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The problem of consensus control of linear discrete-time multi-agent systems (MASs) with switching topology is considered in the presence of a leader. The goal of consensus control is to bring the states of all agents to the leader state while providing stability for local agents, as well as the MAS...
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| author | Dorofieiev, Yurii Lyubchyk, Leonid Malko, Maxim |
| author_facet | Dorofieiev, Yurii Lyubchyk, Leonid Malko, Maxim |
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| description | The problem of consensus control of linear discrete-time multi-agent systems (MASs) with switching topology is considered in the presence of a leader. The goal of consensus control is to bring the states of all agents to the leader state while providing stability for local agents, as well as the MAS as a whole. In contrast to the traditional approach, which uses the concept of an extended dynamic multi-agent system model and communication topology graph Laplacian, this paper proposes a decomposition approach, which provides a separate design of local controllers. The control law is chosen in the form of distributed feedback with discrete PID controllers. The problem of local controllers’ design is reduced to a set of semidefinite programming problems using the method of invariant ellipsoids. Sufficient conditions for agents’ stabilization and global consensus condition fulfillment are obtained using the linear matrix inequality technique. The availability of information about a finite set of possible configurations between agents allows us to design local controllers offline at the design stage. A numerical example demonstrates the effectiveness of the proposed approach. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.2.08 |
| first_indexed | 2025-07-17T10:28:14Z |
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Y.I. Dorofieiev, L.M. Lyubchyk, M.M. Malko, 2024
100 ISSN 1681–6048 System Research & Information Technologies, 2024, № 2
TIДC
МАТЕМАТИЧНІ МЕТОДИ, МОДЕЛІ,
ПРОБЛЕМИ І ТЕХНОЛОГІЇ ДОСЛІДЖЕННЯ
СКЛАДНИХ СИСТЕМ
UDC 681.513.1+681.515.8
DOI: 10.20535/SRIT.2308-8893.2024.2.08
DECENTRALIZED LEADER-FOLLOWING CONSENSUS
CONTROL DESIGN FOR DISCRETE-TIME MULTI-AGENT
SYSTEMS WITH SWITCHING TOPOLOGY
Y.I. DOROFIEIEV, L.M. LYUBCHYK, M.M. MALKO
Abstact. The problem of consensus control of linear discrete-time multi-agent sys-
tems (MASs) with switching topology is considered in the presence of a leader. The
goal of consensus control is to bring the states of all agents to the leader state while
providing stability for local agents, as well as the MAS as a whole. In contrast to the
traditional approach, which uses the concept of an extended dynamic multi-agent
system model and communication topology graph Laplacian, this paper proposes a
decomposition approach, which provides a separate design of local controllers. The
control law is chosen in the form of distributed feedback with discrete PID control-
lers. The problem of local controllers’ design is reduced to a set of semidefinite pro-
gramming problems using the method of invariant ellipsoids. Sufficient conditions
for agents’ stabilization and global consensus condition fulfillment are obtained us-
ing the linear matrix inequality technique. The availability of information about a fi-
nite set of possible configurations between agents allows us to design local control-
lers offline at the design stage. A numerical example demonstrates the effectiveness
of the proposed approach.
Keywords: multi-agent system, consensus control, switching topology, PID controller,
invariant ellipsoids method, linear matrix inequality, semidefinite programming problem.
INTRODUCTION
Recently, consensus control of multi-agent systems (MAS) with networked struc-
tures has attracted the great attention of many researchers from different fields of
science and engineering [1; 2]. In the field of automatic control, the development
of consensus control theory is stimulated, in particular, by the rapid development
of unmanned mobile vehicles and ensuring their coordinated behavior in accor-
dance with the common goal [3]. Similar problems also arise under the control of
large-scale systems with networked structures, such as complex technological and
automated production systems, supply and logistics chains, and energy and trans-
port systems as well.
The main problem of consensus control of MAS is the design of a control
law, which allows all agents to reach the agreed values of their state or output
variables, using the information obtained from other agents. Here, the control law
is constructed based on a consensus protocol [2; 4]. The protocol design assumes
Decentralized leader-following consensus control design for discrete-time multi-agent …
Системні дослідження та інформаційні технології, 2024, № 2 101
that the control for each local agent is formed on the basis of information about
the deviations of the state or output vector of any agent from the corresponding
vectors of neighbouring agents directly related to it. In this case, the
corresponding control system also has a network structure wherein the rules for
information exchange between each local agent and its directly connected
neighbours are determined by the network topology.
A topology model usually describes the MAS structure as connections be-
tween agents in the form of a directed graph, in that the nodes of which corre-
spond to the controlled local agents, where the graph edges describe the informa-
tion transfer channels between them. Usually, a consensus protocol is taken in the
form of a linear deviation feedback between local agent states or outputs and the
weighted average vector of states or outputs of its immediate neighbours. In such
a case, the control problem is reduced to finding a set of feedback gain matrices
from the stability condition for both local controlled agents and the MAS as a
whole, considering the relationship between them, as well as the condition of
reaching a consensus.
Taking into account the peculiarities of practical problems of MAS control
leads to the need to complicate the problems under consideration. In reality, due
to breaks in communication channels, the topology of connections can be arbitrar-
ily changed by switching between elements of a finite set of possible configura-
tions, which leads to the need for consensus control design under switching topol-
ogy conditions.
REVIEW AND ANALYSIS OF INFORMATION SOURCES
In the last years, consensus problem research has developed very rapidly and nu-
merous results have been obtained concerning distributed consensus protocols for
MAS design (see [5; 6] and references therein).
The usual approach to solving consensus control problems is based on a dy-
namic model of MAS with an extended state vector composed of the state vectors
of local agents, thus constructing a model using the concept of the Laplacian of
the communication topology graph [7]. At that, the Kronecker product of the dy-
namic matrices of local agents describes the matrix of the extended MAS model
dynamics.
Efficient methods for studying the stability of such systems have been de-
veloped; for the synthesis of consensus control, modern methods for controllers’
design in state space, including the methods of linear matrix inequalities (LMIs),
are widely used. In [8], using the LMI technique, a new form of state-feedback
consensus control based on the aggregate Laplacian is proposed and sufficient
conditions of stabilization are established using the Lyapunov stability theory.
Early work in this area dealt mainly with the problem of ensuring consensus,
represented in the form of balance ratios of an agent’s state vectors so that all
agents are driven to converge to a common state, determined by the consensus
conditions. Further development of consensus control is associated with an
additional condition of following the leader, which is considered an agent that
imposes the desired behavior on others [5]. A number of works have been
devoted to the consensus control problems in multi-agent systems with a leader, see
links in [9].
Y.I. Dorofieiev, L.M. Lyubchyk, M.M. Malko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 102
A large number of works are devoted to the consensus control design prob-
lem under switching topology conditions; systematized results are given in [10].
In [11], it was shown that under certain assumptions, consensus can be achieved
asymptotically under dynamically changing interaction topologies if the union of
the collection of interaction graphs across some time intervals has a spanning tree
frequently enough. Consensus of MAS in a continuous time domain under fixed
and switching topology was studied in [12], where the dynamics of local and
leader agents are considered linear; the design technique was based on Riccati
inequality and Lyapunov inequality. These results have also been generalized to
discrete-time systems. In [13], sufficient conditions for the solvability of consen-
sus problems for discrete-time multi-agent systems with switching topology and
time-varying delays have been presented. In [14], the consensus problem for
MAS was also considered in the discrete-time domain, the topology of interac-
tions between agents was assumed to be switched and undirected. In [15], the co-
operative control problem of discrete-time multi-agent systems is discussed,
bounded uncertain time delay and directed switching topology are considered, and
sufficient conditions for asymptotic consensus of the system under directed
switching topology are obtained. State-of-the-art survey on consensus control of
network systems with switching network topologies, presented in [16] with em-
phasis on the relationships between the switching among different topology can-
didates and the networked control stability.
The architectural features of networked consensus control systems, com-
bined with a natural desire to reduce the computational resources required to cal-
culate controls in real time, stimulated an increase in interest in building distrib-
uted multi-agent systems with decentralized consensus control. From the
viewpoint of the practical implementation of consensus control, a decentralized
approach is of great interest, in which the local control for each agent is designed
only using locally available information, so it requires less computational effort
and is relatively more scalable with respect to the number of agents. A decentral-
ized approach to asymptotic consensus control design for discrete MAS, where
local agents exchange information only with their nearest neighbours is studied in
detail by a number of researchers. In [17], for agents of the first order, a heuristic
approach is proposed based on an analogy with the Vicsek model of the motion of
a group of particles on a plane. In [18], an adaptive procedure for constructing
control for each agent is proposed using only information from its neighbours in
the network topology. The consensus problem for a multi-agent system with high-
order linear dynamics and a fully decentralized consensus algorithm is proposed
in [19], which allows each agent to reach the consensus value using only a finite
number of steps of its past state information, but the solution is obtained under the
condition that the system has a time-invariant topology. A fully decentralized al-
gorithm that allows any agent to compute the consensus value of the whole net-
work in a finite time using only the minimal number of successive values of its
own history was proposed in [20], where was shown, that this minimal number of
steps is related to a Jordan block decomposition of the network dynamics. How-
ever, this minimum number of steps is related to graph theoretical notions that can
be directly computed from the Laplacian matrix of the graph and from the mini-
mum external equitable partition. Decentralized event-triggered finite-time con-
sensus control under a directed graph was investigated in [21], and an adaptive
law is designed to counteract the effect of uncertainties and external disturbances.
Decentralized leader-following consensus control design for discrete-time multi-agent …
Системні дослідження та інформаційні технології, 2024, № 2 103
The quality of consensus control in transients, which is especially important
in leader-following tracking problems, can be significantly improved using more
complex dynamic consensus protocols, in which control actions depend not only
on the current but also on previous deviations. As such control laws, multivariable
Proportional-Integral-Derivative (PID) controllers are widely used, which make it
possible to significantly improve the quality of consensus control in comparison
with static protocols.
Despite its well-known benefits, PID control is lightly addressed in the de-
centralized MAS control and they are mainly dealt with homogeneous MASs (see
e.g. [22] and references therein). For instance, a robust PID consensus control
strategy has been proposed in [23] for a system of linear high-order agents under
the restrictive assumption of an undirected communication graph. A PD protocol
is proposed in [24] to solve, instead, the problem of the average consensus under
a fast arbitrarily switching topology for the case of first-order nonlinear homoge-
neous MASs with Lipschitz dynamics. To solve the leader tracking for uncertain
high-order homogeneous MASs, robust PID protocols have been investigated
[25]. Nevertheless, practical applications of the networked MASs require hetero-
geneous models due to the presence of mismatches and differences among the
agents. In this context, only a few decentralized protocols aim to extend the PID-
control advantages to the heterogeneous MAS framework. In [26] it was investi-
gated the use of distributed PID actions to achieve consensus in networks of ho-
mogeneous and heterogeneous linear systems. The convergence of the strategy is
proven for both cases using appropriate state transformations and Lyapunov func-
tions. A multiplex proportional–integral approach for solving consensus problems
in networks with heterogeneous node dynamics affected by constant disturbances
was proposed in [27]. The proportional and integral actions are deployed on two
different layers across the network, each with its own topology. Sufficient condi-
tions for convergence are derived that depend upon the structure of the network,
the parameters and topologies characterizing the control layers, and the node dy-
namics. Fully-distributed PID control strategy was proposed in [28], whose stabil-
ity is analytically proven by exploiting the controller equations and the Static
Output Feedback procedure adapted to the MASs framework.
In this paper, the problem of decentralized PID controller design for leader-
following consensus control of networked heterogeneous MASs is considered
under the assumption that sharing information between agents is carried out via
switching communication topology. In this work, we use a distributed consensus
protocol in the form of local dynamic feedback with a PID controller using the
signal of deviation of local agent states from the weighted average states of its
neighbours, with which the agent exchanges information in the current period.
The synthesis of local PID controllers is based on the method of invariant ellip-
soids. To analyze the stability of closed-loop controlled local agents, as well as
the stability of the whole MAS, the second Lyapunov method and LMI technique
were used. This allowed us to reduce a local controller problem design to a prob-
lem of semidefinite programming to find the optimal gain values for each agent
by numerically solving the optimization problem. This, in turn, makes it possible
to simplify the solution in comparison with other well-known approaches, for ex-
ample, with the descriptor method [29].
In summary, the main features of the proposed approach and contributions of
the work are:
Y.I. Dorofieiev, L.M. Lyubchyk, M.M. Malko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 104
Within the framework of the principle of decentralized consensus control,
a method of local controllers’ design is proposed, which ensures the stability of
individually controlled agents and the networked multi-agent systems as a whole
under conditions of a switching communication network topology.
Based on the sufficient conditions of local-controlled agents’ stability and
reaching consensus as well as leader-following tracking in LMI form, a computa-
tional procedure for optimizing the parameters of local controllers by solving a set
of semidefinite programming problems is proposed; while controller parameters
optimization is performed offline during the design phase.
A decentralized optimal dynamic consensus control strategy with a recur-
rent form of the PID control law is proposed, which makes it possible to abandon
the construction of an extended model of a closed local agent, which is usually
used for discrete systems with PID controllers, which makes it possible to reduce
the optimization problem dimension.
Availability of information about a finite set of topology variations of
connections between agents allows solving the problem of calculating the feed-
back gain matrix before the start of the control process. Thus, to determine the
control action at each step of the MAS operation from the resulting set of solu-
tions, the appropriate one is selected depending on which neighbours the agent
exchange information in the current step.
PROBLEM FORMULATION
Consider a MAS represented by a network of N agents as a set of multivariable
discrete-time linear dynamic systems, where each i -th agent is described by the
difference equation
NikuBkxAkx iiiii ,...,1),()()1( , (1)
where ,...2,1,0k is time instant; m
i
n
i kukx RR )(,)( are state and control
vectors of i -th agent at time k ; ii BA , are constant matrices of appropriate di-
mensions such that the system (1) is controllable.
The connection topology of a networked MAS is described by an undirected
graph ),( EVG , where }...,,1{ NV is a set of nodes (i.e., agents), and
VVE is the set of edges. The presence of an edge ),( ji in the graph G
means that agents i and j exchange information. ][ ijdD is the adjacency ma-
trix of graph G and has dimension NN .
Since the topology of the connections between agents can change during the
system dynamics, the graph G can switch at arbitrary time moments among a
finite set KGGG ,...,, 21 , each of which is an undirected graph, containing a spanning
tree. This means that the graph G will have the form 1G during a certain time in-
terval, then },...,1{, KjG j , and so on, moreover, switching occurs arbitrarily.
A set of agents achieves consensus if the agents’ states satisfy the consensus
condition
Njikxkx ji
k
,...,1,,0))()((lim
. (2)
The control law )(kui , hereafter the consensus protocol, solves the consen-
sus control problem if all agents achieve consensus under this control.
Decentralized leader-following consensus control design for discrete-time multi-agent …
Системні дослідження та інформаційні технології, 2024, № 2 105
We construct a distributed consensus protocol in the form of local feedback
to a PID controller using the signal of deviation of local agent states from the
weighted average states of its neighbours
N
ijj
k
l
jiijiI
N
ijj
jiijiPi lxlxkKkxkxkKku
,1
1
0,1
))()(()())()(()()(
))),1()1(())()((()(
,1
kxkxkxkxkK jiji
N
ijj
ijiD (3)
where N
ijj iji d,11 ; nm
DIP KKK R,, are the gain matrices of the pro-
portional, integral, and differential parts of the controller, respectively; )(kij is
binary variable, the value of which determines whether information about the
state of the j -th agent is available to the i -th agent at a time k :
.otherwise,1
,agent fromn informatio receivenot does agent,0
)(
ji
kij
The recurrent form of control law (3) is more convenient for practical im-
plementation, so the current value of the control action is determined by its previ-
ous value and the correction
))()(()()1()(
,1
0 kxkxkdKkuku ji
N
ijj
ijijiiii
))1()1(()(
,1
1 kxkxkdK jiij
N
ijj
ijii
)),2()2(()(
,1
2
kxkxkdK jiij
N
ijj
ijii (4)
where nm
iii KKK R210 ,, are feedback coefficient matrices.
Introduce block matrix ],,[ 210 iiii KKKK and the composite vectors
,}))2()2((,))1()1((,))()(({col)( kxkxkxkxkxkxkv jijijiij ji,
,,...,1 N ij . Then, the consensus protocol (4) takes the form
N
ijj
ijijiiii kvkKkuku
,1
)()()1()( . (5)
The control law for the agent, acting as the leader, differs from (5) by addi-
tion the deviation term between the leader’s state and the set point *x
))(()()( *
0 xkxKkuku iii
leader
i .
The model of a closed-loop local agent with the consensus protocol (5)
will take the form with one-step control lag
)1()()()()()()1(
,1,1
kuBkxkAkvkKBAkx iij
N
ijj
ijiij
N
ijj
ijiiiiCi , (6)
Y.I. Dorofieiev, L.M. Lyubchyk, M.M. Malko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 106
where ]00[ nnnniiC AA is a block matrix, mn0 is the null matrix of the
corresponding dimension. For the agent that is a leader, equation (6) differs by
presence of the term ))(( *
0 xkxKB iii .
For a linear closed-loop discrete MAS with local feedback described by
equation (6) under the conditions of an arbitrarily switching topology described
by a finite set of undirected connected graphs KGGG ,...,, 21 , the problem of de-
centralized consensus control is considered. Such a problem is reduced to the
choice of feedback gain matrices NiKi ,...,1, that ensure the stabilization of
closed-loop local agents, the stability of the controlled MAS as a whole, as well
as the fulfilment of the consensus conditions (2).
CONSENSUS CONTROL DESIGN
The approach based on the second Lyapunov method, which allows us to obtain
sufficient stability conditions for closed-loop local subsystems, is used to calcu-
late the gain matrices of local controllers. The development of the theory of linear
matrix inequalities [30] makes it possible to apply a similar approach to the syn-
thesis of consensus control of multi-agent systems.
The main idea of linear feedback controller design using LMI is as follows.
The control goal is formulated as an inequality with respect to the quadratic
Lyapunov function built on the solutions of the closed-loop system. The resulting
inequality is reduced to the LMI form with respect to the unknown matrix of the
controller parameters. The specified constraints are also reduced to the LMI form.
A certain criterion of optimality is used and the corresponding convex optimiza-
tion problem is solved numerically, because of which the optimal parameters of
the controllers are determined.
The considered technique is implemented based on the invariant ellipsoid’s
method [31]. Consider the ellipsoid described by the equation
}1)()(:{)( T kxQkxxQ iii
n
ii R , (7)
where nn
iQ R0 is ellipsoid matrix; 0M )0( M means that the matrix
M is positive (nonnegative) definite.
The ellipsoid (7) is called state invariant for the system (6) if any trajectory
of the system, having started in the ellipsoid, remains in it for any time moment
0k .
The stabilization problem of system (6) comes down to calculating the block
matrices of feedback gain NiKi ,...,1, such that the consensus protocol (5)
provides minimization of ellipsoid (7) by some optimality criterion. We choose as
a criterion the length squares sum of the ellipsoid semiaxes which is equal to the
trace of its matrix iQ .
Consider a quadratic function constructed from the solutions of the system (6)
nn
iiiiii PPkxPkxkV RTT 0),()()( . (8)
It is well known that function (8) is a Lyapunov function for system (6) if the
conditions are fulfilled:
Decentralized leader-following consensus control design for discrete-time multi-agent …
Системні дослідження та інформаційні технології, 2024, № 2 107
(i) the function values are non-negative for any 0)( kxi ;
(ii) the function values decrease monotonically over time.
If equality holds
1 ii PQ , (9)
then the invariant ellipsoid (7) is the level set of Lyapunov function candidate (8).
It was shown in [38] that for a stable and controllable discrete-time
dynamical system, the solution of the minimization problem by some criterion of
quadratic Lyapunov function under the constraint specified by the Lyapunov
inequality, is achieved by the solution of the Lyapunov equation. Thus, such an
approach allows reducing the robust control design problems with respect to the
described class of system topology uncertainty, to solve the problem of
minimizing a linear function under constraints that can be represented in the form
of linear matrix inequalities, that is, to solve a semidefinite programming
problem.
The hypothesis on the basis of which the design problem of consensus con-
trol for MAS with switching topology in the presence of a leader is solved is that
for any version of the topology described by a finite set of undirected graphs
KGGG ,...,, 21 , each of which contains a spanning tree, the statements of the fol-
lowing theorem are fulfilled.
Theorem. If for linear stable discrete-time system (6) matrices iQ̂ ,
iii YYY 210
ˆ,ˆ,ˆ are obtained by solving the optimization problem
min)(trace iQ (10)
subject to
0
0********
00*******
000*****
0000****
00000**
00000*
)()(
3
33
33
33
33
11
mm
mnnn
mnnnnn
mnnnnnnn
mnnnnnnnnn
mnnnnnnnnni
i
N
iiQiijiiQiij
N
iiiiiii
Q
BYBAYBAQAQAQAQ
(11)
on matrix variables nn
iQ R0 , ],,[ 210 iiii YYYY , nm
iii YYY R210 ,, , where
ijiij , ]00[ nnnniiiQ QAA , «*» denotes the symmetric terms in then
inequality matrix, then:
(i) for any initial state )ˆ()0( iii Qx closed-loop system (6) is asymptoti-
cally stable;
(ii) among all consensus protocols of the form (5), the protocol with gain
matrices
Y.I. Dorofieiev, L.M. Lyubchyk, M.M. Malko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 108
1
22
1
11
1
00
ˆˆ,ˆˆ,ˆˆ QYKQYKQYK iiiiii (12)
delivers the minimum of the matrix trace criterion for the invariant ellipsoid (7) of
the closed-loop system (6) and hence guaranties the fulfillment of consensus con-
dition (2).
Proof. The first of the conditions, which are necessary for a candidate (8) to
be a Lyapunov function for the system (6), is satisfied due to the positive defi-
niteness of the matrix iP . Hence, the consensus protocol (5) should ensure that
the second property is satisfied.
We calculate the difference of the candidates in the Lyapunov function (8)
with respect to k and require that the value of the function decrease over time
0)()()()1( T ksMkskVkV iiiii , (13)
where mnNn
iiNijNii kukvkvkxkxks 34})1(),(...,),(),(...,),({col)( R ,
iM
iii
iiiiNiiNiiiN
iiiijiiNiiij
iiiijiiNiiijiijiiij
iiiiiNiiiijiiiii
iiiiiNiiiijiiiiiiii
iii
N
iiNiiiijii
N
iiiiiiiiii
BPB
BPP
BPP
BPPP
BPAPAPAAPA
BPAPAPAAPAAPA
BPAPAPAAPAAPAPAPA
T
TT
TT
TTT
TTTT
TTTTT
T
1
TT
1
TTT
*******
******
*****
****
***
*
,
iiiCi KBA .
Inequality (13) is equivalent to the matrix inequality 0iM . Let us repre-
sent the inequality matrix in the form
iiiNiijiii
i
iiN
iij
i
i
i BAAP
B
A
A
M
T
T
T
T
T
))1(4())1(4(
))1(4(
0*
0
mnNmnN
mnNniP
.
Using the Schur complement [32], inequality takes the form
Decentralized leader-following consensus control design for discrete-time multi-agent …
Системні дослідження та інформаційні технології, 2024, № 2 109
0
0********
00*******
000*****
0000****
00000**
00000*
3
33
33
33
33
11
1
mm
mnnn
mnnnnn
mnnnnnnn
mnnnnnnnnn
mnnnnnnnnni
i
N
iijiij
N
iiii
P
BAAAP
.
By multiplying the left and right parts of the inequality by 1 , performing
substitution (9) and applying a congruent transformation to the inequality matrix
with },...,,,diag{block
)1(4
m
N
iin IQQI
, where nI is identity matrix of the correspond-
ing dimension, we obtain the matrix inequality, which is nonlinear with respect to
the matrix variables iQ and iK . We introduce matrix variables ,00 iii QKY
iiiiii QKYQKY 2211 , . Whence, by virtue of 0iQ the matrices ,0iK ,1iK
iK2 recovers uniquely in accordance with (12). Then, we finally obtain the linear
matrix inequality (11).
Thus, if there exist matrices iQ̂ , iii YYY 210
ˆ,ˆ,ˆ , being a solution of the optimi-
zation problem (10) subject to (11), then (8) is a Lyapunov function for system
(6), and the consensus protocol (5) with matrices calculated in accordance with
(12), provides the fulfilment of stabilization (13) and consensus (2) conditions for
the system (6). The theorem is proved.
Remark. The optimization problem (10) subject to (11) is a semidefinite
programming problem that is solved numerically using freely distributed software
packages developed based on MATLAB, for example, cvx [33] or SeDuMi [34].
NUMERICAL EXAMPLE
As an example, we consider a linear discrete-time MAS of 6 homogeneous
agents, which was studied in [35] and solve the consensus control problem using
the proposed approach. We deliberately consider a homogeneous MAC to reduce
the amount of calculation.
During the system dynamics, the connection topology between agents
switches randomly among the set of options represented by connected graphs
51,...,GG , as shown in Fig. 1. The reference set point *x is received externally at
the input of agent 1, which is a leader.
The dimensions of the agent model are 2n , 1m . The dynamics and con-
trol matrices are
025,0
10
iA , 6,...,1,
1
1
iBi . All agents are Shur stable
and controllable. The initial states of the agents are chosen:
Y.I. Dorofieiev, L.M. Lyubchyk, M.M. Malko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 110
}100,50{col)0(1 x , }60,30{col)0(2 x , }20,10{col)0(3 x ,
}20,10{col)0(4 x , }60,30{col)0(5 x , }100,50{col)0(6 x .
We calculate the numerical solution of problem (10) subject to (11) for all
versions of the MAS topology, which is presented in Fig. 1. As a result, the
matrices of feedback gains, which determine the consensus protocol (5), were
calculated for all agents. The analysis of the obtained results allowed to conclude
that the hypothesis, put forward about the fulfilment of the statements of the
proved theorem for any version of topology, described by a finite set of connected
directed graphs, was experimentally confirmed, since the values of the local feed-
back matrices depend only on the number of neighbours with which the agent ex-
changes information in the current period.
For example, in a graph 1G agents 2 and 6 are connected with two
neighbours, while agents 3, 4, and 5 are connected with three neighbours.
Therefore, we obtain )()( 1612 GKGK , )()()( 151413 GKGKGK . In a graph
4G , agents 1, 2, 3 and 5 are connected with two neighbours, and only agent 4 — with
three neighbours. Accordingly, we obtain )()()()( 45434241 GKGKGKGK
)( 12 GK , )()( 1344 GKGK . Thus, for the considered network, we obtained:
]558,0324,0[0 iK , ]206,0099,0[1 iK , ]057,0047,0[2 iK , (14)
]480,0301,0[0 iK , ]059,0023,0[1 iK , ]046,0031,0[2 iK ,
]762,0439,0[0 iK , ]027,0019,0[1 iK , ]041,0024,0[2 iK ,
]059,1634,0[0 iK , ]019,0012,0[1 iK , ]037,0019,0[2 iK ,
]342,1752,0[0 iK , ]017,010,0[1 iK , ]026,0010,0[2 iK , (15)
where (14) corresponds to an agent exchanging information with one neighbour in
the network, and (15) — to an agent with five neighbours, respectively.
Thus, it is sufficient to solve the optimization problem (10) subject to (11)
for only one agent, sequentially changing the values of the binary variables ,ij
ijNj ,,...,1 that determine the number of neighbours with which the agent
exchanges information, to calculate the values of the feedback matrices for all
possible cases: from having one neighbour before having )1( N neighbours.
Fig. 1. The set of graphs describing the topology of connections in the MAS
Decentralized leader-following consensus control design for discrete-time multi-agent …
Системні дослідження та інформаційні технології, 2024, № 2 111
The plots of the changes in the values of the first and second components of
the agent’s state vectors with fixed topology in the absence of setting action are
shown in Fig. 2. The simulation results provided that the variant of the connection
topology between agents at each step was chosen randomly among the versions
presented in Fig. 1, are shown in Fig. 3. Clearly in both experiments, all agents
achieve a consensus, but the consensus value in the second case differs from zero,
that is, there is a static error.
a
b
Fig. 2. Values of agents’ states with fixed topology in the absence of setting action:
a — first components of state vectors; b — second components of state vectors
k0 2 4 6 8 10 12 14 16 18 20
x i1
, i
=
1
,..
.,6
-100
-80
-60
-40
-20
0
20
40
60
80
100
x
11
x
21
x
31
x
41
x
51
x
61
a
Fig. 3. Values of agents’ states with switching topology in the absence of setting action:
а — first components of state vectors; b — second components of state vectors
Y.I. Dorofieiev, L.M. Lyubchyk, M.M. Malko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 112
The simulation results of MAS with fixed topology, when the input of
agent1, which is a leader, is supplied with a constant setting action
}500,500{col* x , is shown in Fig. 4. Fig. 5 shows the results obtained for MAS
with switching topology in the presence of a reference setting.
x i2
, i
=
1
,..
.,6
b
Fig. 3. Values of agents’ states with switching topology in the absence of setting action:
а — first components of state vectors; b—– second components of state vectors. End
Fig. 4. Values of agents’ states with fixed topology in the presence of setting action:
а — first components of state vectors; b – second components of state vectors
a
b
Decentralized leader-following consensus control design for discrete-time multi-agent …
Системні дослідження та інформаційні технології, 2024, № 2 113
The analysis of the obtained results makes it possible to conclude that the
consensus protocol (5) with the gain matrices calculated in accordance with the
proposed decentralized method, can achieve consensus in a finite number of steps for
any variant of the interconnection topology between agents from a given finite set.
CONCLUSIONS
The problem of decentralized consensus control of linear discrete-time multi-
agent systems with switching topology in the presence of a leader is solved in this
paper. A consensus protocol providing coordinating control is constructed in a
feedback form with a PID controller using the deviation signal of the local agent
state vector from the weighted average state vector of its neighbours. The discrete
PID controller equation is presented in a recurrent form. The sufficient conditions
for stabilization of the closed-loop local agent and the global consensus by con-
structing the quadratic Lyapunov function are obtained. Based on the invariant
ellipsoid’s method, the problem of local controller design is reduced to the prob-
lem of semidefinite programming, which is solved numerically. Analysis of the
obtained results allowed us to conclude that the values of the gain matrices of lo-
cal controllers depend on the number of neighbours with which the agent ex-
changes information in the current period. The significance of this study is to de-
velop a practically realizable method for solving the consensus control problem of
k0 2 4 6 8 10 12 14 16 18 20
-100
-50
0
50
100
150
x
11
x
21
x
31
x
41
x
51
x
61
a
b
k0 2 4 6 8 10 12 14 16 18 20
-100
-50
0
50
100
150
x
12
x
22
x
32
x
42
x
52
x
62
Fig. 5. Values of agent’s states with switching topology in the presence of setting action:
a — first components of state vectors; b — second components of state vectors
Y.I. Dorofieiev, L.M. Lyubchyk, M.M. Malko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 114
discrete-time multi-agent systems with switching topology based on a decentral-
ized approach, which does not require using the graph Laplacian that describes
the connection topology between agents. The proposed approach may be further
expanded to solve the consensus control problems of multi-agent systems in the
presence of delays in measurements or in the process of information exchange
between agents.
REFERENCES
1. J. Lunze, Networked control of multi-agent systems; 2nd edition, 2022, 725 p.
2. M. Mahmoud, M. Oyedeji, and Y. Xia, Advanced distributed consensus for multi-
agent systems, 2020, 394 p.
3. Z. Qu, Cooperative control of dynamical systems: applications to autonomous vehi-
cles. Springer, 2009, 325 p.
4. R.O. Saber, R.M. Murray, “Consensus protocols for networks of dynamic agents,”
Proceedings of the 2003 American Control Conference, pp. 951–956, 2003. doi:
10.1109/ACC.2003.1239709.
5. Z. Li, Z. Duan, Cooperative control of multi-agent systems: a consensus region ap-
proach. CRC Press, 2017, 262 p.
6. Y. Li, C. Tan, “A survey of the consensus for multi-agent systems,” Systems Sci-
ence and Control Engineering, vol. 7, no. 1, pp. 468–482, 2019. doi:
10.1080/21642583.2019.1695689.
7. M. Mesbahi, M. Egerstedt, Graph theoretic methods in multi-agent networks. Prince-
ton University Press, 2010, 424 p.
8. M.S. Mahmoud, G.D. Khan, “LMI consensus condition foe discrete-time multi-agent
systems, IEEE/CAA Journal of Automatica Sinica, vol. 5, no. 2, pp. 509–513, 2018.
doi: 10.1109/JAS.2016.7510016.
9. X. Deng, X. Sun, and S. Liu, “Leader-following consensus control of nonlinear
multi-agent systems with input constraint,” International Journal of Aeronautical
and Space Sciences, vol. 20, pp. 195–203, 2019. doi: 10.1007/s42405-018-0100-9.
10. L. Dong, S.K. Nguang, (Eds.) Consensus tracking of multi-agent systems with
switching topologies. Academic Press, 2020, 257 p.
11. W. Ren, R.W. Beard, “Consensus seeking in multi-agent systems under dynamically
changing interaction topologies,” IEEE Transactions on Automatic Control, vol. 50,
no. 5, pp. 655–661, 2005. doi: 10.1109/TAC.2005.846556.
12. W. Ni, D.Z. Cheng, “Leader-following consensus of multi-agent systems under and
fixedn and switching topologies,” System and Control Letters, vol. 59. no. 3–4,
pp. 209–217. 2010. doi: 10.1016/j.sysconle.2010.01.006.
13. F. Xiao, L. Wang, “State consensus for multi-agent systems with switching topolo-
gies and time-varying delays,” International Journal of Control, vol. 79, no. 10,
pp. 1277–1284, 2006. doi: 10.1080/00207170600825097.
14. L.X. Gao, C.F. Tong, and L.Y. Wang, “H∞ dynamic output feedback consensus con-
trol for discrete-time multi-agent systems with switching topologies,” Arabian Jour-
nal for Science and Engineering, vol. 39, no. 2, pp. 1477–1478, 2013. doi:
10.1007/s13369-013-0807-7.
15. C. Wang, J. Wang, P. Wu, and J. Gao, “Consensus problem and formation control
for heterogeneous multi-agent systems with switching topologies,” Electronics, vol.
11, no. 16, 2598, 2022. doi: 10.3390/electronics11162598.
16. G. Wen, X. Yu, W. Yu, and J. Lu, “Coordination and control of complex network
systems with switching topologies: a survey,” IEEE Transactiions on Systems, Man,
and Cybernetics: Systems, vol. 51, no. 10, pp. 6242–6357, 2021. doi:
10.1109/TSMC.2019.2961753.
Decentralized leader-following consensus control design for discrete-time multi-agent …
Системні дослідження та інформаційні технології, 2024, № 2 115
17. Q. Li, Z.P. Jiang, “Two decentralized heading consensus algorithms for nonlinear
multi-agent systems,” Asian Journal of Control, vol. 10, no. 2, pp. 187–200, 2008.
doi: doi.org/10.1002/asjc.18.
18. S. Ge, C. Yang, Y. Li, and T.-H. Lee, “Decentralized adaptive control of a class of
discrete-time multi-agent systems for hidden leader following problem,” 2009
IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 5065–
5070, 2009. doi: 10.1109/IROS.2009.5354393.
19. Z. Hu, D. Fu, and H.-T. Zhang, “Decentralized finite time consensus for discrete-
time high-order linear multi-agent system,” 2019 Chinese Control Conference,
pp. 6154–6159, 2019. doi: 10.23919/ChiCC.2019.8866099.
20. Y. Yuan, G.-B. Stan, L. Shi, M. Barahona, and J. Goncalves, “Decentralized mini-
mum-time consensus,” Automatica, vol. 49, pp. 1227–1235, 2013. doi:
10.1016/j.automatica.2013.02.015.
21. C. Xu, B. Wu, D. Wang, and Y. Zhang, “Decentralized event-triggered finite-time
attitude consensus control of multiple spacecraft under directed graph”, Journal
of the Franklin Institute, vol. 358, no. 18, pp. 9794–9817, 2021. doi:
10.1016/j.jfranklin.2021.10.019.
22. H. Gu, P. Liu, J. Lü, and Z. Lin, “PID control for synchronization of complex dy-
namical networks with directed topologies,” IEEE Transactions on Cybernetics,
vol. 51, no. 3, pp. 1334–1346, 2021. doi: 10.1109/TCYB.2019.2902810.
23. C.-X. Shi, G.-H. Yang, “Robust consensus control for a class of multi-agent systems
via distributed PID algorithm and weighted edge dynamics,” Applied Mathematics
and Computation, vol. 316, pp. 73–88, 2018. doi: 10.1016/j.amc.2017.07.069.
24. D. Wang, N. Zhang, J. Wang, and W. Wang, “A PD-like protocol with a time delay
to average consensus control for multi-agent systems under an arbitrarily fast switch-
ing topology,” IEEE Transactions on Cybernetics, vol. 47, pp. 898–907, 2017. doi:
10.1109/TCYB.2016.2532898.
25. G. Fiengo, D.G. Lui, A. Petrillo, S. Santini, and M. Tufo, “Distributed robust PID
control for leader tracking in uncertain connected ground vehicles with v2v commu-
nication delay,” IEEE/ASME Transactions on Mechatronics, vol. 24, pp. 1153–1165,
2019. doi: 10.1109/TMECH.2019.2907053.
26. D.A. Burbano Lombana, M. di Bernardo, “Distributed PID control for consensus
of homogeneous and heterogeneous networks,” IEEE Transactions on Control
of Network Systems, vol. 2, no. 2, pp. 154–163, 2015. doi: 10.1109/TCNS.2014.2378914.
27. D.A. Burbano Lombana, M. di Bernardo, “Multiplex PI control for consensus in
networks of heterogeneous linear agents,” Automatica, vol. 67, pp. 310–320, 2016.
doi: 10.1016/j.automatica.2016.01.039.
28. D.G. Lui, A. Petrillo, and S. Santini, “An optimal distributed PID-like control
for the output containment and leader-following of heterogeneous high-order
multi-agent systems,” Information Sciences, vol. 541, pp. 166–184, 2020. doi:
10.1016/j.ins.2020.06.049.
29. L. Lyubchyk, Y. Dorofieiev, “Consensus control of multi-agent systems with input
delays: a descriptor model approach,” Mathematical Modeling and Computing,
vol. 6, no. 2, pp. 333–343, 2019. doi: 10.23939/mmc2019.02.333.
30. G. Herrmann, M.C. Turner, and I. Postlethwaite, “Linear matrix inequalities in con-
trol,” in Turner M.C., Bates D.G. (eds), Lecture Notes in Control and Information
Sciences, vol. 367, pp. 123–142, 2007.
31. A. Poznyak, A. Polyakov, and V. Azhmyakov, Attractive ellipsoids in robust con-
trol. Birkhäuser. Springer International Publishing, 2014, 348 p.
32. F. Zhang, “The Schur complement and its applications,” Numerical Methods and
Algorithms, vol. 4, Springer, 2005. doi: 10.1007/b105056.
33. M. Grant, S. Boyd, CVX: MATLAB software for disciplined convex programming,
version 2.2, January 2020. Available: https://cvxr.com/cvx
Y.I. Dorofieiev, L.M. Lyubchyk, M.M. Malko
ISSN 1681–6048 System Research & Information Technologies, 2024, № 2 116
34. J.F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over sym-
metric cones,” Optimisation Methods and Software, vol. 11, pp. 625–653, 1999. doi:
10.1080/10556789908805766.
35. Y.I. Dorofieiev, L.M. Lyubchyk, and A.A. Nikulchenko, “Consensus decentralized
control of multi-agent networked systems using vector Lyapunov functions,” 9th IEEE
Intern. Conf. on Intelligent Data Acquisition and Advanced Computing Systems:
Technolgy Application, vol. 1, pp. 60–65, 2017. doi: 10.1109/IDAACS.2017.8095050.
Received 28.07.2023
INFORMATION ON THE ARTICLE
Yurii I. Dorofieiev, ORCID: 0000-0002-7964-1286, National Technical University
“Kharkiv Polytechnic Institute”, Ukraine, e-mail: yurii.dorofieiev@khpi.edu.ua
Leonid M. Lyubchyk, ORCID: 0000-0003-0237-8915, National Technical University
“Kharkiv Polytechnic Institute”, Ukraine, e-mail: leonid.liubchyk@khpi.edu.ua
Maxim M. Malko, ORCID: 0000-0002-0125-2141, National Technical University
“Kharkiv Polytechnic Institute”, Ukraine, e-mail: maxim.malko@khpi.edu.ua
СИНТЕЗ ДЕЦЕНТРАЛІЗОВАНОГО КОНСЕНСУСНОГО КЕРУВАННЯ
ДЛЯ МУЛЬТИАГЕНТНИХ ДИСКРЕТНИХ СИСТЕМ З КОМУТАЦІЙНОЮ
ТОПОЛОГІЄЮ ЗА НАЯВНОСТІ ЛІДЕРА / Ю.І. Дорофєєв, Л.М. Любчик,
М.М. Малько
Анотація. Розглянуто задачу консенсусного керування лінійними мультиагент-
ними дискретними системами (МАС) з комутаційною топологією за наявності
лідера. Мета консенсусного керування полягає у зведенні станів усіх агентів
до стану лідера з одночасним забезпеченням стійкості локальних агентів, а та-
кож MAС у цілому. На відміну від традиційного підходу, який використовує
концепцію розширеної динамічної моделі мультиагентної системи та лапласі-
ан графу комунікаційної топології, запропоновано підхід на основі декомпози-
ції, який передбачає незалежне проєктування локальних регуляторів. Закон
керування вибирається у вигляді розподіленого зворотного зв’язку з дискрет-
ними ПІД-регуляторами. Задачу синтезу локальних регуляторів за допомогою
методу інваріантних еліпсоїдів зведено до набору задач напіввизначеного про-
грамування. Достатні умови стабілізації агентів та досягнення глобального
консенсусу отримано за допомогою техніки лінійних матричних нерівностей.
Наявність інформації про кінцевий набір можливих конфігурацій зв’язків між
агентами дозволяє синтезувати локальні регулятори в автономному режимі на
етапі проєктування. Ефективність запропонованого підходу продемонстровано
за допомогою числового прикладу.
Ключові слова: мультиагентна система, консенсусне керування, комутаційна
топологія, ПІД регулятор, метод інваріантних еліпсоїдів, лінійна матрична не-
рівність, задача напіввизначеного програмування.
|
| id | journaliasakpiua-article-284844 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:14Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/fa/8a2c2bfe35f1cea01c1e21d9f836f2fa.pdf |
| spelling | journaliasakpiua-article-2848442024-08-11T01:12:49Z Decentralized leader-following consensus control design for discrete-time multi-agent systems with switching topology Синтез децентралізованого консенсусного керування для мультиагентних дискретних систем з комутаційною топологією за наявності лідера Dorofieiev, Yurii Lyubchyk, Leonid Malko, Maxim multi-agent system consensus control switching topology PID controller invariant ellipsoids method linear matrix inequality semidefinite programming problem мультиагентна система консенсусне керування комутаційна топологія ПІД регулятор метод інваріантних еліпсоїдів лінійна матрична нерівність задача напіввизначеного програмування The problem of consensus control of linear discrete-time multi-agent systems (MASs) with switching topology is considered in the presence of a leader. The goal of consensus control is to bring the states of all agents to the leader state while providing stability for local agents, as well as the MAS as a whole. In contrast to the traditional approach, which uses the concept of an extended dynamic multi-agent system model and communication topology graph Laplacian, this paper proposes a decomposition approach, which provides a separate design of local controllers. The control law is chosen in the form of distributed feedback with discrete PID controllers. The problem of local controllers’ design is reduced to a set of semidefinite programming problems using the method of invariant ellipsoids. Sufficient conditions for agents’ stabilization and global consensus condition fulfillment are obtained using the linear matrix inequality technique. The availability of information about a finite set of possible configurations between agents allows us to design local controllers offline at the design stage. A numerical example demonstrates the effectiveness of the proposed approach. Розглянуто задачу консенсусного керування лінійними мультиагентними дискретними системами (МАС) з комутаційною топологією за наявності лідера. Мета консенсусного керування полягає у зведенні станів усіх агентів до стану лідера з одночасним забезпеченням стійкості локальних агентів, а також MAС у цілому. На відміну від традиційного підходу, який використовує концепцію розширеної динамічної моделі мультиагентної системи та лапласіан графу комунікаційної топології, запропоновано підхід на основі декомпозиції, який передбачає незалежне проєктування локальних регуляторів. Закон керування вибирається у вигляді розподіленого зворотного зв’язку з дискретними ПІД-регуляторами. Задачу синтезу локальних регуляторів за допомогою методу інваріантних еліпсоїдів зведено до набору задач напіввизначеного програмування. Достатні умови стабілізації агентів та досягнення глобального консенсусу отримано за допомогою техніки лінійних матричних нерівностей. Наявність інформації про кінцевий набір можливих конфігурацій зв’язків між агентами дозволяє синтезувати локальні регулятори в автономному режимі на етапі проєктування. Ефективність запропонованого підходу продемонстровано за допомогою числового прикладу. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-06-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/284844 10.20535/SRIT.2308-8893.2024.2.08 System research and information technologies; No. 2 (2024); 100-116 Системные исследования и информационные технологии; № 2 (2024); 100-116 Системні дослідження та інформаційні технології; № 2 (2024); 100-116 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/284844/301162 |
| spellingShingle | мультиагентна система консенсусне керування комутаційна топологія ПІД регулятор метод інваріантних еліпсоїдів лінійна матрична нерівність задача напіввизначеного програмування Dorofieiev, Yurii Lyubchyk, Leonid Malko, Maxim Синтез децентралізованого консенсусного керування для мультиагентних дискретних систем з комутаційною топологією за наявності лідера |
| title | Синтез децентралізованого консенсусного керування для мультиагентних дискретних систем з комутаційною топологією за наявності лідера |
| title_alt | Decentralized leader-following consensus control design for discrete-time multi-agent systems with switching topology |
| title_full | Синтез децентралізованого консенсусного керування для мультиагентних дискретних систем з комутаційною топологією за наявності лідера |
| title_fullStr | Синтез децентралізованого консенсусного керування для мультиагентних дискретних систем з комутаційною топологією за наявності лідера |
| title_full_unstemmed | Синтез децентралізованого консенсусного керування для мультиагентних дискретних систем з комутаційною топологією за наявності лідера |
| title_short | Синтез децентралізованого консенсусного керування для мультиагентних дискретних систем з комутаційною топологією за наявності лідера |
| title_sort | синтез децентралізованого консенсусного керування для мультиагентних дискретних систем з комутаційною топологією за наявності лідера |
| topic | мультиагентна система консенсусне керування комутаційна топологія ПІД регулятор метод інваріантних еліпсоїдів лінійна матрична нерівність задача напіввизначеного програмування |
| topic_facet | multi-agent system consensus control switching topology PID controller invariant ellipsoids method linear matrix inequality semidefinite programming problem мультиагентна система консенсусне керування комутаційна топологія ПІД регулятор метод інваріантних еліпсоїдів лінійна матрична нерівність задача напіввизначеного програмування |
| url | https://journal.iasa.kpi.ua/article/view/284844 |
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