International Journal of Computer Trends and Technology (IJCTT) – volume 9 number 1– Mar 2014
Distributed Observer Design for Leader following Control of Multi-Agent System with Pinning Technique Namrata v lade#1, Prajakta Borole*2 #1
Lecturer, *2Assistant Professor Department of Electronics. Atharva college of Engineering, Charkopnaka, malad marve road, malad (west), Mumbai-4000 95. Maharashtra,india.
Abstract--This paper is concerned with a leader– follower problem for a multi-agent system designed by the pinning control technique without assuming that the interaction graph is connected. Distributed observers are designed for the second-order follower-agents, under the common assumption that the velocity of the active leader cannot be measured in real time. Some dynamic neighbor-based rules and protocols, consisting of distributed controllers and observers for the autonomous agents, are developed to keep updating the information of the leader. Also it is proved that each agent can follow the active leader using common Lyapunov function (CLF). Finally, a numerical example is given for illustration
Index Terms- Leader-following, Multi-agent networks, Pinning control technique, Active leader, Distributed control, Distributed observer, Common Lyapunov function I. Introduction THE consensus problem of multi-agent system has attracted great attention in many fields, such as biology, physics, robotics and control engineering. As a type of collective behavior, the consensus of multiple dynamic agents means that the states of all the agents reach agreement on a common quantity by implementing an appropriate consensus protocol A multi-agent network provides an excellent model for describing and analyzing complex interconnecting behaviors [1]–[11] Distributed estimation via observers design for multi-agent coordination is an important topic in the study of multi-agent networks, with wide applications especially in sensor networks and robot networks, among many others. Yet, very few theoretic results have been obtained to date on distributed observers design and measurement-based dynamic neighbor-based control design.
ISSN: 2231-2803
Nevertheless, one may find in the literature that Fax and Murray [12] reported some results concerning with distributed dynamic feedback of special multiagent networks, and Hong, Hu, and Gao [13] proposed an algorithm for distributed estimation of the active leader’s un-measurable state variables, to name just a couple Form. Also Yiguang Hong [24] stated distributed observer design for second order multi-agent system using CLF with switching topology. The motivation of this work is to expand the conventional observers design to the distributed observers design for a multi-agent system where an active leader to be followed moves in an unknown velocity. The continuous-time agent models considered here are second-order, different from that first order [13]. With this background, we consider a consensus problem with an active leader with an underlying dynamics. Here, some variables (that is, the velocity and maybe the acceleration) of an active leader cannot be measured, and each agent only gets the measured information (that is, the position) of the leader once there is a connection between them. In this paper, we propose an “observer” by inserting an integrator into the loop for each agent to estimate the leader’s velocity. To analyze the problem, a Lyapunov-based approach is developed. As it is practically impossible to control all agents in a large-scale multi-agent system, the consensus algorithm based on pinning control is proposed, which means that we have to control only a small fraction of agents. Literatures [15], [16] studied the first-order consensus of multi-agent systems via pinning control. Ren [17] considered the secondorder consensus of coupled double integrators with partial access to a time-varying reference state. Since, how to choose pinned nodes is one of the most difficult problems in the pinning control of complex
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International Journal of Computer Trends and Technology (IJCTT) – volume 9 number 1– Mar 2014
networks, Wang and Chen [18] pointed out that it is better to specifically pin the most-highly connected nodes for the undirected complex networks. Li et al.[19] proved that there is no significant difference between specific and random pinning schemes for random networks. Yu et al.[23] state that the nodes with low degree should be pinned first when they are weakly coupled. Chen et al.[20] showed out that it is possible to pin a strongly coupled complex network by a single controller. Lu et al.[21] explore a method to select pinned nodes by finding the strongly connected components which have no edges with heads in and tails out. Song et al. [22] revised problem and provided solution that the node whose out-degree is bigger than in-degree should be chosen as a pinned candidate. Inspired by the existing results, we study distributed observer design problem for the leader-following second-order multi-agent systems based on the pinning control technique without assuming that the interaction graph is connected with active leader. That is to say, compared to the existing results, we get consensus of second order system using distributed observer under a weaker assumption. Consequently, the main results of this paper have less conservation and more practical applications. The rest of this paper is organized as follows. The necessary notions and preliminary results are given in Section II. Distributed observer design with the pinning control technique for leader-following second-order multi-agent systems are provided in Section III. Some examples and simulation results are given in Section IV to illustrate the effectiveness of the obtained theoretical results. The conclusions and the related topics are given in Section V. II. Preliminaries Throughout this paper, Rn denotes the set of n dimensional real column vectors .Let G={V,E,A} be a weighted graph with the vertex set, V={v1,v2…vn} Where node i , ∈ , represents the ith agent, and the finite index set I = { i1,i2,…, in} is the node indexes of G ,E ⊆ V x V is the set of edges, whose elements denote the communication links between the agents. A = [aij] is the weighted adjacency matrix of the weighted graph with nonnegative adjacency elements aij ≥ 0. An edge in G is denoted by an unordered pair ( i, j). The neighborhood set of the ith agent is denoted by Ni={ J ∈ V | ( i , J)∈ E } .If there is an edge between agent i and agent j , i.e.( i , j)∈ E then, aij = aij > 0 otherwise aij = aij = 0 . An undirected network
ISSN: 2231-2803
G is connected, if there is a path between any two distinct nodes i and j in G . For the leaderfollowing system, we consider another graph Ḡ associated with the system consisting of n agents (which are called followers) and one leader denoted by vo. Let vector B={b1,b2,….bn} with the adjacency element bi > 0 if agent is i a neighbor of the leader, otherwise bi = 0. The node v0 is said to be globally reachable in Ḡ , if there is a path from every other node to v0 in Ḡ . Let vector P={p1,p2,….,pn}, where pi is the pinning control gain satisfying pi > 0 if agent i is pinned and pi = 0 otherwise. For simplicity, let vector = P + B = ( 1,
2, … ,
)T
where = pi + bi ,i = 1,2,…,n. In practice the relationships among neighboring agents may vary over time, and their interconnection topology may also be dynamically changing. Suppose that there is an infinite sequence of bounded, non over lapping, continuous time-intervals [tj,tj+1 ),j=0,1,...,say starting at t0 =0, over which N:[0,∞)→ƥ={1,2,...,N}is a piecewise constant switching signal for each N, defined at successive switching times. To avoid infinite switching during a finite time interval, assume that there is a constant with tj+1 – tj ≥ for all j ≥ 0 . III. Main Result Assume that the leader is active, which means state of the leader keep changing throughout entire process. Dynamics of leader is as follows: ̇o =
o
xo ∈ Rn
̇o =
o
vo ∈ Rn
y = xo
(1)
Where xo is the position, o is the velocity, and y is the only measurable variable. This work is to expand the conventional observer design (where the input is somehow known) to a neighbor-based observer design. In some practical cases, the velocity o is hard to measure in real time, but the input o (t) may be regarded as some given policy known to all the agents. Dynamics of i followers are ∈
n
i
∈
n
i
1
̇i =
i+
ℕ i
̇i =
i+
ℕ i
2
, = 1,2, … . . , . (2)
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International Journal of Computer Trends and Technology (IJCTT) – volume 9 number 1– Mar 2014
j
Where ℕ i ( ) ( = 1,2 ) are disturbance and ui (i = 1,2,…,n), is interaction inputs. We assume that j
| ℕ i | ≤
≤ ∞ for all i , j = 1,2,….,n.
Since all followers cannot evaluate the velocity of leader in real time, they have to estimate it all through process. Here i is estimate of o by follower agent. To track active leader following neighbor-based rule with pinning technique is proposed: i
=
o
− [ i− + ∑
i
] – w[∑
∈ i( )
i (xi
∈ i( ) ij (xi
-xo ) ]
-x j ) (3)
And distributed observer ̇ =
i
o –
∑
∈
[∑ ( )
∈ i( ) i (xi
ij (xi
-x j ) +
-xo ) ]
Theorem 1: Consider leader and follower in noise free environment .In each time interval [tj ,tj+1 ) if the entire graph is connected, then there are constants w and q such that controller (3) with observer (4) together yields lim
xi ( )-xo( ) | = 0 ,
→∞ |
lim
→∞
| vi ( )- vo (t) | = 0
Namely, the agents can follow the leader (in the sense of both position and velocity). Proof: Set = ( 1 … . . n)T - xo , = ( 1 … . . n)T - vo and = ( 1 … . . n)T - xo λ where λ = (1 …. t) T ∈ Rn, then in case of =0 , the closed loop system (3) and (4) can be written as ̇=
(4)
Where q ,w > 0and the “observer” is of the firstorder. It is preferred to have a one-dimensional reduced-order “observer” (4) instead of second-order “observers”, regarding possible technical difficulty in constructing a CLF for the higher-order system later on. So the “observer” has the same dimension as the agents in a single-integrator form, stated by Hong al[13].Consensus proof of distributed observer is proved by Yiguang Hong al[24].
̇ = − (
σ
+
̇ = − ( (
σ
∈
i.e. =
σ)
−
+
σ)
or in compact form ̇ =
The following lemma (Horn & Johnson, [13]) will be useful later.
σ
=
σ
σ
.
0 − 0
σ σ
+
0 (6)
σ
Where the switching signal N:[0,∞)→ ƥ ={1,2,...,N} is a piecewise constant, vector P= {p1,p2,……,pn} , where pi is the pinning control gain satisfying pi > 0 if agent is pinned and pi = 0 otherwise.and σ is the laplacian of n agents. In each time interval, LK and PK are time-invariant for some k ∈ ƥ. We have lemma of Hong et al[13] σ = σ + σ is positive definite since the switching graph remains being connected. Moreover, once n is given, and ,denoting the maximum and minimum positive Eigen
Lemma 1: Consider a symmetric matrix T
Where A and C are square .Then D is positive definite if and only if both A and C – ET A-1E are positive definite
ISSN: 2231-2803
σ=
− −
+
3n
0 Assumption1: Assume that under protocol (3) with the feed-back gain satisfying P= {p1,p2,……,pn}T ,I=1,2,…n. where pi is the pinning control gain satisfying pi > 0 if agent is pinned pi = 0 and otherwise, the states of all agents can be directly or indirectly affected by the state of the leader. For simplicity consider system in a noise free environment i.e. =0.
D=
(5)
values of all the positive definite matrices HK , k ∈ ƥ, are fixed and depend directly on the given constants ij and pi, i=1,...,n; j=1,...,n. Select, w = 2/ , q≥ 4 + w (7) Here, for system (6), a CLF is constructed as V(z) = zT(t)Kz(t) , with
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International Journal of Computer Trends and Technology (IJCTT) – volume 9 number 1– Mar 2014
−
qI ⎛ K=⎜ ⎝
− −
−
⎞ ⎟
(8)
⎠
This is positive definite due to (7) Take an interval ( tj,tj+1 ) into consideration. According to the assumed conditions, the graph associated with HK for some fixed k ∈ ƥ is connected and time-invariant. The derivative of V(z) is given by V̇ ( )|(6) = T( + Fk )z - zTCkz
(9)
second order finite time consensus. From Assumption 1, one can obtain the following conclusions: (i) The isolated agents must be pinned because their states cannot be influenced by any other agent; (ii) The agents with small degree should be considered to be pinned in priority, because these agents receive very little information from the other agents. (iii) With the pinning controlled gains, the states of all the agents can be directly or indirectly affected by the states of the leader. Remark 2: Compared to the existing results, different from [24], we study the distributed observer design for the leader-following second-order multi-agent systems by the pinning control technique without assuming that the interaction graph is connected or the leader is globally reachable.
Where IV. Examples and Simulation 2
k
- wHk
Ck= wHk −
wHk −
Hk
2( − 1)
Hk
−
− In this section, some numerical simulations are presented to illustrate the effectiveness of the obtained theoretical results.
−
−
Set 2
Qk = 2( − 1) −
−
-⎛
(
)
k
(
)
⎝
Example 1: Consider the leader-following multiagent system with four followers (denoted by vi , i=1,2,3,4 ), and one leader (denoted by v0), Under the following unconnected graph Ḡ s ,s=1,2 . The initial states of the followers are x (0)=( x1(0), x2(0), x3(0), x4(0) )T
-1 k
⎞ ⎠
= (-200,50,-50,150) T 2
=⎛ ⎝
2( − 1) − −
(
k
)
−
(
v(0)=( v1 (0),v2(0), v3(0), v4(0) )T
)
⎞
=(70 ,-100,60,90) T
-1
−
k
⎠
According to lemma 1 and by in (7) Qk is positive definite matrix and same is true for Ck . It follows that there is a constant ∝ , independent of the selection of the time intervals, such that V (z(t)) ≤ V z(t) e ∝( i) , ∀ t ∈ (tj, tj+1 ). Consequently, V (z(t)) ≤ V z(0) e ∝( ) , t0=0 (10) Which implies (5). Remark 1: This paper adopts the pinning control technique to study the distributed observer design for the leader-following second-order multi-agent systems. Then the key issues for the pinning control technique are what kind of agents and how many agents should be pinned to achieve a leader-following
ISSN: 2231-2803
The initial states of the leader are x0 = -150, v0 (o) = 5, and v0(t) = v0 (0) for all t ≥ 0. Design the following distributed observer with pinning technique protocol i = o− [ i− i]
–w[
ij (xi
-x j ) +
∈ i( )
i (xi
-xo ) ]
∈ i( )
Form fig 1.one can find that B1 =(1,0,0,0)T B2 =(0,0,0,0)T Thus we can choose the pinning gain PS=(p1,p2,p3,p4)T Since P1=(0,0,1,0)T P2=(1,0,0,1)T For the graph ḠS, s = (1,2) we have = ( 1, 2, 3, 4)T With = pi + bi , i=1,2,3,4 . s = (1,2).
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International Journal of Computer Trends and Technology (IJCTT) – volume 9 number 1– Mar 2014
In all the simulations, the curves in green and black denote the position and velocity states of the leader, and those blue and red denote the position and velocity states of the followers, respectively.
order system with pinning protocol efficiently to track position and velocity of active leader. For simplicity, here we take disturbance less system i.e.δ=0 .
Fig 1: Graph Ḡ 1 and Ḡ 2.
Fig
Fig
4: Position tracking of followers for Ḡ 2
Fig
5: Velocity tracking of followers for Ḡ 2
2: Position tracking of followers for Ḡ 1
V. Conclusion
Fig
3: Velocity tracking of followers for Ḡ 2
The numerical results are obtained with q=200,w=35 , and uo=cos (t) . Fig.1 to 5 shows that the followeragents can track the leader in the noise-free case Remark 3: one can find that despite of connectivity problem we can use distributive observer for second
ISSN: 2231-2803
The distributed observer design problem of leaderfollowing second-order multi-agent systems with pinning technique is studied in this paper. Based on the graph theory, matrix theory and common Lyapunov function, the distributed observer with pinning control technique is designed without assuming that the interaction graph is connected or the leader is globally reachable. Moreover, some examples and simulation results are given to illustrate the effectiveness of the obtained theoretical results. The finite-time consensus for observer, time-delay and noise environment will be investigated in the future.
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International Journal of Computer Trends and Technology (IJCTT) – volume 9 number 1– Mar 2014
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