Cooperative fault estimation for linear multi-agent systems with

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The Journal of Engineering

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Cooperative fault estimation for linear multi-agent systems with undirected graphs Hao Lia , Ying Yanga a State

Key Lab for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China

Abstract This paper studies fault estimation problem for linear multi-agent systems with undirected graphs. A twolayer observer based on the relative measured output information is proposed, and specially, it utilizes the transformation of relative output rather than the absolute one. The integral action is used to construct an adaptive fault estimator for improving the estimation performance. The fault estimation is introduced back into the observer so that fault estimation problem can be considered as stabilization problem of the observer error dynamics. Finally, simulations are undertaken for validating the effectiveness of the theoretical results. Keywords: Fault estimation, multi-agent systems, adaptive observer, fault diagnosis 1. Introduction

tributed state monitoring and fault detection. Observer design plays an important role in the state monitoring and fault estimation. For multiIn the last decade, multi-agent systems has agent system, there exist some outstanding works for received intensive research attention from large distributed observer design. In [3] distributed obamounts of scientific communication for sake of its server is designed for the second-order leader-follower increasing application fields, such as formation flysystem. In [4, 5], the distributed consensus tracking of satellites, intelligent mobile robots, vehicle plaing problem with unknown dynamics under directed toons, unmanned air vehicles(UAV), distributed comgraph is studied by designing a two-layer observer, plex network, smart grid, and so on [1, 2]. The multiincluding a local observer and an adaptive estimaagent systems can be classified into leader-follower tor. A distributed reduced-order observer is proposed and leaderless system in terms of whether or not there for achieving consensus based on the relative outare leaders who indicate the final objective. The maputs in [6]. [7] designs a new class of observer-based jor feature of multi-agent systems is that a group control algorithms in order to solve the finite-time of dynamic systems use relative information between consensus tracking problem in the leader-follower agents to achieve a shared common objective over multi-agent systems. The containment control proba network which can be described by a directed or lem for multi-agent systems is solved by applying undirected graph. With the increasing system size distributed observer-based containment controllers and complexity, demands for safety and reliability of based on the relative state estimation in [8]. Furthersystem attract compelling research interest on dismore, observer-based scheme also has potential applications in other research fields, such as consensus based on sampled position data [9], network security Email addresses: [email protected] (Hao Li), [10], and so on. Fault estimation technique for [email protected] (Ying Yang)

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2. Preliminaries and problem statement

eral linear system is relatively mature. [11] proposes a robust state-space observer in order to achieve estimation of system states and actuator faults simultaneously for a descriptor system. A integrated design of observer-based fault detection for nonlinear system is addressed in [12]. [13] presents a robust fault estimation methodology with respect to sensor faults for a class of perturbed system. [14] improves the rapidity of fault estimation by designing a adaptive observer and fault estimator including an integral action. Distributed observer builds the bridge between the general system and the multi-agent system aiming at achieving the fault estimation task. However, limited research results have been proposed in the field of fault estimation for multi-agent systems. Recently, the sliding mode observer has been used for robust fault estimation in linear multi-agent networks [15]. [16] discusses the problem of simultaneous fault detection and consensus control.

2.1. Notations Through this paper, the notation is standard. IN and Rn×m represent the identity matrix of dimension N and the set of n × m real matrices, respectively. The superscript T denotes transpose for real matrices. Also, let 1 be a column vector with all entries equal to one. The matrix inequality A > B where A and B are symmetric matrices means that A − B is positive definite. X ⊗ Y is the Kronecker product of matrices X and Y , which owns the properties that (C ⊗ X)(D ⊗ Y ) = (CD ⊗ XY ) and (X ⊗ Y )T = X T ⊗ Y T . 2.2. Graph theory Network researched in this paper is the leaderfollower system. Its communication relation can be represented by a graph G = (V, E, A) where V = {v1 , · · · , vN } denotes the set of nodes , E ⊆ V × V is the set of edges and A = [aij ] ∈ RN ×N represents the adjacency matrix. Each node in set V denotes an agent and the edge written as an ordered pair (vi , vj ) represents the information flow from agent i to agent j. Graph G is called undirected if (vi , vj ) ∈ E(G) implies (vj , vi ) ∈ E(G). The adjacency matrix A = [aij ] is defined as aii = 0 and aij > 0 if (vj , vi ) ∈ E(G) where i 6= j. The P Laplacian matrix L = [lij ] is defined as lii = j6=i aij and lij = −aij . For the leader-follower system, ai0 ,i = 1, · · · , N denotes the information flow between the leader and the follower. ai0 > 0 if the follower i can get the information of the leader, otherwise, ai0 = 0. A subgraph G1 (V1 , E1 ) of G is a graph such that V1 ⊆ V and E1 ⊆ E ∩ (V1 × V1 ). In this paper, the leader is indexed by 0 and the followers are marked by 1, · · · , N . The communication relation in the graph topology satisfies the following assumption.

In this paper, our contribution is design of a distributed observer and fault estimator based on the relative measured output information for the leaderfollower multi-agent system. We propose a two-layer observer, which does not use the absolute measured output but transformation of the relative output. Moreover, the fault estimation part is introduced into the observer so that the fault estimation can be transfered into the stabilization problem of the observer error dynamics. Inspired by the adaptive fault estimator for the general linear system, the integral action in the fault estimator can effectively improve the estimation performance. Finally, the Lyapunov function method is utilized to prove that the observer error tends to zero and the linear matrix inequality (LMI) technique can be used to solve the corresponding parameter. The remainder of the paper is arranged as follows. In Section 2, some basic knowledge and the problem formulation are provided. The main results including the design of observer and fault estimator are studied in Section 3. Section 4 provides an example to illustrate the theoretical results. Finally, some concluding remarks and potential work are given in Section 5.

Assumption 1. Suppose that the leader-follower communication topology G is connected and fixed. The communication subgraph among N followers is undirected, and at least one ai0 > 0.

2


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3. Main results

Lemma 1 ([17]). Under Assumption 1, the Laplacian matrix L has a simple zero eigenvalue corresponding to a right eigenvector 1 and all non-zero eigenvalues are positive real numbers.

In this section, we proposed a fault estimator design scheme with the relative measured output information of neighboring agents. Before moving ahead, we define

It is assumed that there exists only one leader. The Laplacian matrix L can be written as: 0 01×N (1) Ls G

ζi (t) =

N X

aij (xi (t) − xj (t))

(4)

N X

aij (yi (t) − yj (t))

(5)

j=0

where G ∈ RN ×N and Ls ∈ RN . In terms of Lemma 1, one can get the important property that G is symmetric and positive definite.

ηi (t) =

j=0

The dynamics of the diagnose observer can be written 2.3. Problem statement Consider a group of N + 1 identical agents with as the following structure: linear dynamic systems, including one leader and N N X followers, and the dynamics of ith agent is depicted aij (θi (t) − θj (t)) θ˙i (t) =Aθi (t) + LC as: j=0 x˙ i (t) = Axi (t) + Bui (t) + Efi (t)

(2)

yi (t) = Cxi (t)

(3)

i = 0, 1, · · · , N,

+L

N X

aij (ηi (t) − ηj (t))

j=0

where xi (t) ∈ Rn is the system state, ui (t) ∈ Rm is the control input, yi (t) ∈ Rp denotes the measured output. A, B, E and C are constant matrices with compatible dimensions. fi (t) ∈ Rr represents the fault: if fi (t) 6= 0, there exists a fault in the ith agent, otherwise, the system is fault free. Without loss of generality, it is assumed that leader’s control input is zero, i.e., u0 (t) = 0, f0 (t) = 0. The pair (A,C)is observable and E is of full column rank. The major objective in this paper is to design the observer and fault estimator in oder to achieve the accuracy fault estimation with the relative measured output information. We design a novel dual-layer observer which needs the estimated fault produced by the adaptive fault estimator. To meet the need of the subsequent proof, the existence conditions of the observer are given as follows:

+E

N X j=0

fî (t) − fˆj (t) , i = 1, · · · , N

(6)

where θ0 (t) = 0 and θi (t) ∈ Rn , i = 1, · · · , N are observer states, fˆ0 (t) = 0 and fî (t) ∈ Rn , i = 1, · · · , N denote fault estimations, and matrix L ∈ Rn×p represents the observer gain which will be designed in the sequel. Note that θi (t) is not the estimation of the original state xi (t) but ζi (t). The signal of the fault estimation is introduced into the designed observer so that we transfer the fault estimation problem into stabilization problem of the observer error dynamics. Cooperating with the idea of the exceptional work in [14], the fault estimator in (7) consists of two parts. Moreover, the integral part is introduced for eliminating the estimation error.  N X ˙ aij (si (t) − sj (t)) fî (t) = − F C 

Assumption 2. rank(CE) =rank(E)=r. Assumption 3. The invariant zeros (if any) of (A, E, C) are Hurwitz.

j=0

These two assumptions are quite general conditions in the literature (e.g.[14, 15, 18]) and usually addressed in the observer design.

+

N X j=1

3




(s˙ i (t) − s˙ j (t)) ,

(7)


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where λi , i = 1, · · · , N is the eigenvalue of symmetric positive matrix G associated with graph G and * denotes the symmetric element in the matrix. Then the proposed fault estimator (7) can achieve that lim f˜(t) = 0.

where si (t) is the error which is defined below, F ∈ Rr×p denotes the fault estimator gain designed in the sequel. In terms of ζi (t), the compact form can be written as: ζ(t) = (G ⊗ In ) (x(t) − 1x0 )

(8)

t→∞

Proof. We consider a Lyapunov function candidate as follows:

T where ζ(t) = [ζ1T (t), · · · , ζN (t)]T , x0 (t) ∈ Rn denotes the leader’s state. Utilizing Lemma 1, one can obtain G > 0, so ζ(t) can be treated as the transformed error between the leader and followers. Furthermore, using the transform (8), (2) can be written into the compact structure as follows:

V (t) = sT (t) (IN ⊗ P ) s(t) + f˜T (t)f˜(t)

Taking the time derivative of V (t) along the trajectory of Equation (11) can obtain V˙ (t) =s˙ T (t) (IN ⊗ P ) s(t) + sT (t) (IN ⊗ P ) s(t) ˙ ˙ + 2f˜T (t)f˜(t) =sT (t) IN ⊗ AT P + (IN ⊗ P A) s(t) T + sT (t) (G ⊗ LC) (IN ⊗ P ) + (IN ⊗ P ) (G ⊗ LC) s(t)

˙ =(IN ⊗ A)ζ(t) + (G ⊗ B)u(t) ζ(t) + (G ⊗ E)f (t)

(14)

(9)

where u(t) = [uT1 (t), · · · , uTN (t)]T , f (t) = T [f1T (t), · · · , fN (t)]T . The observer dynamics (6) can also be transformed into the compact form: ˙ =(IN ⊗ A)θ(t) + (G ⊗ LC)θ(t) + (G ⊗ B)u(t) θ(t) − (G ⊗ LC)ζ(t) + (G ⊗ E)fˆ(t) (10)

+ 2sT (t) (IN ⊗ P ) (G ⊗ E) f˜(t) ˙ + 2f˜T (t)f˜(t)

(15)

fˆ(t) = where θ(t) = T T T ˆ ˆ [f1 (t), · · · , fN (t)] . Let s(t) = θ(t) − ζ(t) and estimation error f˜(t) = ˆ f (t) − f (t), then we can obtain the error dynamics as follows:

Let the observer gain L = P −1 C T where P can be obtained by solving the inequality (13), then (IN ⊗ P )(G ⊗ LC) = (G ⊗ C T C) is symmetric matrix. In this proof, the constant fault or the fault with slowly varying rate(i.e.,f˙ ≈ 0) is considered. Thus,

s(t) ˙ = (IN ⊗ A) s(t) + (G ⊗ LC) s(t) + (G ⊗ E) f˜(t)

˙ 2f˜T (t)f˜(t) = − 2f˜T (t) (G ⊗ F C) (s(t) + s(t)) ˙ = − 2f˜T (t) (G ⊗ F C) (IN ⊗ A) s(t) + (G ⊗ LC) s(t) + (G ⊗ E) f˜(t)

[θ1T (t), · · ·

T , θN (t)]T ,

(11)

In the following content, we address a theorem to indicate the designed parameter and show the stability of the error dynamics (11).

− 2f˜T (t) (G ⊗ F C) s(t)

Theorem 1. Suppose that Assumptions 1-3 hold. If there exist symmetric positive definite matrix P ∈ Rn×n and matrices F ∈ Rr×p , L = P −1 C T ∈ Rn×p satisfying the following conditions:

(16)

By using the Equation (12), substituting Equation (16) into Equation (15) can obtain the fact that ˙ (t) =sT (t) IN ⊗ AT P + (IN ⊗ P A) V T E P = FC (12) + 2G ⊗ C T C s(t) − 2f˜T (G ⊗ F CA) s(t) T A P + P A + 2λi C T C −λ2i C T CE − λi AT P E − 2f˜T (t) GT G ⊗ F CLC s(t) < 0, ∗ −2λ2i E T P E − 2f˜T (t) GT G ⊗ F CE f˜(t) (17) (13) 4


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Remark 1. The equation constraint E T P = F C can be transfered into the following optimization issue: ϕI E T P − F C > 0, min ϕ (24) ∗ ϕI

Since G is symmetric positive definite, let U ∈ R be such an unitary matrix, and it follows that   λ1   .. U T GU =  (18)  . N ×N

λN

So the estimator parameters L, F can be obtained by solving linear matrix inequality (13) and (24).

where 0 < λ1 ≤ · · · ≤ λN denotes eigenvalues of G. Before moving on, we define two following transformations s(t) = (U ⊗ In ) s¯(t) ¯ ˜ f (t) = (U ⊗ In ) f˜(t)

Remark 2. The slowly varying fault is considered in the proof. On one hand, the fault is not only discussed in some literature[18, 19], but also can be applied in some practical problem, such as fault resulting from actuator stuck. On the other hand, the proposed distributed fault estimator can also realize the varying fault estimation, although the estimation error can be uniformly bounded.

(19) (20)

¯ ¯ ¯T where s¯(t) = [¯ sT1 , · · · , s¯TN ], f˜(t) = [f˜1T , · · · , fÑ ]. Using Equation (19) and (20), Equation (17) can be transformed into the decompressed form: V˙ (t) =

N X i=1

−2

s¯Ti (t) AT P + P A + 2λi C T C s¯i

4. Simulation Results

N X

We give an example to validate the effectiveness of the proposed distributed fault estimator. It is assumed that there exist one leader indexed by 0 and 3 followers indexed by 1, 2, 3. The dynamics of each agent is defined in (2)-(3) where matrix coefficients are described as     −2 1 0 1 A =  0 −1 0  , E = 0 , 0 1 −1 0 1 0 0 C= . (25) 0 1 0

i=1

−2

¯ sTi (t) λ2i C T CE + λi AT P E f˜i (t)

N X ¯ ¯ f˜iT (t) λ2i E T P E f˜i (t) i=1

=

N X

φTi (t)Πi φi (t)

(21)

i=1

where, φi (t) =

Πi = T A P + P A + 2λi C T C ∗

s¯Ti (t) ¯ f˜i (t)

(22)

The communication graph is shown in Figure 1. In virtue of Lemma 1, it is easy to see that G has the following form:

−λ2i C T CE − λi AT P E −2λ2i E T P E (23)

 3 −1 −1 G = −1 2 −1 −1 −1 2 

It can be seen that if Πi < 0, then V˙ (t) < 0, which ¯ means lim φi (t) = 0. Since f˜(t) = (U ⊗ In ) f˜(t) t→∞ where U is nonsingular, it follows that lim f˜(t) = 0.

Note that rank(CE)=rank(E)=1 and the invariant zero of (A,E,C) is −1. Therefore, Assumption 2-3 is satisfied.

t→∞

This ends the proof.

(26)

5



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2

fˆ1 1 0 -1 0

5

10

15

20

25

15

20

25

15

20

25

Time(t) fˆ2

2 0

-2 0

5

10

Time(t)

Figure 1: Communication graph

2

fˆ3 1 0

By solving the inequality (13) and (24), we can obtain the related parameter as follows:

-1 0

5

10

Time(t)

ϕ = 2.9230 × 10−12   49.4810 −6.6888 −0.0000 0.4092  P = −6.6888 69.4035 −0.0000 0.4092 49.1342 F = 49.4810 −6.6888   0.0205 0.0020 0.0146  L =  0.0020 −0.0000 −0.0001

Figure 2: Fault f (t) (dotted line)and fault estimation fˆ(t) (solid line)

2

f˜1

We randomly choose the initial states of the agents. It is assumed that the fault signal f (t) has the following form:

0

-2 0

5

10

15

20

25

15

20

25

15

20

25

Time(t) 2

f1 (t) =

0, 1,

0 ≤ t ≤ 2s; 2s ≤ t ≤ 25s.

(27)

f2 (t) =

0, 2,

0 ≤ t ≤ 6s; 6s ≤ t ≤ 25s.

(28)

f3 (t) =

0, 1,

0 ≤ t ≤ 8s; 8s ≤ t ≤ 25s.

f˜2 0 -2 0

5

10

Time(t) 1

f˜3 0

(29)

-1 -2 0

Note that the proposed estimator can effectively estimate the fault signal with the relative measured output between agents. Figure 2 shows the explicit estimation results from different agents. Figure 3 illustrates the trajectory of estimation error f˜(t). Note that f˜(t) tends to zero as time envolves.

5

10

Time(t)

Figure 3: Estimation error of fault f˜(t)

6


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5. Conclusions

[4] J. Sun, Z. Geng, and Y. Lv, “Adaptive output feedback consensus tracking for heterogeneous multi-agent systems with unknown dynamics under directed graphs,” Systems & Control Letters, vol. 87, pp. 16–22, 2016.

In this paper, the fault estimation problem is studied for the linear leader-follower system with undirected graph. A new observer based on adaptive method is proposed for achieving fault estimation. The integral action is introduced into the fault estimator in order to obtain better estimation performance. Furthermore, some potential research areas consist of how to extend the result into the directed graph and how to deal with the fault-tolerant control problem.

[5] J. Sun and Z. Geng, “Adaptive output feedback consensus tracking for linear multi-agent systems with unknown dynamics,” International Journal of Control, vol. 88, no. 9, pp. 1735–1745, 2015. [6] Z. Li, X. Liu, P. Lin, and W. Ren, “Consensus of linear multi-agent systems with reduced-order observer-based protocols,” Systems & Control Letters, vol. 60, no. 7, pp. 510–516, 2011.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

[7] Y. Zhao, Z. Duan, G. Wen, and Y. Zhang, “Distributed finite-time tracking control for multiagent systems: an observer-based approach,” Systems & Control Letters, vol. 62, no. 1, pp. 22–28, 2013.

Acknowledgements This work is supported by the National Basic Research Program of China (973 program) 2012CB821202 and by the National Natural Science Foundation of China under grant 61174052.

[8] G. Wen, Y. Zhao, D. Zhisheng, W. Yu, and G. Chen, “Containment of higher-order multileader multi-agent,” IEEE Transactions on Automatic Control, 2015. [9] N. Huang, Z. Duan, and G. R. Chen, “Some necessary and sufficient conditions for consensus of second-order multi-agent systems with sampled position data,” Automatica, vol. 63, pp. 148–155, 2016.

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