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Distributed estimation and control of multiple nonholonomic mobile agents with external disturbances
Accepted Manuscript
Distributed estimation and control of multiple nonholonomic mobile agents with external disturbances Ahmadreza Jenabzadeh , Behrouz Safarinejadian PII: DOI: Reference:
S0016-0032(18)30117-0 10.1016/j.jfranklin.2018.02.008 FI 3333
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
30 January 2017 30 October 2017 16 February 2018
Please cite this article as: Ahmadreza Jenabzadeh , Behrouz Safarinejadian , Distributed estimation and control of multiple nonholonomic mobile agents with external disturbances, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.02.008
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ACCEPTED MANUSCRIPT Distributed estimation and control of multiple nonholonomic mobile agents with external disturbances Ahmadreza Jenabzadeh, Behrouz Safarinejadian School of Electrical and Electronic Engineering, Shiraz University of Technology, Shiraz, Iran
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Abstract This paper investigates the tracking control problem of nonholonomic multigent systems with external disturbances. For this purpose, distributed finite time controllers (DFCs) based on the terminal sliding mode method are proposed to ensure that states of the agents track the
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states of the target in a finite time. Furthermore, a distributed estimator (DE) is designed for each agent to estimate the target’s states. The stability analysis of DFCs and DE is also considered. Simulation examples demonstrate the promising performance of the proposed
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algorithms.
target tracking.
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1. Introduction
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Keywords: Distributed control, distributed estimator, nonholonomic multiagent systems,
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Study of distributed tracking control problem in multiagent systems has received great attention in the recent decade due to its applications in different areas such as flocking,
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intelligent transportation, surveillance, cooperative control and so on [1-7]. The tracking control of multiagent systems includes two aspects of estimation and control. The estimation aspect aims to design a distributed estimator for each agent to estimate the states of the target. The control aspect plans to design a distributed controller for every agent to track the states of the target.
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ACCEPTED MANUSCRIPT The tracking control problem of multiagent systems can be considered in two classes: agents with linear and nonlinear dynamics. Some algorithms have been presented for target tracking of agents that have linear dynamics [8-14]. [8] has considered the coordinated tracking problem of linear multiagent systems with a directed communication graph. In [9], the authors established a distributed estimation and control algorithm such that the mobile
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agents estimate the states of a linear target and track it in a flocking manner. [10] studied a leader-follower cooperative control of multiagent systems under the directed graph and transmission time delay. This paper has assumed that some agents (followers) are connected to the target (leader) and designs estimators and controllers for each follower to track the
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leader. In [11], some tracking and formation control protocols have been suggested for a second order multiagent system in which the interaction topology among agents was invariant. In [12], an event-based control law has been proposed for a second-order
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multiagent system which ensures that each agent with input delay can track a linear target. In this work, the suggested controller in each agent used the discrete states of the neighboring
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agents and has been derived based on the matrix and graph theories and inequality technique.
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[13] designed a fully distributed controller for multiagent systems with input saturation to track a target. The proposed control law did not require global information such as
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information of the Laplacian matrix or network size unlike most of the existing tracking control algorithms. In [14], an observer-based controller has been suggested for linear
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multiagent systems with external disturbances. The proposed algorithm guaranteed that each agent can track the target whose input was unavailable and unknown to the agents. In practical target tracking applications, many multiagent systems and targets have
nonlinear dynamics. One of the most practical multiagent systems is nonholonomic ones. Thus, it is necessary to investigate distributed tracking algorithms of nonholonomic multiagent systems. In [15], a finite time control law has been designed for some
2
ACCEPTED MANUSCRIPT nonholonomic mobile agents. In this paper, agents can track a target in a finite time when velocitiy conditions of the target are satisfied. [16] presented a distributed controller for nonholonomic agents that use local measurements to track a target with a time varying velocity. Furthermore, the topology of the multiagent system is switched based on the distance among agents and the target. In [17], distributed continuous control laws have been
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proposed for single and multiple nonholonomic agents based on sliding mode control and artificial potentials methods. The suggested controllers make agents converge to a maneuvering target. In [18], the problem of finite-time tracking control of multiple nonholonomic mobile robots was solved. A distributed finite-time controller has been
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proposed for each agent to track a target with nonholonomic dynamics in this paper. [19] proposed distributed state feedback controllers for nonholonomic multiagent systems such that the states of each agent converge to the states of a target with an exponential rate. This
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paper has also extended the results for multiagent systems with a time-varying communication graph. In [20], two discontinuous control laws have been designed so that
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nonholonomic agents can track a target with unknown dynamics in a flocking maner.
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Suggested controllers ensure collision avoidance among agents and can be used for both fixed and switching topologies. In [21], a sampled-data controller has been presented for a
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group of nonholonomic agents to track a mobile target. In this study, a dwell time is assumed for the feasibility of information sensing and processing to avoid chattering caused by abrupt
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changes of the neighboring relations. The aformantionted nonholonomics multiagent systems have dynamic models. In the recent decay, study of chained form of nonholonomic multiagent systems has attracted researchers’ attentions because this form includes any kinematic model of mechanical nonholonomics systems. Actually, [22] proved that nonholonomic systems can be converted into the chained form by state and input transformations. [23] has considered the leader-follower problem of nonholonomic chained-
3
ACCEPTED MANUSCRIPT form systems. This paper proposed a distributed finite time estimator and controller for each follower to estimate and track the leader’s states in a finite time. The targets in the above tracking control algorithms of nonholonomic multiagent systems have continuous dynamics and their states are assumed to be available or at least one of the agents is connected with the target. But, these assumptions do not hold in practical
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environments and target’s states must be estimated by a distributed estimation algorithm. Since the targets with nonholonomic dynamics are included in nonlinear systems. Therefore, a nonlinear distributed estimator is required for each agent to estimate the states of these types of targets. There are some distributed estimators for state estimation of targets with
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nonlinear dynamics [24-27]. [24] proposed a distributed estimator which estimated the states of a continuous linear system whose input is generated by another linear system. In [25, 26], some nonlinear filters have been presented for targets with discrete nonlinear dynamics based
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on extended and unscented Kalman filters. In [27], the distributed state estimation problem of discrete nonlinear systems with randomly delayed measurements was considered and a
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Gaussian-consensus filter was proposed to pursue a trade off between estimation accuracy
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and computing time. It should be mentioned that the distributed estimators of [25-27] were suggested for state estimation of the targets with discrete nonlinear dynamics and cannot be
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used for state estimation of continuous nonholonomic systems. Moreover, [24] has presented a continuous estimator for targets with a very special structure which do not include
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nonholonomic systems. Therefore, it is necessary to present a novel distributed estimator to estimate the states of the targets with continuous nonholonomic dynamics. In almost all practical multiagent systems, model uncertainties and disturbances are
inevitable. If no appropriate controller is used for multiagent systems to deal with disturbances, the existence of disturbances may affect system performance and lead to instability. Recently, the problem of tracking control has been solved for nonholonomic
4
ACCEPTED MANUSCRIPT multiagent systems in dynamic models subject to external disturbances [28, 29]. This problem has not been considered for chained-form nonholonomic multiagent systems in the literature according to the best of the authors’ knowledge. Motivated by the above analysis, this paper will focus on solving the target tracking problem for chained-form nonholonomic multiagent systems with external disturbances. In
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the first step, each agent is assumed to have a third-order nonholonomic dynamics in chainedform subject to external disturbances. Based on the terminal sliding mode (TSM) method, A DFC is designed for each agent to guarantee that the states of the nonholonomic multiagent systems track the states of the target in a finite time. In the second step, a DE is designed for
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each agent to estimate the states of a noisy nonholonomic target. The main contributions of this paper are as follows:
In contrast to the tracking control algorithms of nonholonomic multiagent
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systems [15-21, 28, 29], this paper considers the estimation aspect, in addition to the control aspect, of tracking control and proposes a distributed estimator
A finite time distributed controller is obtained for
chained-form
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to estimate the states of the target.
nonholonomic multiagent systems that makes each agent track appropriately
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the states of the target in the presence of external disturbance, whereas the
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results of [15-21, 23] have been designed for disturbance free case and were
not robust against external disturbance. A distributed estimator is suggested in this paper that can be applied for targets with a general class of continuous nonlinear dynamics, whereas the results in [24-27] are designed for continuous nonlinear systems with a special structure and discrete nonlinear dynamics, and cannot be used for state estimation of continuous nonlinear systems.
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ACCEPTED MANUSCRIPT The rest of this paper is organized as follows: Section 2 presents the preliminaries and problem definition. Section 3 proposes the distributed control laws and DE method for nonholonomic multiagent systems and their stability analysis is given. The effectiveness of the proposed algorithms is studied with simulations in Section 4. Finally, the conclusions are drawn in Section 5.
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Notation: R n and R n m denote the n dimensional Euclidean space and the set of all n m real matrices, respectively. I M denotes an M M identity matrix. The symbol represents the Kronecker product. . denotes the expectation operator. max (.) and min (.) denote the
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2,1 biggest and the smallest eigenvalues, respectively. C denotes the family of all nonnegative
functions V (x (t ),t ) that are continuously twice differentiable in x and once differentiable in t. 2. Preliminaries and problem definition
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Graph Theory. Every multiagent system can be modeled by an undirected graph
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G (v , , A ) , where v 1, 2,..., N is the agent (node) set, v v (i , j ) : i , j v is the
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edge set and A [aij ] R N N is the adjacent matrix. If nodes i and j are connected, then the node i is the neighbor of node j and aij a ji 0 . The Laplacian matrix of the graph G is
a
j N i
ij
.
AC
di
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defined as L D A , in which D is a diagonal matrix with the diagonal elements The
1 (L ) 2 (L )
eigenvalues
of
a
Laplacian
matrix
can
be
ordered
as
N (L ) in which the second smallest eigenvalue, 2 (L ) is called the
algebraic connectivity of the network. N i j v : (i , j ) , j i denotes the set of neighbors of node i. If there is a link between nodes i and j , then aij 0 . G is called a connected graph if and only if there is at least one path between every two arbitrary nodes. It
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ACCEPTED MANUSCRIPT is a critical point that an undirected graph is connected if and only if its algebraic connectivity is positive: 2 (L ) 0 . Consider a nonholonomic multiagent system with N mobile agents. The dynamics of each agent is described by:
u1i d 1i u 2i d 2 i i 1,..., N x 2i (u1i d 1i )
(1)
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x 1i x 2i x 3i where
x 1i , x 2i and x 3i are states of the ith mobile agent and u 1i , u 2i are the control inputs
of agent i. d i 1 and d i 2 are bounded disturbances. The objective is to track the states of a target
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with the following dynamics u1d x d (t ) u 2d B dw d (t ), x (u ) 2d 1d
(2)
x d (t ), B d R 3
where x d (t ) and (u1d ,u 2d ) are state and control inputs of the target, respectively. The
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sensing model of the ith agent (measurement equation) is described by: y id (t ) hid (x d (t )) Didv id (t ), i 1,..., N
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(3)
d where w d (t ) and v id (t ) are Gaussian white noises and hi (.) is a nonlinear function. It is
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assumed that all of the involved matrices have compatible dimensions. To solve tracking
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control problem, a distributed estimator and controller must be designed such that each agent with dynamics (1) can estimate and track the states of the target (2).
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To track the target (2), the DFCs based on a switching strategy are proposed for
multiagent system (1) as follows: u1i u1d ,
t T * , 1
2
u1i 5 sgn S i 1 4 ([ aij (x 1i x 1 j ) (x 1i )]q [ aij (x 1i x 1 j ) (x 1i )] j N i
j N i
2 1 q
u 2i 3 sgn S i 2 1[(x 2i ) ( aij (x 3i x 3 j ) (x 3i ))] q
q 2
j N i
with 7
u 2d ,
1 q
) u1d , t T * ,
)4(
)5(
ACCEPTED MANUSCRIPT 1
t
2
S i 1 x 1i ( 4 ([ aij (x 1i x 1 j ) (x 1i )]q [ aij (x 1i x 1 j ) (x 1i )] j N i
0
1 q
)d
)6(
j N i
2 1 q
t
S i 2 x 2i ( 1[(x 2i ) ( aij (x 3i x 3 j ) (x 3i ))] q
0
q 2
)7(
)d
j N i
where x 1i x 1i x 1d , x 2i x 2i x 2d and x 3i x 3i x 3d . 1 q q1 q 2 2 , 1 q q1 q 2 2 , q1 , q 2 , q1 , q 2 are positive odd integers, 1 , 2 , 4 , 3 d 0 , 5 d 0 are positive constants,
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T * t 1 t 2 is switched time, and S i 1 , S i 2 are ideal sliding surfaces. Controllers (4) and (5) are
nonsingular terminal sliding mode controllers because the parameters q1 , q 2 , q1 and q 2 are positive odd integers and 1 q q1 q 2 2 , 1 q q1 q 2 2 . In this case, the system dynamics
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with controllers (4) and (5) does not have any terms with negative (fractional) powers in sliding and reaching phases and when S i 1 , S i 2 0 . Therefore, no singularity occurs in sliding surfaces and system dynamics when the controllers (4) and (5) are used.
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max Assumption 1. There exist positive constants d 0 , u1dmin , u1dmax and u 2d such that d 1i d 0 , max d 2i d 0 , u1min and u 2d u 2max d . d u1d u1d
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Assumption 2. The graph G related to the nonholonomic mobile agents is connected and
3. Main results
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undirected.
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In this section, the stability analysis of the DFCs (4) and (5) is presented. Then, a DE is introduced and analyzed for each agent to estimate the states of the target with dynamics (2).
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To analyze the stability of the DFCs (4) and (5), the tracking error is defined as
follows:
x 1i x 1i x 1d x 2 i x 2 i x 2d x 3i x 3i x 3d u1i u1i u 1d u u u . 2d 2i 2i
)8(
where u1i and u 2i are the DFCs given in (4) and (5). Then, one can obtain (without noise) 8
ACCEPTED MANUSCRIPT x 1i (t ) u1i d 1i x 2i (t ) u 2i d 2i x 3i (t ) (x 2i x 2d )(u1i d 1i ) x 2i u1d .
)9(
To achieve the finite-time tracking, it should be shown that the states of the error system (9) with u1i and u 2i converge to zero in a finite time. For this aim, the system (9) is divided into two subsystems, a first order subsystem as
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x 1i u1i d1i and a second order subsystem x 2 i u 2i d 2 i x 3i (x 2i x 2d )(u1i d 1i ) x 2i u1d .
)10(
)11(
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Theorem 1. Consider the multiagent system (9). Under Assumptions 1 and 2, the u1i and u 2i ensure that the states of the system (9) can reach zero in a finite time.
x 2 i u 2i d 2 i x 3i (x 2i x 2d )(d 1i ) x 2i u1d .
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Proof. When t T * , one has u1i 0 and the subsystem (11) is converted to
)12(
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We prove that the states of the system (12) with controller u 2i can reach zero in a finite time.
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To this goal, it must firstly be proved that S i 2 0 for any t t 1 where t 1 is a finite time. Then, it must be shown when S i 2 0 is obtained, the states of subsystem (12) converge to zero in
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the finite time t 2 . Choose the following Lyapunov function 1 N 2 S i 2 , 2 i 1
)13(
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U 1 (t )
taking the derivative of U 1 (t ) leads to U 1 (t )
N 1 N S i 2 ( 3 sgn S i 2 d i 2 ) ( 3 d 0 ) S i 2 . 2 i 1 i 1
)14(
By Lemma A.4, one has U 1 (t ) 1U 11/2 (
N 1 ( 3 d 0 )) S i 2 0 2 i 1
)15(
9
ACCEPTED MANUSCRIPT where 1 (0, 2(3 d 0 )) . From Lemma A.3, it can be resulted from (15) that the manifold S i 2 can reach zero in the finite time t 1 satisfying t 1
2U 11/ 2 (0)
1
. For t t 1 , we have S i 2 0 and
system (12) can be written as
)16(
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2 1 q q q x [( x ) ( a ( x x ) ( x ))] 2i 1 2 ij 2i 3i 3j 3i j N i x 3i (x 2i x 2d )(d 1i ) x 2i u 1d .
It will be proved that the states of the system (16) can reach zero in the finite time t 2 .The proof includes the following two steps.
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Step 1: Choose a Lyapunov function 𝑉0 as 1 2
V 0 (t ) x 3Ti (L I )x 3i .
)17(
By Lemma A.2, taking derivation of V 0 along system (16) yields N
V 0 (t ) x 3Ti (L I )x 3i (d 1i u1d )x 2i ( aij (x 3i x 3 j ) x 3i ) j N i
M
i 1
N
)18(
(d 1i ) x 2d ( aij (x 3i x 3 j ) x 3i ). i 1
a
j N i
ij
(x 3i x 3 j ) x 3i and taking x 2*i 2e i1/3q as a virtual control law, one can
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Defining e i 3
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obtain N
j N i
N
N
i 1
i 1
V 0 (t ) 2 (d 1i u1d )e 3(1i q )/q (d 1i u1d )e 3i (x 2i x 2*i ) (d 1i ) x 2d e 3i .
CE
i 1
)19(
AC
Step 2: In this step, we consider another Lyapunov function as N
V (t ) V 0 (t ) V i (t )
)20(
i 1
with
x 2i
V i (t ) x (s q x 2*i q )
2 1/ q
* 2i
)21(
ds .
By taking the derivative of V (t ) , one gets N
V (t ) V 0 (t ) V i (t ).
)22(
i 1
10
ACCEPTED MANUSCRIPT N
To obtain the terms
V i 1
1 dx 2*i q q dt
V i (t ) (2 )
i
x 2i
x 2*i
(t ) , we define i x 2qi x 2*i q . The derivative of V i (t ) is given by 11/ q
(s q x 2*i q )
)23(
ds i21/q x 2i .
Note that dx 2*i q 2q e i 3 2q aij ((d 1i (x 2i x 2d ) x 2i u 1d ) (d 1 j (x 2 j x 2d ) x 2i u 1d )) (d 1i (x 2i x 2d ) x 2i u 1d ) dt j N i
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2q aij (d 1i x 2i d 1 j x 2 j ) d 1i x 2i j N i 2q aij (x 2i u1d x 2i u1d ) x 2i u1d j N i 2q aij (d 1i x 2d d 1 j x 2d ) d 1i x 2d ) . j N i
By using the following inequalities
2q ( aij (x 2i u 1d x 2 j u 1d ) x 2i u 1d ) 2q ( aij 1) x 2i u 1d j N i j N i
a
ij
j N i
)24(
x 2 j u 1d
N
M
2q ( 1 x 2i u 1d 2 x 2 m u 1d ) m 1
N
2q c1 x 2 m u1d ,
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m 1
CE
PT
2q aij (d 1i x 2i d 1 j x 2 j ) d 1i x 2i 2q d 0 ( aij 1) x 2i j N i j N i
x 2 j
a
j N i
ij
)25(
N
2q d 0 ( 1 x 2i 2 x 2 m ) m 1
N
2q d 0c1 x 2 m , m 1
AC
2q aij (d 1i x 2d d 1 j x 2d ) d 1i x 2d ) 2q d 0 ( aij 1) x 2d j N i j N i
a
j N i
ij
x 2d
N
2q d 0 ( 1 x 2d 2 x 2d ) m 1
N
2q d 0c1 x 2d , m 1
j N i
j N i
where 1 max i aij 1 , 2 max i , j aij and c1 is a positive constant, one has N N N dx 2*i q 2q c1 x 2 m u1d 2q d 0c1 ( x 2 m x 2d ). dt m 1 m 1 m 1
)26(
11
ACCEPTED MANUSCRIPT Furthermore, it can be also verified from proposition 8 of [30] that
x 2i
x 2*i
11/ q
(s q x 2*i q )
11/ q
ds x 2i x 2*i x 2qi x 2*i q 11/ q
x 2i x 2*i i
)27(
.
Substituting (26) and (27) into (23) yields 1
N
N
N
m 1
m 1
m 1
11/ q
i21/q x 2i .
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V i (t ) (2 ) 2q c1 x 2 m u1d 2q d 0c1 ( x 2 m x 2d ) x 2i x 2*i i q
)28(
By using Lemma A.4 and proposition 9 of [30], one can obtain 11/ q
( 2 1) i
i
11/ q
x 2d
x 2d x 2 i x
* 2i
c 2 m
d
x
x
1/ q
q 2i
d
2c 3 e m 3 ,
11/ q
d
* q 1/ q 2i
i
1
2
1 q
)29(
x 2d i
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x 2 m x 2i x 2*i i
where c 2 and c 3 are positive constants and d 1 1/ q . By the definition of 𝜉𝑖 and (16), we have x 2i 1i(2q )/q . Substituting (29) into (28), one has
2q c1 (2q 1)( 2 1) i q
d
N
u m 1
d 0c1 (2q 1)( 2 1)N i q q 2
1d
d
c c (2q 1) 2q 1 N 2q c1c 2 (2q 1) N d d m u1d 1 3 e m 3 u1d q q m 1 m 1
2q d 0c1c 2 (2q 1) N d 0c1c 3 (2q 1) 2q 1 N d d em 3 m q q m 1 m 1
M
V i (t )
)30(
q d c (2q 1) 1 q N 2 0 1 (2 ) x 2d i 1id . q m 1 To continue proof, from (19), one gets
PT
ED
1
N
N
N
V 0 (t ) 2 u1d e 3(1i q )/q 2d 0 e 3(1i q )/q u1d e 3i (x 2i x 2*i ) i 1
i 1
CE
N
d0
e i 1
3i
)31(
i 1
N
(x 2i x 2*i ) d 0 x 2d e 3i . i 1
AC
By Lemma A.4, one has N
1
N
u1d e 3i (x 2i x 2*i ) u1d e 3i 2 i 1
1 q
i 1 N
u1d 3 e 3i i 1
d
x 2qi x 2*i q 4 i
d
1 q
1
2
1 N q
u i 1
1d
c u e
e 3i i
N
4
i 1
1d
d 3i
i
1 q
)32( d
where 3 , 4 and c 4 are positive constants. Substituting (32) into (31), one can obtain
12
ACCEPTED MANUSCRIPT N
N
N
i 1
i 1
i 1
V 0 (t ) 2 u1d e 3(1i q )/q 2d 0 e 3(1i q )/q c 4 u1d e 3i N
c 4d 0 e 3i i 1
i
d
d
d x
d
i
d
)33(
N
0
i 1
2d
e 3i .
Putting (33) and (30) into (22) and considering equation x 2d u 2d yield V (t ) (u1max d (
q d c (2q 1)( 2 1)N 2q c1 (2q 1)( 2 1) 2q c1c 2 (2q 1) c4 ) 2 0 1 q q q
N 2q d 0c1c 2 (2q 1) q d c (2q 1) 1 q c 4d 0 1 2 0 1 (2 ) (u 2max )) id d q q i 1
(u1max d (
c1c 3 (2q 1) q
q 1 2
c4 )
d 0c1c 3 (2q 1) q
q 1 2
N
max d 2d 0 c 4d 0 2u 1min d d 0 (u 2d )) e 3i . i 1
N
N
i 1
i 1
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Parameters 1 and 2 must be selected such that
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1
V (t ) e 3di id
)34(
)35(
for positive constant . By using proposition 12 of [30] and Lemma A.1, one can obtain
e i 1
2 3i
2min (L I )V 0
M
N
N
V i (t ) i 1
ED
and
N 1 x 2i x 2*i i (2 1 / q )211/q 2q 1 i 1
N 1 i (2 1 / q ) 2q 1 i 1
N 1 2 i . (2 1 / q ) 2q 1 i 1
i
2 1/ q
2 1/ q
)37(
CE
PT
1/ q
)36(
AC
By (37) and (36), one gets N
N
i 1
i 1
V (t ) c 5 (e 32i i2 )
)38(
1 1 , . By using Lemma A.4 and for 0 d / 2 1 , we q 1 2min (L I ) (2 1 / q ) 2
where c 5 max have
V
d /2
N
N
i 1
i 1
(t ) c 5d /2 (e 3di id ).
)39(
13
ACCEPTED MANUSCRIPT By using (39), (35) and for 2 (0, / c5d /2 ) , one can obtain V (t ) 2V
d /2
N
N
i 1
i 1
(t ) ( 2c 5d /2 )(e 3di id ) 0.
)40(
By Lemma A.3, one concludes that V (t ) reaches zero in the following finite time t2
V
1d /2
(t 1 ) . 2 (1 d / 2)
)41(
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Therefore, from (15) and (40), for T * t1 t 2 one has V (t ) 0 . It means that N
V 0 (t ) V i (t ) 0 and one obtains i 1
x 3i x 3 j 0 * 1/ q x 2i x 2i 2e i 3 0.
,N
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i , j 1,
From (42), one can obtain x 2i x 3i 0,
i 1,
,N
)42(
)43(
for T * t1 t 2 . Therefore, the states of subsystem (12) under u 2i can reach zero in a finite
M
time.
ED
When t T * , we have x 2i x 3i 0 and should only consider system (10). It is proved that state ( x 1i ) of system (10) with u1i can converge zero in a finite time. Proof is
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divided into the following two steps.
CE
Step 1. In this step, we prove that manifold S i 1 can converge zero in a finite time. To this 1 N 2 S i 1 , whose derivative is 2 i 1
AC
end, one chooses a Lyapunov function U 2 (t ) U 2 (t ) S i 1 (5 sgn S i 1 d i 1 ) (5 d 0 ) S i 1 .
)44(
By Lemma A.4, one can obtain )45(
U 2 (t ) 3U 21/2 0
where 3 (0, 2(5 d 0 )) . By using Lemma A.3, the manifold S i 2 can reach zero in a finite time t 3 .
14
ACCEPTED MANUSCRIPT Step 2. In this step, we prove that when S i 1 0 is obtained; the states of subsystem (10) are converged to zero in a finite time t 4 . In this case, system (10) is converted to: 1
2
x 1i 4 ([ aij (x 1i x 1 j ) (x 1i )]q [ aij (x 1i x 1 j ) (x 1i )] j N i
1 q
)46(
)
j N i
where 4 0 . To prove finite time stability of system (46), one selects the following
1 U 3 (t ) x 1Ti (L I )x 1i . 2 Taking derivation of U 3 (t ) along system (46) leads to N
N
CR IP T
Lyapunov function
N
U 3 (t ) x 1Ti (L I )x 1i x 1i ( aij (x 1i x 1 j ) (x 1i )) 4 (e i111/q e i311/q ) i 1
a
j N i
ij
i 1
q 1
N 2q N U 3 (t ) 4 e i21 4 e i21 i 1 i 1
3q 1 2q
N
i 1
2 i1
M
and
e
)48(
i 1
(x 1i x 1 j ) (x 1i ) . By using Lemmas A.1 and A.3, one has
AN US
where e i 1
j N i
)47(
2min (L I )U 3 (t ).
)49(
)50(
q 1 2q
q 1 2q 3
3q 1 2q
3q 1 2q 3
)51(
(t ) 4 (2min (L I )) U (t ) 0. By using results of [31], it is concluded that U 3 (t ) reaches zero in a finite time which is U
PT
U 3 (t ) 4 (2min (L I ))
ED
By (49) and (50), one can obtain
CE
independent of the x 1i (0) . It implies x 1i x 1 j 0, i , j 1,
)52(
,N .
AC
Therefore, the states of subsystem (10) with u1i can reach zero in a finite time. Using (43) and (52), one concludes that x 1i x 2i x 3i 0 for t T * t 3 t 4 . This completes the proof.■ DFCs (4) and (5) are designed based on this assumption that the states and inputs of target (2) are available. But, in reality, this assumption does not hold and a DE is required to estimate the states of the target (2). Since the target (2) has a nonlinear dynamics, then a class
15
ACCEPTED MANUSCRIPT of nonlinear systems that includes target (2) is considered as the target with the following dynamic equations:
p (t ) f ( p (t )) Bw (t ),
(53) (54)
y i (t ) hi ( p (t )) D iv i (t ), i 1,..., N
y i (t ) R S is the i-th sensor measurement vector. f (.) : R M
CR IP T
where w (t ) and v i (t ) are Gaussian white noises. p (t ) R M is the state vector and M R M and hi (.) : R
R S are
nonlinear functions. It is worth noting that the states and nonlinear function of target (2) in
AN US
u1d terms of equation (53) are p (t ) x d (t ) and f ( p (t )) u 2d . To estimate the states of p (u ) 2d 1d
the nonlinear system (53), based on the suggested filter in [32], the proposed DE for each agent is given as:
M
pˆ i (t ) f ( pˆ i (t )) K i ( y i (t ) hi ( pˆ i (t ))) T i i1 aij [ pˆ j (t ) pˆ i (t )], i 1,..., N
)55(
j N i
ED
M S where pˆ i (t ) is the estimation of p (t ) . K i R , T i 0 and i R M M are the
estimator gain matrix, the consensus gain and the estimator matrix of agent i, respectively. It
PT
should be noted that DE (55) has the following equation for estimating the states of the target
CE
(2):
AC
u1d pˆ id (t ) u 2d pˆ (t )u 1d 2id
d d 1 K i ( y i (t ) hi ( pˆ id (t ))) T i i aij [ pˆ jd (t ) pˆ id (t )], i 1,..., N j N i
)56(
where pˆ id (t ) is the estimation of x d (t ) . The gain matrix K i and the matrix i formulas will be provided for the DE (55) in this section. The following assumptions for the DE (55) should be mentioned.
16
ACCEPTED MANUSCRIPT
Assumption 3. There exists constant matrices A i , C i and positive constants
, 1 , 2 ,
3 , 4 such that the following conditions are safisfied for every vector p : 1) Boundedness condition
B
F
.
f ( p ) f ( p ) Ai 1 2 hi ( p ) hi ( p ) C i 3 4 where . is the Euclidean vector norm and B
F
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2) The following pseudo Lipschitz conditions
tr (B T B ) is the Frobenius norm.
Ai , C i
AN US
are constant matrices. Constants ,…, 4 should be selected such that the deviations or the upper bounds in Assumption 3 are made as small as possible [33].
Assumption 4. There are matrices i 0 and K i such that satisfy the following two
M
conditions:
2T i 2 (L ) max (Q i ) 0.
ED
Q i : i (Ai K i C i ) (Ai K i C i )T i ( 1 2 )i i ( 3 4 )i K i K iT i ( 1 3 )I M 0,
)57( )58(
PT
Remark 1. Inequality (57) implies that (A, C) must be detectable. This condition exists in
CE
most of the nonlinear observers and is known as a general condition. Moreover, according to the positivity of max (Qi ) and i , the inequality (58) holds when 2 (L ) is positive, i.e.
AC
when the graph of the multiagent system is connected. Assumption 5. All agents are identical in every aspect and the same gain is designed for all of them.
Theorem 2. To estimate the states of the target (53), under Assumptions 3-5, a DE with the dynamic equation (55) is obtained, for each agent, in which estimation error
pi (t ) p (t ) pˆ i (t ) is exponentially ultimately bounded in mean square.
17
ACCEPTED MANUSCRIPT ■
Proof. See Appendix B.
As mentioned in Theorem 2, the estimation error of the proposed DE is exponentially ultimately bounded in mean square under the Assumptions 3-5. In this case, in order to obtain the DE parameters including i 0 and K i , it is necessary that the Assumptions 3-5 are satisfied. Since it is assumed that the agents are connected throughout this paper, then
CR IP T
2 (L ) 0 and the inequality (58) holds. Therefore, it is only necessary to solve the inequality (57). Thus, it is assumed that the matrix Q i in inequality (57) is such that
Qi I where 0 is a given number. To solve this inequality, the LMI technique is used.
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In the following, the LMI for obtaining parameters of DE (55) is presented.
The LMI related to the boundedness: The matrix inequality Qi I is converted to the following matrix inequality by using Lemma A.5,
ED
M
i (Ai K i C i ) (Ai K i C i )T i ( 1 3 )I ( 1 2 )i ( 3 4 )K iT i
( 1 2 )i I 0
( 3 4 )i K i 0 0 )59( I
where i 0 . Supposing X i i K i , the following LMI problem is created
CE
PT
i Ai X i C i AiT i C iT X iT ( 1 3 )I ( 1 2 )i ( 3 4 )X iT
( 1 2 )i I 0
( 3 4 ) X i 0 0. I
)60(
AC
By solving the LMI (60) using the YALMIP toolbox in Matlab, the DE parameters are obtained.
Remark 2. Theorem 2 proved that the states of the target (2) can be estimated by DE (56) with a bounded error. Therefore, the estimated states of the target (2) ( ( pˆ1id (t ), pˆ 2id (t ), pˆ 3id (t )) ) can be used instead of (x 1d , x 2d , x 3d ) in DFCs (4) and (5). In this case, the finite time
18
ACCEPTED MANUSCRIPT stability of DFCs (4) and (5) still holds and the states of multiagent system (1) can track the estimated states of target 2. To show this fact, the DE (56) is firstly rewritten as follows: u1d d d pˆ id (t ) u 2d K i (hi (x d (t )) hi ( pˆ id (t )) C i p i (t )) pˆ (t )u 1d 2id C i p i (t ) T i i1 aij [ p i (t ) p j (t )] D idv id (t ).
)61(
j N i
CR IP T
From the pseudo Lipschitz conditions, boundedness of measurement noise, and estimation errors pi (t ) and p j (t ) , one can conclude
d 1d (t ) )62( d 2d (t ) d (t ) 3d where d 1d (t ) , d 2d (t ) and d 3d (t ) are bounded by a positive constant. Then, the tracking
AN US
u1d pˆ id (t ) u 2d pˆ (t )u 1d 2id
errors are defined as follows:
)63(
M
x 1it x 1i pˆ1id (t ) x 2it x 2i pˆ 2id (t ) x x pˆ (t ) 3i 3id 3it
ED
Then, one has
x 1it (t ) u1i d 1i d 1d (t ) x 2it (t ) u 2i d 2i d 2d (t ) x 3it (t ) x 2i (t )(u1i d 1i ) x 2i u 1d d 3d (t ).
PT
)63(
CE
Defining d 1 d1i d1d (t ) , d 2 d 2i d 2d (t ) and d 3 d 1i x 1it (t ) u1i d 1 x 2it (t ) u 2i d 2 x 3it (t ) (x 2i x 2d )(u1i d 3 ) x 2i u 1d .
d 3d (t ) x 2i (t )
, one gets
AC
)64(
It can be shown that the states of the system (64) converge to zero in a finite time similar to the system (9). Therefore, one can conclude that the states of the multiagent system (1) can track the estimated states of target (2) in a finite time.
19
ACCEPTED MANUSCRIPT 4. Simulation examples In this section, the performance of the proposed DE and DFCs is validated in target tracking for a nonholonomic multiagent system with 4 agents. The communication among agents with aij 0.5 is illustrated in Figure 1. Each agent is a wheeled mobile robot and its equations are
described by [30]:
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x 1i u1i d 1i x 2i u 2i d 2i x 3i (u1i d 1i )x 2i
where state and control transformations are defined by: x 1i i
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x 2i x i cos i y i sin i x 3i x i sin i y i cos i v i x 2i u 1i u 2i
i u1i ,
)65(
)66(
where x i and y i are the Cartesian coordinates of the center of the rear wheel and i
M
measures the heading angle from the x-axis. v i and i are the linear and angular velocities
ED
and d 1i 0.8 , d 2i 0.9sin t are disturbances. The target has the following equation
PT
u1d x d (t ) u 2d B dw d (t ), x (u ) 2d 1d
x d (t ), B d R 3
(67)
CE
where u1d 1 and u 2d x 2d x 3d . The sensing model of the ith agent (measurement of
AC
target’s states) is described by:
2
3
1
4
Figure 1. A multiagent system with N 4 agents.
20
ACCEPTED MANUSCRIPT y id (t ) x 1d (t ) x 2d (t ) x 3d (t ) 0.1v id (t ),
(68)
where w d (t ) and v id (t ) are independent white noises with variances 0.1 and 0.01, respectively. The parameters considered for DFCs are
1 =40, 2 =2.7, 3 =3, 4 =2.7, 5 =1, q q =9/7, T * =7.
(69)
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To validate the proposed DFCs, we simulate a multiagent system in which each agent has the dynamics given in (65) with DFCs (4, 5). Furthermore, it is assumed that target’s states are available and the target is noise free ( B d 0 (0 0 0)T ). Tracking errors of four agents are demonstrated in Figure 2. These results show the states of all agents converge to
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the target’s states in a finite time.
To validate the proposed DE and DFCs simultaneously, we assume that target’s states are not available and target has noise ( B d 0 ). In this case, target’s states must be estimated
M
by the DE (56). The parameters considered for DE and B d are
ED
0 0 0.2 0.01 A 0 1 0.8 , B d 0.1 , C 1 1 1 , 0.2 1 0.1 0 0.1414, 1 0.2, 2 0, 3 0, 4 0, 0.01, T i 0.5,
CE
PT
0.7546 -0.2224 -0.0407 0.6039 i -0.2224 0.7779 0.1534 , K i 0.0162 . -0.0407 0.1534 0.9467 0.3352
AC
Parameters of the DFCs are the same as (69) except T * (T * 10 ). Tracking errors of agents are illustrated in Figure 3 which shows that the states of all agents converge to estimate states of the target in a finite time. Furthermore, Figure 4 shows the estimation errors of the DE (56) for each agent and verifies the successful performance of DE in estimation and reaching consensus.
21
ACCEPTED MANUSCRIPT
TE of x1d
20 0
TE of x2d
-20 0 20
2
4
6
8
10
12
2
4
6
8
10
12
TE of x3d
-20 0 5 0
Agent 1 -5 0
2
4
6 Time(s)
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0
Agent 2 8
Agent 3 10
Agent 4
12
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Figure 2. Tracking errors (TEs) of agents without the DE.
TE of x1d
10
2
4
2
0
0
2
4
6
8
10
12
14
8
10
12
14
Agent 1 4
6 Time(s) 8
CE
-5 0
PT
TE of x3d
-5 0 5
6
M
TE of x2d
-10 0 5
ED
0
Agent 2 10
Agent 3 12
AC
Figure 3. Tracking errors (TEs) of agents with the DE (56).
22
Agent 4 14
ACCEPTED MANUSCRIPT
EE of x1d
5 0
EE of x2d
-5 0 5
2
4
6
8
10
12
14
2
4
6
8
10
12
14
-5 0 5 0
Agent 1 -5 0
2
4
6 Time(s) 8
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EE of x3d
0
Agent 2
10
Agent 3
12
Agent 4
14
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Figure 4. Estimation errors (EEs) of the DE (56) for every agent. 5. Conclusions
In this paper, a DE and DFCs have been proposed for nonholonomic multiagent systems with external disturbances to track a target with nonholonomic dynamics. The DE was an
M
exponentially ultimately bounded estimator that has been obtained based on a Lyapunov
ED
theory and consensus technique. Furthermore, DFCs have been designed for nonholonomic agents to track a target in a finite time. The sufficient conditions for the convergence of the
PT
proposed algorithms were also analyzed. The simulations have been performed according to
time.
CE
the DE and DFCs that show the suitable state estimation and target tracking at the desired
Appendix A
AC
Lemma A.1 ([34]). If Assumption 2 holds, then the matrix L I is positive definite where I is an identity matrix with proper dimensions. Lemma A.2 ([35]). If 𝐿 is the Laplacian matrix of a connected undirected graph 𝐺, one has x T Lx
1 n 1 n aij (x i x j )2 aij (x i x j )2 2 i , j 1 2 i 1 j N i
23
ACCEPTED MANUSCRIPT for any x x 1 ,..., x n . 0 is a simple eigenvalue of 𝐿 and 1 is the associated eigenvector, T
where 1= 1, ,1 with appropriate dimensions. T
Lemma A.3 ([36]). Consider the following nonlinear system x F (x ), dddddddF (0) 0, x R n .
continuous function V (x )
such that
CR IP T
Suppose there exists a positive definite
V (x ) cV (x ) 0 , where c 0 and 0 1 . Then V (x ) approaches 0 in a finite time.
Furthermore, the finite convergence time T satisfies T x (x 0 )
V (x 0 )1 . c (1 )
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Lemma A.4 ([37]). For any a R , b R , the following inequalities hold: a b 2z 1 a z b z z
( a b )1/ z a
1/ z
b
1/ z
a b 2z 1 a z b z z
( a b ) z a b a z 1 b z 1 . z
ED
z
M
when z 1 is a constant. If z 1 is odd, then
X
X
PT
12 Lemma A.5 ([38]): (Schur Complement): For a given symmetric matrix X 11 , X X 22 12
the following conditions are equivalent:
CE
(a ) X 0,
T (b ) X 11 0, X 22 X 12 X 111X 12 0,
AC
(c ) X 22 0, X 11 X 12 X 221X 12T 0.
Appendix B. Proof of Theorem 1 Before proof of Theorem 1, some definition and sufficient conditions for exponentially ultimately boundedness in mean square is presented. Definition B.1 [39]: Consider the stochastic differential equation:
dp (t ) f ( p (t ))dt g ( p (t ))d(t ).
)B.1( 24
ACCEPTED MANUSCRIPT p (t ) is said to be exponentially ultimately bounded in mean square if there exist positive constants C 1 , C 2 , C 3 such that for all t 0 , 2 E p (t ) C 1 exp(C 2t ) C 3 .
)B.2(
Lemma B.1 [39]: Consider the stochastic differential equation (B.1). Assume that there exists a function V ( p ,t ) C 2,1 and positive constants C 1 , k 1 , k 2 such that C 1 p (t )
2
)B.3(
V ( p , t )
CR IP T
and V ( p , t ) k 1V ( p , t ) k 2 where V ( p , t ) V ( p , t ) 2V ( p , t ) V ( p ,t ) f ( p , t ) tr g ( p , t ) g ( p ,t ) . 2 t p p
)B.4(
AN US
Furthermore, suppose that E [V ( p0 ,0)] . In this case, p (t ) is called exponentially ultimately bounded in mean square. In addition, it can be said that: E [V ( p , t )] (E [V ( p 0 ,0)] k 2 k 1 ) exp(k 1t ) k 2 k 1 . )B.5( Proof of Theorem 1. The proof of Theorem 1 will be provided in the Ito form [40]. Therefore, equations (53) and (54) are firstly rewritten in Ito form as:
M
dp (t ) f ( p (t ))dt BdW (t ), p (t ) R M dz i (t ) hi ( p (t ))dt D i dV i (t ), i 1,..., N
ED
)B.6(
where t
0
PT
z i (t ) y i (s )ds , z i (0) 0,
CE
W (t ) and V i (t ) are standard Brownian motions. Furthermore, the DE equation (55) should
be written in Ito form as:
AC
dpˆ i (t ) f ( pˆ i (t ))dt K i (dz i (t ) hi ( pˆ i (t ))dt ) T i i1 aij [ pˆ j (t ) pˆ i (t )]dt , i 1,..., N .
)B.7(
j N i
By defining the error of each estimator as pi (t ) pˆ (t ) pˆ i (t ) , we have
dpi (t ) dp (t ) dpˆi (t ) f ( p (t ), pˆi (t ))dt B (t )dw (t ).
)B.8(
T T T Defining the error vector as p (t ) [ p1 (t ),..., p N (t )] , we prove that the state estimation error
of the DE (55) is bounded. To achieve this purpose, a stochastic Lyapunov function is considered as 25
ACCEPTED MANUSCRIPT N
V 1 ( p (t )) p iT (t )i p i (t ).
)B.9(
i 1
of V 1 with respect to (B.8), one can obtain
By calculating the differential generator
N V 1 V 1 ( p (t )) T f ( p (t ), pˆ i (t )) tr BB T i 2 i 1 p i (t ) N N 1 N 2 p iT (t )i f ( p (t ), pˆ i (t )) tr (BB T K i D i D iT K iT )i 2 p iT (t )i f ( p (t ), pˆ i (t )) c1. 2 i 1 i 1 i 1
)B.10(
N
N
i 1
i 1
CR IP T
Extending the equation (B.10) gives V 1 ( p (t )) 2 piT (t )i f ( p (t ), pˆ i (t )) c1
N 2 piT (t )i f ( p (t )) f ( pˆ i (t )) K i (hi ( p (t )) hi ( pˆ i (t ))) T i i1 aij [ pˆ j (t ) pˆ i (t )] c1. i 1 j N i i 1 By adding and reducing Ai e i (t ) and C i e i (t ) , the equation (B.11) is converted to N
AN US
N
)B.11(
V 1 ( p (t )) 2 p iT (t )i (A i p i (t ) f ( p (t )) f ( pˆ i (t )) A i e i (t ) i 1
K i (hi ( p (t )) hi ( pˆ i (t ))) C i p i (t ) K i C i p i (t ) T i i1 N
a
j N i
ij
[ pˆ j (t ) pˆ i (t )] c 1 )
)B.12(
N
p (t ) i (A i K i C i ) (A i K i C i ) i p i (t ) 2 p (t )i f ( p (t )) f ( pˆ i (t )) A i p i (t ) T
M
i 1
T i
N
T i
i 1 N
N
2 p iT (t )i K i (hi ( p (t )) hi ( pˆ i (t ))) C i p i (t ) 2T i p iT (t ) aij [ pˆ j (t ) pˆ i (t )] c 1.
ED
i 1
i 1
j N i
i 1
By using the inequalities x y x y , 2 x y x x y y and Assumption 5, the second T
T
T
PT
and third terms in equation (B.12) are simplified as
CE
2 p iT (t )i f ( p (t )) f ( pˆ i (t )) A i p i (t ) 2 i p i (t ) f ( p (t )) f ( pˆ i (t )) A i p i (t ) 2 i p i (t ) ( 1 p i (t ) 2 ) p iT (t ) ( 1 2 )i i 1I M p i (t ) 2
AC
2 p iT (t )i K i (hi ( p (t )) hi ( pˆ i (t ))) C i p i (t ) 2 K iT i p i (t ) hi ( p (t )) hi ( pˆ i (t ))) C i p i (t )
)B.13(
2 K iT i p i (t ) ( 3 p i (t ) 4 ) p iT (t ) ( 3 4 )i K i K iT i 3 I M p i (t ) 4 .
By substituting (B.13) in equations (B.12) and using pˆ j (t ) pˆi (t ) pi (t ) p j (t ) , Assumption 4 and equation (B.12) results in,
26
ACCEPTED MANUSCRIPT N
V 1 ( p (t )) p iT (t )[i (A i K i C i ) (A i K i C i )T i (t ) i 1
N
( 1 2 )i i ( 3 4 )i K i K iT i ( 1 3 )I M ] p i (t ) 2T i p T ( L I N ) p (t ) ( 2 4 c1 ) )B.14( i 1
N
p iT (t )Q i p i (t ) 2T i 2 (L ) p T (t ) p (t ) k 2 i 1
( i 1
2
4 c1 ) . Considering Assumption 4 and matrix properties yield
N
V 1 ( p (t )) p iT (t )Q i p i (t ) 2T i 2 (L ) p T (t ) p (t ) k 2 i 1
max (Q i ) p T (t ) p (t ) 2T i 2 (L ) p T (t ) p (t ) k 2
max
)B.15(
(Q i ) 2T i 2 (L ) V 1 ( p (t )) k 2 k 1V 1 ( p (t )) k 2 . max (i )
AN US
CR IP T
where k 2
N
In which, constants k 1 and k 2 are defined as k1
2T i 2 (L ) max (Q i ) max (i )
)B.16(
1 k 2 ( 2 4 c1 ) 2 4 tr (BB T K i D i D iT K iT )i . 2 i 1 i 1 N
M
N
ED
According to Assumption 4, constants k 1 and k 2 are positive. By using Lemma B.1, it is proven that
)B.17(
PT
E [V 1 ( p (t ))] E [V 1 ( p (0))]exp(k 1t ) k 2 k 1 (1 exp(k 1t )) k 3 exp(k 1t ) k 2 k 1 .
E [V 1 ( p (t ))] min (i )
CE
Since
E [ p (t ) ] 2
)B.18(
AC
Consequently, it is proven from Lemma B.1 that estimation error is exponentially ultimately ■
bounded in mean square. References [1]
[2]
H. Pei, S. Chen, and Q. Lai, "Multi-target consensus circle pursuit for multi-agent systems via a distributed multi-flocking method," International Journal of Systems Science, pp. 1-8, 2015. S. Satunin and E. Babkin, "A multi-agent approach to Intelligent Transportation Systems modeling with combinatorial auctions," Expert Systems with Applications, vol. 41, pp. 6622-6633, 2014. 27
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[9]
[10]
[11]
[12]
[13]
[14]
[15] [16]
AC
[17]
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[8]
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[7]
M
[6]
ED
[5]
PT
[4]
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Distributed estimation and control of multiple nonholonomic mobile agents with external disturbances Ahmadreza Jenabzadeh , Behrouz Safarinejadian PII: DOI: Reference:
S0016-0032(18)30117-0 10.1016/j.jfranklin.2018.02.008 FI 3333
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
30 January 2017 30 October 2017 16 February 2018
Please cite this article as: Ahmadreza Jenabzadeh , Behrouz Safarinejadian , Distributed estimation and control of multiple nonholonomic mobile agents with external disturbances, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.02.008
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ACCEPTED MANUSCRIPT Distributed estimation and control of multiple nonholonomic mobile agents with external disturbances Ahmadreza Jenabzadeh, Behrouz Safarinejadian School of Electrical and Electronic Engineering, Shiraz University of Technology, Shiraz, Iran
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Abstract This paper investigates the tracking control problem of nonholonomic multigent systems with external disturbances. For this purpose, distributed finite time controllers (DFCs) based on the terminal sliding mode method are proposed to ensure that states of the agents track the
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states of the target in a finite time. Furthermore, a distributed estimator (DE) is designed for each agent to estimate the target’s states. The stability analysis of DFCs and DE is also considered. Simulation examples demonstrate the promising performance of the proposed
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algorithms.
target tracking.
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1. Introduction
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Keywords: Distributed control, distributed estimator, nonholonomic multiagent systems,
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Study of distributed tracking control problem in multiagent systems has received great attention in the recent decade due to its applications in different areas such as flocking,
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intelligent transportation, surveillance, cooperative control and so on [1-7]. The tracking control of multiagent systems includes two aspects of estimation and control. The estimation aspect aims to design a distributed estimator for each agent to estimate the states of the target. The control aspect plans to design a distributed controller for every agent to track the states of the target.
1
ACCEPTED MANUSCRIPT The tracking control problem of multiagent systems can be considered in two classes: agents with linear and nonlinear dynamics. Some algorithms have been presented for target tracking of agents that have linear dynamics [8-14]. [8] has considered the coordinated tracking problem of linear multiagent systems with a directed communication graph. In [9], the authors established a distributed estimation and control algorithm such that the mobile
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agents estimate the states of a linear target and track it in a flocking manner. [10] studied a leader-follower cooperative control of multiagent systems under the directed graph and transmission time delay. This paper has assumed that some agents (followers) are connected to the target (leader) and designs estimators and controllers for each follower to track the
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leader. In [11], some tracking and formation control protocols have been suggested for a second order multiagent system in which the interaction topology among agents was invariant. In [12], an event-based control law has been proposed for a second-order
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multiagent system which ensures that each agent with input delay can track a linear target. In this work, the suggested controller in each agent used the discrete states of the neighboring
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agents and has been derived based on the matrix and graph theories and inequality technique.
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[13] designed a fully distributed controller for multiagent systems with input saturation to track a target. The proposed control law did not require global information such as
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information of the Laplacian matrix or network size unlike most of the existing tracking control algorithms. In [14], an observer-based controller has been suggested for linear
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multiagent systems with external disturbances. The proposed algorithm guaranteed that each agent can track the target whose input was unavailable and unknown to the agents. In practical target tracking applications, many multiagent systems and targets have
nonlinear dynamics. One of the most practical multiagent systems is nonholonomic ones. Thus, it is necessary to investigate distributed tracking algorithms of nonholonomic multiagent systems. In [15], a finite time control law has been designed for some
2
ACCEPTED MANUSCRIPT nonholonomic mobile agents. In this paper, agents can track a target in a finite time when velocitiy conditions of the target are satisfied. [16] presented a distributed controller for nonholonomic agents that use local measurements to track a target with a time varying velocity. Furthermore, the topology of the multiagent system is switched based on the distance among agents and the target. In [17], distributed continuous control laws have been
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proposed for single and multiple nonholonomic agents based on sliding mode control and artificial potentials methods. The suggested controllers make agents converge to a maneuvering target. In [18], the problem of finite-time tracking control of multiple nonholonomic mobile robots was solved. A distributed finite-time controller has been
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proposed for each agent to track a target with nonholonomic dynamics in this paper. [19] proposed distributed state feedback controllers for nonholonomic multiagent systems such that the states of each agent converge to the states of a target with an exponential rate. This
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paper has also extended the results for multiagent systems with a time-varying communication graph. In [20], two discontinuous control laws have been designed so that
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nonholonomic agents can track a target with unknown dynamics in a flocking maner.
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Suggested controllers ensure collision avoidance among agents and can be used for both fixed and switching topologies. In [21], a sampled-data controller has been presented for a
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group of nonholonomic agents to track a mobile target. In this study, a dwell time is assumed for the feasibility of information sensing and processing to avoid chattering caused by abrupt
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changes of the neighboring relations. The aformantionted nonholonomics multiagent systems have dynamic models. In the recent decay, study of chained form of nonholonomic multiagent systems has attracted researchers’ attentions because this form includes any kinematic model of mechanical nonholonomics systems. Actually, [22] proved that nonholonomic systems can be converted into the chained form by state and input transformations. [23] has considered the leader-follower problem of nonholonomic chained-
3
ACCEPTED MANUSCRIPT form systems. This paper proposed a distributed finite time estimator and controller for each follower to estimate and track the leader’s states in a finite time. The targets in the above tracking control algorithms of nonholonomic multiagent systems have continuous dynamics and their states are assumed to be available or at least one of the agents is connected with the target. But, these assumptions do not hold in practical
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environments and target’s states must be estimated by a distributed estimation algorithm. Since the targets with nonholonomic dynamics are included in nonlinear systems. Therefore, a nonlinear distributed estimator is required for each agent to estimate the states of these types of targets. There are some distributed estimators for state estimation of targets with
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nonlinear dynamics [24-27]. [24] proposed a distributed estimator which estimated the states of a continuous linear system whose input is generated by another linear system. In [25, 26], some nonlinear filters have been presented for targets with discrete nonlinear dynamics based
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on extended and unscented Kalman filters. In [27], the distributed state estimation problem of discrete nonlinear systems with randomly delayed measurements was considered and a
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Gaussian-consensus filter was proposed to pursue a trade off between estimation accuracy
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and computing time. It should be mentioned that the distributed estimators of [25-27] were suggested for state estimation of the targets with discrete nonlinear dynamics and cannot be
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used for state estimation of continuous nonholonomic systems. Moreover, [24] has presented a continuous estimator for targets with a very special structure which do not include
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nonholonomic systems. Therefore, it is necessary to present a novel distributed estimator to estimate the states of the targets with continuous nonholonomic dynamics. In almost all practical multiagent systems, model uncertainties and disturbances are
inevitable. If no appropriate controller is used for multiagent systems to deal with disturbances, the existence of disturbances may affect system performance and lead to instability. Recently, the problem of tracking control has been solved for nonholonomic
4
ACCEPTED MANUSCRIPT multiagent systems in dynamic models subject to external disturbances [28, 29]. This problem has not been considered for chained-form nonholonomic multiagent systems in the literature according to the best of the authors’ knowledge. Motivated by the above analysis, this paper will focus on solving the target tracking problem for chained-form nonholonomic multiagent systems with external disturbances. In
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the first step, each agent is assumed to have a third-order nonholonomic dynamics in chainedform subject to external disturbances. Based on the terminal sliding mode (TSM) method, A DFC is designed for each agent to guarantee that the states of the nonholonomic multiagent systems track the states of the target in a finite time. In the second step, a DE is designed for
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each agent to estimate the states of a noisy nonholonomic target. The main contributions of this paper are as follows:
In contrast to the tracking control algorithms of nonholonomic multiagent
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systems [15-21, 28, 29], this paper considers the estimation aspect, in addition to the control aspect, of tracking control and proposes a distributed estimator
A finite time distributed controller is obtained for
chained-form
PT
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to estimate the states of the target.
nonholonomic multiagent systems that makes each agent track appropriately
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the states of the target in the presence of external disturbance, whereas the
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results of [15-21, 23] have been designed for disturbance free case and were
not robust against external disturbance. A distributed estimator is suggested in this paper that can be applied for targets with a general class of continuous nonlinear dynamics, whereas the results in [24-27] are designed for continuous nonlinear systems with a special structure and discrete nonlinear dynamics, and cannot be used for state estimation of continuous nonlinear systems.
5
ACCEPTED MANUSCRIPT The rest of this paper is organized as follows: Section 2 presents the preliminaries and problem definition. Section 3 proposes the distributed control laws and DE method for nonholonomic multiagent systems and their stability analysis is given. The effectiveness of the proposed algorithms is studied with simulations in Section 4. Finally, the conclusions are drawn in Section 5.
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Notation: R n and R n m denote the n dimensional Euclidean space and the set of all n m real matrices, respectively. I M denotes an M M identity matrix. The symbol represents the Kronecker product. . denotes the expectation operator. max (.) and min (.) denote the
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2,1 biggest and the smallest eigenvalues, respectively. C denotes the family of all nonnegative
functions V (x (t ),t ) that are continuously twice differentiable in x and once differentiable in t. 2. Preliminaries and problem definition
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Graph Theory. Every multiagent system can be modeled by an undirected graph
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G (v , , A ) , where v 1, 2,..., N is the agent (node) set, v v (i , j ) : i , j v is the
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edge set and A [aij ] R N N is the adjacent matrix. If nodes i and j are connected, then the node i is the neighbor of node j and aij a ji 0 . The Laplacian matrix of the graph G is
a
j N i
ij
.
AC
di
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defined as L D A , in which D is a diagonal matrix with the diagonal elements The
1 (L ) 2 (L )
eigenvalues
of
a
Laplacian
matrix
can
be
ordered
as
N (L ) in which the second smallest eigenvalue, 2 (L ) is called the
algebraic connectivity of the network. N i j v : (i , j ) , j i denotes the set of neighbors of node i. If there is a link between nodes i and j , then aij 0 . G is called a connected graph if and only if there is at least one path between every two arbitrary nodes. It
6
ACCEPTED MANUSCRIPT is a critical point that an undirected graph is connected if and only if its algebraic connectivity is positive: 2 (L ) 0 . Consider a nonholonomic multiagent system with N mobile agents. The dynamics of each agent is described by:
u1i d 1i u 2i d 2 i i 1,..., N x 2i (u1i d 1i )
(1)
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x 1i x 2i x 3i where
x 1i , x 2i and x 3i are states of the ith mobile agent and u 1i , u 2i are the control inputs
of agent i. d i 1 and d i 2 are bounded disturbances. The objective is to track the states of a target
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with the following dynamics u1d x d (t ) u 2d B dw d (t ), x (u ) 2d 1d
(2)
x d (t ), B d R 3
where x d (t ) and (u1d ,u 2d ) are state and control inputs of the target, respectively. The
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sensing model of the ith agent (measurement equation) is described by: y id (t ) hid (x d (t )) Didv id (t ), i 1,..., N
ED
(3)
d where w d (t ) and v id (t ) are Gaussian white noises and hi (.) is a nonlinear function. It is
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assumed that all of the involved matrices have compatible dimensions. To solve tracking
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control problem, a distributed estimator and controller must be designed such that each agent with dynamics (1) can estimate and track the states of the target (2).
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To track the target (2), the DFCs based on a switching strategy are proposed for
multiagent system (1) as follows: u1i u1d ,
t T * , 1
2
u1i 5 sgn S i 1 4 ([ aij (x 1i x 1 j ) (x 1i )]q [ aij (x 1i x 1 j ) (x 1i )] j N i
j N i
2 1 q
u 2i 3 sgn S i 2 1[(x 2i ) ( aij (x 3i x 3 j ) (x 3i ))] q
q 2
j N i
with 7
u 2d ,
1 q
) u1d , t T * ,
)4(
)5(
ACCEPTED MANUSCRIPT 1
t
2
S i 1 x 1i ( 4 ([ aij (x 1i x 1 j ) (x 1i )]q [ aij (x 1i x 1 j ) (x 1i )] j N i
0
1 q
)d
)6(
j N i
2 1 q
t
S i 2 x 2i ( 1[(x 2i ) ( aij (x 3i x 3 j ) (x 3i ))] q
0
q 2
)7(
)d
j N i
where x 1i x 1i x 1d , x 2i x 2i x 2d and x 3i x 3i x 3d . 1 q q1 q 2 2 , 1 q q1 q 2 2 , q1 , q 2 , q1 , q 2 are positive odd integers, 1 , 2 , 4 , 3 d 0 , 5 d 0 are positive constants,
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T * t 1 t 2 is switched time, and S i 1 , S i 2 are ideal sliding surfaces. Controllers (4) and (5) are
nonsingular terminal sliding mode controllers because the parameters q1 , q 2 , q1 and q 2 are positive odd integers and 1 q q1 q 2 2 , 1 q q1 q 2 2 . In this case, the system dynamics
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with controllers (4) and (5) does not have any terms with negative (fractional) powers in sliding and reaching phases and when S i 1 , S i 2 0 . Therefore, no singularity occurs in sliding surfaces and system dynamics when the controllers (4) and (5) are used.
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max Assumption 1. There exist positive constants d 0 , u1dmin , u1dmax and u 2d such that d 1i d 0 , max d 2i d 0 , u1min and u 2d u 2max d . d u1d u1d
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Assumption 2. The graph G related to the nonholonomic mobile agents is connected and
3. Main results
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undirected.
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In this section, the stability analysis of the DFCs (4) and (5) is presented. Then, a DE is introduced and analyzed for each agent to estimate the states of the target with dynamics (2).
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To analyze the stability of the DFCs (4) and (5), the tracking error is defined as
follows:
x 1i x 1i x 1d x 2 i x 2 i x 2d x 3i x 3i x 3d u1i u1i u 1d u u u . 2d 2i 2i
)8(
where u1i and u 2i are the DFCs given in (4) and (5). Then, one can obtain (without noise) 8
ACCEPTED MANUSCRIPT x 1i (t ) u1i d 1i x 2i (t ) u 2i d 2i x 3i (t ) (x 2i x 2d )(u1i d 1i ) x 2i u1d .
)9(
To achieve the finite-time tracking, it should be shown that the states of the error system (9) with u1i and u 2i converge to zero in a finite time. For this aim, the system (9) is divided into two subsystems, a first order subsystem as
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x 1i u1i d1i and a second order subsystem x 2 i u 2i d 2 i x 3i (x 2i x 2d )(u1i d 1i ) x 2i u1d .
)10(
)11(
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Theorem 1. Consider the multiagent system (9). Under Assumptions 1 and 2, the u1i and u 2i ensure that the states of the system (9) can reach zero in a finite time.
x 2 i u 2i d 2 i x 3i (x 2i x 2d )(d 1i ) x 2i u1d .
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Proof. When t T * , one has u1i 0 and the subsystem (11) is converted to
)12(
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We prove that the states of the system (12) with controller u 2i can reach zero in a finite time.
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To this goal, it must firstly be proved that S i 2 0 for any t t 1 where t 1 is a finite time. Then, it must be shown when S i 2 0 is obtained, the states of subsystem (12) converge to zero in
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the finite time t 2 . Choose the following Lyapunov function 1 N 2 S i 2 , 2 i 1
)13(
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U 1 (t )
taking the derivative of U 1 (t ) leads to U 1 (t )
N 1 N S i 2 ( 3 sgn S i 2 d i 2 ) ( 3 d 0 ) S i 2 . 2 i 1 i 1
)14(
By Lemma A.4, one has U 1 (t ) 1U 11/2 (
N 1 ( 3 d 0 )) S i 2 0 2 i 1
)15(
9
ACCEPTED MANUSCRIPT where 1 (0, 2(3 d 0 )) . From Lemma A.3, it can be resulted from (15) that the manifold S i 2 can reach zero in the finite time t 1 satisfying t 1
2U 11/ 2 (0)
1
. For t t 1 , we have S i 2 0 and
system (12) can be written as
)16(
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2 1 q q q x [( x ) ( a ( x x ) ( x ))] 2i 1 2 ij 2i 3i 3j 3i j N i x 3i (x 2i x 2d )(d 1i ) x 2i u 1d .
It will be proved that the states of the system (16) can reach zero in the finite time t 2 .The proof includes the following two steps.
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Step 1: Choose a Lyapunov function 𝑉0 as 1 2
V 0 (t ) x 3Ti (L I )x 3i .
)17(
By Lemma A.2, taking derivation of V 0 along system (16) yields N
V 0 (t ) x 3Ti (L I )x 3i (d 1i u1d )x 2i ( aij (x 3i x 3 j ) x 3i ) j N i
M
i 1
N
)18(
(d 1i ) x 2d ( aij (x 3i x 3 j ) x 3i ). i 1
a
j N i
ij
(x 3i x 3 j ) x 3i and taking x 2*i 2e i1/3q as a virtual control law, one can
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Defining e i 3
PT
obtain N
j N i
N
N
i 1
i 1
V 0 (t ) 2 (d 1i u1d )e 3(1i q )/q (d 1i u1d )e 3i (x 2i x 2*i ) (d 1i ) x 2d e 3i .
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i 1
)19(
AC
Step 2: In this step, we consider another Lyapunov function as N
V (t ) V 0 (t ) V i (t )
)20(
i 1
with
x 2i
V i (t ) x (s q x 2*i q )
2 1/ q
* 2i
)21(
ds .
By taking the derivative of V (t ) , one gets N
V (t ) V 0 (t ) V i (t ).
)22(
i 1
10
ACCEPTED MANUSCRIPT N
To obtain the terms
V i 1
1 dx 2*i q q dt
V i (t ) (2 )
i
x 2i
x 2*i
(t ) , we define i x 2qi x 2*i q . The derivative of V i (t ) is given by 11/ q
(s q x 2*i q )
)23(
ds i21/q x 2i .
Note that dx 2*i q 2q e i 3 2q aij ((d 1i (x 2i x 2d ) x 2i u 1d ) (d 1 j (x 2 j x 2d ) x 2i u 1d )) (d 1i (x 2i x 2d ) x 2i u 1d ) dt j N i
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2q aij (d 1i x 2i d 1 j x 2 j ) d 1i x 2i j N i 2q aij (x 2i u1d x 2i u1d ) x 2i u1d j N i 2q aij (d 1i x 2d d 1 j x 2d ) d 1i x 2d ) . j N i
By using the following inequalities
2q ( aij (x 2i u 1d x 2 j u 1d ) x 2i u 1d ) 2q ( aij 1) x 2i u 1d j N i j N i
a
ij
j N i
)24(
x 2 j u 1d
N
M
2q ( 1 x 2i u 1d 2 x 2 m u 1d ) m 1
N
2q c1 x 2 m u1d ,
ED
m 1
CE
PT
2q aij (d 1i x 2i d 1 j x 2 j ) d 1i x 2i 2q d 0 ( aij 1) x 2i j N i j N i
x 2 j
a
j N i
ij
)25(
N
2q d 0 ( 1 x 2i 2 x 2 m ) m 1
N
2q d 0c1 x 2 m , m 1
AC
2q aij (d 1i x 2d d 1 j x 2d ) d 1i x 2d ) 2q d 0 ( aij 1) x 2d j N i j N i
a
j N i
ij
x 2d
N
2q d 0 ( 1 x 2d 2 x 2d ) m 1
N
2q d 0c1 x 2d , m 1
j N i
j N i
where 1 max i aij 1 , 2 max i , j aij and c1 is a positive constant, one has N N N dx 2*i q 2q c1 x 2 m u1d 2q d 0c1 ( x 2 m x 2d ). dt m 1 m 1 m 1
)26(
11
ACCEPTED MANUSCRIPT Furthermore, it can be also verified from proposition 8 of [30] that
x 2i
x 2*i
11/ q
(s q x 2*i q )
11/ q
ds x 2i x 2*i x 2qi x 2*i q 11/ q
x 2i x 2*i i
)27(
.
Substituting (26) and (27) into (23) yields 1
N
N
N
m 1
m 1
m 1
11/ q
i21/q x 2i .
CR IP T
V i (t ) (2 ) 2q c1 x 2 m u1d 2q d 0c1 ( x 2 m x 2d ) x 2i x 2*i i q
)28(
By using Lemma A.4 and proposition 9 of [30], one can obtain 11/ q
( 2 1) i
i
11/ q
x 2d
x 2d x 2 i x
* 2i
c 2 m
d
x
x
1/ q
q 2i
d
2c 3 e m 3 ,
11/ q
d
* q 1/ q 2i
i
1
2
1 q
)29(
x 2d i
AN US
x 2 m x 2i x 2*i i
where c 2 and c 3 are positive constants and d 1 1/ q . By the definition of 𝜉𝑖 and (16), we have x 2i 1i(2q )/q . Substituting (29) into (28), one has
2q c1 (2q 1)( 2 1) i q
d
N
u m 1
d 0c1 (2q 1)( 2 1)N i q q 2
1d
d
c c (2q 1) 2q 1 N 2q c1c 2 (2q 1) N d d m u1d 1 3 e m 3 u1d q q m 1 m 1
2q d 0c1c 2 (2q 1) N d 0c1c 3 (2q 1) 2q 1 N d d em 3 m q q m 1 m 1
M
V i (t )
)30(
q d c (2q 1) 1 q N 2 0 1 (2 ) x 2d i 1id . q m 1 To continue proof, from (19), one gets
PT
ED
1
N
N
N
V 0 (t ) 2 u1d e 3(1i q )/q 2d 0 e 3(1i q )/q u1d e 3i (x 2i x 2*i ) i 1
i 1
CE
N
d0
e i 1
3i
)31(
i 1
N
(x 2i x 2*i ) d 0 x 2d e 3i . i 1
AC
By Lemma A.4, one has N
1
N
u1d e 3i (x 2i x 2*i ) u1d e 3i 2 i 1
1 q
i 1 N
u1d 3 e 3i i 1
d
x 2qi x 2*i q 4 i
d
1 q
1
2
1 N q
u i 1
1d
c u e
e 3i i
N
4
i 1
1d
d 3i
i
1 q
)32( d
where 3 , 4 and c 4 are positive constants. Substituting (32) into (31), one can obtain
12
ACCEPTED MANUSCRIPT N
N
N
i 1
i 1
i 1
V 0 (t ) 2 u1d e 3(1i q )/q 2d 0 e 3(1i q )/q c 4 u1d e 3i N
c 4d 0 e 3i i 1
i
d
d
d x
d
i
d
)33(
N
0
i 1
2d
e 3i .
Putting (33) and (30) into (22) and considering equation x 2d u 2d yield V (t ) (u1max d (
q d c (2q 1)( 2 1)N 2q c1 (2q 1)( 2 1) 2q c1c 2 (2q 1) c4 ) 2 0 1 q q q
N 2q d 0c1c 2 (2q 1) q d c (2q 1) 1 q c 4d 0 1 2 0 1 (2 ) (u 2max )) id d q q i 1
(u1max d (
c1c 3 (2q 1) q
q 1 2
c4 )
d 0c1c 3 (2q 1) q
q 1 2
N
max d 2d 0 c 4d 0 2u 1min d d 0 (u 2d )) e 3i . i 1
N
N
i 1
i 1
AN US
Parameters 1 and 2 must be selected such that
CR IP T
1
V (t ) e 3di id
)34(
)35(
for positive constant . By using proposition 12 of [30] and Lemma A.1, one can obtain
e i 1
2 3i
2min (L I )V 0
M
N
N
V i (t ) i 1
ED
and
N 1 x 2i x 2*i i (2 1 / q )211/q 2q 1 i 1
N 1 i (2 1 / q ) 2q 1 i 1
N 1 2 i . (2 1 / q ) 2q 1 i 1
i
2 1/ q
2 1/ q
)37(
CE
PT
1/ q
)36(
AC
By (37) and (36), one gets N
N
i 1
i 1
V (t ) c 5 (e 32i i2 )
)38(
1 1 , . By using Lemma A.4 and for 0 d / 2 1 , we q 1 2min (L I ) (2 1 / q ) 2
where c 5 max have
V
d /2
N
N
i 1
i 1
(t ) c 5d /2 (e 3di id ).
)39(
13
ACCEPTED MANUSCRIPT By using (39), (35) and for 2 (0, / c5d /2 ) , one can obtain V (t ) 2V
d /2
N
N
i 1
i 1
(t ) ( 2c 5d /2 )(e 3di id ) 0.
)40(
By Lemma A.3, one concludes that V (t ) reaches zero in the following finite time t2
V
1d /2
(t 1 ) . 2 (1 d / 2)
)41(
CR IP T
Therefore, from (15) and (40), for T * t1 t 2 one has V (t ) 0 . It means that N
V 0 (t ) V i (t ) 0 and one obtains i 1
x 3i x 3 j 0 * 1/ q x 2i x 2i 2e i 3 0.
,N
AN US
i , j 1,
From (42), one can obtain x 2i x 3i 0,
i 1,
,N
)42(
)43(
for T * t1 t 2 . Therefore, the states of subsystem (12) under u 2i can reach zero in a finite
M
time.
ED
When t T * , we have x 2i x 3i 0 and should only consider system (10). It is proved that state ( x 1i ) of system (10) with u1i can converge zero in a finite time. Proof is
PT
divided into the following two steps.
CE
Step 1. In this step, we prove that manifold S i 1 can converge zero in a finite time. To this 1 N 2 S i 1 , whose derivative is 2 i 1
AC
end, one chooses a Lyapunov function U 2 (t ) U 2 (t ) S i 1 (5 sgn S i 1 d i 1 ) (5 d 0 ) S i 1 .
)44(
By Lemma A.4, one can obtain )45(
U 2 (t ) 3U 21/2 0
where 3 (0, 2(5 d 0 )) . By using Lemma A.3, the manifold S i 2 can reach zero in a finite time t 3 .
14
ACCEPTED MANUSCRIPT Step 2. In this step, we prove that when S i 1 0 is obtained; the states of subsystem (10) are converged to zero in a finite time t 4 . In this case, system (10) is converted to: 1
2
x 1i 4 ([ aij (x 1i x 1 j ) (x 1i )]q [ aij (x 1i x 1 j ) (x 1i )] j N i
1 q
)46(
)
j N i
where 4 0 . To prove finite time stability of system (46), one selects the following
1 U 3 (t ) x 1Ti (L I )x 1i . 2 Taking derivation of U 3 (t ) along system (46) leads to N
N
CR IP T
Lyapunov function
N
U 3 (t ) x 1Ti (L I )x 1i x 1i ( aij (x 1i x 1 j ) (x 1i )) 4 (e i111/q e i311/q ) i 1
a
j N i
ij
i 1
q 1
N 2q N U 3 (t ) 4 e i21 4 e i21 i 1 i 1
3q 1 2q
N
i 1
2 i1
M
and
e
)48(
i 1
(x 1i x 1 j ) (x 1i ) . By using Lemmas A.1 and A.3, one has
AN US
where e i 1
j N i
)47(
2min (L I )U 3 (t ).
)49(
)50(
q 1 2q
q 1 2q 3
3q 1 2q
3q 1 2q 3
)51(
(t ) 4 (2min (L I )) U (t ) 0. By using results of [31], it is concluded that U 3 (t ) reaches zero in a finite time which is U
PT
U 3 (t ) 4 (2min (L I ))
ED
By (49) and (50), one can obtain
CE
independent of the x 1i (0) . It implies x 1i x 1 j 0, i , j 1,
)52(
,N .
AC
Therefore, the states of subsystem (10) with u1i can reach zero in a finite time. Using (43) and (52), one concludes that x 1i x 2i x 3i 0 for t T * t 3 t 4 . This completes the proof.■ DFCs (4) and (5) are designed based on this assumption that the states and inputs of target (2) are available. But, in reality, this assumption does not hold and a DE is required to estimate the states of the target (2). Since the target (2) has a nonlinear dynamics, then a class
15
ACCEPTED MANUSCRIPT of nonlinear systems that includes target (2) is considered as the target with the following dynamic equations:
p (t ) f ( p (t )) Bw (t ),
(53) (54)
y i (t ) hi ( p (t )) D iv i (t ), i 1,..., N
y i (t ) R S is the i-th sensor measurement vector. f (.) : R M
CR IP T
where w (t ) and v i (t ) are Gaussian white noises. p (t ) R M is the state vector and M R M and hi (.) : R
R S are
nonlinear functions. It is worth noting that the states and nonlinear function of target (2) in
AN US
u1d terms of equation (53) are p (t ) x d (t ) and f ( p (t )) u 2d . To estimate the states of p (u ) 2d 1d
the nonlinear system (53), based on the suggested filter in [32], the proposed DE for each agent is given as:
M
pˆ i (t ) f ( pˆ i (t )) K i ( y i (t ) hi ( pˆ i (t ))) T i i1 aij [ pˆ j (t ) pˆ i (t )], i 1,..., N
)55(
j N i
ED
M S where pˆ i (t ) is the estimation of p (t ) . K i R , T i 0 and i R M M are the
estimator gain matrix, the consensus gain and the estimator matrix of agent i, respectively. It
PT
should be noted that DE (55) has the following equation for estimating the states of the target
CE
(2):
AC
u1d pˆ id (t ) u 2d pˆ (t )u 1d 2id
d d 1 K i ( y i (t ) hi ( pˆ id (t ))) T i i aij [ pˆ jd (t ) pˆ id (t )], i 1,..., N j N i
)56(
where pˆ id (t ) is the estimation of x d (t ) . The gain matrix K i and the matrix i formulas will be provided for the DE (55) in this section. The following assumptions for the DE (55) should be mentioned.
16
ACCEPTED MANUSCRIPT
Assumption 3. There exists constant matrices A i , C i and positive constants
, 1 , 2 ,
3 , 4 such that the following conditions are safisfied for every vector p : 1) Boundedness condition
B
F
.
f ( p ) f ( p ) Ai 1 2 hi ( p ) hi ( p ) C i 3 4 where . is the Euclidean vector norm and B
F
CR IP T
2) The following pseudo Lipschitz conditions
tr (B T B ) is the Frobenius norm.
Ai , C i
AN US
are constant matrices. Constants ,…, 4 should be selected such that the deviations or the upper bounds in Assumption 3 are made as small as possible [33].
Assumption 4. There are matrices i 0 and K i such that satisfy the following two
M
conditions:
2T i 2 (L ) max (Q i ) 0.
ED
Q i : i (Ai K i C i ) (Ai K i C i )T i ( 1 2 )i i ( 3 4 )i K i K iT i ( 1 3 )I M 0,
)57( )58(
PT
Remark 1. Inequality (57) implies that (A, C) must be detectable. This condition exists in
CE
most of the nonlinear observers and is known as a general condition. Moreover, according to the positivity of max (Qi ) and i , the inequality (58) holds when 2 (L ) is positive, i.e.
AC
when the graph of the multiagent system is connected. Assumption 5. All agents are identical in every aspect and the same gain is designed for all of them.
Theorem 2. To estimate the states of the target (53), under Assumptions 3-5, a DE with the dynamic equation (55) is obtained, for each agent, in which estimation error
pi (t ) p (t ) pˆ i (t ) is exponentially ultimately bounded in mean square.
17
ACCEPTED MANUSCRIPT ■
Proof. See Appendix B.
As mentioned in Theorem 2, the estimation error of the proposed DE is exponentially ultimately bounded in mean square under the Assumptions 3-5. In this case, in order to obtain the DE parameters including i 0 and K i , it is necessary that the Assumptions 3-5 are satisfied. Since it is assumed that the agents are connected throughout this paper, then
CR IP T
2 (L ) 0 and the inequality (58) holds. Therefore, it is only necessary to solve the inequality (57). Thus, it is assumed that the matrix Q i in inequality (57) is such that
Qi I where 0 is a given number. To solve this inequality, the LMI technique is used.
AN US
In the following, the LMI for obtaining parameters of DE (55) is presented.
The LMI related to the boundedness: The matrix inequality Qi I is converted to the following matrix inequality by using Lemma A.5,
ED
M
i (Ai K i C i ) (Ai K i C i )T i ( 1 3 )I ( 1 2 )i ( 3 4 )K iT i
( 1 2 )i I 0
( 3 4 )i K i 0 0 )59( I
where i 0 . Supposing X i i K i , the following LMI problem is created
CE
PT
i Ai X i C i AiT i C iT X iT ( 1 3 )I ( 1 2 )i ( 3 4 )X iT
( 1 2 )i I 0
( 3 4 ) X i 0 0. I
)60(
AC
By solving the LMI (60) using the YALMIP toolbox in Matlab, the DE parameters are obtained.
Remark 2. Theorem 2 proved that the states of the target (2) can be estimated by DE (56) with a bounded error. Therefore, the estimated states of the target (2) ( ( pˆ1id (t ), pˆ 2id (t ), pˆ 3id (t )) ) can be used instead of (x 1d , x 2d , x 3d ) in DFCs (4) and (5). In this case, the finite time
18
ACCEPTED MANUSCRIPT stability of DFCs (4) and (5) still holds and the states of multiagent system (1) can track the estimated states of target 2. To show this fact, the DE (56) is firstly rewritten as follows: u1d d d pˆ id (t ) u 2d K i (hi (x d (t )) hi ( pˆ id (t )) C i p i (t )) pˆ (t )u 1d 2id C i p i (t ) T i i1 aij [ p i (t ) p j (t )] D idv id (t ).
)61(
j N i
CR IP T
From the pseudo Lipschitz conditions, boundedness of measurement noise, and estimation errors pi (t ) and p j (t ) , one can conclude
d 1d (t ) )62( d 2d (t ) d (t ) 3d where d 1d (t ) , d 2d (t ) and d 3d (t ) are bounded by a positive constant. Then, the tracking
AN US
u1d pˆ id (t ) u 2d pˆ (t )u 1d 2id
errors are defined as follows:
)63(
M
x 1it x 1i pˆ1id (t ) x 2it x 2i pˆ 2id (t ) x x pˆ (t ) 3i 3id 3it
ED
Then, one has
x 1it (t ) u1i d 1i d 1d (t ) x 2it (t ) u 2i d 2i d 2d (t ) x 3it (t ) x 2i (t )(u1i d 1i ) x 2i u 1d d 3d (t ).
PT
)63(
CE
Defining d 1 d1i d1d (t ) , d 2 d 2i d 2d (t ) and d 3 d 1i x 1it (t ) u1i d 1 x 2it (t ) u 2i d 2 x 3it (t ) (x 2i x 2d )(u1i d 3 ) x 2i u 1d .
d 3d (t ) x 2i (t )
, one gets
AC
)64(
It can be shown that the states of the system (64) converge to zero in a finite time similar to the system (9). Therefore, one can conclude that the states of the multiagent system (1) can track the estimated states of target (2) in a finite time.
19
ACCEPTED MANUSCRIPT 4. Simulation examples In this section, the performance of the proposed DE and DFCs is validated in target tracking for a nonholonomic multiagent system with 4 agents. The communication among agents with aij 0.5 is illustrated in Figure 1. Each agent is a wheeled mobile robot and its equations are
described by [30]:
CR IP T
x 1i u1i d 1i x 2i u 2i d 2i x 3i (u1i d 1i )x 2i
where state and control transformations are defined by: x 1i i
AN US
x 2i x i cos i y i sin i x 3i x i sin i y i cos i v i x 2i u 1i u 2i
i u1i ,
)65(
)66(
where x i and y i are the Cartesian coordinates of the center of the rear wheel and i
M
measures the heading angle from the x-axis. v i and i are the linear and angular velocities
ED
and d 1i 0.8 , d 2i 0.9sin t are disturbances. The target has the following equation
PT
u1d x d (t ) u 2d B dw d (t ), x (u ) 2d 1d
x d (t ), B d R 3
(67)
CE
where u1d 1 and u 2d x 2d x 3d . The sensing model of the ith agent (measurement of
AC
target’s states) is described by:
2
3
1
4
Figure 1. A multiagent system with N 4 agents.
20
ACCEPTED MANUSCRIPT y id (t ) x 1d (t ) x 2d (t ) x 3d (t ) 0.1v id (t ),
(68)
where w d (t ) and v id (t ) are independent white noises with variances 0.1 and 0.01, respectively. The parameters considered for DFCs are
1 =40, 2 =2.7, 3 =3, 4 =2.7, 5 =1, q q =9/7, T * =7.
(69)
CR IP T
To validate the proposed DFCs, we simulate a multiagent system in which each agent has the dynamics given in (65) with DFCs (4, 5). Furthermore, it is assumed that target’s states are available and the target is noise free ( B d 0 (0 0 0)T ). Tracking errors of four agents are demonstrated in Figure 2. These results show the states of all agents converge to
AN US
the target’s states in a finite time.
To validate the proposed DE and DFCs simultaneously, we assume that target’s states are not available and target has noise ( B d 0 ). In this case, target’s states must be estimated
M
by the DE (56). The parameters considered for DE and B d are
ED
0 0 0.2 0.01 A 0 1 0.8 , B d 0.1 , C 1 1 1 , 0.2 1 0.1 0 0.1414, 1 0.2, 2 0, 3 0, 4 0, 0.01, T i 0.5,
CE
PT
0.7546 -0.2224 -0.0407 0.6039 i -0.2224 0.7779 0.1534 , K i 0.0162 . -0.0407 0.1534 0.9467 0.3352
AC
Parameters of the DFCs are the same as (69) except T * (T * 10 ). Tracking errors of agents are illustrated in Figure 3 which shows that the states of all agents converge to estimate states of the target in a finite time. Furthermore, Figure 4 shows the estimation errors of the DE (56) for each agent and verifies the successful performance of DE in estimation and reaching consensus.
21
ACCEPTED MANUSCRIPT
TE of x1d
20 0
TE of x2d
-20 0 20
2
4
6
8
10
12
2
4
6
8
10
12
TE of x3d
-20 0 5 0
Agent 1 -5 0
2
4
6 Time(s)
CR IP T
0
Agent 2 8
Agent 3 10
Agent 4
12
AN US
Figure 2. Tracking errors (TEs) of agents without the DE.
TE of x1d
10
2
4
2
0
0
2
4
6
8
10
12
14
8
10
12
14
Agent 1 4
6 Time(s) 8
CE
-5 0
PT
TE of x3d
-5 0 5
6
M
TE of x2d
-10 0 5
ED
0
Agent 2 10
Agent 3 12
AC
Figure 3. Tracking errors (TEs) of agents with the DE (56).
22
Agent 4 14
ACCEPTED MANUSCRIPT
EE of x1d
5 0
EE of x2d
-5 0 5
2
4
6
8
10
12
14
2
4
6
8
10
12
14
-5 0 5 0
Agent 1 -5 0
2
4
6 Time(s) 8
CR IP T
EE of x3d
0
Agent 2
10
Agent 3
12
Agent 4
14
AN US
Figure 4. Estimation errors (EEs) of the DE (56) for every agent. 5. Conclusions
In this paper, a DE and DFCs have been proposed for nonholonomic multiagent systems with external disturbances to track a target with nonholonomic dynamics. The DE was an
M
exponentially ultimately bounded estimator that has been obtained based on a Lyapunov
ED
theory and consensus technique. Furthermore, DFCs have been designed for nonholonomic agents to track a target in a finite time. The sufficient conditions for the convergence of the
PT
proposed algorithms were also analyzed. The simulations have been performed according to
time.
CE
the DE and DFCs that show the suitable state estimation and target tracking at the desired
Appendix A
AC
Lemma A.1 ([34]). If Assumption 2 holds, then the matrix L I is positive definite where I is an identity matrix with proper dimensions. Lemma A.2 ([35]). If 𝐿 is the Laplacian matrix of a connected undirected graph 𝐺, one has x T Lx
1 n 1 n aij (x i x j )2 aij (x i x j )2 2 i , j 1 2 i 1 j N i
23
ACCEPTED MANUSCRIPT for any x x 1 ,..., x n . 0 is a simple eigenvalue of 𝐿 and 1 is the associated eigenvector, T
where 1= 1, ,1 with appropriate dimensions. T
Lemma A.3 ([36]). Consider the following nonlinear system x F (x ), dddddddF (0) 0, x R n .
continuous function V (x )
such that
CR IP T
Suppose there exists a positive definite
V (x ) cV (x ) 0 , where c 0 and 0 1 . Then V (x ) approaches 0 in a finite time.
Furthermore, the finite convergence time T satisfies T x (x 0 )
V (x 0 )1 . c (1 )
AN US
Lemma A.4 ([37]). For any a R , b R , the following inequalities hold: a b 2z 1 a z b z z
( a b )1/ z a
1/ z
b
1/ z
a b 2z 1 a z b z z
( a b ) z a b a z 1 b z 1 . z
ED
z
M
when z 1 is a constant. If z 1 is odd, then
X
X
PT
12 Lemma A.5 ([38]): (Schur Complement): For a given symmetric matrix X 11 , X X 22 12
the following conditions are equivalent:
CE
(a ) X 0,
T (b ) X 11 0, X 22 X 12 X 111X 12 0,
AC
(c ) X 22 0, X 11 X 12 X 221X 12T 0.
Appendix B. Proof of Theorem 1 Before proof of Theorem 1, some definition and sufficient conditions for exponentially ultimately boundedness in mean square is presented. Definition B.1 [39]: Consider the stochastic differential equation:
dp (t ) f ( p (t ))dt g ( p (t ))d(t ).
)B.1( 24
ACCEPTED MANUSCRIPT p (t ) is said to be exponentially ultimately bounded in mean square if there exist positive constants C 1 , C 2 , C 3 such that for all t 0 , 2 E p (t ) C 1 exp(C 2t ) C 3 .
)B.2(
Lemma B.1 [39]: Consider the stochastic differential equation (B.1). Assume that there exists a function V ( p ,t ) C 2,1 and positive constants C 1 , k 1 , k 2 such that C 1 p (t )
2
)B.3(
V ( p , t )
CR IP T
and V ( p , t ) k 1V ( p , t ) k 2 where V ( p , t ) V ( p , t ) 2V ( p , t ) V ( p ,t ) f ( p , t ) tr g ( p , t ) g ( p ,t ) . 2 t p p
)B.4(
AN US
Furthermore, suppose that E [V ( p0 ,0)] . In this case, p (t ) is called exponentially ultimately bounded in mean square. In addition, it can be said that: E [V ( p , t )] (E [V ( p 0 ,0)] k 2 k 1 ) exp(k 1t ) k 2 k 1 . )B.5( Proof of Theorem 1. The proof of Theorem 1 will be provided in the Ito form [40]. Therefore, equations (53) and (54) are firstly rewritten in Ito form as:
M
dp (t ) f ( p (t ))dt BdW (t ), p (t ) R M dz i (t ) hi ( p (t ))dt D i dV i (t ), i 1,..., N
ED
)B.6(
where t
0
PT
z i (t ) y i (s )ds , z i (0) 0,
CE
W (t ) and V i (t ) are standard Brownian motions. Furthermore, the DE equation (55) should
be written in Ito form as:
AC
dpˆ i (t ) f ( pˆ i (t ))dt K i (dz i (t ) hi ( pˆ i (t ))dt ) T i i1 aij [ pˆ j (t ) pˆ i (t )]dt , i 1,..., N .
)B.7(
j N i
By defining the error of each estimator as pi (t ) pˆ (t ) pˆ i (t ) , we have
dpi (t ) dp (t ) dpˆi (t ) f ( p (t ), pˆi (t ))dt B (t )dw (t ).
)B.8(
T T T Defining the error vector as p (t ) [ p1 (t ),..., p N (t )] , we prove that the state estimation error
of the DE (55) is bounded. To achieve this purpose, a stochastic Lyapunov function is considered as 25
ACCEPTED MANUSCRIPT N
V 1 ( p (t )) p iT (t )i p i (t ).
)B.9(
i 1
of V 1 with respect to (B.8), one can obtain
By calculating the differential generator
N V 1 V 1 ( p (t )) T f ( p (t ), pˆ i (t )) tr BB T i 2 i 1 p i (t ) N N 1 N 2 p iT (t )i f ( p (t ), pˆ i (t )) tr (BB T K i D i D iT K iT )i 2 p iT (t )i f ( p (t ), pˆ i (t )) c1. 2 i 1 i 1 i 1
)B.10(
N
N
i 1
i 1
CR IP T
Extending the equation (B.10) gives V 1 ( p (t )) 2 piT (t )i f ( p (t ), pˆ i (t )) c1
N 2 piT (t )i f ( p (t )) f ( pˆ i (t )) K i (hi ( p (t )) hi ( pˆ i (t ))) T i i1 aij [ pˆ j (t ) pˆ i (t )] c1. i 1 j N i i 1 By adding and reducing Ai e i (t ) and C i e i (t ) , the equation (B.11) is converted to N
AN US
N
)B.11(
V 1 ( p (t )) 2 p iT (t )i (A i p i (t ) f ( p (t )) f ( pˆ i (t )) A i e i (t ) i 1
K i (hi ( p (t )) hi ( pˆ i (t ))) C i p i (t ) K i C i p i (t ) T i i1 N
a
j N i
ij
[ pˆ j (t ) pˆ i (t )] c 1 )
)B.12(
N
p (t ) i (A i K i C i ) (A i K i C i ) i p i (t ) 2 p (t )i f ( p (t )) f ( pˆ i (t )) A i p i (t ) T
M
i 1
T i
N
T i
i 1 N
N
2 p iT (t )i K i (hi ( p (t )) hi ( pˆ i (t ))) C i p i (t ) 2T i p iT (t ) aij [ pˆ j (t ) pˆ i (t )] c 1.
ED
i 1
i 1
j N i
i 1
By using the inequalities x y x y , 2 x y x x y y and Assumption 5, the second T
T
T
PT
and third terms in equation (B.12) are simplified as
CE
2 p iT (t )i f ( p (t )) f ( pˆ i (t )) A i p i (t ) 2 i p i (t ) f ( p (t )) f ( pˆ i (t )) A i p i (t ) 2 i p i (t ) ( 1 p i (t ) 2 ) p iT (t ) ( 1 2 )i i 1I M p i (t ) 2
AC
2 p iT (t )i K i (hi ( p (t )) hi ( pˆ i (t ))) C i p i (t ) 2 K iT i p i (t ) hi ( p (t )) hi ( pˆ i (t ))) C i p i (t )
)B.13(
2 K iT i p i (t ) ( 3 p i (t ) 4 ) p iT (t ) ( 3 4 )i K i K iT i 3 I M p i (t ) 4 .
By substituting (B.13) in equations (B.12) and using pˆ j (t ) pˆi (t ) pi (t ) p j (t ) , Assumption 4 and equation (B.12) results in,
26
ACCEPTED MANUSCRIPT N
V 1 ( p (t )) p iT (t )[i (A i K i C i ) (A i K i C i )T i (t ) i 1
N
( 1 2 )i i ( 3 4 )i K i K iT i ( 1 3 )I M ] p i (t ) 2T i p T ( L I N ) p (t ) ( 2 4 c1 ) )B.14( i 1
N
p iT (t )Q i p i (t ) 2T i 2 (L ) p T (t ) p (t ) k 2 i 1
( i 1
2
4 c1 ) . Considering Assumption 4 and matrix properties yield
N
V 1 ( p (t )) p iT (t )Q i p i (t ) 2T i 2 (L ) p T (t ) p (t ) k 2 i 1
max (Q i ) p T (t ) p (t ) 2T i 2 (L ) p T (t ) p (t ) k 2
max
)B.15(
(Q i ) 2T i 2 (L ) V 1 ( p (t )) k 2 k 1V 1 ( p (t )) k 2 . max (i )
AN US
CR IP T
where k 2
N
In which, constants k 1 and k 2 are defined as k1
2T i 2 (L ) max (Q i ) max (i )
)B.16(
1 k 2 ( 2 4 c1 ) 2 4 tr (BB T K i D i D iT K iT )i . 2 i 1 i 1 N
M
N
ED
According to Assumption 4, constants k 1 and k 2 are positive. By using Lemma B.1, it is proven that
)B.17(
PT
E [V 1 ( p (t ))] E [V 1 ( p (0))]exp(k 1t ) k 2 k 1 (1 exp(k 1t )) k 3 exp(k 1t ) k 2 k 1 .
E [V 1 ( p (t ))] min (i )
CE
Since
E [ p (t ) ] 2
)B.18(
AC
Consequently, it is proven from Lemma B.1 that estimation error is exponentially ultimately ■
bounded in mean square. References [1]
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