Adaptive output feedback consensus tracking for linear multi-agent ...
To appear in the International Journal of Control Vol. 00, No. 00, Month 20XX, 1–18
Publisher: Taylor & Francis Journal: International Journal of Control DOI: http://dx.doi.org/10.1080/00207179.2015.1015171
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RESEARCH ARTICLE
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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In this paper, the consensus tracking problem with unknown dynamics in the leader for the linear multiagent systems is addressed. Based on the relative output information among the agents, decentralized adaptive consensus protocols with static coupling gains are designed to guarantee that the consensus tracking errors converge to a small neighborhood around the origin and all the signals in the closed-loop dynamics are uniformly ultimately bounded. Moreover, the result is extended to the case with dynamic coupling gains which are independent of the eigenvalues of the Laplacian matrix. Both of the protocols with static and dynamic coupling gains are designed by using the relative outputs, which are more practical than the state-feedback ones. Finally, the theoretical results are verified through an example.
1.
Introduction
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Keywords: consensus tracking; cooperative control; output feedback; adaptive control; linear multi-agent systems
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In the past few years, distributed cooperative control of multi-agent systems has gained compelling interest partially motivated by its great potential applications in many scientific and engineering fields including flocking, coverage control, formation flying of satellites and unmanned air vehicles, cooperative manipulation of multiple robots ( Olfati-Saber (2006), Cortes et al. (2002), Xu (2003), Kuriki & Namerikawa (2014), Chung & Slotine (2009)), and so forth. Among the above research works, consensus problem is a fundamental issue meaning that a group of agents reach an agreement regarding some common value of interest which is the premise for many cooperative tasks. The main objective of the consensus control is to design distributed control protocols to achieve certain global task based on the local information. In the early pioneering works ( Vicsek et al. (1995), Olfati-Saber & Murray (2004)), Vicsek et al. proposed a simple model of autonomous agents, and Olfati et al. established a framework of consensus control for the first-order integrators with different topologies. Since then, many results have been done under different conditions such as finite-time consensus ( Khoo et al. (2009), Zhao et al. (2013), Meng & Lin (2013)), quantized
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Junyong Suna , Zhiyong Genga∗
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Adaptive output feedback consensus tracking for linear multi-agent systems with unknown dynamics
∗ Corresponding
author. Email: [email protected]
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consensus (Kashyap et al. (2007), Carli & Bullo (2009), Zhu & Martinez (2011)), consensus with time delay (Wang & Slotine (2006), Lin & Jia (2011), Wen et al. (2013), Wen et al. (2013b)), consensus via pinning control( Yu et al. (2012), Yu et al. (2013)) and consensus with different dynamics from the early first-order integrators to the recent Euler-Lagrangian systems ( Hu (2011), Chen et al. (2014) Wen et al. (2012), Yu et al. (2010), Li et al. (2010), Li et al. (2013), Wang et al. (2013), Diao et al. (2014), Mei et al. (2011), Zhao et al. (2014)), just to name a few. Among the above results, the consensus problems roughly fall into two cases, the case where there is no leader called leaderless consensus and the case where there exists a leader which determining the consensus value named leader-following consensus or consensus tracking. In this paper, we focus on the consensus tracking problem for the general linear multi-agent systems. Among the literature of consensus tracking, the authors in Hong et al. (2008) designed observer-based protocols for the double integrators to track a dynamic leader. In Peng & Yang (2009), the authors proposed an estimator for each single integrator to track a double integrator with time-varying velocity. In Hu (2011), the author considered the robust consensus tracking for the first-order dynamics with disturbances and unmodelled dynamics. Reference Cao & Ren (2012) studied the consensus tracking problem for first-order and second-order dynamics, respectively, via variable structure approach which results in discontinuous controllers. In Zhang et al. (2011), a framework of consensus tracking was proposed with both state feedback and dynamic output feedback. In Wen et al. (2014), the consensus tracking problem was addressed for the linear multiagent systems with Lipschitz-type nonlinear dynamics. Note that the dynamics of the agents in the references Hong et al. (2008), Peng & Yang (2009), Hu (2011), Cao & Ren (2012) focus on the first-order and second-order integrators which are special cases of the general linear multiagent systems. Moreover, in the aforementioned references including Zhang et al. (2011), Wen et al. (2014), the leader’s control input is assumed to be zero which may limit the applications, for example, it is not applicable for the leader to avoid hazardous obstacles. In addition, most of the controllers are relative state-feedback which is less practical compared with the output-feedback protocols. Motivated by the aforementioned observations, we consider the consensus tracking problem of linear multi-agent systems with unknown dynamics in the leader. Similarly, in Yu & Xia (2012) , an adaptive consensus protocol is proposed for the single-integrator systems by exactly linearly parameterizing the control input of the leader. The authors in Zhang & Lewis (2012) addressed the consensus tracking problem for high-order integrator-type systems with unknown dynamics which was parameterized by neural network, and the consensus errors were uniformly ultimately bounded. Also, by virtue of the neural network approach, the unknown dynamics was linearly parameterized, and both state-feedback and output feedback consensus protocols were designed in Peng et al. (2014) to solve the tracking problem. Recently, in Hu & Zheng (2014), an adaptive dynamic output-feedback consensus protocol was developed for the second-order integrators with unknown dynamics in the leader which was exactly linearly parameterized. The contributions of the present paper mainly lie in three aspects. Firstly, two new outputfeedback consensus protocols with adaptive update laws are proposed for general linear multiagent systems. Compared with Yu & Xia (2012), Zhang & Lewis (2012), Hu & Zheng (2014), the similar problems with unknown dynamics in the leader are mainly for the single, double, and high-order integrator-type multi-agent systems. In addition, the nonzero control input of the leader is in a linearly parameterized form resulting that the leader’s dynamics is different from the followers. Secondly, the consensus tracking protocols in most of the above references except Hu & Zheng (2014), Peng et al. (2014) are designed based on relative states of the neighbors which is not practical enough for that the state information is not always available in many situations. Based on the parameterized model, we have designed two different observers to obtain the state estimation for each follower which is used to develop the controller. Then, both the tracking error and estimation error are guaranteed to be uniformly ultimately bounded via the Lyapunov function method. Finally, the consensus protocol with dynamic coupling gains is in fully decentralized fashion. In references Hu & Zheng (2014), Peng et al. (2014), although the consensus protocols are developed
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by using the output information, the parameters in the controller depend on the eigenvalues of the Laplacian matrix associated with the whole information-exchange graph which is the global information and difficult to be known prior in the case of very large networks. The paper is organized as follows. In section 2, we introduce some preliminaries and the consensus tracking problem. In section 3, we present two adaptive dynamic output-feedback consensus protocols with static and dynamic coupling gains, respectively. A simulation is provided in section 4 and the concluding remarks are given in section 5.
2. 2.1
Preliminaries and problem statement
Notations
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2.2
Graph theory
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In this paper, the network consists of one leader and N followers among which the communication topology can be described by a graph G (V, E, A) in which V {0, 1, · · · , N } is the node set, E is the edge set and A = [aij ] ∈ R(N +1)×(N +1) is the associated adjacent matrix. In the directed graph G, (i, j) ∈ E means that agent j has access to the information from agent i, but not vice versa, while agent i and agent j can get information from each other, i.e. (i, j) ∈ E implies (j, i) ∈ E for the undirected graph. As to the adjacent matrix A = [aij ], aij = 1 if (j, i) ∈ E, otherwise is zero. In addition, it is assumed that aii = 0 for all nodes. The Laplacian matrix L = [Lij ] of the graph G is defined as Lii = N j=0,j=i aij and Lij = −aij , i = j. Obviously, the Laplacian matrix associated with an undirected graph is symmetric. Throughout this paper, let the agent indexed by 0 be the leader and the agents indexed by 1, · · · , N be the followers, and the information graph among them satisfies the following assumption. Assumption 1: Suppose that the graph G is fixed and connected. The communication topology among the N followers is undirected, and at least one follower has access to the information of the leader. Lemma 1: (Ren & Beard. (2005)) Under the Assumption 1, 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 the non-zero eigenvalues are positive real numbers.
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Let Rn×m and IN denote a set of n × m real matrices and the identity matrix of dimension N , respectively. Also, let 1 be a column vector with all entries equal to one. diag(Ai ) denotes a blockdiagonal matrix of Ai , i = 1, · · · , N. For a vector x ∈ Rn , x2 is the 2-norm, while for a matrix A, XF represents the Frobenius norm which has the definition that AF = tr(AT A) in which tr(·) is the trace of a matrix. Moreover, λmin (·), λmax (·) denote, respectively, the minimum and maximum eigenvalue of a matrix while σ(·), σ(·) represent the minimum and maximum singular value of a matrix, respectively. X ⊗ Y denotes the Kronecker product of matrices X and Y , and it has the following properties that λ(X ⊗ Y ) = {λi (X)λj (Y )} and σ(X ⊗ Y ) = {σi (X)σj (Y )}. The notation A > B, in which A and B are symmetric real matrices, denotes that A − B is positive definite.
It is assumed that the leader has no neighbors. Therefore, the Laplacian matrix L of G has the following structure 0 01×N , L= L2 L1
(1)
where L1 ∈ RN ×N and L2 ∈ RN ×1 . By virtue of Lemma 1, the Laplacian matrix L has only one
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zero eigenvalue, then we can see that L1 is symmetric and positive definite. 2.3
Problem statement
Consider a group of N + 1 linear systems, consisting of one leader labeled by 0 and N followers, whose dynamics are modeled as
x˙ i = Axi + Bui yi = Cxi
i = 0, 1, 2, · · · , N.
(2)
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where xi ∈ Rn denotes the state, ui ∈ Rm is the control input, yi ∈ Rq is the measured output, and A, B, C are constant matrices with compatible dimensions. For the leader-following systems, the control input u0 of the leader is parameterized as
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where W0 ∈ Rm×p is an unknown constant matrix to be adaptively estimated that satisfies W0 ≤ MW , ϕ0 (t) ∈ Rp is the base function vector with ϕ0 ≤ Mϕ , and ω0 is an unknown bounded vector with ω0 ≤ ε0 . The control objective of tracking problem in this paper is to design decentralized 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.
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Remark 1: The unknown control input of the leader can be seen as a nonlinear function 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 Slotine & Li. (1991), Marino & Tomei (1996) in classical adaptive control and Yu & Xia (2012), Hu & Zheng (2014) in multi-agent systems.
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3.1
Main results
Adaptive output feedback with static coupling
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In this section, based on the relative output information of neighboring agents, a distributed dynamic output-feedback controller with static coupling gain is proposed as follows N N ˙θi = (A + LC)θi + cBK[ aij (θi − θj ) + ai0 θi ] − L[ aij (yi − yj ) + ai0 (yi − y0 )] j=1
j=1
N ˆ i0 − W ˆ j0 ) + ai0 W ˆ i0 ]ϕ0 (t), + B[ aij (W
(3)
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u0 (t) = W0 ϕ0 (t) + ω0 ,
j=1
ˆ i0 ϕ0 , ui = cKθi + W
ˆ i0 is the ith follower’s where c ∈ R is the constant coupling gain, θi is the internal state, and W m×n n×q estimation of the unknown matrix W0 . K ∈ R and L ∈ R are the feedback gain matrices to be designed as K = −B T Q and Q is a positive definite solution to the following linear matrix inequality(LMI): AQ−1 + Q−1 AT − 2BB T < 0.
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(4)
Moreover, the update laws are designed as N
ˆ i0 ], ˆ˙ i0 = τi [−(QB)T ( W Lij θj )ϕT0 − kWi W
(5)
j=1
where kWi > 0 is a small design parameter and τi is the adaptive gain. Before moving forward, let δi = xi − x0 as the global tracking error, then with (2), we can get ˆ i0 ϕ0 − u0 ]. δ˙i (t) = Aδi (t) + B[cKθi + W N j=1
aij (xi − xj ) + ai0 (xi − x0 )
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where L1 is defined in (1). As L1 is nonsingular according to Lemma 1, which means ξ(t) is equivalent to δ(t), we use ξ(t) as the global tracking error hereafter. Then we obtain the dynamics of ξi based on (6) and (7) as
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N N ˆ i0 − W ˆ j0 ) + ai0 (W ˆ i0 − W0 )]ϕ0 − ai0 Bω0 . ξ˙i = Aξi + cBK[ aij (θi − θj ) + ai0 θi ] + B[ aij (W j=1
j=1
ξi Let ei = , i = 1, · · · , N. Using the controller above, we can obtain the closed-loop network θi dynamics as
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e˙ i = Mei + c
where
Lij Hej +
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j=1
N ¯ i, ˜ j0 ϕ0 ) − ai0 BΔ Lij B(W j=1
ω0 A 0 0 BK B B 0 ¯ M= ,H = ,B = ,B = , Δi = −W0 ϕ0 −LC A + LC 0 BK B 0 B
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ˆ j0 − W0 denotes the estimate error. ˜ j0 = W and W T Let e = [e1 , · · · , eTN ]T . Then the above equation can be written in the following compact form:
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ξ(t) = (L1 ⊗ In )δ(t),
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denotes the relative state measurement. Then we have the fact that
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Δ
T (t)]T , ξ(t) = [ξ T (t), · · · , ξ T (t)]T , where ξ = Let δ(t) = [δ1T (t), · · · , δN i 1 N
(6)
˜ 0 Φ0 ) − (G ⊗ B)Δ, ¯ e˙ = (IN ⊗ M + cL1 ⊗ H)e + (L1 ⊗ B)(W
(8)
˜ 0 = diag(W ˜ 10 , . . . , W ˜ N 0 ), Φ0 = col(ϕ0 , · · · , ϕ0 )∈ RpN ×1 , G = col(a10 , · · · , aN 0 ) and where W Δ = col(Δ1 , · · · , ΔN ).
Theorem 1: Suppose the Assumption 1 holds, and the feedback gain matrix L is designed such that A + LC is Hurwitz, the coupling gain c satisfies that c > λ1i , where λi is the ith eigenvalue of L1 with 0 < λ1 ≤ · · · ≤ λN , then with the dynamic output-feedback controller (3) and the adaptive ˆ i0 are uniformly ultimately bounded update law (5), all the closed-loop signals including ξi , θi , W
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and the consensus tracking error converges to a small neighborhood around the origin as
lim δ ≤
t→∞
h1 R1 , σ(L1 )σ(H1 ) r1 σ(P )
(9)
where H1 , h1 , R1 and r1 are defined in the following proof.
N i=1
1 ˜T ˜ trace( W Wi0 ), τi i0
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˜ −τ Q ˜ τQ T ˜ ˜ ˜ where P = ˜ Q + τQ ˜ , Q > 0 satisfies that Q(A + LC) + (A + LC) Q < 0 and τ is a positive −τ Q constant to be determined later. And it is easy to verify that P > 0 according to Schur Complement Lemma. Thus, V1 is positive definite. The time derivative of V1 along the trajectory of (8) is obtained as ˜ 0 Φ0 ) V˙ 1 = 2eT (IN ⊗ P )(IN ⊗ M + cL1 ⊗ H)e + 2eT (L1 ⊗ P B)(W N i=1
1 ˜ T ˜˙ ¯ Wi0 ) + 2eT (G ⊗ P B)Δ. trace( W τi i0
(11)
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I −I Let e˜ = (IN ⊗ T )e with T = . Then equation (11) can be rewritten as 0 I
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˜ + cL1 ⊗ P˜ H)˜ ¯ e + 2eT (L1 ⊗ P B)(W ˜ 0 Φ0 ) eT (IN ⊗ P˜ M V˙ 1 = 2˜ +2
N i=1
1 ˜ T ˜˙ ¯ trace( W Wi0 ) + 2eT (G ⊗ P B)Δ, τi i0
(12)
˜ 0 A + LC 0 ˜ 0 0 τQ −T −1 −1 −1 ˜ ˜ ,M = T MT = ,H = T HT = . where P = T P T = −LC A 0 BK 0 Q 0 And it is easy to note that P B = , then we can obtain QB
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(10)
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V1 = eT (IN ⊗ P )e +
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Proof. Consider the following Lyapunov function candidate
˜ 0 Φ0 ) = 2θT (L1 ⊗ QB)(W ˜ 0 Φ0 ) 2eT (L1 ⊗ P B)(W =2
N N ˜ i0 ϕ0 . ( Lij θjT )QB W i=1 j=1
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(13)
Then with the adaptive law (5), we have
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N N N 1 ˜ T ˜˙ T ˜ TW ˆ ˜ T (QB)T ( trace( W ) = −2 trace[ W L θ )ϕ ] − 2 trace[kWi W W i0 ij j i0 0 i0 i0 ] τi i0 i=1
j=1
i=1
N N ˜ i0 ϕ0 = −2 ( Lij θjT )QB W
(14)
i=1 j=1
−2
N
˜ TW ˜ trace[kWi W i0 i0 ] − 2
i=1
N
˜ T W0 ]. trace[kWi W i0
i=1
N
˜ TW ˜ trace[kWi W i0 i0 ] − 2
N
˜ T W0 ]. trace[kWi W i0
i=1
(15)
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Let U be such an unitary matrix that U T L1 U = ∧ = diag(λ1 , · · · , λN ), where 0 < λ1 ≤ · · · ≤ λN are the eigenvalues of L1 . Let e¯ = [¯ eT1 , · · · , e¯TN ]T = (U T ⊗ In )˜ e, then it follows ˜ + cL1 ⊗ P˜ H)˜ ¯ e = 2¯ ˜ + c ∧ ⊗P˜ H)¯ ˜ e 2˜ eT (IN ⊗ P˜ M eT (IN ⊗ P˜ M =
N
˜ ei . ˜ +M ˜ T P˜ + 2cλi P˜ H)¯ e¯Ti (P˜ M
(16)
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Note that
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˜ +M ˜ T P˜ + 2cλi P˜ H ˜ P˜ M −C T LT Q Π1 = = Ω1 , −QLC QA + AT Q − 2cλi QBB T Q
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(17)
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˜ If cλi ≥ 1, i = 1, · · · , N, it can be obtained that ˜ where Π1 = τ [Q(A + LC) + (A + LC)T Q]. T T QA+A Q−2cλi QBB Q < 0. By choosing τ > 0 suffciently large and in light of Schur Complement ˜ +M ˜ T P˜ + 2cλi P˜ H ˜ < 0. Lemma, we can obtain from (17) that P˜ M By virtue of (15) and (16), we get that ¯ eT e¯ + 2eT (G ⊗ P B)Δ V˙ 1 ≤ λmax (Ω1 )¯
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˜ + cL1 ⊗ P˜ H)˜ ¯ e + 2eT (G ⊗ P B)Δ ¯ eT (IN ⊗ P˜ M V˙ 1 = 2˜
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Substituting (13) and (14) into (12) gives
−2
N i=1
˜ TW ˜ trace[kWi W i0 i0 ] − 2
N
˜ T W0 ] trace[kWi W i0
(18)
i=1
˜ 0 2 + 2kW W ˜ 0 F MW + 2e¯ ¯ ≤ λmax (Ω1 )¯ eT e¯ − 2kW W σ (G)¯ σ (P B)Δ, F
where kW = mini=1,··· ,N {kWi }, kW = maxi=1,··· ,N {kWi }.
N T T In addition, because of Δi = ω0 (−W0 ϕ0 )T , we can obtain that Δ ≤ (ε0 + MW Mϕ )2 , i=1
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I −I , and e¯ = (U T ⊗ In )˜ and according to e˜ = (IN ⊗ T )e with T = e, (18) becomes 0 I
˜ 0 2 V˙ 1 ≤ λmax (Ω1 )σ(T T T )e2 − 2kW W F N
˜ ¯ + 2kW W0 F MW + 2e¯ σ (G)¯ σ (P B) (ε0 + MW Mϕ )2 −λmax (Ω1 )σ(T T T ) 0 e ˜ 0 F 0 2kW W
¯ + 2¯ σ (G)¯ σ (P B)
H1
N
i=1
(ε0 + MW Mϕ )2 2kW MW
h1
e ˜ 0 F . W
(19)
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˜ 0 F = − e W
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˜ 0 F T , it can It is easy to see that H1 is positive definite with kW > 0. Then define z1 = e W be obtained that V˙ 1 ≤ −z1T H1 z1 + h1 z1 ≤ −σ(H1 )z1 2 + h1 z1 .
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According to the equation (10), one can get that
r1 z1 2 ≤ V1 ≤ R1 z1 2 ,
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where
⎧ ⎨ r1 =
min (σ(P ), τ1i )
i=1,··· ,N
⎩ R1 = max (σ(P ), τ1i )
.
(20)
(21)
(22)
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i=1,··· ,N
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Then substituting (21) into (20) yields
Let l1 =
σ(H1 ) R1 , l2
=
h1 √ , r1
h1 σ(H1 ) V1 + √ V1 . V˙ 1 ≤ − r1 R1
(23)
and integrating both sides of (23), we have
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V1 ≤
l1
V1 (0)e− 2 t +
l1 l2 (1 − e− 2 t ). l1
(24)
ˆ i0 are It can be seen that limt→∞ V1 (t) = ll21 , thus all the closed-loop signals including ξi , θi , W uniformly ultimately bounded. Based on the Lyapunov function (10) and equation (7), the tracking error δ converges to a small neighborhood around the origin as lim δ ≤
t→∞
h1 R1 . σ(L1 )σ(H1 ) r1 σ(P )
This completes the proof.
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Remark 2: In the dynamic controller (3), a sufficient condition for the existence of the feedback gain K and L is that (A, B) is stabilizable and (A, C) is detectable to guarantee the existence of LMI(4) and A + LC to be Hurwitz. In addition, the parameters K,L,c in (3) are independent with each other which provides some freedom in designing the protocols. Note that the protocol (3) is essentially nonlinear which results in that the separation principle is not valid anymore, thus it is more challenging for the consensus tracking using output feedback. 3.2
Adaptive output feedback with dynamic coupling
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Note that in the above section, the parameter c in the dynamic controller (3) depends on the eigenvalues of L1 , which is not easy to get if the communication network is very large. Thus, in this section, we extend the results to the output feedback with dynamic coupling gains without knowing the eigenvalues priori. Then, a different adaptive consensus protocol with time-varying coupling gains is proposed as follows
i = 1, · · · , N.
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and the update law is
˜˙ i0 = τi [−(QB)T ( ˆ˙ i0 = W ˆ i0 ], Lij θj )ϕT0 − kWi W W
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j=1
(26)
(27)
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ˆ i0 is the ith where kWi > 0, σi > 0 are small design parameters and τi , νi are the adaptive gains, W N Δ ˜ i0 = W ˆ i0 −W0 , Θi = follower’s estimation of the unknown matrix W0 , and W aij (θi − θj ) + ai0 θi , j=1
in which θi is the state of the following observer based on the neighboring output information as
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N ˙θi = (A + LC)θi + Si − L[ aij (yi − yj ) + ai0 (yi − y0 )] N
j=1
(28)
te + B[
ˆ i0 ]ϕ0 (t) ˆ i0 − W ˆ j0 ) + ai0 W aij (W
where
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j=1
Si =
N j=1
aij [ci BK
N
Lil θl − cj BK
l=1
N k=1
Ljk θk ] + ai0 [ci BK
N
Lil θl ].
(29)
l=1
As in the above section, we define δi = xi − x0 as the global tracking error, then with the consensus protocol (26), we can get
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c˙i = νi [ΘTi ΓΘi − σi ci ],
cr
ˆ i0 ϕ0 (t) ui = ci (t)KΘi + W
δ˙i (t) = Aδi (t) + ci BK[
N
˜ i0 ϕ0 ] − Bω0 . aij (θi − θj ) + ai0 θi ] + B[W
(30)
j=1
From the equation (7), we get the dynamics of ξ as
ˆ 1 ⊗ BK)θ + (L1 ⊗ B)[W ˜ 0 Φ0 ] − (G ⊗ B)ω0 , ξ˙ = [IN ⊗ A]ξ + (L1 CL
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(31)
and the dynamics of the observer can be written in the compact form as
ˆ 1 ⊗ BK)θ − (IN ⊗ LC)ξ θ˙ = [IN ⊗ (A + LC)]θ + (L1 CL
(32)
˜ 0 Φ0 ]) + (G ⊗ B)[W0 Φ0 ], + (L1 ⊗ B)[W
˜ 0 , G and Φ0 are the same as those in the where W0 = diag(W0 , . . . , W0 ), and the definition of W last section. T Let e = ξ T θT , then we get the closed-loop network dynamics as ˜ 0 Φ0 + Ξ, e˙ = Me + He + J W where
(34)
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0 IN ⊗ A L1 ⊗ B M= ,J = , −IN ⊗ LC IN ⊗ (A + LC) L1 ⊗ B ˆ 1 ⊗ BK −(G ⊗ B)ω0 0 L1 CL , Ξ = . H= ˆ 1 ⊗ BK (G ⊗ B)[W0 Φ0 ] 0 L1 CL
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Theorem 2: Suppose the Assumption 1 holds, the parameters L, K, Γ in the dynamic outputfeedback controller are designed such that A + LC is Hurwitz, and K = −B T Q, Γ = QBB T Q in which Q > 0 is a solution of the LMI(4), then under the consensus protocol (26)(28) and adaptive ˆ i0 and ci are uniformly ultimately update law (27), all the closed-loop signals including ξi , θi , W bounded and the consensus tracking error δ converges to a small neighborhood around the origin as h2 R2 , σ (L1 )σ(H2 ) r2 σ(P )
(35)
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lim δ ≤
te
t→∞
where H2 , h2 , R2 and r2 are defined in the following proof.
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Proof. Consider the following Lyapunov function candidate
V2 = eT P e +
N i=1
N
1 1 ˜T ˜ trace( W (ci − α)2 , i0 Wi0 ) + τi νi
(36)
i=1
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(33)
˜ ˜ −τ IN ⊗ Q τ IN ⊗ Q T ˜ ˜ ˜ where P = ˜ IN ⊗ (Q + τ Q) ˜ , Q > 0 satisfies that Q(A + LC) + (A + LC) Q < 0 for −τ IN ⊗ Q A + LC is Hurwitz, and τ, α are positive constants. In light of Schur Complement Lemma, it can be obtained that matrix P > 0 which guarantees that the Lyapunov function V2 is positive definite. The time derivative of V2 along the trajectory of (33) is obtained as
˜ 0 Φ0 + Ξ] + 2 V˙ 2 = 2eT P [Me + He + J W
N i=1
10
N
1 1 ˜ T ˜˙ trace( W (ci − α)c˙i . i0 Wi0 ) + 2 τi νi i=1
(37)
IN ⊗ I −IN ⊗ I . Then equation (37) can be rewritten as Let e˜ = T e with T = 0 IN ⊗ I
˜ e + 2eT P J (W ˜ +M ˜ T P˜ + 2P˜ H]˜ ˜ 0 Φ0 ) + 2eT P Ξ V˙ 2 = e˜T [P˜ M +2
N i=1
N
1 1 ˜ T ˜˙ trace( W (ci − α)c˙i , i0 Wi0 ) + 2 νi τi
(38)
i=1
where
T
˜ e = e˜ ˜ + M P˜ + 2P˜ H]˜ e˜ [P˜ M
−IN ⊗ C T LT Q Π2 ˆ 1 ⊗ QBK e˜. −IN ⊗ QLC IN ⊗ (QA + AT Q) + 2L1 CL
us
˜T
N 1 (ci − α)c˙i νi i=1
N
ci ΘTi ΓΘi − 2α
i=1
ΘTi ΓΘi − 2
i=1
N
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≤2
N
N
d
=2
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2
j=1
N
σi (ci − α)ci
i=1
Lij θj )T QBB T Q(
N
Lij θj ) − 2
j=1
ce p
i=1
ci (
(40)
an
˜ ˜ + LC) + τ (A + LC)T Q]. where Π2 = IN ⊗ [τ Q(A Moreover, according to the protocol (26), we get
N
σi c˜2i − 2
N
i=1
(41) σi α˜ ci
i=1
N N N T T − 2α ( Lij θj ) QBB Q( Lij θj ), i=1 j=1
j=1
where c˜i = ci − α. On the other hand, from (40), we have
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Based on (39), it can be obtained that T
(39)
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˜ 0 0 I ⊗ τQ ˜ = T MT −1 = IN ⊗ (A + LC) ,M P˜ = T −T P T −1 = N , −IN ⊗ LC IN ⊗ A 0 IN ⊗ Q 0 0 −1 ˜ H = T HT = ˆ 1 ⊗ BK . 0 L1 CL
T
e˜
0 0 0 T T 0 ˆ 1 ⊗ QBK e˜ = e T ˆ 1 ⊗ QBK T e 0 2L1 CL 0 2L1 CL 0 T 0 =e ˆ 1 ⊗ QBK e 0 2L1 CL = −2
N
ci (
i=1
11
N j=1
Lij θj )T QBB T Q(
N j=1
(42) Lij θj ).
Substituting (41)(42) into (38) yields that
˜ 0 Φ0 ) + 2eT P Ξ V˙ 2 = e˜T Υ˜ e + 2eT P J (W +2
N
N
i=1
i=1
(43)
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i=1
N
1 ˜ T ˜˙ 2 trace( W ) − 2 σ c ˜ − 2 σi α˜ ci . W i0 i i τi i0
Π2 −IN ⊗ C T LT Q e¯ = e¯T Ω2 e¯, −IN ⊗ QLC IN ⊗ (QA + AT Q − 2αλ2i QBB T Q)
(44)
M
e = e¯ e˜ Υ˜
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−IN ⊗ C T LT Q Π2 where Υ = . If αλ2i ≥ 1, i = 1, · · · , N, it can −IN ⊗ QLC IN ⊗ (QA + AT Q) − 2αL21 ⊗ QBB T Q be obtained that QA + AT Q − 2αλ2i QBB T Q < 0. By choosing τ > 0 sufficiently large and in the light of Schur Complement Lemma, we can obtain that ¯eT Ω2 e¯ < 0. 0 , and by using the adaptive law (27), it is not difficult to obtain Note that P J = L1 ⊗ QB that
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Let U be such an unitary matrix that U T L1 U = ∧= diag(λ1 , · · · , λN), where 0 < λ1 ≤ · · · ≤ λN 0 U T ⊗ In e˜, then we can get are the eigenvalues of L1 . Let e¯ = [¯ eT1 , · · · , e¯TN ]T = 0 U T ⊗ In
˜ 0 Φ0 ) + 2 2eT P J (W
N i=1
=2
N
(
N
1 ˜ T ˜˙ trace( W Wi0 ) τi i0
˜ i0 ϕ0 Lij θjT )QB W
N N ˜ i0 ϕ0 −2 ( Lij θjT )QB W
i=1 j=1
−2
N
i=1 j=1
T ˜ ˜ i0 Wi0 ] − 2 trace[kWi W
N
i=1
i=1
12
T ˜ i0 trace[kWi W W0 ].
(45)
By using (43)(44)(45), one can conclude that V˙ 2 = e¯T Ω2 e¯ + 2eT P Ξ − 2
N
˜ TW ˜ trace[kWi W i0 i0 ]
i=1
−2
N
˜ T W0 ] − 2 trace[kWi W i0
i=1
N
σi c˜2i
−2
i=1
N
σi α˜ ci
i=1
˜ 0 2 + 2kW W ˜ 0 F MW ≤ λmax (Ω2 )σ(T T T )e2 − 2kW W F N + 2e¯ σ (P )¯ σ (G ⊗ B) (ε0 + MW Mϕ )2 − 2σ˜ c2 + 2σα˜ c ⎡ ⎤⎡ ⎤ T e −λmax (Ω2 )σ(T T ) 0 0 ˜ 0 F ⎦ ˜ 0 F ˜ = − e W 0 2kW 0 ⎦ ⎣ W c ⎣ 0 0 2σ ˜ c
ip
+ 2¯ σ (P )¯ σ (G ⊗ B)
N
i=1
(ε0 + MW Mϕ )2 2kW MW
⎡ e ⎤ ˜ 0 F ⎦ . 2σα ⎣ W ˜ c
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H2
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h2
Since λmax (Ω2 ) < 0, it is easy to obtain that matrix H2 is positive definite. Let z2 = T ˜ 0 F ˜ c , then we have e W
M
V˙ 2 ≤ −z2T H2 z2 + h2 z2 ≤ −σ(H2 )z2 2 + h2 z2 ,
(47)
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By virtue of the Lyapunov function V2 , one can obtain that
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r2 z2 2 ≤ V2 ≤ R2 z2 2 ,
⎧ ⎨ r2 =
ce p
where
min (σ(P ), τ1i , ν1i )
i=1,··· ,N
⎩ R2 = max (σ(P ), τ1i , ν1i )
(48)
.
i=1,··· ,N
Combining (47) and (48) yields
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(46)
t
i=1
Let s1 =
σ(H2 ) R2 ,
s2 =
h2 √ , r2
σ(H2 ) h2 V˙ 2 ≤ − V2 . V2 + √ r2 R2
(49)
and integrating the inequality (49), one can get
V2 ≤
s1
V2 (0)e− 2 t +
s1 s2 (1 − e− 2 t ). s1
(50)
So we have V2 (t) → ss21 as t → ∞. Based on the Lyapunov function V2 , it can be concluded that ˆ i0 and ci are uniformly ultimately bounded. all the signals in the closed loop such as ξi , θi , W
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Moreover, in light of the fact that V2 ≥ σ(P )e2 ≥ σ(P )ξ2 and ξ(t) = (L1 ⊗ In )δ(t), it is not difficult to get that the tracking error converges to a small neighborhood around the origin as lim δ ≤
t→∞
h2 R2 . σ(L1 )σ(H2 ) r2 σ(P )
(51)
Thus the proof is completed.
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Remark 3: Compared with the related references Yu & Xia (2012), Hu & Zheng (2014) in which the leader’s control inputs 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 relative output information which is more practical than the ones with relative states.
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4.
Simulation
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In this section, an example is given to demonstrate the validity of the above theoretical results. In the example, it is assumed that there are five followers and one leader whose dynamics are presented as 01 0 A= ,B = ,C = 1 0 . 00 0.1
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The communication topology among the agents is illustrated in Figure 1 in which one leader indexed by 0 and five followers are involved. The adaptive consensus protocols in Theorem 2 are used in this example. The base function is defined by ϕ0 = [sin(t), cos(t)]T for simplicity. The matrix L are T designed as L = −16.00 −60.00 to guarantee that A + LC is Hurwitz. By solving the matrix inequality (4), the feedback gain matrices Γ and K are given as
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Remark 4: By virtue of the relative output information among the agents, the protocols (26)(27) with dynamic coupling gains in this section are designed in fully decentralized fashion. In references Hu & Zheng (2014), Peng et al. (2014) the consensus tracking protocols are designed using output information. However, a common limit of them is that the coupling gain in the controller of each agent is an identical constant, which is also determined by the eigenvalues of the Laplacian matrix. Compared with these two references, the coupling gain ci in the protocol (26) is adaptively updated based on the local information, independent of the global information that the eigenvalues of the Laplacian matrix associated with the whole communication graph. But the tradeoff is that the observer (28) needs the two-hop neighbors’ information, which means that more information interaction and communication load are required. In the framework of this paper, the reason why the two-hop information is needed is that the observer (28) keeps similar structure with the dynamics of tracking error ξ. It is interesting to extend the result to the one-hop case with dynamic coupling gains in the future work.
0.0270 0.4314 Γ= , K = −0.1644 −2.6247 . 0.4314 6.8890
The initial states of the six agents are chosen randomly within [−25, 25], and the parameters in the consensus protocol are taken as τi = 10, kwi = 0.08, σi = 0.008 and νi = 1, i = 1, · · · , 5. Clearly, it can be seen that the coupling gain ci is bounded from the Figure 2. The evolution of the tracking errors δi1 = x01 − xi1 and δi2 = x02 − xi2 are described in Figure3 and Figure 4, respectively, which show that the tracking errors are bounded by a small neighborhood around the origin. In addition,
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ˆ i0 , i = 1, · · · , 5 of the unknown Figure 5 and Figure 6 tell clearly that the parameter estimations W constant W0 are bounded.
5.
Conclusion
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This paper has addressed the consensus tracking problem for the linear multi-agent systems with unknown dynamics in the leader. Firstly, a decentralized adaptive output-feedback consensus protocol with static coupling gains has been proposed which is more practical than the state-feedback case for that only the relative output information has been utilized. Then, the result has been extended to the case with dynamic coupling gains which is in fully decentralized fashion without using the eigenvalues of the Laplacian matrix which is actually the global information of the whole network. Both of the protocols have guaranteed that the tracking errors converge to a small neighborhood of the origin. Finally, an example was given to illustrate the effectiveness of the methods proposed in this paper.
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The authors would like to thank the editor and all the anonymous reviewers for their valuable comments and suggestions. This work is supported by the National Nature Science Foundation(NNSF) of China under Grant 61374033. The authors would like to acknowledge Dr. Zhongkui Li for his inspiring discussions.
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References
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Cortes, J., Martinez, S., Karatas, T., & Bullo, F. (2002). Coverage control for mobile sensing networks. IEEE International Conference on Robotics and Automation, 2 , 1327-1332. Carli, R., & Bullo, F. (2009). Quantized coordination algorithms for rendezvous and deployment. SIAM Journal on Control and Optimization, 48 (3), 1251-1274. Chung, S., & Slotine, J. (2009). Cooperative robot control and concurrent synchronization of Lagrangian systems. IEEE Transactions on Robotics, 25 (3), 686-700. Cao, Y., & Ren, W. (2012). Distributed coordinated tracking with reduced interaction via a variable structure approach. IEEE Transactions on Automatic Control, 57 (1), 33-48. Chen, X., Hao, F., & Rahmani, A. (2014). Asynchronous decentralised event-triggered control of multi-agent systems. International Journal of Control, 87 (10), 2130-2139. Diao, M., Duan, Z., & Wen, G. (2014). Consensus tracking of linear multi-agent systems under networked observability conditions. International Journal of Control, 87 (8), 1478-1486. Hong, Y., Chen, G., & Bushnell, L. (2008). Distributed observers design for leader-following control of multi-agent networks. Automatica, 44 (3), 846-850. Hu, G. (2011). Robust consensus tracking for an integrator-type multi-agent system with disturbances and unmodelled dynamics. International Journal of control, 84 (1), 1-8. Hu, J., & Zheng, W. X. (2014). Adaptive tracking control of leader-follower systems with unknown dynamics and partial measurements. Automatica, 50 (5), 1416-1423. Kashyap, A., Basar, T., & Srikant, R. (2007). Quantized consensus. Automatica, 43 (7), 1192-1203. Khoo, S., Xie, L., & Man, Z. (2009). Robust finite-time consensus tracking algorithm for multirobot systems. IEEE/ASME Transactions on Mechatronics, 14 (2), 219-228. Kuriki, Y., & Namerikawa, T. (2014). Consensus-based cooperative formation control with collision avoidance for a multi-UAV system. In American Control Conference, 2077-2082. Lin, P., & Jia, Y. (2011). Multi-agent consensus with diverse time-delays and jointly-connected topologies. Automatica, 47 (4), 848-856. Li, Z., Duan, Z., Chen, G., & Huang, L. (2010). Consensus of multiagent systems and synchronization of com-
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plex networks: a unified viewpoint. IEEE Transactions on Circuits and Systems I: Regular Papers,57 (1), 213-224. Li, Z., Ren, W., Liu, X., & Fu, M. (2013). Consensus of multi-agent systems with general linear and Lipschitz nonlinear dynamics using distributed adaptive protocols. IEEE Transactions on Automatic Control, 58 (7), 1786-1791. Marino, R., & Tomei, P. (1996). Nonlinear control design: geometric, adaptive and robust. Prentice Hall International (UK) Ltd.. Mei, J., Ren, W., & Ma, G. (2011). Distributed coordinated tracking with a dynamic leader for multiple Euler-Lagrange systems. IEEE Transactions on Automatic Control, 56 (6), 1415-1421. Meng, Z., & Lin, Z. (2013). On distributed finite-time observer design and finite-time coordinated tracking of multiple double integrator systems via local interactions. International Journal of Robust and Nonlinear Control. DOI: 10.1002/rnc.3004 Olfati-Saber, R., & Murray, R. (2004) Consensu problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49 (9), 1520-1533. Olfati-Saber, R. (2006). Flocking for multi-agent dynamic systems: Algorithms and theory. IEEE Transactions on Automatic Control, 51 (3), 401-420. Peng, K., & Yang, Y. (2009). Leader-following consensus problem with a varying-velocity leader and timevarying delays. Physica A: Statistical Mechanics and its Applications, 388 (2), 193-208. Peng, Z., Wang, D., Zhang, H., & Sun, G. (2014). Distributed Neural Network Control for Adaptive Synchronization of Uncertain Dynamical Multiagent Systems. IEEE Transactions on Neural Networks and Learning Systems, 25 (8), 1508-1519. Ren, W., & Beard, R. W. (2005). Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Transactions on Automatic Control, 50 (5), 655-661. Slotine, J. J. E., & Li, W. (1991). Applied nonlinear control. Englewood Cliffs, NJ: Prentice-Hall. Vicsek, T., Czirk, A., Ben-Jacob, E., Cohen, I., & Shochet, O. (1995). Novel type of phase transition in a system of self-driven particles. Physical Review Letters, 75 (6), 1226-1229. Wang, W., & Slotine, J. (2006). Contraction analysis of time-delayed communications and group cooperation. IEEE Transactions on Automatic Control, 51 (4), 712-717. Wen, G., Duan, Z., Li, Z., & Chen, G. (2012). Consensus and its L2 -gain performance of multi-agent systems with intermittent information transmissions. International Journal of Control, 85 (4), 384-396. Wen, G., Duan, Z., Yu, W., & Chen, G. (2013). Consensus of second-order multi-agent systems with delayed nonlinear dynamics and intermittent communications. International Journal of Control, 86 (2), 322-331. Wang, J., Duan, Z., Zhao, Y., Qin, G., & Yan, Y. (2013). H∞ and H2 control of multi-agent systems with transient performance improvement. International Journal of Control, 86 (12), 2131-2145. Wen, G., Duan, Z., Yu, W., & Chen, G. (2013). Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach. International Journal of Robust and Nonlinear Control, 23 (6), 602-619. Wen, G., Duan, Z., Chen, G., & Yu, W. (2014). Consensus tracking of multi-agent systems with Lipschitztype node dynamics and switching topologies. IEEE Transactions on Circuits and Systems I: Regular Papers, 61 (2), 499-511. Xu Y. (2003). Dynamics and control for formation flying system. University of Florida, USA. Yu, W., Chen, G., & Cao, M. (2010). Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems. Automatica, 46 (6), 1089-1095. Yu, W., DeLellis, P., Chen, G., di Bernardo, M., & Kurths, J. (2012). Distributed adaptive control of synchronization in complex networks. IEEE Transactions on Automatic Control, 57 (8), 2153-2158. Yu, H., & Xia, X. (2012). Adaptive consensus of multi-agents in networks with jointly connected topologies. Automatica, 48 (8), 1783-1790. Yu, W., Chen, G., Lu, J., & Kurths, J. (2013). Synchronization via pinning control on general complex networks. SIAM Journal on Control and Optimization, 51 (2), 1395-1416. Zhang, H., Lewis, F. L., & Das, A. (2011). Optimal design for synchronization of cooperative systems: state feedback, observer and output feedback. IEEE Transactions on Automatic Control, 56 (8), 1948-1952. Zhang, H., & Lewis, F. L. (2012). Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics. Automatica, 48 (7), 1432-1439. Zhu, M., & Martinez, S. (2011). On the convergence time of asynchronous distributed quantized averaging algorithms. IEEE Transactions on Automatic Control, 56 (2), 386-390. Zhao, Y., Duan, Z., Wen, G., & Zhang, Y. (2013). Distributed finite-time tracking control for multi-agent
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systems: an observer-based approach. Systems & Control Letters, 62 (1), 22-28. Zhao, Y., Duan, Z., & Wen, G. (2014). Distributed finite-time tracking of multiple Euler-Lagrange systems without velocity measurements. International Journal of Robust and Nonlinear Control, DOI: 10.1002/rnc.3170.
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Figure 1. Communication topology.
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Figure 2. The evolutions of coupling gains.
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Figure 3. The evolutions of tracking errors x01 − xi1 .
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Figure 4. The evolutions of tracking errors x02 − xi2 .
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ˆ i0 . Figure 5. The evolutions of parameter estimation W 1
ˆ i0 . Figure 6. The evolutions of parameter estimation W 2
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Publisher: Taylor & Francis Journal: International Journal of Control DOI: http://dx.doi.org/10.1080/00207179.2015.1015171
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RESEARCH ARTICLE
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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In this paper, the consensus tracking problem with unknown dynamics in the leader for the linear multiagent systems is addressed. Based on the relative output information among the agents, decentralized adaptive consensus protocols with static coupling gains are designed to guarantee that the consensus tracking errors converge to a small neighborhood around the origin and all the signals in the closed-loop dynamics are uniformly ultimately bounded. Moreover, the result is extended to the case with dynamic coupling gains which are independent of the eigenvalues of the Laplacian matrix. Both of the protocols with static and dynamic coupling gains are designed by using the relative outputs, which are more practical than the state-feedback ones. Finally, the theoretical results are verified through an example.
1.
Introduction
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Keywords: consensus tracking; cooperative control; output feedback; adaptive control; linear multi-agent systems
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In the past few years, distributed cooperative control of multi-agent systems has gained compelling interest partially motivated by its great potential applications in many scientific and engineering fields including flocking, coverage control, formation flying of satellites and unmanned air vehicles, cooperative manipulation of multiple robots ( Olfati-Saber (2006), Cortes et al. (2002), Xu (2003), Kuriki & Namerikawa (2014), Chung & Slotine (2009)), and so forth. Among the above research works, consensus problem is a fundamental issue meaning that a group of agents reach an agreement regarding some common value of interest which is the premise for many cooperative tasks. The main objective of the consensus control is to design distributed control protocols to achieve certain global task based on the local information. In the early pioneering works ( Vicsek et al. (1995), Olfati-Saber & Murray (2004)), Vicsek et al. proposed a simple model of autonomous agents, and Olfati et al. established a framework of consensus control for the first-order integrators with different topologies. Since then, many results have been done under different conditions such as finite-time consensus ( Khoo et al. (2009), Zhao et al. (2013), Meng & Lin (2013)), quantized
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Junyong Suna , Zhiyong Genga∗
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Adaptive output feedback consensus tracking for linear multi-agent systems with unknown dynamics
∗ Corresponding
author. Email: [email protected]
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consensus (Kashyap et al. (2007), Carli & Bullo (2009), Zhu & Martinez (2011)), consensus with time delay (Wang & Slotine (2006), Lin & Jia (2011), Wen et al. (2013), Wen et al. (2013b)), consensus via pinning control( Yu et al. (2012), Yu et al. (2013)) and consensus with different dynamics from the early first-order integrators to the recent Euler-Lagrangian systems ( Hu (2011), Chen et al. (2014) Wen et al. (2012), Yu et al. (2010), Li et al. (2010), Li et al. (2013), Wang et al. (2013), Diao et al. (2014), Mei et al. (2011), Zhao et al. (2014)), just to name a few. Among the above results, the consensus problems roughly fall into two cases, the case where there is no leader called leaderless consensus and the case where there exists a leader which determining the consensus value named leader-following consensus or consensus tracking. In this paper, we focus on the consensus tracking problem for the general linear multi-agent systems. Among the literature of consensus tracking, the authors in Hong et al. (2008) designed observer-based protocols for the double integrators to track a dynamic leader. In Peng & Yang (2009), the authors proposed an estimator for each single integrator to track a double integrator with time-varying velocity. In Hu (2011), the author considered the robust consensus tracking for the first-order dynamics with disturbances and unmodelled dynamics. Reference Cao & Ren (2012) studied the consensus tracking problem for first-order and second-order dynamics, respectively, via variable structure approach which results in discontinuous controllers. In Zhang et al. (2011), a framework of consensus tracking was proposed with both state feedback and dynamic output feedback. In Wen et al. (2014), the consensus tracking problem was addressed for the linear multiagent systems with Lipschitz-type nonlinear dynamics. Note that the dynamics of the agents in the references Hong et al. (2008), Peng & Yang (2009), Hu (2011), Cao & Ren (2012) focus on the first-order and second-order integrators which are special cases of the general linear multiagent systems. Moreover, in the aforementioned references including Zhang et al. (2011), Wen et al. (2014), the leader’s control input is assumed to be zero which may limit the applications, for example, it is not applicable for the leader to avoid hazardous obstacles. In addition, most of the controllers are relative state-feedback which is less practical compared with the output-feedback protocols. Motivated by the aforementioned observations, we consider the consensus tracking problem of linear multi-agent systems with unknown dynamics in the leader. Similarly, in Yu & Xia (2012) , an adaptive consensus protocol is proposed for the single-integrator systems by exactly linearly parameterizing the control input of the leader. The authors in Zhang & Lewis (2012) addressed the consensus tracking problem for high-order integrator-type systems with unknown dynamics which was parameterized by neural network, and the consensus errors were uniformly ultimately bounded. Also, by virtue of the neural network approach, the unknown dynamics was linearly parameterized, and both state-feedback and output feedback consensus protocols were designed in Peng et al. (2014) to solve the tracking problem. Recently, in Hu & Zheng (2014), an adaptive dynamic output-feedback consensus protocol was developed for the second-order integrators with unknown dynamics in the leader which was exactly linearly parameterized. The contributions of the present paper mainly lie in three aspects. Firstly, two new outputfeedback consensus protocols with adaptive update laws are proposed for general linear multiagent systems. Compared with Yu & Xia (2012), Zhang & Lewis (2012), Hu & Zheng (2014), the similar problems with unknown dynamics in the leader are mainly for the single, double, and high-order integrator-type multi-agent systems. In addition, the nonzero control input of the leader is in a linearly parameterized form resulting that the leader’s dynamics is different from the followers. Secondly, the consensus tracking protocols in most of the above references except Hu & Zheng (2014), Peng et al. (2014) are designed based on relative states of the neighbors which is not practical enough for that the state information is not always available in many situations. Based on the parameterized model, we have designed two different observers to obtain the state estimation for each follower which is used to develop the controller. Then, both the tracking error and estimation error are guaranteed to be uniformly ultimately bounded via the Lyapunov function method. Finally, the consensus protocol with dynamic coupling gains is in fully decentralized fashion. In references Hu & Zheng (2014), Peng et al. (2014), although the consensus protocols are developed
2
by using the output information, the parameters in the controller depend on the eigenvalues of the Laplacian matrix associated with the whole information-exchange graph which is the global information and difficult to be known prior in the case of very large networks. The paper is organized as follows. In section 2, we introduce some preliminaries and the consensus tracking problem. In section 3, we present two adaptive dynamic output-feedback consensus protocols with static and dynamic coupling gains, respectively. A simulation is provided in section 4 and the concluding remarks are given in section 5.
2. 2.1
Preliminaries and problem statement
Notations
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2.2
Graph theory
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In this paper, the network consists of one leader and N followers among which the communication topology can be described by a graph G (V, E, A) in which V {0, 1, · · · , N } is the node set, E is the edge set and A = [aij ] ∈ R(N +1)×(N +1) is the associated adjacent matrix. In the directed graph G, (i, j) ∈ E means that agent j has access to the information from agent i, but not vice versa, while agent i and agent j can get information from each other, i.e. (i, j) ∈ E implies (j, i) ∈ E for the undirected graph. As to the adjacent matrix A = [aij ], aij = 1 if (j, i) ∈ E, otherwise is zero. In addition, it is assumed that aii = 0 for all nodes. The Laplacian matrix L = [Lij ] of the graph G is defined as Lii = N j=0,j=i aij and Lij = −aij , i = j. Obviously, the Laplacian matrix associated with an undirected graph is symmetric. Throughout this paper, let the agent indexed by 0 be the leader and the agents indexed by 1, · · · , N be the followers, and the information graph among them satisfies the following assumption. Assumption 1: Suppose that the graph G is fixed and connected. The communication topology among the N followers is undirected, and at least one follower has access to the information of the leader. Lemma 1: (Ren & Beard. (2005)) Under the Assumption 1, 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 the non-zero eigenvalues are positive real numbers.
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Let Rn×m and IN denote a set of n × m real matrices and the identity matrix of dimension N , respectively. Also, let 1 be a column vector with all entries equal to one. diag(Ai ) denotes a blockdiagonal matrix of Ai , i = 1, · · · , N. For a vector x ∈ Rn , x2 is the 2-norm, while for a matrix A, XF represents the Frobenius norm which has the definition that AF = tr(AT A) in which tr(·) is the trace of a matrix. Moreover, λmin (·), λmax (·) denote, respectively, the minimum and maximum eigenvalue of a matrix while σ(·), σ(·) represent the minimum and maximum singular value of a matrix, respectively. X ⊗ Y denotes the Kronecker product of matrices X and Y , and it has the following properties that λ(X ⊗ Y ) = {λi (X)λj (Y )} and σ(X ⊗ Y ) = {σi (X)σj (Y )}. The notation A > B, in which A and B are symmetric real matrices, denotes that A − B is positive definite.
It is assumed that the leader has no neighbors. Therefore, the Laplacian matrix L of G has the following structure 0 01×N , L= L2 L1
(1)
where L1 ∈ RN ×N and L2 ∈ RN ×1 . By virtue of Lemma 1, the Laplacian matrix L has only one
3
zero eigenvalue, then we can see that L1 is symmetric and positive definite. 2.3
Problem statement
Consider a group of N + 1 linear systems, consisting of one leader labeled by 0 and N followers, whose dynamics are modeled as
x˙ i = Axi + Bui yi = Cxi
i = 0, 1, 2, · · · , N.
(2)
ip
t
where xi ∈ Rn denotes the state, ui ∈ Rm is the control input, yi ∈ Rq is the measured output, and A, B, C are constant matrices with compatible dimensions. For the leader-following systems, the control input u0 of the leader is parameterized as
an
us
cr
where W0 ∈ Rm×p is an unknown constant matrix to be adaptively estimated that satisfies W0 ≤ MW , ϕ0 (t) ∈ Rp is the base function vector with ϕ0 ≤ Mϕ , and ω0 is an unknown bounded vector with ω0 ≤ ε0 . The control objective of tracking problem in this paper is to design decentralized 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.
M
Remark 1: The unknown control input of the leader can be seen as a nonlinear function 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 Slotine & Li. (1991), Marino & Tomei (1996) in classical adaptive control and Yu & Xia (2012), Hu & Zheng (2014) in multi-agent systems.
d
3.1
Main results
Adaptive output feedback with static coupling
te
3.
ce p
In this section, based on the relative output information of neighboring agents, a distributed dynamic output-feedback controller with static coupling gain is proposed as follows N N ˙θi = (A + LC)θi + cBK[ aij (θi − θj ) + ai0 θi ] − L[ aij (yi − yj ) + ai0 (yi − y0 )] j=1
j=1
N ˆ i0 − W ˆ j0 ) + ai0 W ˆ i0 ]ϕ0 (t), + B[ aij (W
(3)
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u0 (t) = W0 ϕ0 (t) + ω0 ,
j=1
ˆ i0 ϕ0 , ui = cKθi + W
ˆ i0 is the ith follower’s where c ∈ R is the constant coupling gain, θi is the internal state, and W m×n n×q estimation of the unknown matrix W0 . K ∈ R and L ∈ R are the feedback gain matrices to be designed as K = −B T Q and Q is a positive definite solution to the following linear matrix inequality(LMI): AQ−1 + Q−1 AT − 2BB T < 0.
4
(4)
Moreover, the update laws are designed as N
ˆ i0 ], ˆ˙ i0 = τi [−(QB)T ( W Lij θj )ϕT0 − kWi W
(5)
j=1
where kWi > 0 is a small design parameter and τi is the adaptive gain. Before moving forward, let δi = xi − x0 as the global tracking error, then with (2), we can get ˆ i0 ϕ0 − u0 ]. δ˙i (t) = Aδi (t) + B[cKθi + W N j=1
aij (xi − xj ) + ai0 (xi − x0 )
cr
(7)
us
where L1 is defined in (1). As L1 is nonsingular according to Lemma 1, which means ξ(t) is equivalent to δ(t), we use ξ(t) as the global tracking error hereafter. Then we obtain the dynamics of ξi based on (6) and (7) as
an
N N ˆ i0 − W ˆ j0 ) + ai0 (W ˆ i0 − W0 )]ϕ0 − ai0 Bω0 . ξ˙i = Aξi + cBK[ aij (θi − θj ) + ai0 θi ] + B[ aij (W j=1
j=1
ξi Let ei = , i = 1, · · · , N. Using the controller above, we can obtain the closed-loop network θi dynamics as
M
N
d
e˙ i = Mei + c
where
Lij Hej +
te
j=1
N ¯ i, ˜ j0 ϕ0 ) − ai0 BΔ Lij B(W j=1
ω0 A 0 0 BK B B 0 ¯ M= ,H = ,B = ,B = , Δi = −W0 ϕ0 −LC A + LC 0 BK B 0 B
ce p
ˆ j0 − W0 denotes the estimate error. ˜ j0 = W and W T Let e = [e1 , · · · , eTN ]T . Then the above equation can be written in the following compact form:
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ξ(t) = (L1 ⊗ In )δ(t),
ip
denotes the relative state measurement. Then we have the fact that
t
Δ
T (t)]T , ξ(t) = [ξ T (t), · · · , ξ T (t)]T , where ξ = Let δ(t) = [δ1T (t), · · · , δN i 1 N
(6)
˜ 0 Φ0 ) − (G ⊗ B)Δ, ¯ e˙ = (IN ⊗ M + cL1 ⊗ H)e + (L1 ⊗ B)(W
(8)
˜ 0 = diag(W ˜ 10 , . . . , W ˜ N 0 ), Φ0 = col(ϕ0 , · · · , ϕ0 )∈ RpN ×1 , G = col(a10 , · · · , aN 0 ) and where W Δ = col(Δ1 , · · · , ΔN ).
Theorem 1: Suppose the Assumption 1 holds, and the feedback gain matrix L is designed such that A + LC is Hurwitz, the coupling gain c satisfies that c > λ1i , where λi is the ith eigenvalue of L1 with 0 < λ1 ≤ · · · ≤ λN , then with the dynamic output-feedback controller (3) and the adaptive ˆ i0 are uniformly ultimately bounded update law (5), all the closed-loop signals including ξi , θi , W
5
and the consensus tracking error converges to a small neighborhood around the origin as
lim δ ≤
t→∞
h1 R1 , σ(L1 )σ(H1 ) r1 σ(P )
(9)
where H1 , h1 , R1 and r1 are defined in the following proof.
N i=1
1 ˜T ˜ trace( W Wi0 ), τi i0
an
us
cr
˜ −τ Q ˜ τQ T ˜ ˜ ˜ where P = ˜ Q + τQ ˜ , Q > 0 satisfies that Q(A + LC) + (A + LC) Q < 0 and τ is a positive −τ Q constant to be determined later. And it is easy to verify that P > 0 according to Schur Complement Lemma. Thus, V1 is positive definite. The time derivative of V1 along the trajectory of (8) is obtained as ˜ 0 Φ0 ) V˙ 1 = 2eT (IN ⊗ P )(IN ⊗ M + cL1 ⊗ H)e + 2eT (L1 ⊗ P B)(W N i=1
1 ˜ T ˜˙ ¯ Wi0 ) + 2eT (G ⊗ P B)Δ. trace( W τi i0
(11)
M
+2
te
d
I −I Let e˜ = (IN ⊗ T )e with T = . Then equation (11) can be rewritten as 0 I
ce p
˜ + cL1 ⊗ P˜ H)˜ ¯ e + 2eT (L1 ⊗ P B)(W ˜ 0 Φ0 ) eT (IN ⊗ P˜ M V˙ 1 = 2˜ +2
N i=1
1 ˜ T ˜˙ ¯ trace( W Wi0 ) + 2eT (G ⊗ P B)Δ, τi i0
(12)
˜ 0 A + LC 0 ˜ 0 0 τQ −T −1 −1 −1 ˜ ˜ ,M = T MT = ,H = T HT = . where P = T P T = −LC A 0 BK 0 Q 0 And it is easy to note that P B = , then we can obtain QB
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(10)
ip
V1 = eT (IN ⊗ P )e +
t
Proof. Consider the following Lyapunov function candidate
˜ 0 Φ0 ) = 2θT (L1 ⊗ QB)(W ˜ 0 Φ0 ) 2eT (L1 ⊗ P B)(W =2
N N ˜ i0 ϕ0 . ( Lij θjT )QB W i=1 j=1
6
(13)
Then with the adaptive law (5), we have
2
N i=1
N N N 1 ˜ T ˜˙ T ˜ TW ˆ ˜ T (QB)T ( trace( W ) = −2 trace[ W L θ )ϕ ] − 2 trace[kWi W W i0 ij j i0 0 i0 i0 ] τi i0 i=1
j=1
i=1
N N ˜ i0 ϕ0 = −2 ( Lij θjT )QB W
(14)
i=1 j=1
−2
N
˜ TW ˜ trace[kWi W i0 i0 ] − 2
i=1
N
˜ T W0 ]. trace[kWi W i0
i=1
N
˜ TW ˜ trace[kWi W i0 i0 ] − 2
N
˜ T W0 ]. trace[kWi W i0
i=1
(15)
us
i=1
an
Let U be such an unitary matrix that U T L1 U = ∧ = diag(λ1 , · · · , λN ), where 0 < λ1 ≤ · · · ≤ λN are the eigenvalues of L1 . Let e¯ = [¯ eT1 , · · · , e¯TN ]T = (U T ⊗ In )˜ e, then it follows ˜ + cL1 ⊗ P˜ H)˜ ¯ e = 2¯ ˜ + c ∧ ⊗P˜ H)¯ ˜ e 2˜ eT (IN ⊗ P˜ M eT (IN ⊗ P˜ M =
N
˜ ei . ˜ +M ˜ T P˜ + 2cλi P˜ H)¯ e¯Ti (P˜ M
(16)
M
j=1
Note that
d
˜ +M ˜ T P˜ + 2cλi P˜ H ˜ P˜ M −C T LT Q Π1 = = Ω1 , −QLC QA + AT Q − 2cλi QBB T Q
te
(17)
ce p
˜ If cλi ≥ 1, i = 1, · · · , N, it can be obtained that ˜ where Π1 = τ [Q(A + LC) + (A + LC)T Q]. T T QA+A Q−2cλi QBB Q < 0. By choosing τ > 0 suffciently large and in light of Schur Complement ˜ +M ˜ T P˜ + 2cλi P˜ H ˜ < 0. Lemma, we can obtain from (17) that P˜ M By virtue of (15) and (16), we get that ¯ eT e¯ + 2eT (G ⊗ P B)Δ V˙ 1 ≤ λmax (Ω1 )¯
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−2
cr
˜ + cL1 ⊗ P˜ H)˜ ¯ e + 2eT (G ⊗ P B)Δ ¯ eT (IN ⊗ P˜ M V˙ 1 = 2˜
ip
t
Substituting (13) and (14) into (12) gives
−2
N i=1
˜ TW ˜ trace[kWi W i0 i0 ] − 2
N
˜ T W0 ] trace[kWi W i0
(18)
i=1
˜ 0 2 + 2kW W ˜ 0 F MW + 2e¯ ¯ ≤ λmax (Ω1 )¯ eT e¯ − 2kW W σ (G)¯ σ (P B)Δ, F
where kW = mini=1,··· ,N {kWi }, kW = maxi=1,··· ,N {kWi }.
N T T In addition, because of Δi = ω0 (−W0 ϕ0 )T , we can obtain that Δ ≤ (ε0 + MW Mϕ )2 , i=1
7
I −I , and e¯ = (U T ⊗ In )˜ and according to e˜ = (IN ⊗ T )e with T = e, (18) becomes 0 I
˜ 0 2 V˙ 1 ≤ λmax (Ω1 )σ(T T T )e2 − 2kW W F N
˜ ¯ + 2kW W0 F MW + 2e¯ σ (G)¯ σ (P B) (ε0 + MW Mϕ )2 −λmax (Ω1 )σ(T T T ) 0 e ˜ 0 F 0 2kW W
¯ + 2¯ σ (G)¯ σ (P B)
H1
N
i=1
(ε0 + MW Mϕ )2 2kW MW
h1
e ˜ 0 F . W
(19)
t
˜ 0 F = − e W
ip
us
cr
˜ 0 F T , it can It is easy to see that H1 is positive definite with kW > 0. Then define z1 = e W be obtained that V˙ 1 ≤ −z1T H1 z1 + h1 z1 ≤ −σ(H1 )z1 2 + h1 z1 .
an
According to the equation (10), one can get that
r1 z1 2 ≤ V1 ≤ R1 z1 2 ,
M
where
⎧ ⎨ r1 =
min (σ(P ), τ1i )
i=1,··· ,N
⎩ R1 = max (σ(P ), τ1i )
.
(20)
(21)
(22)
d
i=1,··· ,N
ce p
te
Then substituting (21) into (20) yields
Let l1 =
σ(H1 ) R1 , l2
=
h1 √ , r1
h1 σ(H1 ) V1 + √ V1 . V˙ 1 ≤ − r1 R1
(23)
and integrating both sides of (23), we have
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i=1
V1 ≤
l1
V1 (0)e− 2 t +
l1 l2 (1 − e− 2 t ). l1
(24)
ˆ i0 are It can be seen that limt→∞ V1 (t) = ll21 , thus all the closed-loop signals including ξi , θi , W uniformly ultimately bounded. Based on the Lyapunov function (10) and equation (7), the tracking error δ converges to a small neighborhood around the origin as lim δ ≤
t→∞
h1 R1 . σ(L1 )σ(H1 ) r1 σ(P )
This completes the proof.
8
(25)
Remark 2: In the dynamic controller (3), a sufficient condition for the existence of the feedback gain K and L is that (A, B) is stabilizable and (A, C) is detectable to guarantee the existence of LMI(4) and A + LC to be Hurwitz. In addition, the parameters K,L,c in (3) are independent with each other which provides some freedom in designing the protocols. Note that the protocol (3) is essentially nonlinear which results in that the separation principle is not valid anymore, thus it is more challenging for the consensus tracking using output feedback. 3.2
Adaptive output feedback with dynamic coupling
ip
t
Note that in the above section, the parameter c in the dynamic controller (3) depends on the eigenvalues of L1 , which is not easy to get if the communication network is very large. Thus, in this section, we extend the results to the output feedback with dynamic coupling gains without knowing the eigenvalues priori. Then, a different adaptive consensus protocol with time-varying coupling gains is proposed as follows
i = 1, · · · , N.
N
us
and the update law is
˜˙ i0 = τi [−(QB)T ( ˆ˙ i0 = W ˆ i0 ], Lij θj )ϕT0 − kWi W W
an
j=1
(26)
(27)
M
ˆ i0 is the ith where kWi > 0, σi > 0 are small design parameters and τi , νi are the adaptive gains, W N Δ ˜ i0 = W ˆ i0 −W0 , Θi = follower’s estimation of the unknown matrix W0 , and W aij (θi − θj ) + ai0 θi , j=1
in which θi is the state of the following observer based on the neighboring output information as
d
N ˙θi = (A + LC)θi + Si − L[ aij (yi − yj ) + ai0 (yi − y0 )] N
j=1
(28)
te + B[
ˆ i0 ]ϕ0 (t) ˆ i0 − W ˆ j0 ) + ai0 W aij (W
where
ce p
j=1
Si =
N j=1
aij [ci BK
N
Lil θl − cj BK
l=1
N k=1
Ljk θk ] + ai0 [ci BK
N
Lil θl ].
(29)
l=1
As in the above section, we define δi = xi − x0 as the global tracking error, then with the consensus protocol (26), we can get
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c˙i = νi [ΘTi ΓΘi − σi ci ],
cr
ˆ i0 ϕ0 (t) ui = ci (t)KΘi + W
δ˙i (t) = Aδi (t) + ci BK[
N
˜ i0 ϕ0 ] − Bω0 . aij (θi − θj ) + ai0 θi ] + B[W
(30)
j=1
From the equation (7), we get the dynamics of ξ as
ˆ 1 ⊗ BK)θ + (L1 ⊗ B)[W ˜ 0 Φ0 ] − (G ⊗ B)ω0 , ξ˙ = [IN ⊗ A]ξ + (L1 CL
9
(31)
and the dynamics of the observer can be written in the compact form as
ˆ 1 ⊗ BK)θ − (IN ⊗ LC)ξ θ˙ = [IN ⊗ (A + LC)]θ + (L1 CL
(32)
˜ 0 Φ0 ]) + (G ⊗ B)[W0 Φ0 ], + (L1 ⊗ B)[W
˜ 0 , G and Φ0 are the same as those in the where W0 = diag(W0 , . . . , W0 ), and the definition of W last section. T Let e = ξ T θT , then we get the closed-loop network dynamics as ˜ 0 Φ0 + Ξ, e˙ = Me + He + J W where
(34)
an
us
cr
0 IN ⊗ A L1 ⊗ B M= ,J = , −IN ⊗ LC IN ⊗ (A + LC) L1 ⊗ B ˆ 1 ⊗ BK −(G ⊗ B)ω0 0 L1 CL , Ξ = . H= ˆ 1 ⊗ BK (G ⊗ B)[W0 Φ0 ] 0 L1 CL
M
Theorem 2: Suppose the Assumption 1 holds, the parameters L, K, Γ in the dynamic outputfeedback controller are designed such that A + LC is Hurwitz, and K = −B T Q, Γ = QBB T Q in which Q > 0 is a solution of the LMI(4), then under the consensus protocol (26)(28) and adaptive ˆ i0 and ci are uniformly ultimately update law (27), all the closed-loop signals including ξi , θi , W bounded and the consensus tracking error δ converges to a small neighborhood around the origin as h2 R2 , σ (L1 )σ(H2 ) r2 σ(P )
(35)
d
lim δ ≤
te
t→∞
where H2 , h2 , R2 and r2 are defined in the following proof.
ce p
Proof. Consider the following Lyapunov function candidate
V2 = eT P e +
N i=1
N
1 1 ˜T ˜ trace( W (ci − α)2 , i0 Wi0 ) + τi νi
(36)
i=1
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ip
t
(33)
˜ ˜ −τ IN ⊗ Q τ IN ⊗ Q T ˜ ˜ ˜ where P = ˜ IN ⊗ (Q + τ Q) ˜ , Q > 0 satisfies that Q(A + LC) + (A + LC) Q < 0 for −τ IN ⊗ Q A + LC is Hurwitz, and τ, α are positive constants. In light of Schur Complement Lemma, it can be obtained that matrix P > 0 which guarantees that the Lyapunov function V2 is positive definite. The time derivative of V2 along the trajectory of (33) is obtained as
˜ 0 Φ0 + Ξ] + 2 V˙ 2 = 2eT P [Me + He + J W
N i=1
10
N
1 1 ˜ T ˜˙ trace( W (ci − α)c˙i . i0 Wi0 ) + 2 τi νi i=1
(37)
IN ⊗ I −IN ⊗ I . Then equation (37) can be rewritten as Let e˜ = T e with T = 0 IN ⊗ I
˜ e + 2eT P J (W ˜ +M ˜ T P˜ + 2P˜ H]˜ ˜ 0 Φ0 ) + 2eT P Ξ V˙ 2 = e˜T [P˜ M +2
N i=1
N
1 1 ˜ T ˜˙ trace( W (ci − α)c˙i , i0 Wi0 ) + 2 νi τi
(38)
i=1
where
T
˜ e = e˜ ˜ + M P˜ + 2P˜ H]˜ e˜ [P˜ M
−IN ⊗ C T LT Q Π2 ˆ 1 ⊗ QBK e˜. −IN ⊗ QLC IN ⊗ (QA + AT Q) + 2L1 CL
us
˜T
N 1 (ci − α)c˙i νi i=1
N
ci ΘTi ΓΘi − 2α
i=1
ΘTi ΓΘi − 2
i=1
N
te
≤2
N
N
d
=2
M
2
j=1
N
σi (ci − α)ci
i=1
Lij θj )T QBB T Q(
N
Lij θj ) − 2
j=1
ce p
i=1
ci (
(40)
an
˜ ˜ + LC) + τ (A + LC)T Q]. where Π2 = IN ⊗ [τ Q(A Moreover, according to the protocol (26), we get
N
σi c˜2i − 2
N
i=1
(41) σi α˜ ci
i=1
N N N T T − 2α ( Lij θj ) QBB Q( Lij θj ), i=1 j=1
j=1
where c˜i = ci − α. On the other hand, from (40), we have
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Based on (39), it can be obtained that T
(39)
cr
ip
t
˜ 0 0 I ⊗ τQ ˜ = T MT −1 = IN ⊗ (A + LC) ,M P˜ = T −T P T −1 = N , −IN ⊗ LC IN ⊗ A 0 IN ⊗ Q 0 0 −1 ˜ H = T HT = ˆ 1 ⊗ BK . 0 L1 CL
T
e˜
0 0 0 T T 0 ˆ 1 ⊗ QBK e˜ = e T ˆ 1 ⊗ QBK T e 0 2L1 CL 0 2L1 CL 0 T 0 =e ˆ 1 ⊗ QBK e 0 2L1 CL = −2
N
ci (
i=1
11
N j=1
Lij θj )T QBB T Q(
N j=1
(42) Lij θj ).
Substituting (41)(42) into (38) yields that
˜ 0 Φ0 ) + 2eT P Ξ V˙ 2 = e˜T Υ˜ e + 2eT P J (W +2
N
N
i=1
i=1
(43)
ip
t
i=1
N
1 ˜ T ˜˙ 2 trace( W ) − 2 σ c ˜ − 2 σi α˜ ci . W i0 i i τi i0
Π2 −IN ⊗ C T LT Q e¯ = e¯T Ω2 e¯, −IN ⊗ QLC IN ⊗ (QA + AT Q − 2αλ2i QBB T Q)
(44)
M
e = e¯ e˜ Υ˜
us
cr
T
an
T
ce p
te
d
−IN ⊗ C T LT Q Π2 where Υ = . If αλ2i ≥ 1, i = 1, · · · , N, it can −IN ⊗ QLC IN ⊗ (QA + AT Q) − 2αL21 ⊗ QBB T Q be obtained that QA + AT Q − 2αλ2i QBB T Q < 0. By choosing τ > 0 sufficiently large and in the light of Schur Complement Lemma, we can obtain that ¯eT Ω2 e¯ < 0. 0 , and by using the adaptive law (27), it is not difficult to obtain Note that P J = L1 ⊗ QB that
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Let U be such an unitary matrix that U T L1 U = ∧= diag(λ1 , · · · , λN), where 0 < λ1 ≤ · · · ≤ λN 0 U T ⊗ In e˜, then we can get are the eigenvalues of L1 . Let e¯ = [¯ eT1 , · · · , e¯TN ]T = 0 U T ⊗ In
˜ 0 Φ0 ) + 2 2eT P J (W
N i=1
=2
N
(
N
1 ˜ T ˜˙ trace( W Wi0 ) τi i0
˜ i0 ϕ0 Lij θjT )QB W
N N ˜ i0 ϕ0 −2 ( Lij θjT )QB W
i=1 j=1
−2
N
i=1 j=1
T ˜ ˜ i0 Wi0 ] − 2 trace[kWi W
N
i=1
i=1
12
T ˜ i0 trace[kWi W W0 ].
(45)
By using (43)(44)(45), one can conclude that V˙ 2 = e¯T Ω2 e¯ + 2eT P Ξ − 2
N
˜ TW ˜ trace[kWi W i0 i0 ]
i=1
−2
N
˜ T W0 ] − 2 trace[kWi W i0
i=1
N
σi c˜2i
−2
i=1
N
σi α˜ ci
i=1
˜ 0 2 + 2kW W ˜ 0 F MW ≤ λmax (Ω2 )σ(T T T )e2 − 2kW W F N + 2e¯ σ (P )¯ σ (G ⊗ B) (ε0 + MW Mϕ )2 − 2σ˜ c2 + 2σα˜ c ⎡ ⎤⎡ ⎤ T e −λmax (Ω2 )σ(T T ) 0 0 ˜ 0 F ⎦ ˜ 0 F ˜ = − e W 0 2kW 0 ⎦ ⎣ W c ⎣ 0 0 2σ ˜ c
ip
+ 2¯ σ (P )¯ σ (G ⊗ B)
N
i=1
(ε0 + MW Mϕ )2 2kW MW
⎡ e ⎤ ˜ 0 F ⎦ . 2σα ⎣ W ˜ c
us
cr
H2
an
h2
Since λmax (Ω2 ) < 0, it is easy to obtain that matrix H2 is positive definite. Let z2 = T ˜ 0 F ˜ c , then we have e W
M
V˙ 2 ≤ −z2T H2 z2 + h2 z2 ≤ −σ(H2 )z2 2 + h2 z2 ,
(47)
d
By virtue of the Lyapunov function V2 , one can obtain that
te
r2 z2 2 ≤ V2 ≤ R2 z2 2 ,
⎧ ⎨ r2 =
ce p
where
min (σ(P ), τ1i , ν1i )
i=1,··· ,N
⎩ R2 = max (σ(P ), τ1i , ν1i )
(48)
.
i=1,··· ,N
Combining (47) and (48) yields
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(46)
t
i=1
Let s1 =
σ(H2 ) R2 ,
s2 =
h2 √ , r2
σ(H2 ) h2 V˙ 2 ≤ − V2 . V2 + √ r2 R2
(49)
and integrating the inequality (49), one can get
V2 ≤
s1
V2 (0)e− 2 t +
s1 s2 (1 − e− 2 t ). s1
(50)
So we have V2 (t) → ss21 as t → ∞. Based on the Lyapunov function V2 , it can be concluded that ˆ i0 and ci are uniformly ultimately bounded. all the signals in the closed loop such as ξi , θi , W
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Moreover, in light of the fact that V2 ≥ σ(P )e2 ≥ σ(P )ξ2 and ξ(t) = (L1 ⊗ In )δ(t), it is not difficult to get that the tracking error converges to a small neighborhood around the origin as lim δ ≤
t→∞
h2 R2 . σ(L1 )σ(H2 ) r2 σ(P )
(51)
Thus the proof is completed.
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Remark 3: Compared with the related references Yu & Xia (2012), Hu & Zheng (2014) in which the leader’s control inputs 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 relative output information which is more practical than the ones with relative states.
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4.
Simulation
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In this section, an example is given to demonstrate the validity of the above theoretical results. In the example, it is assumed that there are five followers and one leader whose dynamics are presented as 01 0 A= ,B = ,C = 1 0 . 00 0.1
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The communication topology among the agents is illustrated in Figure 1 in which one leader indexed by 0 and five followers are involved. The adaptive consensus protocols in Theorem 2 are used in this example. The base function is defined by ϕ0 = [sin(t), cos(t)]T for simplicity. The matrix L are T designed as L = −16.00 −60.00 to guarantee that A + LC is Hurwitz. By solving the matrix inequality (4), the feedback gain matrices Γ and K are given as
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Remark 4: By virtue of the relative output information among the agents, the protocols (26)(27) with dynamic coupling gains in this section are designed in fully decentralized fashion. In references Hu & Zheng (2014), Peng et al. (2014) the consensus tracking protocols are designed using output information. However, a common limit of them is that the coupling gain in the controller of each agent is an identical constant, which is also determined by the eigenvalues of the Laplacian matrix. Compared with these two references, the coupling gain ci in the protocol (26) is adaptively updated based on the local information, independent of the global information that the eigenvalues of the Laplacian matrix associated with the whole communication graph. But the tradeoff is that the observer (28) needs the two-hop neighbors’ information, which means that more information interaction and communication load are required. In the framework of this paper, the reason why the two-hop information is needed is that the observer (28) keeps similar structure with the dynamics of tracking error ξ. It is interesting to extend the result to the one-hop case with dynamic coupling gains in the future work.
0.0270 0.4314 Γ= , K = −0.1644 −2.6247 . 0.4314 6.8890
The initial states of the six agents are chosen randomly within [−25, 25], and the parameters in the consensus protocol are taken as τi = 10, kwi = 0.08, σi = 0.008 and νi = 1, i = 1, · · · , 5. Clearly, it can be seen that the coupling gain ci is bounded from the Figure 2. The evolution of the tracking errors δi1 = x01 − xi1 and δi2 = x02 − xi2 are described in Figure3 and Figure 4, respectively, which show that the tracking errors are bounded by a small neighborhood around the origin. In addition,
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ˆ i0 , i = 1, · · · , 5 of the unknown Figure 5 and Figure 6 tell clearly that the parameter estimations W constant W0 are bounded.
5.
Conclusion
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This paper has addressed the consensus tracking problem for the linear multi-agent systems with unknown dynamics in the leader. Firstly, a decentralized adaptive output-feedback consensus protocol with static coupling gains has been proposed which is more practical than the state-feedback case for that only the relative output information has been utilized. Then, the result has been extended to the case with dynamic coupling gains which is in fully decentralized fashion without using the eigenvalues of the Laplacian matrix which is actually the global information of the whole network. Both of the protocols have guaranteed that the tracking errors converge to a small neighborhood of the origin. Finally, an example was given to illustrate the effectiveness of the methods proposed in this paper.
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The authors would like to thank the editor and all the anonymous reviewers for their valuable comments and suggestions. This work is supported by the National Nature Science Foundation(NNSF) of China under Grant 61374033. The authors would like to acknowledge Dr. Zhongkui Li for his inspiring discussions.
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systems: an observer-based approach. Systems & Control Letters, 62 (1), 22-28. Zhao, Y., Duan, Z., & Wen, G. (2014). Distributed finite-time tracking of multiple Euler-Lagrange systems without velocity measurements. International Journal of Robust and Nonlinear Control, DOI: 10.1002/rnc.3170.
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Figure 1. Communication topology.
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Figure 2. The evolutions of coupling gains.
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Figure 3. The evolutions of tracking errors x01 − xi1 .
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Figure 4. The evolutions of tracking errors x02 − xi2 .
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ˆ i0 . Figure 5. The evolutions of parameter estimation W 1
ˆ i0 . Figure 6. The evolutions of parameter estimation W 2
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