Cooperative fault estimation for linear multi-agent systems with undirected graphs
Hao Li, Ying Yang
Department of Mechanics and Engineering Science, College of Engineering, State Key Laboratory for Turbulence and
Complex Systems, Peking University, Beijing 100871, People’s Republic of China
E-mail: [email protected]
Published in The Journal of Engineering; Received on 8th April 2016; Accepted on 26th May 2016
Abstract: Fault estimation problem for linear multi-agent systems with undirected graphs is studied. A two-layer observer based on the relative
measured output information is proposed, and specially it utilises 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 stabilisation problem of the observer error dynamics. Finally, simulations are
undertaken for validating the effectiveness of the theoretical results.
1
Introduction
In the last decade, multi-agent systems have received intensive research attention from large amounts of scientific communication
for sake of its increasing application fields such as formation
flying of satellites, intelligent mobile robots, vehicle platoons, unmanned air vehicles, distributed complex network, smart grid,
and so on [1, 2]. The multi-agent systems can be classified into
leader–follower and leaderless system in terms of whether or not
there are leaders who indicate the final objective. The major
feature of multi-agent systems is that a group of dynamic systems
use relative information between agents to achieve a shared
common objective over a network which can be described by a
directed or undirected graph. With the increasing system size and
complexity, demands for safety and reliability of system attract
compelling research interest on distributed state monitoring and
fault detection.
Observer design plays an important role in the state monitoring
and fault estimation. For multi-agent system, there exist some outstanding works for distributed observer design. In [3], distributed
observer is designed for the second-order leader–follower system.
In [4, 5], the distributed consensus tracking problem with
unknown dynamics under directed graph is studied by designing
a two-layer observer including a local observer and an adaptive estimator. A distributed reduced-order observer is proposed for
achieving consensus based on the relative outputs in [6]. Zhao
et al. [7] designs a new class of observer-based control algorithms
in order to solve the finite-time consensus tracking problem in the
leader–follower multi-agent systems and [8] develops the observerbased method for tracking problem with non-linear multi-agent
systems under directed graphs. The containment control problem
for multi-agent systems is solved by applying distributed observerbased containment controllers based on the relative state estimation
in [9]. Furthermore, observer-based scheme also has potential applications in other research fields such as consensus based on sampled
position data [10], network security [11], and so on. Fault estimation technique for general linear system is relatively mature. Gao
and Ding [12] propose a robust state-space observer in order to
achieve estimation of system states and actuator faults simultaneously for a descriptor system. An integrated design of observerbased fault detection for non-linear system is addressed in [13].
Liu et al. [14] present a robust fault estimation methodology with
respect to sensor faults for a class of perturbed system. Zhang
et al. [15] improve the rapidity of fault estimation by designing
an adaptive observer and fault estimator including an integral
action. Distributed observer builds the bridge between the general
J Eng 2016
doi: 10.1049/joe.2016.0099
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 [16]. Davoodi et al. [17] discuss the
problem of simultaneous fault detection and consensus control.
In this paper, our contribution is design of a distributed observer
and fault estimator based on the relative measured output information for the leader–follower multi-agent system. We propose a twolayer 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 transferred into the stabilisation 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 utilised 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 this 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.
2
Preliminaries and problem statement
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 leader–follower 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
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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
E(G) where i ≠
A = [aij ] is defined as aii = 0 and aij > 0 if (vj , vi ) [
j. The Laplacian matrix L = [lij ] is defined as lii = j=i aij and lij
= − aij. For the leader–follower system, ai0, i = 1, …, N denotes
the information flow between the leader and the follower. Here,
ai0 > 0 if the follower i can get the information of the leader; otherwise, ai0 = 0. A subgraph G1 (V 1 , E 1 ) of G is a graph such that
V 1 # V and E 1 # E > (V 1 × V 1 ). 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.
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.
Before moving ahead, we define
zi (t) =
0
Ls
01×N
G
(1)
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.
2.3 Problem statement
Consider a group of N + 1 identical agents with linear dynamic
systems including one leader and N followers and the dynamics
of the ith agent is depicted as
ẋi (t) = Axi (t) + Bui (t) + Efi (t)
yi (t) = Cxi (t),
i = 0, 1, . . . , N ,
(2)
hi (t) =
Assumption 2: rank(CE) = rank(E) = r.
Assumption 3: The invariant zeros (if any) of (A, E, C) are Hurwitz.
These two assumptions are quite general conditions in the literature
(e.g. [15, 16, 19]) and usually addressed in the observer design.
3
Main results
In this section, we proposed a fault estimator design scheme with
the relative measured output information of neighbouring agents.
(4)
N
aij yi (t) − yj (t)
(5)
j=0
The dynamics of the diagnose observer can be written as the following structure
u̇i (t) = Aui (t) + Bui (t) + LC
N
aij ui (t) − uj (t)
j=0
+L
N
aij hi (t) − hj (t)
j=0
+E
N fˆ i (t) − fˆ j (t) ,
i = 1, . . . , N
(6)
j=0
where θ0(t) = 0 and ui (t) [ Rn , i = 1, …, N are observer states,
fˆ0 (t) = 0 and fˆi (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 stabilisation problem of the observer
error dynamics. Cooperating with the idea of the exceptional
work in [15], the fault estimator in (7) consists of two parts.
Moreover, the integral part is introduced for eliminating the estimation error
N
N ˙fˆ (t) = −FC a s (t) − s (t) + ṡ (t) − ṡ (t) ,
i
ij i
j
i
j
j=0
(7)
j=1
where si (t) is the error which is defined below and F [ Rr×p
denotes the fault estimator gain designed in the sequel. In terms
of ζi (t), the compact form can be written as
(3)
where xi (t) [ Rn is the system state, ui (t) [ Rm is the control input,
and 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) ≠ 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 order 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:
aij xi (t) − xj (t)
j=0
Lemma 1 [18]: 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.
It is assumed that there exists only one leader. The Laplacian matrix
L can be written as
N
z(t) = G ⊗ I n x(t) − 1x0
(8)
where z(t) = [zT1 (t), . . . , zTN (t)]T , x0 (t) [ Rn denotes the leader’s
state. Utilising 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
ż(t) = (I N ⊗ A)z(t) + (G ⊗ B)u(t) + (G ⊗ E)f (t)
(9)
where u(t) = [uT1 (t), . . . , uTN (t)]T , f (t) = [f1T (t), . . . , fNT (t)]T . The
observer dynamics (6) can also be transformed into the compact
form
u̇(t) = (I N ⊗ A)u(t) + (G ⊗ LC)u(t) + (G ⊗ B)u(t)
− (G ⊗ LC)z(t) + (G ⊗ E)fˆ (t)
(10)
where u(t) = [uT1 (t), . . . , uTN (t)]T , fˆ (t) = [fˆ1T (t), . . . , fˆNT (t)]T .
Let s(t) = θ(t) − ζ(t) and estimation error f˜ (t) = fˆ (t) − f (t), then
we can obtain the error dynamics as follows
ṡ(t) = I N ⊗ A s(t) + (G ⊗ LC )s(t) + (G ⊗ E)f˜ (t)
(11)
In the following content, we address a theorem to indicate the
designed parameter and show the stability of the error dynamics (11).
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doi: 10.1049/joe.2016.0099
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
ET P = FC
A P + PA + 2li C C
T
T
Using (19) and (20), (17) can be transformed into the decompressed form
−l2i C T CE − li AT PE
−2l2i ET PE
−2
(13)
T
V (t) = sT (t) I N ⊗ P s(t) + f˜ (t)f˜ (t)
i=1
Let the observer gain L = P −1C T where P can be obtained by
solving the inequality (13), then (IN ⊗ P)(G ⊗ LC) = (G ⊗ C TC)
is symmetric matrix.
In this proof, the constant fault or the fault with slowly varying
rate (i.e. f˙ (t) ≃ 0) is considered. Thus
T
T
2f˜ (t)f˙˜ (t) = −2f˜ (t)(G ⊗ FC ) s(t) + ṡ(t)
T
= −2f˜ (t)(G ⊗ FC ) I N ⊗ A s(t) + (G ⊗ LC )s(t)
T
+(G ⊗ E)f˜ (t) − 2f˜ (t)(G ⊗ FC )s(t)
(16)
By using (12), substituting (16) into (15) can obtain the fact that
V̇ (t) = sT (t)((I N ⊗ AT P) + (I N ⊗ PA) + 2G ⊗ C T C)s(t)
(17)
Since G is symmetric positive definite, let U [ RN ×N be such a
unitary matrix, and it follows that
where
fi (t) =
..
.
⎥
⎦
(18)
where s(t) =
[sT1 ,
...,
(19)
f˜ (t) = U ⊗ I n f˜ (t)
(20)
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doi: 10.1049/joe.2016.0099
Pi =
AT P + PA + 2li C T C
f˜ (t) = [f˜ T , . . . , f˜ T ].
1
N
(22)
−l2i C T CE − li AT PE
−2l2i ET PE
(23)
It can be seen that if Πi < 0, then V̇ (t) , 0, which means limt→∞φi
(t) = 0. Since f˜ (t) = U ⊗ I n f˜ (t) where U is non-singular, it
□
follows that limt1 f˜ (t) = 0. This ends the proof.
Remark 1: The equation constraint E TP = FC can be transferred
into the following optimisation issue
wI
ET P − FC
wI
. 0,
min w
(24)
So, the estimator parameters L, F can be obtained by solving LMIs
(13) and (24).
Remark 2: The slowly varying fault is considered in the proof. On
one hand, the fault is not only discussed in some literatures [19, 20]
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 realise the varying fault estimation,
though the estimation error can be uniformly bounded.
Simulation results
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 three followers indexed by 1, 2, 3. The dynamics of each agent is defined in (2) and (3) where matrix coefficients are described as
−2 1
⎢
A = ⎣ 0 −1
s(t) = U ⊗ I n s(t)
sTN ],
lN
where 0 < λ1 ≤ · · · ≤ λN denotes eigenvalues of G.
Before moving on, we define two following transformations
sTi (t)
f˜ (t)
i
⎡
⎤
(21)
i=1
4
T
T
− 2f˜ (G ⊗ FCA)s(t) − 2f˜ (t)(G T G ⊗ FCLC)s(t)
⎢
U T GU = ⎣
fTi (t)Pi fi (t)
(14)
T
V̇ (t) = ṡT (t) I N ⊗ P s(t) + sT (t) I N ⊗ P ṡ(t) + 2f˜ (t)f˙˜ (t)
= sT (t) I N ⊗ AT P + I N ⊗ PA s(t)
+ sT (t) (G ⊗ LC )T I N ⊗ P
+ I N ⊗ P (G ⊗ LC ) s(t)
T
+ 2sT (t) I N ⊗ P (G ⊗ E)f˜ (t) + 2f˜ (t)f˙˜ (t)
(15)
l1
N
=
Taking the time derivative of V(t) along the trajectory of (11) can
obtain
⎡
sTi (t) l2i C T CE + li AT PE f˜ i (t)
i=1
Proof: We consider a Lyapunov function candidate as follows
T
N
N
f˜ T (t)
l2 ET PEf˜ (t)
−2
i
i
i
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 limt1 f˜ (t) = 0.
− 2f˜ (t)(GT G ⊗ FCE)f˜ (t)
sTi (t) AT P + PA + 2li C T C si
i=1
, 0,
N
V̇ (t) =
(12)
C=
0
1
⎤
0
⎥
0 ⎦,
−1
⎡ ⎤
1
⎢ ⎥
E = ⎣ 0 ⎦,
0
(25)
1 0 0
.
0 1 0
The communication graph is shown in Fig. 1. In virtue of Lemma 1,
it is easy to see that G has the following form
⎡
⎤
3 −1 −1
G = ⎣ −1 2 −1 ⎦
−1 −1 2
(26)
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Fig. 1 Communication graph
Note that rank(CE) = rank(E) = 1 and the invariant zero of (A, E,
C) is − 1. Therefore, Assumptions 2 and 3 are satisfied.
By solving the inequalities (13) and (24), we can obtain the
related parameter as follows:
Fig. 3 Estimation error of fault f˜ ( t)
−12
w = 2.9230 × 10
⎡
⎤
49.4810 −6.6888 −0.0000
⎢
⎥
P = ⎣ −6.6888 69.4035
0.4092 ⎦
−0.0000 0.4092
49.1342
fault signal with the relative measured output between agents.
Fig. 2 shows the explicit estimation results from different agents.
Fig. 3 illustrates the trajectory of estimation error f˜ (t). Note that
f˜ (t) tends to zero as time evolves.
F = 49.4810 −6.6888
⎡
⎤
0.0205
0.0020
⎢
⎥
L = ⎣ 0.0020
0.0146 ⎦
−0.0000 −0.0001
5
We randomly choose the initial states of the agents. It is assumed
that the fault signal f (t) has the following form
f1 (t) =
f2 (t) =
f3 (t) =
0,
1,
0 ≤ t , 2s;
2s ≤ t ≤ 25s.
(27)
0,
2,
0 ≤ t , 6s;
6s ≤ t ≤ 25s.
(28)
0,
1,
0 ≤ t , 8s;
8s ≤ t ≤ 25s.
(29)
Note that the proposed estimator can effectively estimate the
Fig. 2 Fault f(t) (dotted line) and fault estimation fˆ ( t) (solid line)
Conclusions
In this paper, the fault estimation problem is studied for the linear
leader–follower system with undirected graphs. 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.
6
Acknowledgments
This work was supported by the National Basic Research Program
of China (973 program) 2012CB821202 and by the National
Natural Science Foundation of China under grant 61174052.
7
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