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Adaptive output feedback consensus tracking for heterogeneous multi-agent systems with unknown dynamics under directed graphs
Systems & Control Letters 87 (2016) 16–22
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Adaptive output feedback consensus tracking for heterogeneous multi-agent systems with unknown dynamics under directed graphs Junyong Sun, Zhiyong Geng ∗ , Yuezu Lv State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China
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Article history: Received 2 April 2015 Received in revised form 17 August 2015 Accepted 20 October 2015
Keywords: Cooperative control Output feedback Adaptive control Multi-agent system Directed graph
abstract This paper considers the distributed consensus tracking problem for the linear multi-agent systems with unknown dynamics under general directed graphs. Based on the output information among the agents, distributed adaptive consensus tracking protocols together with two observers, consisting of a local observer and an adaptive estimator, are designed to guarantee that all the signals in the closed-loop dynamics are uniformly ultimately bounded and the tracking errors converge to a small neighborhood around the origin. Moreover, the consensus protocols are in fully distributed fashion in the sense that the coupling gains in the controllers are independent without requiring the global knowledge that the eigenvalues of the Laplacian matrix associated with the whole communication graph. Finally, a simulation example is provided to verify the theoretical results. © 2015 Elsevier B.V. All rights reserved.
1. Introduction In recent years, coordination control of multi-agent systems has gained considerable interest partially due to its potential applications in many areas such as formation flying of unmanned air vehicles, cooperative surveillance, cooperative manipulation of multiple robots, distributed sensor networks [1–4], etc. In the previous literature, consensus control problem is a fundamental issue since the state agreement is often the premise for many coordination behaviors. The key objective of consensus control is to design distributed protocols, meaning that only the local state or output information is required, to achieve certain global task. In the pioneering work [5], a framework of consensus control for the first-order integrators with different topologies and time delays has been established. Since then, the consensus control has been studied intensively in different directions such as quantized consensus [6,7], finite-time consensus [8,9], consensus with input or communication time delays [10,11], consensus with different dynamics [12–16]. In terms of the number of leaders, the above research works roughly fall into two classes, that is, leaderfollowing consensus (or consensus tracking) with a reference to determine the final agreement value, and leaderless consensus
∗
Corresponding author. E-mail addresses: [email protected] (J. Sun), [email protected] (Z. Geng), [email protected] (Y. Lv). http://dx.doi.org/10.1016/j.sysconle.2015.10.007 0167-6911/© 2015 Elsevier B.V. All rights reserved.
whose agreement value depends on the initial states of the agents. Thus, the consensus tracking control has the advantages to determine the consensus value in advance, especially in some tasks, for example, to avoid hazardous obstacle. In this paper, we consider the distributed consensus control problem for the general linear multi-agent systems with unknown dynamics in the agents. Related works on adaptive consensus control with unknown dynamics include [17–26], where the unknown nonlinear dynamics in the leader or the followers are linearly parameterized to design the consensus protocols. In [17], the authors considered the leaderless consensus problem for the single and double integrator-type systems with unknown dynamics under the undirected communication topology. Then in [18], the consensus tracking problem for the single integrator systems was solved with the directed graphs, but the controller required the relative velocity and position simultaneously. Under a stronglyconnected communication graph, [19–21] studied the consensus tracking problem in a similar idea for the first-, second- and high-order integrator-type systems, respectively. In [22], adaptive consensus protocols were proposed for the single-integrator systems by exactly linearly parameterizing the unknown dynamics in the leader and followers. Among the aforementioned works, one common feature is that the consensus protocols are designed based on the state information which is not always available in practice. Moreover, another feature is that the dynamics of the agents focus on the first-order, second-order or high-order integrator-type systems which are special cases of the more general linear multi-agent systems. In [23,24], the output feedback
J. Sun et al. / Systems & Control Letters 87 (2016) 16–22
consensus protocols were designed, respectively, for the double integrators and linear multi-agent systems with unknown dynamics. However, one limitation in [23,24] is that the control gains in the consensus protocols depend on the eigenvalues of the Laplacian matrix associated with the whole communication topology which is actually the global information of the network. Besides, such a limitation also exists in the controllers in [19–21]. By virtue of the adaptive coupling gain technique in [27–29], the statefeedback and output-feedback consensus tracking protocols have been designed in [25,26], respectively, without knowing the Laplacian matrices’ eigenvalues. However, both of the protocols are only applicable to the undirected communication topology. Recently, adaptive consensus protocols have been designed under directed graphs for the second-order integrator-type systems [30] and general linear multi-agent systems [31], but both of them are state-feedback controllers. In [32], the authors proposed a new sequential observer approach to deal with the output feedback consensus for the linear multi-agent systems on directed graphs, but the dynamics focus on the nominal linear systems without considering the heterogeneous unknown dynamics in the followers. Motivated by the aforementioned literature, in this paper, we aim to design fully distributed adaptive output-feedback consensus protocols for the multi-agent systems with unknown dynamics under the directed communication topology. Based on the output information among the agents, the observer-based adaptive protocols are designed for the leader-following consensus under the directed graph with a spanning tree. The main contributions of this paper are fourfold. Firstly, the protocols are designed for the general linear multi-agent systems which include the single, double, and high-order integrator-type systems [17–21] as its special cases. Secondly, compared with [17–21,25], in which the protocols are designed based on the state information among the agents, the observer-based adaptive controllers in this paper are developed by using the output information which is more practical. Moreover, the control gains in the protocols are independent of eigenvalues of the Laplacian matrix associated with the information-exchange graph, thus the protocols are in fully distributed fashion in the sense that only the local information is required without knowing global knowledge of the entire graph a priori. In [23,24], although the output-feedback protocols are developed, the parameters in the controllers are determined by the eigenvalues of the Laplacian matrix. Finally, the protocols in the present paper are applicable to more general directed graphs. The paper is organized as follows. In Section 2, some preliminaries and the consensus problem are introduced. In Section 3, observer-based protocols are developed for the consensus tracking problem. A simulation example is given in Section 4 and the concluding remarks are provided in Section 5. 2. Preliminaries and problem statement
17
2.2. Graph theory In this paper, the communication topology among the agents is denoted by the directed graph G , (V , E , A), where V , {1, . . . , N } is the node set, E ⊆ V × V is the edge set consisting of ordered pair of distinct nodes and A = [aij ] ∈ RN ×N is the adjacent matrix associated with the communication graph. In the directed graph G, (i, j) ∈ E means that the jth agent has access to ith agent’s information, but not vice versa. A directed path from node i1 to node is is a sequence of edges of the form (i1 , i2 ), . . . , (is−1 , is ). The communication graph G is strongly connected if there is a directed path from every node to the other node. A directed graph that contains a spanning tree is that there is a node called the root that has no parent node, and the root has a directed path to every other node of the graph. For the adjacent matrix A = [aij ], aij = 1 if (j, i) ∈ E , otherwise is zero. It is assumed that there is no selfedge, that is, aii = 0 for all nodes. The Laplacian matrix L = [Lij ] associated with the graph G is defined as Lii = Lij = −aij , i ̸= j.
N
j=0,j̸=i
aij and
Lemma 1 ([33]). The Laplacian matrix L associated with graph G has the property that it has a simple zero eigenvalue with vector 1 as a corresponding right eigenvector and all other eigenvalues have positive real parts if and only if G contains a directed spanning tree. Assumption 1. The graph G contains a directed spanning tree with the leader as the root node. Because the leader has no neighbors, the Laplacian matrix L of
G has the following structure
L=
0
01×N
L2
L1
,
(1)
where L1 ∈ RN ×N and L2 ∈ RN ×1 . Under Assumption 1, zero is a simple eigenvalue of the Laplacian matrix L by virtue of Lemma 1. Thus it is obvious that L1 is a nonsingular M-matrix which has the following property: Lemma 2 ([31,34]). For the nonsingular M-matrix L1 , there exists a diagonal matrix G = diag(g1 , . . . , gN ) with gi > 0, i = 1, . . . , N such that GL1 + LT1 G = Lˆ 1 > 0. And the positive definite matrix G can be given with [g1 , . . . , gN ]T = (L1 )−1 1. Lemma 3 ([27,35]). For a system x˙ = g (x, t ), in which g (·) is locally Lipschitz in x and piecewise continuous in t, and it is assumed that there exists a continuously differentiable function V (x, t ) ≥ 0 such that the derivative along the trajectory of the system,
2.1. Notations
K1 (∥x∥) ≤ V (x, t ) ≤ K2 (∥x∥),
In this paper, let IN and 1 denote the identity matrix of dimension N and a column vector with all entries equal to one, respectively. Let Rn×m represent a set of n × m real matrices, and 0n×m denote the matrices with all zeros. Given a real vector x ∈ Rn , ∥x∥ is the Euclidean norm of x, and for a matrixA, ∥A∥F denotes tr(AT A), where the Frobenius norm that is defined by ∥A∥F = tr(·) denotes the trace of a matrix. For a matrix P, λmin (P ), λmax (P ) represent its minimum and maximum eigenvalue, respectively, and σmin (P ), σmax (P ) denote its minimum and maximum singular value, respectively. Given two matrices X and Y , X ⊗ Y denotes the Kronecker product of the matrices with the following properties that λ(X ⊗ Y ) = {λi (X )λj (Y )} and σ (X ⊗ Y ) = {σi (X )σj (Y )}. diag(Ai ) denotes a block-diagonal matrix with Ai , i = 1, . . . , N, on the diagonal. Given two symmetric real matrices A and B, A > B denotes that A − B is positive definite.
V˙ (x, t ) ≤ K3 (∥x∥) + ϵ where the constant ϵ > 0, K1 , K2 belong to class K∞ functions, and K3 belongs to class K function. Then, the solution x(t ) of the system is uniformly ultimately bounded.
ˆ ,W ˜ ∈ Rm×n with the relation that Lemma 4. For the matrices W , W ˜ ˆ W = W − W , then ˜ TW ˆ]= tr[W
1 2
˜ TW ˜ +W ˆ TW ˆ − W T W ]. tr[W
Lemma 5 (Young’s Inequality [36]). For nonnegative real numbers a, p q b, if p, q are real numbers that satisfy 1p + 1q = 1, then ab ≤ ap + bq .
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J. Sun et al. / Systems & Control Letters 87 (2016) 16–22
ai0 (ˆxi − x0 ). Γ = P −1 BBT P −1 , L = −lQ −1 C T , K = −BT P −1 , and P > 0, Q > 0 are the positive solutions of the following LMIs:
2.3. Problem statement Consider a group of N + 1 agents, consisting of N followers and one leader labeled by 0. The dynamics of the ith follower, (i = 1, 2, . . . , N) are modeled as x˙ i = Axi + B[ui + fi (xi )], yi = Cxi ,
i = 1, . . . , N ,
(2)
where xi ∈ Rn denotes the state, ui ∈ Rm is the control input, yi ∈ Rm is the measured output, fi (xi ) is an unknown smooth nonlinear function, and A, B, C are constant matrices with compatible dimensions. Also, the leader’s dynamics are modeled as follows x˙ 0 = Ax0 + Bf0 (x0 ), y0 = Cx0 ,
(3)
where x0 ∈ Rn denotes the state of the leader, y0 ∈ Rm is the leader’s measured output, and f0 (x0 ) is the unknown control input which is bounded with ∥f (x0 )∥∞ ≤ r0 . Assumption 2. For the multi-agent systems, the matched unknown dynamics fi (xi ) in the follower can be linearly parameterized as fi (xi (t )) = Wi ϕi (xi ) + εi (xi ),
(4)
where Wi ∈ Rm×p is an unknown constant matrix to be adaptive estimated that satisfies ∥Wi ∥ ≤ MWi , ϕi (xi ) ∈ Rp is the basis function vector with ∥ϕi ∥ ≤ Mϕi , and εi (xi ) is the residual error vector to be arbitrary small bounded with ∥εi ∥ ≤ Mεi . The control objective of tracking problem in this paper is to design distributed consensus protocols with output information for each follower such that the state of each follower converges to that of the leader, i.e., xi (t ) − x0 (t ) → 0 as t → ∞ for any initial condition of xi (0), i = 1, . . . , N, with bounded errors. Remark 1. The unknown dynamics in the followers can be seen as nonlinear functions which can be approximated by the neural network that has the linear-in-the-weight property. Besides, the parameterized expression is widely adopted in many literature such as [37,38] in classical adaptive control and [21–23] in multiagent systems. 3. Distributed adaptive consensus tracking protocols design In this section, based on the output information of neighboring agents, a distributed dynamic output-feedback controller with the independent coupling gain is proposed as follows
ˆ i ϕi (ˆxi ) ui = cK θi − W
(5)
where c ∈ R is the coupling gain independent of the Laplacian matrix. θi is the state of the following adaptive estimator with adaptive couplings as
θ˙i = Aθi + cBK θi + L(C xˆ i − yi ) + ρ(αi + βi )BK (ei − ξi ), α˙ i = −σi (αi − 1) + (ei − ξi )T Γ (ei − ξi ),
(C − QB) T
T
T
B Q
C T − QB 1 − I 2µ 0
QB
0 <0
(9)
−dI
where µ > 0, d > 0, c > 0, l > 0 are independent positive scalars. Moreover, the update laws are designed based on the output information as
˙
ˆ i = τi [−(ˆyi − yi )ϕiT (ˆxi ) − W
1 2µ
ˆ i ϕi (ˆxi )ϕiT (ˆxi ) − ki W ˆ i ], W
(10)
where τi is the adaptive gain, and ki > 0 is a small design parameter. Before moving forward, let x˜ i = xˆ i − xi , δi = xˆ i − x0 , as the estimation error and estimated tracking error, respectively, then with (2), we can get
˜ i ϕi (ˆxi ) − Wi ϕ˜ i + εi ], x˙˜ i (t ) = (A + LC )˜xi (t ) − B[−W δ˙i (t ) = Aδi (t ) + cBK θi + LC x˜ i − Bf0
(11)
˜i = W ˆ i − Wi . Let x˜ = [˜xT1 (t ), where ϕ˜ i = ϕi (ˆxi ) − ϕi (xi ), W T T T T . . . , x˜ N (t )] , δ(t ) = [δ1 (t ), . . . , δN (t )]T , ξ (t ) = [ξ1T (t ), . . . , ξNT (t )]T , e(t ) = [eT1 (t ), . . . , eTN (t )]T . Then we have the fact that ξ (t ) = (L1 ⊗ In )δ(t ),
e(t ) = (L1 ⊗ In )θ (t )
(12)
where L1 is defined in (1). As L1 is nonsingular according to Lemma 1, ξ (t ), e(t ) are similar to δ(t ), θ (t ), respectively. Then we obtain the dynamics of x˜ i , ξi , ei in the compact form as x˙˜ = [IN ⊗ (A + LC )]˜x − (IN ⊗ B)[−W˜ ϕ(ˆx) − W ϕ˜ + ε],
ξ˙ = (IN ⊗ A)ξ + (IN ⊗ cBK )e + (L1 ⊗ LC )˜x − (L1 1 ⊗ B)f0 , (13) ˆ ⊗ BK ] e˙ = [IN ⊗ (A + cBK )]e + (L1 ⊗ LC )˜x + [ρ L1 (αˆ + β) × (e − ξ ). ˜ 1, . . . , W ˜ N ), W = diag(W1 , . . . , WN ), ϕ(ˆx) = where W˜ = diag(W col(ϕ1 (ˆx1 ), . . . , ϕN (ˆxN )) ∈ RpN ×1 , ϕ˜ = col(ϕ˜ 1 , . . . , ϕ˜ N ) ∈ RpN ×1 , and αˆ = diag(α1 , . . . , αN ), βˆ = diag(β1 , . . . , βN ). Remark 2. The consensus tracking protocols consist of (5)–(7) and (10), in which Eq. (7) is a local observer to estimate the state of the system, Eq. (10) is the update law to estimate the unknown constant matrix Wi . The main idea lies in the adaptive estimator (6) with dynamic coupling gain, which has improved the order of the system and provides extra freedom for the controller design. The following theorem states the main result of this section.
and xˆ i is the state of the following local state observer as
yˆ i = C xˆ i ,
AT Q + QA − 2lC T C
(8)
(6)
βi = (ei − ξi )T P −1 (ei − ξi ), x˙ˆ i = Axˆ i + cBK θi + L(ˆyi − yi )
AP + PAT − 2cBBT < 0
(7)
where σi > 0 is a small design parameter, ρ > 0 is a constant scalar which is introduced to tune the bounded control input of N N the leader. ei = xi − xˆ j ) + j=1 aij (θi − θj ) + ai0 θi , ξi = j=1 aij (ˆ
Theorem 6. Suppose Assumption 1 holds, then with the dynamic output-feedback controller (5)–(7) and the adaptive update law (10), ˆ i are uniformly all the closed-loop signals including ξi , θi , x˜ i , αi , W ultimately bounded and the consensus tracking error converges to a small neighborhood around the origin as lim ∥x − 1 ⊗ x0 ∥ ≤ κ1 +
t →∞
1
σmin (L1 )
(κ2 + κ3 )
where the constants κ1 , κ2 , κ3 are determined in the following proof.
J. Sun et al. / Systems & Control Letters 87 (2016) 16–22
Proof. Consider the following candidate Lyapunov function N γ1
V1 = eT (IN ⊗ P −1 )e +
2 i=1
+ γ2 x˜ T (IN ⊗ Q )˜x + γ2
Observe that
N γ1 gi (2αi + βi )βi + (αi − α)2
2 i =1
N
tr
1
τi
i =1
˜ iT W ˜i W
V˙ 1 = 2e (IN ⊗ P
−1
)˙e + γ1
N
,
(14)
gi (αi + βi )β˙ +
i=1
+ γ1
N
gi α˙ i βi
+ 2γ2
tr
i =1
1
τi
˙˜
WiT W i
˜
.
(15)
i=1
N
gi α˙ i βi
+ γ1
i =1
N
(αi − α)α˙ i
σi α˜ i (α˜ i + α − 1)
i =1
ˆ ⊗ (P −1 A + AT P −1 ) ≤ γ1 (e − ξ )T G(αˆ + β) 4
ˆ 2⊗Γ (αˆ + β)
α¯ is √ ˆ ≥ ρλ α( ¯ αˆ + β)
1
τi
˙
˜ iT W ˜i W
= −γ2 x˜ (IN ⊗ H2 )˜x − 2γ2 x˜ [IN ⊗ (C T − QB)]W˜ ϕ(ˆx) N ˜ iT W ˆ i] + 2γ2 x˜ T (IN ⊗ QB)[W ϕ˜ − ε] − 2γ2 tr[ki W
σi (α − 1)2 +
N γ2 ˜ i ϕi (ˆxi ))T (W ˆ i ϕi (ˆxi ))] − tr[(W µ i=1 ≤ γ2 x˜ T IN ⊗ (−H2 + 2µ(C T − QB)(C T − QB)T
+
1 d
QBBT Q ) x˜ − γ2
N
˜ iT W ˜ i ] + ∆1 tr[ki W
4γ1
ρλ0
N
where AT Q + QA − 2lC T C = −H2 , −H2 + 2µ(C T − QB)(C T − QB)T + 1d QBBT Q = −Qe < 0. The equality is obtained by using Lemmas 4 and 5, and ∆1 = γ2 γ2 2µ
N
i =1
tr[ki WiT Wi ]+γ2 d∥W ϕ˜ −ε∥2 +
xi )) (Wi ϕi (ˆxi ))]. i=1 tr[(Wi ϕi (ˆ Based on (16)–(19), we obtain that
N
T
IN ⊗
1 2
H1
ˆ e + γ1 (e − ξ )T [G(αˆ + β)
⊗ (P −1 A + AT P −1
ρλ0
2 ˆ − 2c Γ )](e − ξ ) − γ1 (e − ξ ) (αˆ + β) ⊗ Γ (e − ξ ) 4 2 2 4ρ σmax (L1 )λmax (Γ ) ˆ 2 ⊗ Γ (e − ξ ) + (e − ξ )T (αˆ + β) λmin (H1 ) N N ˜ iT W ˜ i ] − γ1 − γ2 tr[ki W σi α˜ i2 − γ2 x˜ T (IN ⊗ Qe )˜x T
2 i =1
2
2 max 2 min
2
gi2 a2i0
(19)
i=1
4l σ (L1 )λ2max (C T C ) ˜+∆ + x˜ T IN ⊗ Q e x λmin (H1 )λ (P )λ2min (Q )λmin (Qe ) 1 ˆ ⊗ H1 ](e − ξ ) ≤ −eT IN ⊗ H1 e − γ1 (e − ξ )T [G(αˆ + β)
2 i =1
2 i =1
tr
i=1
γ1 + (Gβˆ ⊗ Γ ) + (αˆ − α IN ) ⊗ Γ (e − ξ ) − σi α˜ i2 N γ1
≥
T
N
+
N
T
i=1
ˆ ⊗ P −1 ](˙e − ξ˙ ) + γ1 (e − ξ )T = 2γ1 (e − ξ ) [G(αˆ + β) × [(Gβˆ ⊗ Γ ) − (Gσˆ αˆ − IN ⊗ P −1 ) + (αˆ − α IN ) ⊗ Γ ](e − ξ )
3ρλ0
2γ2 x˜ T (IN ⊗ Q )x˙˜ + 2γ2
V˙ 1 ≤ −eT
T
−
ˆ + α¯ (αˆ + β)
i=1
gi (αi + βi )β˙i +
N
+
i=1
where the positive definite matrix Qe > 0 will be determined later. By substituting the second and third equation of the adaptive estimator (6) into part of Eq. (15), we have
− γ1
(18)
1
√
= −eT (IN ⊗ H1 )e + 2eT (L1 ⊗ P −1 LC )˜x ˆ ⊗ Γ ](e − ξ ) − 2eT [ρ L1 (αˆ + β) 1 ≤ −eT IN ⊗ H1 e 2 2 4l2 σmax (L1 )λ2max (C T C ) + x˜ T IN ⊗ Q x˜ e λmin (H1 )λ2min (P )λ2min (Q )λmin (Qe ) 2 2 4ρ σmax (L1 )λmax (Γ ) ˆ 2 ⊗ Γ (e − ξ ) (16) + (e − ξ )T (αˆ + β) λmin (H1 )
γ1
1
ˆ if ρλ0 α¯ IN ≥ 2cG. 2cG(αˆ + β) In addition, by using the adaptive update law (10), one can get
2eT (IN ⊗ P −1 )˙e
N
4
α ≥ α+ ¯
i=1
Define P −1 A + AT P −1 − 2cP −1 BBT P −1 = −H1 < 0, by using Eq. (13) and Young’s inequality in Lemma 5, we can obtain
2 ˆ − (αˆ + β) ⊗ Γ (e − ξ ) 2 ρλ0 1 ˆ 2 ⊗ Γ + α IN − ≤ −γ1 (e − ξ )T IN (αˆ + β) 4 ρλ0 1 2 − G ⊗ Γ (e − ξ ) ρλ0 ρλ0 T 2 ˆ ≤ −γ1 (e − ξ ) (αˆ + β) ⊗ Γ + α¯ ⊗ Γ (e − ξ )
where maxi=1,...,N gi2 , and the constant scalar ρλ0 ρλ0 ρλ 2 determined by the fact that 4 0 IN 0
i=1
Gβˆ ⊗ Γ + (αˆ − α IN ) ⊗ Γ
ˆ ⊗ Γ ](e − ξ ) ≤ −γ1 (e − ξ )T [2cG(αˆ + β)
N (αi − α)α˙ i + 2γ2 x˜ T (IN ⊗ Q )x˙˜
N
γ1 (e − ξ )
T
ρλ0
where γ1 , γ2 , α are constants to be determined later, P , Q are positive definite matrices defined in (8) and (9). In addition, with αi (0) ≥ 1, it can be seen from Eq. (6) that αi (t ) ≥ 1. Thus, the candidate Lyapunov function V1 is positive definite. The time derivative of V1 along the trajectory of (6), (10) and (13) is obtained as T
19
r0T r0
(17)
i =1
where we have used the fact that GL1 + LT1 G ≥ λ0 IN , −α˜ i (α˜ i + α − 1) ≤ − 12 α˜ i2 + 12 (α − 1)2 , and Lemma 5 to get the inequality.
− x˜ T (IN ⊗ Qe )˜x − γ2
N i=1
∆
+∆=W
˜ iT W ˜ i] − tr[ki W
N γ1
2 i=1
σi α˜ i2 (20)
20
J. Sun et al. / Systems & Control Letters 87 (2016) 16–22
4γ N where ∆ = ∆1 + ρλ1 ( i=1 gi2 a2i0 )r02 + 0
constants γ1 , γ2 are chosen as γ1 > 2 (L )λ T 4l2 σmax 1 max (C C )
λmin (H1 )λ2min (P )λ2min (Q )λmin (Qe )
γ1
N
σ (α − )2 . The
1 i =1 i 2 2 (L )λ 16ρσmax 1 max (Γ ) , λ0 λmin (H1 )
γ2 >
+ 1 to get the last inequality. In light of
Lemma 3, it can be obtained that all the signals including ξi , θi , x˜ i , ˆ i in the closed-loop network systems are uniformly ultimately αi , W bounded. Moreover, in order to get the residual set of the tracking error, (20) can be rewritten as V˙ 1 ≤ −ηV1 (t ) + ηV1 (t ) + W + ς Z − ς Z
ς = −ηV1 (t ) − eT IN ⊗ H1 − ηP −1 e − γ1 (e − ξ )T [G(αˆ 2
ˆ ⊗ (ς H1 − ηP −1 )](e − ξ ) − x˜ T [IN ⊗ (ς Qe − ηγ2 Q )]˜x + β) N N γ1 η ˜T ˜ Wi Wi − − γ2 tr ki − (σi − η)α˜ i2 + ∆ τi 2 i =1 i =1 1 T + e IN ⊗ (ς − 1)H1 e + x˜ T (IN ⊗ (ς − 1)Qe )˜x 2
ˆ ⊗ (ς − 1)H1 ](e − ξ ) + γ1 (e − ξ )T [G(αˆ + β) η ς − λmin (H1 ) ∥e∥2 ≤ −ηV1 (t ) − 2 λmin (H1 )λmin (P ) η − γ1 min{gi } ς − λmin (H1 ) λmin (H1 )λmin (P ) × ∥(e − ξ )∥2 λmax (Q ) − ς − ηγ2 λmin (Qe ) ∥˜x∥2 + ∆ λmin (Qe )
i=1,...,N
ki τi , σi ,
ς λmin (H1 )λmin (P ) ς λmin (Qe ) , . 2 γ2 λmax (Q )
(21)
(22)
It is not difficult to get V˙ 1 ≤ −ηV1 (t ), if ∥˜x∥2 ≥ κ1 or ∥e∥2 ≥ κ2 or ∥(e − ξ )∥2 ≥ κ3 , where
λmax (Q ) κ1 = ∆/ ς − ηγ2 λmin (Qe ) , λmin (Qe ) ς η κ2 = ∆/ − λmin (H1 ) , 2 λmin (H1 )λmin (P ) η κ3 = ∆/ γ1 min{gi } ς − λmin (H1 ) . λmin (H1 )λmin (P )
4. Simulation
that, the tracking error converges to a small neighborhood around the origin as lim ∥x − 1 ⊗ x0 ∥ ≤ κ1 +
t →∞
σmin (L1 )
(κ2 + κ3 ).
Remark 3. In Theorem 6, the inequality AT Q + QA − 2lC T C + 2µ(C T − QB)(C T − QB)T + 1d QBBT Q = −Qe < 0 can be rewritten in the following parameter-dependent Riccati inequality form [39] AT Q + QA − 2lC T C + N = C T − QB.
1 d
QBBT Q + 2µNN T < 0
Remark 6. By virtue of the output information of the neighboring agents, the consensus protocols are designed in a fully distributed fashion in the sense that only the local information of each agent and its neighbors is utilized. In the related references [23,24], the protocols are also developed based on the output information, but the common limitation is that the coupling gain in the controller requires the eigenvalues of the Laplacian matrix associated with the whole communication topology. In comparison with these two related references, the coupling gain c in the protocol (5) is an independent constant without the knowledge of the Laplacian matrix a priori. Besides, the protocols of this paper are applicable for the general directed graph, while the adaptive protocols in [26,27,29] just work for the undirected graphs.
(23)
Note the fact that ξ = (L1 ⊗ In )δ and ∥x − 1 ⊗ x0 ∥ = ∥x − xˆ + xˆ − 1 ⊗ x0 ∥ ≤ ∥x − xˆ ∥ + ∥ˆx − 1 ⊗ x0 ∥ = ∥˜x∥ + ∥δ∥, and ∥ξ ∥ = ∥ξ − e + e∥ ≤ ∥e∥ + ∥e − ξ ∥, it is not difficult to get
1
Remark 4. In Theorem 6, the consensus protocols with adaptive update laws guarantee that all the signals in the closed-loop sysˆ i are uniformly ultimately tems including the parameter estimate W bounded. The results do not imply that the adaptive parameter estimates will converge to the real unknown parameters. If the persistent excitation(PE) condition and some other assumptions are satisfied, the parameter convergence may be obtained [25]. Without the PE condition, the consensus tracking object still can be achieved with the designed protocols. Remark 5. Compared with the related references [17,22,23], in which the unknown nonlinear functions in the followers are also linearly parameterized, the consensus tracking protocols in this paper are designed for the general linear multi-agent systems including the integrator-type system as a special case. Furthermore, the protocols are developed based on output information which is more practical than the state-feedback ones [19–21,25].
ˆ ⊗ H1 ](e − ξ ) + where Z = eT (IN ⊗ 12 H1 )e + γ1 (e − ξ )T [G(αˆ + β) x˜ T (IN ⊗ Qe )˜x, ς is a constant that satisfies 0 < ς < 1, and η ≤ min
Once the independent constants l, d have been chosen, Q > 0 depends continuously on the parameter µ. Note that µ = 0 corresponds to the Riccati inequality which can be easily computed to obtain the positive definite solution Q . Thus, there exists solution Q (µ) > 0 on the interval µ ∈ (0, µmax ), which defines the largest set within which there exists a positive definite solution for Q . By virtue of the Schur Complement Lemma, the parameterdependent Riccati inequality (24) can be transformed into the LMI (9). Moreover, the adaptive estimator (6) in Theorem 6 has been designed partially motivated by [32].
(24)
In this section, a simulation example is provided to illustrate the validity of the theoretical results. Consider a network of five followers and one leader whose dynamics are third-order integrators as 0 0 0
A=
1 0 0
0 1 , 0
0 0 , 1
B=
C = 1
0
0 .
The communication topology is given by Fig. 1 in which the node indexed by 0 is the leader and only the follower indexed by 1 is available to the leader’s information. It is easy to see that the communication graph contains a directed spanning tree. The basis functions in the linearly parameterized model (4) are defined by
ϕi (xi (t )) = [sin((i + 1)D1 xi (t )), cos((i + 1)D2 xi (t ))]T , i = 1, . . . , 5 where D1 = [1, 0], D2 = [0, 1]. Based on Section 3, we can get a full picture of the consensus tracking protocol with local observer and estimator for the ith
J. Sun et al. / Systems & Control Letters 87 (2016) 16–22
21
Fig. 1. Communication graph.
Fig. 3. The evolutions of tracking error xi2 − x02 .
Fig. 2. The evolutions of tracking error xi1 − x01 .
agent as
ˆ i ϕi (ˆxi ) ui = cK θi − W ˙ xˆ i = Axˆ i + cBK θi + L(C xˆ i − yi ) θ˙i = Aθi + cBK θi + L(C xˆ i − yi ) + ρ(αi + βi )BK (ei − ξi ), α˙ i = −σi (αi − 1) + (ei − ξi )T Γ (ei − ξi ), βi = (ei − ξi )T P −1 (ei − ξi ), W ˆ i ϕi (ˆxi )ϕiT (ˆxi ) − ki W ˆi ˆ˙ i = τi −(ˆyi − yi )ϕiT (ˆxi ) − 1 W 2µ
Fig. 4. The evolutions of tracking error xi3 − x03 .
where the parameters K , L, Γ , c , l, ρ, µ, d, σi , τi , ki in the consensus protocol are determined in the following part. By solving the matrix inequality (8) and (9) with the LMI toolbox of Matlab, the matrices K , L and Γ are given as K = −0.9194
−1.9056
−45.2909 −191.0680 , −112.3833 0.8453 1.7520 Γ = 1.7520 3.6314 2.3021 4.7715
−2.5039 , (25)
L=
2.3021 4.7715 , 6.2696
where the constants c = 1, l = 25, ρ = 1, µ = 20, d = 0.05. The initial states of agents are chosen randomly within [−20, 20], and the initial states of αi are chosen to satisfy αi (0) ≥ 1. The parameters in the adaptive protocol are taken as τi = 100, σi = 0.005, ki = 0.0008, i = 1, . . . , 5. The tracking consensus errors xi − x0 , i = 1, . . . , 5 are depicted in Figs. 2–4, denoting that the errors converge to a small neighborhood around the origin. Fig. 5 shows clearly that the coupling gains are bounded with αi ≥ 1, i = 1, . . . , 5.
Fig. 5. The evolutions of coupling gains αi (t ).
output-feedback consensus tracking protocols have been designed to guarantee that the tracking errors converge to a small neighborhood around the origin, and all the signals in the closedloop system are uniformly ultimately bounded. In addition, the consensus protocols are in fully distributed fashion for the control gains which are independent of the eigenvalues of the Laplacian matrix associated with the whole communication graph which is actually the global information. Finally, the validity of the theoretical results was illustrated through an example.
5. Conclusion Acknowledgments This paper has addressed the distributed consensus tracking problem for the linear multi-agent systems with unknown dynamics under general directed graphs. Distributed adaptive
The authors would like to thank the editor and all the anonymous reviewers for their valuable comments and suggestions. This
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work is supported by the National Natural Science Foundation (NNSF) of China under Grant 61374033. The authors would like to acknowledge Dr. Zhongkui Li for his inspiring discussions. References [1] Y. Kuriki, T. Namerikawa, Consensus-based cooperative formation control with collision avoidance for a multi-UAV system, in: American Control Conference, 2014, pp. 2077–2082. [2] S. Chung, J. Slotine, Cooperative robot control and concurrent synchronization of Lagrangian systems, IEEE Trans. Robot. 25 (3) (2009) 686–700. [3] W. Ren, R.W. Beard, E.M. Atkins, Information consensus in multivehicle cooperative control, IEEE Control Syst. Mag. 27 (2) (2007) 71–82. [4] J. Cortes, F. Bullo, Coordination and geometric optimization via distributed dynamical systems, SIAM J. Control Optim. 44 (5) (2005) 1543–1574. [5] R. Olfati-Saber, R. Murray, Consensu problems in networks of agents with switching topology and time-delays, IEEE Trans. Automat. Control 49 (9) (2004) 1520–1533. [6] A. Kashyap, T. Basar, R. Srikant, Quantized consensus, Automatica 43 (7) (2007) 1192–1203. [7] M. Zhu, S. Martinez, On the convergence time of asynchronous distributed quantized averaging algorithms, IEEE Trans. Automat. Control 56 (2) (2011) 386–390. [8] S. Khoo, L. Xie, Z. Man, Robust finite-time consensus tracking algorithm for multirobot systems, IEEE/ASME Trans. Mechatronics 14 (2) (2009) 219–228. [9] Y. Zhao, Z. Duan, G. Wen, Y. Zhang, Distributed finite-time tracking control for multi-agent systems: an observer-based approach, Systems Control Lett. 62 (1) (2013) 22–28. [10] P. Lin, Y. Jia, Multi-agent consensus with diverse time-delays and jointlyconnected topologies, Automatica 47 (4) (2011) 848–856. [11] G. Wen, Z. Duan, W. Yu, G. Chen, Consensus of second-order multi-agent systems with delayed nonlinear dynamics and intermittent communications, Internat. J. Control 86 (2) (2013) 322–331. [12] G. Hu, Robust consensus tracking for an integrator-type multi-agent system with disturbances and unmodelled dynamics, Internat. J. Control 84 (1) (2011) 1–8. [13] Z. Li, Z. Duan, G. Chen, L. Huang, Consensus of multiagent systems and synchronization of complex networks: a unified viewpoint, IEEE Trans. Circuits Syst. I. Regul. Pap. 57 (1) (2010) 213–224. [14] Z. Li, X. Liu, P. Lin, W. Ren, Consensus of linear multi-agent systems with reduced-order observer-based protocols, Systems Control Lett. 60 (7) (2011) 510–516. [15] G. Wen, G. Hu, W. Yu, J. Cao, G. Chen, Consensus tracking for higher-order multi-agent systems with switching directed topologies and occasionally missing control inputs, Systems Control Lett. 62 (12) (2013) 1151–1158. [16] J. Mei, W. Ren, J. Chen, G. Ma, Distributed adaptive coordination for multiple Lagrangian systems under a directed graph without using neighbors velocity information, Automatica 49 (6) (2013) 1723–1731. [17] Z. Hou, L. Cheng, M. Tan, Decentralized robust adaptive control for the multiagent system consensus problem using neural networks, IEEE Trans. Syst. Man Cybern. B 39 (3) (2009) 636–647. [18] L. Cheng, Z. Hou, M. Tan, Y. Lin, W. Zhang, Neural-network-based adaptive leader-following control for multiagent systems with uncertainties, IEEE Trans. Neural Netw. 21 (8) (2010) 1351–1358. [19] A. Das, F.L. Lewis, Distributed adaptive control for synchronization of unknown nonlinear networked systems, Automatica 46 (12) (2010) 2014–2021.
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Contents lists available at ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Adaptive output feedback consensus tracking for heterogeneous multi-agent systems with unknown dynamics under directed graphs Junyong Sun, Zhiyong Geng ∗ , Yuezu Lv State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China
article
info
Article history: Received 2 April 2015 Received in revised form 17 August 2015 Accepted 20 October 2015
Keywords: Cooperative control Output feedback Adaptive control Multi-agent system Directed graph
abstract This paper considers the distributed consensus tracking problem for the linear multi-agent systems with unknown dynamics under general directed graphs. Based on the output information among the agents, distributed adaptive consensus tracking protocols together with two observers, consisting of a local observer and an adaptive estimator, are designed to guarantee that all the signals in the closed-loop dynamics are uniformly ultimately bounded and the tracking errors converge to a small neighborhood around the origin. Moreover, the consensus protocols are in fully distributed fashion in the sense that the coupling gains in the controllers are independent without requiring the global knowledge that the eigenvalues of the Laplacian matrix associated with the whole communication graph. Finally, a simulation example is provided to verify the theoretical results. © 2015 Elsevier B.V. All rights reserved.
1. Introduction In recent years, coordination control of multi-agent systems has gained considerable interest partially due to its potential applications in many areas such as formation flying of unmanned air vehicles, cooperative surveillance, cooperative manipulation of multiple robots, distributed sensor networks [1–4], etc. In the previous literature, consensus control problem is a fundamental issue since the state agreement is often the premise for many coordination behaviors. The key objective of consensus control is to design distributed protocols, meaning that only the local state or output information is required, to achieve certain global task. In the pioneering work [5], a framework of consensus control for the first-order integrators with different topologies and time delays has been established. Since then, the consensus control has been studied intensively in different directions such as quantized consensus [6,7], finite-time consensus [8,9], consensus with input or communication time delays [10,11], consensus with different dynamics [12–16]. In terms of the number of leaders, the above research works roughly fall into two classes, that is, leaderfollowing consensus (or consensus tracking) with a reference to determine the final agreement value, and leaderless consensus
∗
Corresponding author. E-mail addresses: [email protected] (J. Sun), [email protected] (Z. Geng), [email protected] (Y. Lv). http://dx.doi.org/10.1016/j.sysconle.2015.10.007 0167-6911/© 2015 Elsevier B.V. All rights reserved.
whose agreement value depends on the initial states of the agents. Thus, the consensus tracking control has the advantages to determine the consensus value in advance, especially in some tasks, for example, to avoid hazardous obstacle. In this paper, we consider the distributed consensus control problem for the general linear multi-agent systems with unknown dynamics in the agents. Related works on adaptive consensus control with unknown dynamics include [17–26], where the unknown nonlinear dynamics in the leader or the followers are linearly parameterized to design the consensus protocols. In [17], the authors considered the leaderless consensus problem for the single and double integrator-type systems with unknown dynamics under the undirected communication topology. Then in [18], the consensus tracking problem for the single integrator systems was solved with the directed graphs, but the controller required the relative velocity and position simultaneously. Under a stronglyconnected communication graph, [19–21] studied the consensus tracking problem in a similar idea for the first-, second- and high-order integrator-type systems, respectively. In [22], adaptive consensus protocols were proposed for the single-integrator systems by exactly linearly parameterizing the unknown dynamics in the leader and followers. Among the aforementioned works, one common feature is that the consensus protocols are designed based on the state information which is not always available in practice. Moreover, another feature is that the dynamics of the agents focus on the first-order, second-order or high-order integrator-type systems which are special cases of the more general linear multi-agent systems. In [23,24], the output feedback
J. Sun et al. / Systems & Control Letters 87 (2016) 16–22
consensus protocols were designed, respectively, for the double integrators and linear multi-agent systems with unknown dynamics. However, one limitation in [23,24] is that the control gains in the consensus protocols depend on the eigenvalues of the Laplacian matrix associated with the whole communication topology which is actually the global information of the network. Besides, such a limitation also exists in the controllers in [19–21]. By virtue of the adaptive coupling gain technique in [27–29], the statefeedback and output-feedback consensus tracking protocols have been designed in [25,26], respectively, without knowing the Laplacian matrices’ eigenvalues. However, both of the protocols are only applicable to the undirected communication topology. Recently, adaptive consensus protocols have been designed under directed graphs for the second-order integrator-type systems [30] and general linear multi-agent systems [31], but both of them are state-feedback controllers. In [32], the authors proposed a new sequential observer approach to deal with the output feedback consensus for the linear multi-agent systems on directed graphs, but the dynamics focus on the nominal linear systems without considering the heterogeneous unknown dynamics in the followers. Motivated by the aforementioned literature, in this paper, we aim to design fully distributed adaptive output-feedback consensus protocols for the multi-agent systems with unknown dynamics under the directed communication topology. Based on the output information among the agents, the observer-based adaptive protocols are designed for the leader-following consensus under the directed graph with a spanning tree. The main contributions of this paper are fourfold. Firstly, the protocols are designed for the general linear multi-agent systems which include the single, double, and high-order integrator-type systems [17–21] as its special cases. Secondly, compared with [17–21,25], in which the protocols are designed based on the state information among the agents, the observer-based adaptive controllers in this paper are developed by using the output information which is more practical. Moreover, the control gains in the protocols are independent of eigenvalues of the Laplacian matrix associated with the information-exchange graph, thus the protocols are in fully distributed fashion in the sense that only the local information is required without knowing global knowledge of the entire graph a priori. In [23,24], although the output-feedback protocols are developed, the parameters in the controllers are determined by the eigenvalues of the Laplacian matrix. Finally, the protocols in the present paper are applicable to more general directed graphs. The paper is organized as follows. In Section 2, some preliminaries and the consensus problem are introduced. In Section 3, observer-based protocols are developed for the consensus tracking problem. A simulation example is given in Section 4 and the concluding remarks are provided in Section 5. 2. Preliminaries and problem statement
17
2.2. Graph theory In this paper, the communication topology among the agents is denoted by the directed graph G , (V , E , A), where V , {1, . . . , N } is the node set, E ⊆ V × V is the edge set consisting of ordered pair of distinct nodes and A = [aij ] ∈ RN ×N is the adjacent matrix associated with the communication graph. In the directed graph G, (i, j) ∈ E means that the jth agent has access to ith agent’s information, but not vice versa. A directed path from node i1 to node is is a sequence of edges of the form (i1 , i2 ), . . . , (is−1 , is ). The communication graph G is strongly connected if there is a directed path from every node to the other node. A directed graph that contains a spanning tree is that there is a node called the root that has no parent node, and the root has a directed path to every other node of the graph. For the adjacent matrix A = [aij ], aij = 1 if (j, i) ∈ E , otherwise is zero. It is assumed that there is no selfedge, that is, aii = 0 for all nodes. The Laplacian matrix L = [Lij ] associated with the graph G is defined as Lii = Lij = −aij , i ̸= j.
N
j=0,j̸=i
aij and
Lemma 1 ([33]). The Laplacian matrix L associated with graph G has the property that it has a simple zero eigenvalue with vector 1 as a corresponding right eigenvector and all other eigenvalues have positive real parts if and only if G contains a directed spanning tree. Assumption 1. The graph G contains a directed spanning tree with the leader as the root node. Because the leader has no neighbors, the Laplacian matrix L of
G has the following structure
L=
0
01×N
L2
L1
,
(1)
where L1 ∈ RN ×N and L2 ∈ RN ×1 . Under Assumption 1, zero is a simple eigenvalue of the Laplacian matrix L by virtue of Lemma 1. Thus it is obvious that L1 is a nonsingular M-matrix which has the following property: Lemma 2 ([31,34]). For the nonsingular M-matrix L1 , there exists a diagonal matrix G = diag(g1 , . . . , gN ) with gi > 0, i = 1, . . . , N such that GL1 + LT1 G = Lˆ 1 > 0. And the positive definite matrix G can be given with [g1 , . . . , gN ]T = (L1 )−1 1. Lemma 3 ([27,35]). For a system x˙ = g (x, t ), in which g (·) is locally Lipschitz in x and piecewise continuous in t, and it is assumed that there exists a continuously differentiable function V (x, t ) ≥ 0 such that the derivative along the trajectory of the system,
2.1. Notations
K1 (∥x∥) ≤ V (x, t ) ≤ K2 (∥x∥),
In this paper, let IN and 1 denote the identity matrix of dimension N and a column vector with all entries equal to one, respectively. Let Rn×m represent a set of n × m real matrices, and 0n×m denote the matrices with all zeros. Given a real vector x ∈ Rn , ∥x∥ is the Euclidean norm of x, and for a matrixA, ∥A∥F denotes tr(AT A), where the Frobenius norm that is defined by ∥A∥F = tr(·) denotes the trace of a matrix. For a matrix P, λmin (P ), λmax (P ) represent its minimum and maximum eigenvalue, respectively, and σmin (P ), σmax (P ) denote its minimum and maximum singular value, respectively. Given two matrices X and Y , X ⊗ Y denotes the Kronecker product of the matrices with the following properties that λ(X ⊗ Y ) = {λi (X )λj (Y )} and σ (X ⊗ Y ) = {σi (X )σj (Y )}. diag(Ai ) denotes a block-diagonal matrix with Ai , i = 1, . . . , N, on the diagonal. Given two symmetric real matrices A and B, A > B denotes that A − B is positive definite.
V˙ (x, t ) ≤ K3 (∥x∥) + ϵ where the constant ϵ > 0, K1 , K2 belong to class K∞ functions, and K3 belongs to class K function. Then, the solution x(t ) of the system is uniformly ultimately bounded.
ˆ ,W ˜ ∈ Rm×n with the relation that Lemma 4. For the matrices W , W ˜ ˆ W = W − W , then ˜ TW ˆ]= tr[W
1 2
˜ TW ˜ +W ˆ TW ˆ − W T W ]. tr[W
Lemma 5 (Young’s Inequality [36]). For nonnegative real numbers a, p q b, if p, q are real numbers that satisfy 1p + 1q = 1, then ab ≤ ap + bq .
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J. Sun et al. / Systems & Control Letters 87 (2016) 16–22
ai0 (ˆxi − x0 ). Γ = P −1 BBT P −1 , L = −lQ −1 C T , K = −BT P −1 , and P > 0, Q > 0 are the positive solutions of the following LMIs:
2.3. Problem statement Consider a group of N + 1 agents, consisting of N followers and one leader labeled by 0. The dynamics of the ith follower, (i = 1, 2, . . . , N) are modeled as x˙ i = Axi + B[ui + fi (xi )], yi = Cxi ,
i = 1, . . . , N ,
(2)
where xi ∈ Rn denotes the state, ui ∈ Rm is the control input, yi ∈ Rm is the measured output, fi (xi ) is an unknown smooth nonlinear function, and A, B, C are constant matrices with compatible dimensions. Also, the leader’s dynamics are modeled as follows x˙ 0 = Ax0 + Bf0 (x0 ), y0 = Cx0 ,
(3)
where x0 ∈ Rn denotes the state of the leader, y0 ∈ Rm is the leader’s measured output, and f0 (x0 ) is the unknown control input which is bounded with ∥f (x0 )∥∞ ≤ r0 . Assumption 2. For the multi-agent systems, the matched unknown dynamics fi (xi ) in the follower can be linearly parameterized as fi (xi (t )) = Wi ϕi (xi ) + εi (xi ),
(4)
where Wi ∈ Rm×p is an unknown constant matrix to be adaptive estimated that satisfies ∥Wi ∥ ≤ MWi , ϕi (xi ) ∈ Rp is the basis function vector with ∥ϕi ∥ ≤ Mϕi , and εi (xi ) is the residual error vector to be arbitrary small bounded with ∥εi ∥ ≤ Mεi . The control objective of tracking problem in this paper is to design distributed consensus protocols with output information for each follower such that the state of each follower converges to that of the leader, i.e., xi (t ) − x0 (t ) → 0 as t → ∞ for any initial condition of xi (0), i = 1, . . . , N, with bounded errors. Remark 1. The unknown dynamics in the followers can be seen as nonlinear functions which can be approximated by the neural network that has the linear-in-the-weight property. Besides, the parameterized expression is widely adopted in many literature such as [37,38] in classical adaptive control and [21–23] in multiagent systems. 3. Distributed adaptive consensus tracking protocols design In this section, based on the output information of neighboring agents, a distributed dynamic output-feedback controller with the independent coupling gain is proposed as follows
ˆ i ϕi (ˆxi ) ui = cK θi − W
(5)
where c ∈ R is the coupling gain independent of the Laplacian matrix. θi is the state of the following adaptive estimator with adaptive couplings as
θ˙i = Aθi + cBK θi + L(C xˆ i − yi ) + ρ(αi + βi )BK (ei − ξi ), α˙ i = −σi (αi − 1) + (ei − ξi )T Γ (ei − ξi ),
(C − QB) T
T
T
B Q
C T − QB 1 − I 2µ 0
QB
0 <0
(9)
−dI
where µ > 0, d > 0, c > 0, l > 0 are independent positive scalars. Moreover, the update laws are designed based on the output information as
˙
ˆ i = τi [−(ˆyi − yi )ϕiT (ˆxi ) − W
1 2µ
ˆ i ϕi (ˆxi )ϕiT (ˆxi ) − ki W ˆ i ], W
(10)
where τi is the adaptive gain, and ki > 0 is a small design parameter. Before moving forward, let x˜ i = xˆ i − xi , δi = xˆ i − x0 , as the estimation error and estimated tracking error, respectively, then with (2), we can get
˜ i ϕi (ˆxi ) − Wi ϕ˜ i + εi ], x˙˜ i (t ) = (A + LC )˜xi (t ) − B[−W δ˙i (t ) = Aδi (t ) + cBK θi + LC x˜ i − Bf0
(11)
˜i = W ˆ i − Wi . Let x˜ = [˜xT1 (t ), where ϕ˜ i = ϕi (ˆxi ) − ϕi (xi ), W T T T T . . . , x˜ N (t )] , δ(t ) = [δ1 (t ), . . . , δN (t )]T , ξ (t ) = [ξ1T (t ), . . . , ξNT (t )]T , e(t ) = [eT1 (t ), . . . , eTN (t )]T . Then we have the fact that ξ (t ) = (L1 ⊗ In )δ(t ),
e(t ) = (L1 ⊗ In )θ (t )
(12)
where L1 is defined in (1). As L1 is nonsingular according to Lemma 1, ξ (t ), e(t ) are similar to δ(t ), θ (t ), respectively. Then we obtain the dynamics of x˜ i , ξi , ei in the compact form as x˙˜ = [IN ⊗ (A + LC )]˜x − (IN ⊗ B)[−W˜ ϕ(ˆx) − W ϕ˜ + ε],
ξ˙ = (IN ⊗ A)ξ + (IN ⊗ cBK )e + (L1 ⊗ LC )˜x − (L1 1 ⊗ B)f0 , (13) ˆ ⊗ BK ] e˙ = [IN ⊗ (A + cBK )]e + (L1 ⊗ LC )˜x + [ρ L1 (αˆ + β) × (e − ξ ). ˜ 1, . . . , W ˜ N ), W = diag(W1 , . . . , WN ), ϕ(ˆx) = where W˜ = diag(W col(ϕ1 (ˆx1 ), . . . , ϕN (ˆxN )) ∈ RpN ×1 , ϕ˜ = col(ϕ˜ 1 , . . . , ϕ˜ N ) ∈ RpN ×1 , and αˆ = diag(α1 , . . . , αN ), βˆ = diag(β1 , . . . , βN ). Remark 2. The consensus tracking protocols consist of (5)–(7) and (10), in which Eq. (7) is a local observer to estimate the state of the system, Eq. (10) is the update law to estimate the unknown constant matrix Wi . The main idea lies in the adaptive estimator (6) with dynamic coupling gain, which has improved the order of the system and provides extra freedom for the controller design. The following theorem states the main result of this section.
and xˆ i is the state of the following local state observer as
yˆ i = C xˆ i ,
AT Q + QA − 2lC T C
(8)
(6)
βi = (ei − ξi )T P −1 (ei − ξi ), x˙ˆ i = Axˆ i + cBK θi + L(ˆyi − yi )
AP + PAT − 2cBBT < 0
(7)
where σi > 0 is a small design parameter, ρ > 0 is a constant scalar which is introduced to tune the bounded control input of N N the leader. ei = xi − xˆ j ) + j=1 aij (θi − θj ) + ai0 θi , ξi = j=1 aij (ˆ
Theorem 6. Suppose Assumption 1 holds, then with the dynamic output-feedback controller (5)–(7) and the adaptive update law (10), ˆ i are uniformly all the closed-loop signals including ξi , θi , x˜ i , αi , W ultimately bounded and the consensus tracking error converges to a small neighborhood around the origin as lim ∥x − 1 ⊗ x0 ∥ ≤ κ1 +
t →∞
1
σmin (L1 )
(κ2 + κ3 )
where the constants κ1 , κ2 , κ3 are determined in the following proof.
J. Sun et al. / Systems & Control Letters 87 (2016) 16–22
Proof. Consider the following candidate Lyapunov function N γ1
V1 = eT (IN ⊗ P −1 )e +
2 i=1
+ γ2 x˜ T (IN ⊗ Q )˜x + γ2
Observe that
N γ1 gi (2αi + βi )βi + (αi − α)2
2 i =1
N
tr
1
τi
i =1
˜ iT W ˜i W
V˙ 1 = 2e (IN ⊗ P
−1
)˙e + γ1
N
,
(14)
gi (αi + βi )β˙ +
i=1
+ γ1
N
gi α˙ i βi
+ 2γ2
tr
i =1
1
τi
˙˜
WiT W i
˜
.
(15)
i=1
N
gi α˙ i βi
+ γ1
i =1
N
(αi − α)α˙ i
σi α˜ i (α˜ i + α − 1)
i =1
ˆ ⊗ (P −1 A + AT P −1 ) ≤ γ1 (e − ξ )T G(αˆ + β) 4
ˆ 2⊗Γ (αˆ + β)
α¯ is √ ˆ ≥ ρλ α( ¯ αˆ + β)
1
τi
˙
˜ iT W ˜i W
= −γ2 x˜ (IN ⊗ H2 )˜x − 2γ2 x˜ [IN ⊗ (C T − QB)]W˜ ϕ(ˆx) N ˜ iT W ˆ i] + 2γ2 x˜ T (IN ⊗ QB)[W ϕ˜ − ε] − 2γ2 tr[ki W
σi (α − 1)2 +
N γ2 ˜ i ϕi (ˆxi ))T (W ˆ i ϕi (ˆxi ))] − tr[(W µ i=1 ≤ γ2 x˜ T IN ⊗ (−H2 + 2µ(C T − QB)(C T − QB)T
+
1 d
QBBT Q ) x˜ − γ2
N
˜ iT W ˜ i ] + ∆1 tr[ki W
4γ1
ρλ0
N
where AT Q + QA − 2lC T C = −H2 , −H2 + 2µ(C T − QB)(C T − QB)T + 1d QBBT Q = −Qe < 0. The equality is obtained by using Lemmas 4 and 5, and ∆1 = γ2 γ2 2µ
N
i =1
tr[ki WiT Wi ]+γ2 d∥W ϕ˜ −ε∥2 +
xi )) (Wi ϕi (ˆxi ))]. i=1 tr[(Wi ϕi (ˆ Based on (16)–(19), we obtain that
N
T
IN ⊗
1 2
H1
ˆ e + γ1 (e − ξ )T [G(αˆ + β)
⊗ (P −1 A + AT P −1
ρλ0
2 ˆ − 2c Γ )](e − ξ ) − γ1 (e − ξ ) (αˆ + β) ⊗ Γ (e − ξ ) 4 2 2 4ρ σmax (L1 )λmax (Γ ) ˆ 2 ⊗ Γ (e − ξ ) + (e − ξ )T (αˆ + β) λmin (H1 ) N N ˜ iT W ˜ i ] − γ1 − γ2 tr[ki W σi α˜ i2 − γ2 x˜ T (IN ⊗ Qe )˜x T
2 i =1
2
2 max 2 min
2
gi2 a2i0
(19)
i=1
4l σ (L1 )λ2max (C T C ) ˜+∆ + x˜ T IN ⊗ Q e x λmin (H1 )λ (P )λ2min (Q )λmin (Qe ) 1 ˆ ⊗ H1 ](e − ξ ) ≤ −eT IN ⊗ H1 e − γ1 (e − ξ )T [G(αˆ + β)
2 i =1
2 i =1
tr
i=1
γ1 + (Gβˆ ⊗ Γ ) + (αˆ − α IN ) ⊗ Γ (e − ξ ) − σi α˜ i2 N γ1
≥
T
N
+
N
T
i=1
ˆ ⊗ P −1 ](˙e − ξ˙ ) + γ1 (e − ξ )T = 2γ1 (e − ξ ) [G(αˆ + β) × [(Gβˆ ⊗ Γ ) − (Gσˆ αˆ − IN ⊗ P −1 ) + (αˆ − α IN ) ⊗ Γ ](e − ξ )
3ρλ0
2γ2 x˜ T (IN ⊗ Q )x˙˜ + 2γ2
V˙ 1 ≤ −eT
T
−
ˆ + α¯ (αˆ + β)
i=1
gi (αi + βi )β˙i +
N
+
i=1
where the positive definite matrix Qe > 0 will be determined later. By substituting the second and third equation of the adaptive estimator (6) into part of Eq. (15), we have
− γ1
(18)
1
√
= −eT (IN ⊗ H1 )e + 2eT (L1 ⊗ P −1 LC )˜x ˆ ⊗ Γ ](e − ξ ) − 2eT [ρ L1 (αˆ + β) 1 ≤ −eT IN ⊗ H1 e 2 2 4l2 σmax (L1 )λ2max (C T C ) + x˜ T IN ⊗ Q x˜ e λmin (H1 )λ2min (P )λ2min (Q )λmin (Qe ) 2 2 4ρ σmax (L1 )λmax (Γ ) ˆ 2 ⊗ Γ (e − ξ ) (16) + (e − ξ )T (αˆ + β) λmin (H1 )
γ1
1
ˆ if ρλ0 α¯ IN ≥ 2cG. 2cG(αˆ + β) In addition, by using the adaptive update law (10), one can get
2eT (IN ⊗ P −1 )˙e
N
4
α ≥ α+ ¯
i=1
Define P −1 A + AT P −1 − 2cP −1 BBT P −1 = −H1 < 0, by using Eq. (13) and Young’s inequality in Lemma 5, we can obtain
2 ˆ − (αˆ + β) ⊗ Γ (e − ξ ) 2 ρλ0 1 ˆ 2 ⊗ Γ + α IN − ≤ −γ1 (e − ξ )T IN (αˆ + β) 4 ρλ0 1 2 − G ⊗ Γ (e − ξ ) ρλ0 ρλ0 T 2 ˆ ≤ −γ1 (e − ξ ) (αˆ + β) ⊗ Γ + α¯ ⊗ Γ (e − ξ )
where maxi=1,...,N gi2 , and the constant scalar ρλ0 ρλ0 ρλ 2 determined by the fact that 4 0 IN 0
i=1
Gβˆ ⊗ Γ + (αˆ − α IN ) ⊗ Γ
ˆ ⊗ Γ ](e − ξ ) ≤ −γ1 (e − ξ )T [2cG(αˆ + β)
N (αi − α)α˙ i + 2γ2 x˜ T (IN ⊗ Q )x˙˜
N
γ1 (e − ξ )
T
ρλ0
where γ1 , γ2 , α are constants to be determined later, P , Q are positive definite matrices defined in (8) and (9). In addition, with αi (0) ≥ 1, it can be seen from Eq. (6) that αi (t ) ≥ 1. Thus, the candidate Lyapunov function V1 is positive definite. The time derivative of V1 along the trajectory of (6), (10) and (13) is obtained as T
19
r0T r0
(17)
i =1
where we have used the fact that GL1 + LT1 G ≥ λ0 IN , −α˜ i (α˜ i + α − 1) ≤ − 12 α˜ i2 + 12 (α − 1)2 , and Lemma 5 to get the inequality.
− x˜ T (IN ⊗ Qe )˜x − γ2
N i=1
∆
+∆=W
˜ iT W ˜ i] − tr[ki W
N γ1
2 i=1
σi α˜ i2 (20)
20
J. Sun et al. / Systems & Control Letters 87 (2016) 16–22
4γ N where ∆ = ∆1 + ρλ1 ( i=1 gi2 a2i0 )r02 + 0
constants γ1 , γ2 are chosen as γ1 > 2 (L )λ T 4l2 σmax 1 max (C C )
λmin (H1 )λ2min (P )λ2min (Q )λmin (Qe )
γ1
N
σ (α − )2 . The
1 i =1 i 2 2 (L )λ 16ρσmax 1 max (Γ ) , λ0 λmin (H1 )
γ2 >
+ 1 to get the last inequality. In light of
Lemma 3, it can be obtained that all the signals including ξi , θi , x˜ i , ˆ i in the closed-loop network systems are uniformly ultimately αi , W bounded. Moreover, in order to get the residual set of the tracking error, (20) can be rewritten as V˙ 1 ≤ −ηV1 (t ) + ηV1 (t ) + W + ς Z − ς Z
ς = −ηV1 (t ) − eT IN ⊗ H1 − ηP −1 e − γ1 (e − ξ )T [G(αˆ 2
ˆ ⊗ (ς H1 − ηP −1 )](e − ξ ) − x˜ T [IN ⊗ (ς Qe − ηγ2 Q )]˜x + β) N N γ1 η ˜T ˜ Wi Wi − − γ2 tr ki − (σi − η)α˜ i2 + ∆ τi 2 i =1 i =1 1 T + e IN ⊗ (ς − 1)H1 e + x˜ T (IN ⊗ (ς − 1)Qe )˜x 2
ˆ ⊗ (ς − 1)H1 ](e − ξ ) + γ1 (e − ξ )T [G(αˆ + β) η ς − λmin (H1 ) ∥e∥2 ≤ −ηV1 (t ) − 2 λmin (H1 )λmin (P ) η − γ1 min{gi } ς − λmin (H1 ) λmin (H1 )λmin (P ) × ∥(e − ξ )∥2 λmax (Q ) − ς − ηγ2 λmin (Qe ) ∥˜x∥2 + ∆ λmin (Qe )
i=1,...,N
ki τi , σi ,
ς λmin (H1 )λmin (P ) ς λmin (Qe ) , . 2 γ2 λmax (Q )
(21)
(22)
It is not difficult to get V˙ 1 ≤ −ηV1 (t ), if ∥˜x∥2 ≥ κ1 or ∥e∥2 ≥ κ2 or ∥(e − ξ )∥2 ≥ κ3 , where
λmax (Q ) κ1 = ∆/ ς − ηγ2 λmin (Qe ) , λmin (Qe ) ς η κ2 = ∆/ − λmin (H1 ) , 2 λmin (H1 )λmin (P ) η κ3 = ∆/ γ1 min{gi } ς − λmin (H1 ) . λmin (H1 )λmin (P )
4. Simulation
that, the tracking error converges to a small neighborhood around the origin as lim ∥x − 1 ⊗ x0 ∥ ≤ κ1 +
t →∞
σmin (L1 )
(κ2 + κ3 ).
Remark 3. In Theorem 6, the inequality AT Q + QA − 2lC T C + 2µ(C T − QB)(C T − QB)T + 1d QBBT Q = −Qe < 0 can be rewritten in the following parameter-dependent Riccati inequality form [39] AT Q + QA − 2lC T C + N = C T − QB.
1 d
QBBT Q + 2µNN T < 0
Remark 6. By virtue of the output information of the neighboring agents, the consensus protocols are designed in a fully distributed fashion in the sense that only the local information of each agent and its neighbors is utilized. In the related references [23,24], the protocols are also developed based on the output information, but the common limitation is that the coupling gain in the controller requires the eigenvalues of the Laplacian matrix associated with the whole communication topology. In comparison with these two related references, the coupling gain c in the protocol (5) is an independent constant without the knowledge of the Laplacian matrix a priori. Besides, the protocols of this paper are applicable for the general directed graph, while the adaptive protocols in [26,27,29] just work for the undirected graphs.
(23)
Note the fact that ξ = (L1 ⊗ In )δ and ∥x − 1 ⊗ x0 ∥ = ∥x − xˆ + xˆ − 1 ⊗ x0 ∥ ≤ ∥x − xˆ ∥ + ∥ˆx − 1 ⊗ x0 ∥ = ∥˜x∥ + ∥δ∥, and ∥ξ ∥ = ∥ξ − e + e∥ ≤ ∥e∥ + ∥e − ξ ∥, it is not difficult to get
1
Remark 4. In Theorem 6, the consensus protocols with adaptive update laws guarantee that all the signals in the closed-loop sysˆ i are uniformly ultimately tems including the parameter estimate W bounded. The results do not imply that the adaptive parameter estimates will converge to the real unknown parameters. If the persistent excitation(PE) condition and some other assumptions are satisfied, the parameter convergence may be obtained [25]. Without the PE condition, the consensus tracking object still can be achieved with the designed protocols. Remark 5. Compared with the related references [17,22,23], in which the unknown nonlinear functions in the followers are also linearly parameterized, the consensus tracking protocols in this paper are designed for the general linear multi-agent systems including the integrator-type system as a special case. Furthermore, the protocols are developed based on output information which is more practical than the state-feedback ones [19–21,25].
ˆ ⊗ H1 ](e − ξ ) + where Z = eT (IN ⊗ 12 H1 )e + γ1 (e − ξ )T [G(αˆ + β) x˜ T (IN ⊗ Qe )˜x, ς is a constant that satisfies 0 < ς < 1, and η ≤ min
Once the independent constants l, d have been chosen, Q > 0 depends continuously on the parameter µ. Note that µ = 0 corresponds to the Riccati inequality which can be easily computed to obtain the positive definite solution Q . Thus, there exists solution Q (µ) > 0 on the interval µ ∈ (0, µmax ), which defines the largest set within which there exists a positive definite solution for Q . By virtue of the Schur Complement Lemma, the parameterdependent Riccati inequality (24) can be transformed into the LMI (9). Moreover, the adaptive estimator (6) in Theorem 6 has been designed partially motivated by [32].
(24)
In this section, a simulation example is provided to illustrate the validity of the theoretical results. Consider a network of five followers and one leader whose dynamics are third-order integrators as 0 0 0
A=
1 0 0
0 1 , 0
0 0 , 1
B=
C = 1
0
0 .
The communication topology is given by Fig. 1 in which the node indexed by 0 is the leader and only the follower indexed by 1 is available to the leader’s information. It is easy to see that the communication graph contains a directed spanning tree. The basis functions in the linearly parameterized model (4) are defined by
ϕi (xi (t )) = [sin((i + 1)D1 xi (t )), cos((i + 1)D2 xi (t ))]T , i = 1, . . . , 5 where D1 = [1, 0], D2 = [0, 1]. Based on Section 3, we can get a full picture of the consensus tracking protocol with local observer and estimator for the ith
J. Sun et al. / Systems & Control Letters 87 (2016) 16–22
21
Fig. 1. Communication graph.
Fig. 3. The evolutions of tracking error xi2 − x02 .
Fig. 2. The evolutions of tracking error xi1 − x01 .
agent as
ˆ i ϕi (ˆxi ) ui = cK θi − W ˙ xˆ i = Axˆ i + cBK θi + L(C xˆ i − yi ) θ˙i = Aθi + cBK θi + L(C xˆ i − yi ) + ρ(αi + βi )BK (ei − ξi ), α˙ i = −σi (αi − 1) + (ei − ξi )T Γ (ei − ξi ), βi = (ei − ξi )T P −1 (ei − ξi ), W ˆ i ϕi (ˆxi )ϕiT (ˆxi ) − ki W ˆi ˆ˙ i = τi −(ˆyi − yi )ϕiT (ˆxi ) − 1 W 2µ
Fig. 4. The evolutions of tracking error xi3 − x03 .
where the parameters K , L, Γ , c , l, ρ, µ, d, σi , τi , ki in the consensus protocol are determined in the following part. By solving the matrix inequality (8) and (9) with the LMI toolbox of Matlab, the matrices K , L and Γ are given as K = −0.9194
−1.9056
−45.2909 −191.0680 , −112.3833 0.8453 1.7520 Γ = 1.7520 3.6314 2.3021 4.7715
−2.5039 , (25)
L=
2.3021 4.7715 , 6.2696
where the constants c = 1, l = 25, ρ = 1, µ = 20, d = 0.05. The initial states of agents are chosen randomly within [−20, 20], and the initial states of αi are chosen to satisfy αi (0) ≥ 1. The parameters in the adaptive protocol are taken as τi = 100, σi = 0.005, ki = 0.0008, i = 1, . . . , 5. The tracking consensus errors xi − x0 , i = 1, . . . , 5 are depicted in Figs. 2–4, denoting that the errors converge to a small neighborhood around the origin. Fig. 5 shows clearly that the coupling gains are bounded with αi ≥ 1, i = 1, . . . , 5.
Fig. 5. The evolutions of coupling gains αi (t ).
output-feedback consensus tracking protocols have been designed to guarantee that the tracking errors converge to a small neighborhood around the origin, and all the signals in the closedloop system are uniformly ultimately bounded. In addition, the consensus protocols are in fully distributed fashion for the control gains which are independent of the eigenvalues of the Laplacian matrix associated with the whole communication graph which is actually the global information. Finally, the validity of the theoretical results was illustrated through an example.
5. Conclusion Acknowledgments This paper has addressed the distributed consensus tracking problem for the linear multi-agent systems with unknown dynamics under general directed graphs. Distributed adaptive
The authors would like to thank the editor and all the anonymous reviewers for their valuable comments and suggestions. This
22
J. Sun et al. / Systems & Control Letters 87 (2016) 16–22
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