < 2|sin( )|σ¯ (M (jω)) 2 < 1. Hence, the equation (34) has its roots on the open left half complex plane, i.e., the agents in system (23) converge to the scaled consensus asymptotically. □ Furthermore, we present a concrete delay bound for the achievement of asymptotic scaled consensus seeking. Corollary 1. It is assumed that the agents in system (23) without communication delay converges to the scaled consensus asymptotically. Then, all the agents in system (23) reach an asymptotic scaled consensus, if
τ<
1 ^ σ¯ ( jωM (jω))
holds with arbitrary communication delay if the assumption in Theorem 3 holds. Thus, asymptotic scaled consensus convergence in this case is delay-independent.
^ holds with ω ∈ R , where M (jω) is defined in (30).
It should be pointed out that the delay-independent scaled consensus condition in Theorem 3 is relatively conservative, and does not hold for arbitrary topology, coupling weights and control parameters according to Remark 3, Remark 5 and Remark 7 in [29].
5. Numerical simulation
(35)
5.1. Scaled consensus convergence without time delay
4.2. Delay-dependent case
Investigate a heterogeneous second-order multi-agent network given by
In this section, we will present a delay-dependent condition that is suitable for an arbitrary connected leader-following topology.
⎡ 0⎤ ⎡ 0 1 ⎤ ẋi (t ) = ⎢ x (t ) + ⎢ ⎥ (ui (t ) + ni ), ⎣ −αi0 −αi1⎦⎥ i ⎣ 1⎦ y (t ) = ⎣⎡ 1 0⎦⎤ xi (t ),
Theorem 4. Suppose that the agents in system (23) without communication delay converges to the scaled consensus asymptotically. Let
and we consider the agents (36) driven by the algorithm (8) under an undirected topology shown in Fig. 1 firstly. The undirected topology is connected and the leader is the globally reachable node.
i
(36)
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Fig. 1. Undirected topology of five agents and a leader.
The agents’ parameters are α10 = 1.5, α11 = 0.5, α20 = 1.6, α21 = 0.6, α30 = 1.7, α31 = 0.7, α40 = 1.8, α41 = 0.8, α50 = 1.9, α51 = 0.9 , b1= 0.8, b2 = 1.2, b3 = 0.5, b4 = 1, b5 = 1.5 , and the constant disturbances are n1 = 0.5, n2 = 0.3, n3 = 0.7, n4 = 1.0, n5 = − 0.2. The weights with respect to the edges are a12 = a21 = 0.5, a14 = a41 = 0.5, a23 =a32 = 0.4, a34 = a43 = 0.2, a45 = a54 = 0.3, p1 = 0.5, p5 = 0.2. . The scaled ratios for each agent are chosen as r1 = 0.3, r2 = 0.4, r3 = − 0.2, r4 = 0.5, r5 = − 0.4 , and the control parameters are set as κ = 3, γ = 1.2. Besides, the static leader's output is y0 = 1. In y addition, we define ei = ⎣⎡ 1 0⎦⎤x i − 0 as the errors of agents’ states. ri
Firstly, we take into account the usual scaled consensus algorithm (5). With the above coupling weights and control parameters of agents (36), the condition (7) is not satisfied, so the algorithm (5) cannot drive the agents (36) to achieve the assigned scaled consensus seeking (see Fig. 2). Then, we study the agents (36) driven by our proposed algorithm (8). Now, we analyze the Nyquist curves of
gi( jω) =
jκω + γ jω(( jω)2 + αi1( jω) + αi0)
γ κ
κ (jω + )
=
jω( jω + si )( jω + 1)
with the above assigned
parameters, and s1 = 0.5, s2 = 0.6, s3 = 0.7, s4 = 0.8, s5 = 0.9. Since γ = 0.4 < si , i = 1, 2, … , 5, we get arg(gi(jω)) ∈ ( − 2π , − π ), i.e., the κ curves of gi(jω) just lie in the third and fourth quadrants (see Fig. 3). In Theorem 1, Γ is an infinitesimal positive number, and it is concluded from Theorem 1 that the agents (36) with (8) converge to the scaled consensus asymptotically (see Fig. 4). Next, we study the scaled consensus seeking of second-order agents with a directed and asymmetric topology shown in Fig. 5. The adjacent weights are chosen as: a13 = 0.4, a21 = 0.5, a32 = 0.4, a34 = 0.2, a45 = 0.3, a51 = 0.5, p1 = 0.2 , and the eigenva^ lues of λ1 = 1.232, λ2 = 0.7334 + j0.0991, λ3 = 0.7334 L are
Fig. 3. Nyquist curves of gi(jω) . 0 1 −j0.0991, λ4 = 0.1519, λ5 = 0.3792 . Then, we set A = ⎡⎣ −1 −2 ⎤⎦, i.e.,
d(s ) = s2 + 2 s + 1, and other parameters are same as the above, and we will design ΔAi for each agent. For simplicity, we just in⎡ 0 0⎤ vestigate the case that ΔAi = ⎣⎢ −Δa 0 ⎦⎥, i.e., Δdi(s ) = Δαi0 . i0 According to the Routh-like criterion, the roots of Eq. (19) are proved to be lie on the open left half complex plane, and we obtain σ¯(jω(jωd(jω)I from the numerical simulation that ^ +(jκω + γ ) L )−1) < 1.0136 hold for ω ∈ R . Based on the condition 1
(20), we get |Δdi(jω)| = |Δαi0| < 1.0136 ≃ 0.9866, i = 1, 2, 3, 4, 5, so we choose Δα10 = − 0.2, Δα20 = 0.3, Δα30 = 0.2, Δα40 = − 0.3, Δα50 = 0.8 . Therefore, the agents (36) with (8) converge to the scaled consensus asymptotically (see Fig. 6). 5.2. Scaled consensus convergence with time delay Remark 2 has demonstrated a delay-independent scaled consensus seeking case based on Theorem 3, so this section just cares about the delay-dependent scaled consensus convergence in Theorem 4 and Corollary 1. Study second-order multi-agent system (36) with delayed scaled consensus algorithm (22) under the same directed and asymmetric topology shown in Fig. 5, and we choose the same
Fig. 2. Convergence under usual scaled consensus algorithm with undirected topology.
Please cite this article as: Liu C-L. Scaled consensus seeking in multiple non-identical linear autonomous agents. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.022i
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Fig. 4. Scaled consensus convergence under our proposed algorithm with undirected topology.
Fig. 5. Directed topology.
coupling weights, agents’ parameters αi0, αi1, bi , scaled gains ri, and control parameters κ , γ as those of the directed topology case in above subsection. By numerical computing, then, we get ^ σ¯( jωM (jω)) < 2.044 for ω ∈ R , so the time delay bound is τ < 0.4892(s) from condition (35), i.e., the agents in system (36) with (22) converge to the scaled consensus asymptotically if τ < 0.4892(s) (see Fig. 7 with τ = 0.3(s)).
6. Conclusion In this paper, we propose an adaptive scaled consensus algorithm, which is in the leader-following form, to solve the scaled
consensus problem of a class of heterogeneous linear stable multiple agents with distinct constant disturbances. Based on frequency-domain analysis, we obtain the scaled consensus criteria for the multi-agent systems under the undirected and directed topologies, respectively. Moreover, the scaled consensus seeking is also analyzed for the multi-agent systems with communication delay under the general directed topology. With proper topologies and parameters, the asymptotic scaled consensus convergence can be delay-independent. Furthermore, the delay-dependent consensus condition, which is less conservative, is also provided for the agents converging to the scaled consensus asymptotically under general directed topology. Compared with current results on scaled consensus problem [23–26], we focus on a heterogeneous linear multi-agent system, which is relatively complicated, and scaled consensus algorithm is designed and analyzed for the agents with diverse disturbances and communication delay. However, the results herein are just for the agents under fixed topology and time-invariant identical communication delay. In our future work, thus, we will propose some proper analysis methods to gain the convergence conditions for the heterogeneous multiagent systems with time-varying disturbances, switching topologies and time-varying communication delays.
Fig. 6. Scaled consensus convergence under our proposed algorithm with directed topology.
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Fig. 7. Delay-dependent scaled consensus seeking under our proposed algorithm.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 61473138 and 61104092), Natural Science Foundation of Jiangsu Province (Grant No. BK20151130), Six Talent Peaks Project in Jiangsu Province (Grant no. 2015-DZXX011), and Fundamental Research Funds for the Central Universities (Grant No. JUSRP51407B).
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Please cite this article as: Liu C-L. Scaled consensus seeking in multiple non-identical linear autonomous agents. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.06.022i