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Consensus control of multi-agent systems with input and communication delay: A frequency domain perspective
Journal Pre-proof Consensus control of multi-agent systems with input and communication delay: A frequency domain perspective Zahoor Ahmed, Muhammad Mansoor Khan, Muhammad Abid Saeed, Zhang Weidong
PII: DOI: Reference:
S0019-0578(20)30055-0 https://doi.org/10.1016/j.isatra.2020.02.005 ISATRA 3485
To appear in:
ISA Transactions
Received date : 19 November 2018 Revised date : 7 October 2019 Accepted date : 3 February 2020 Please cite this article as: Z. Ahmed, M.M. Khan, M.A. Saeed et al., Consensus control of multi-agent systems with input and communication delay: A frequency domain perspective. ISA Transactions (2020), doi: https://doi.org/10.1016/j.isatra.2020.02.005. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier Ltd on behalf of ISA.
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*Title page showing Author Details
CONSENSUS CONTROL OF MULTI-AGENT SYSTEMS WITH INPUT AND
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COMMUNICATION DELAY: A FREQUENCY DOMAIN PERSPECTIVE
Zahoor Ahmed1, Muhammad Mansoor Khan2, Muhammad Abid Saeed1, Zhang Weidong1* 1
Department of Automation, Shanghai Jiaotong University, Shanghai 200240, PRC. Department of Electrical Engineering, Shanghai Jiaotong University, Shanghai, PRC. *Corresponding Author: [email protected]
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2
1
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*Highlights (for review)
Highlights
Consensus control of Multi-agent system (MAS) with input and communication delay in frequency domain.
The time delay is an inherent feature of practical systems.
A sufficient criterion for gain and delay margin.
Homogenous linear dynamic model of each agent involves multiple input delays.
The performance tracking, decomposition performance index and internal stability are
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used to design H2 controller.
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*Blinded Manuscript - without Author Details Click here to view linked References
CONSENSUS CONTROL OF MULTI-AGENT SYSTEMS WITH INPUT AND
2
COMMUNICATION DELAY: A FREQUENCY DOMAIN PERSPECTIVE
3 4
Abstract: This paper describes consensus control of multi-agent systems (MAS) with input
5
and communication delay in the frequency domain. Each agent of MAS is a linear
6
continuous-time system. The considered linear dynamic model of each agent involves
7
multiple input delays. An H2 controller is proposed for optimal performance and robustness
8
of the MAS. The internal stability approach is employed to compute the H2 controller and the
9
performance index of the overall MAS. A sufficient criterion is derived for gain and delay
10
margin to reach convergence. The simulation paradigm shows the effectiveness of the
11
proposed control scheme.
12
Keywords: Consensus, Multi-agent Systems, Communication Delay, Input Delay, Frequency
13
Domain.
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1. Introduction
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During the past twenty years of research, it has been determined that the distributed
16
coordination of MASs [1] have comprehensive applications such as flocking [2] autonomous
17
vehicles [3-5], formation control of spacecraft [6], multi-robot [7] and DC microgrid [8]. The
18
key objective of all these studies was to design a control protocol based on local information
19
exchange upon which all agents of MAS agree to certain conditions or interests which is
20
known as a consensus problem.
21
The consensus problems of MAS with time delay have been widely addressed. Two types of
22
delay in MAS can be found: input time delay (ITD) and communication time delay (CTD).
23
ITD is related to the processing time of data [9]. It may occur when actuators, controllers or
24
other components are connected to the network while, CTD is associated with the inter-
25
connecting time between two agents. Each agent may receive delayed information from its
26
neighbouring agents because of CTD [10]. CTD has been taken into account, because it exists
28 29 30 31 32
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15
in most distribution system in real applications. For example, PWM and computational delays in an LCL type grid-connected inverter [11,12], induced delay in controller area networks (CAN) bus of distributed control network (DCN) [13]. Thus coexistence of delay influences the convergence, speed, stability and performance of the overall system. In practice, these “two types of delays” coexist at the same time [9,14]. These types of time delays have been addressed in numerous studies, some of which considered time varying delays and some as 1
multiple ITD [1,15-16]. Besides this, a few work on separate CTD and ITD can be also found
34
in the literature [17,18] among different interacting systems. Thus current study discusses
35
cascaded multiple input time delays in addition to communication time delay.
36
This is a more complex case in which multiple time delays are part of either the nominator or
37
denominator of the system along with CTD. Therefore, we will discuss the consensus
38
problems of continuous time linear homogenous MAS with ITD and CTD. The CTDs are
39
single time delays, and ITD is considered as multiple input time delays. We proposed an
40
analytical H2 controller to solve this problem. The communication topology is typically
41
treated as a directed graph [6,9-10,14] but for the sake of convenience, here it is represented
42
by an undirected graph [12,16]. Generally, the communication graph is directed for one-way
43
communication (half duplex mode) and undirected or bidirectional for two-way
44
communication(full-duplex). When systems are connected through communication wires as
45
the medium in half duplex mode, bidirectional communication remains stable as long the
46
medium (a wire in this case) is intact. Under medium failure both way communication is lost
47
simultaneously. This is also substantially true for the half duplexed communication mode
48
where the same medium is used for transmission and reception. In the literature, some other
49
applications related to an undirected connected MASs can also be seen in [19,20]. One
50
example of such system is distributed battery energy storage systems (BESSs) linked with
51
grid for charge balancing with distributed packets level [21], where each BESS is employed
52
as an agent while ac-dc converters are treated as edges i.e. links between batteries and grid.
53
Thus for the frequency and voltage regulation, all BESSs are connected to each other through
54
a communication topology. A virtual leader connected to atleast one BESSs in the system is
55
presumed for the compatibility of all BESSs to track the reference frequency and voltage. In
56
this study, the authors assumed a generalized system with weighted undirected and fixed
57
communication topology.
58
In search of explicit work which can define the overall performance of all kinds of systems
59
having CTD as well ITD, we found that [22-23] solved a problem of MAS with ITD and
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proposed a PID controller in frequency domain. In [24], a model for heterogeneous systems and homogeneous systems has been presented in the frequency domain. Then in [25], the authors have proved that the previous conditions are insufficient for the stability of MAS and they introduced the new necessary and sufficient conditions. Li et al. [26] worked on H 2 and H performance
regions, and described a comparative analysis between the two.
Furthermore, Wang et al. [27] have designed a state feedback controller for transient 2
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performance and have proved that the H 2 performance of uncertain network is equal to the
67
min H index. The main objective of H2 optimal control is finding a controller that makes
68
the system stable and reduces integral square error (ISE). It can be able to quantify the
69
robustness and performance of the system.
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Recently, Ye et al. [28-29] worked on a consensus control for MAS in the frequency domain
71
with input delay and disturbance attenuation. Although the problem of time delay in
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consensus is discussed in detail by using the new technique of controller parametrization in
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the frequency domain by checking the performance tracking and stability, CTD which often
74
appears in the real distributed system applications and causes of uncertainties such as
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message losses, message delays and link failure [12], was not considered . In this work, the
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key purpose is to design a controller that meets the desired H2 performance consensus
77
conditions of MAS with CTD, in which each agent is a general linear dynamical model with
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ITD. The Multi-agent MIMO model is extended to multiple ITDs and CTDs. Then, based on
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the newly obtained MIMO model, we design an analytical controller with multiple time delay
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in the frequency domain by using controller parametrisation. Delay margin criteria is derived
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by using a quasi-polynomial in order to achieve robustness and high nominal performance. A
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sufficient condition clue is knowing that there exists a tradeoff between consensus high
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nominal performance and robustness of MAS under both considered time delays. Hence the
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main focus of this study is to introduce a general framework for MAS which hopefully
85
reveals many properties that facilitate study of the complete systems.
86
In addition, this work is structured follows as: basic concepts of graph theory and
87
mathematics are described in Section-II, Section-III defines problem formulation, Section IV
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represents the controller design by using parametrization, its stability, H 2 performance
89
tracking and robustness; Section V describes the simulation results of an example; and
90
Section VI finalizes the paper with conclusion.
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2. Mathematical Preliminaries
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In this paper, some basic concepts of graph theory have been used to design the MAS model and solve the consensus problem with communication delay. We assumed a MAS network of identical agents, whose communication structure is symbolized by a graph ,
edges,
is the finite vertices set as agents ,i.e. and
, where
is the finite set of
is the adjacency matrix of the graph such that A [ aij ] , 3
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where aij wij , if
98
aij a ji , then it is called the directed graph. The Laplacian L (l ) ij , where (l ) ij aij when
99
i j
and
and aij 0 , otherwise. If aij a ji , the graph is undirected and if
N
(l ) ij
a
i , j 1
ij
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when i j , where wij between agent i and j , is called weighting
100
function and known as communication strength. The L of a graph is positive semi-definite
101
with a simple zero eigenvalue iff the graph is connected. The corresponding right eigenvector
102
to this simple zero eigenvector is 1n i.e., L1n=0n. The authors refer the readers to” Diesetel
103
(2000)” for further concepts or related information on graph theory.
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the eigenvalues. Laplacian is an asymmetric matrix for a directed graph and symmetric
105
matrix for an undirected graph. A directed tree of a graph is called a spanning tree which is
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formed by graph edges such that these edges connect all vertices (nodes) of the graph. If any
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two nodes are linked through communication, then a directed graph should exist between two
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nodes. The directed spanning tree is called the leader-following topology under consensus
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tracking in which the root edge (for example ) is called leader [30-31]. Moreover, ‘tr’ and ‘
110
’ represents the trace of a matrix and direct sum, respectively.
111
Lemma 1.
112
For a directed and an undirected graph, zero is invariably the smallest and simple eigenvalue
113
of L, the laplacian matrix with right eigenvector 1 iff it has a spanning tree or its graph is
114
firmly connected. Specially for an undirected connected graph all eigenvalues of “L” are real
115
and can be arranged as
116
notations in the present paper: R nn is a set of real matrix of order n, I NM diag{1M , 0 N M } is
117
denoted by 1 and 0 with all elements equal to ones and zeros respectively,
118
matrix while 0 is a null matrix. The 2-norm of Y ( s) is defined as:
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( 0 1 2 ... N ) [32].We have used the following further
IN
is identity
N
Y (s)
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(for i = 1,2,3….) are
2
1
1 2 tr Y ( jw)Y ( jw) dw 2
3. Problem Formulation
Consider a homogenous MAS described in an n-first order linear dynamic model with both ITD and CTD. The ITD is the inherent delay of the system. Many systems have intrinsic delay which is more deterministic and can be treated as almost constant. While CTD is associated with communication from one agent to another. In this case, each agent gets 4
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delayed state information from another agent. ITD is applicable to single system. While CTD
126
is considered among agents of MAS because agents are interacting with each other through
127
networks. The traditional topology structure of MAS with communication delay is shown in
128
Fig.1. Subsystem 1 r1 (s)
Subsystem 2
r2 (s)
C1 ( s )
P1 (s)
X 1 ( s )e
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X 2 ( s )e
- 13 s
C2 ( s )
- 12 s
P 2 ( s)
X 2 ( s )e
r3 (s)
-
23 s
rN (s)
C3 ( s )
X N ( s )e
P3 ( s)
-
MN
s
Subsystem 3
129
CN ( s)
PN ( s)
Subsystem N
130
Fig.1: Multi-agent System with ITD and CTD
131
The transfer function for each agent is P ( s) P ( s) ... P ( s) P( s) , while their corresponding controllers are C ( s) C ( s) ... Cn ( s) C ( s) . Some agents have external references (msubsystems), i.e., r ( s) r ( s) ... r ( s) r ( s) , while rest ( n m subsystems) have no external references, i.e., r ( s) r ( s ) ... r ( s ) 0 , but these have access to relative states of their neighbours.
133 134 135
1
1
2
1
2
M 1
2
n
m
M 2
mn
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132
136
For agent i, if
137
model of each agent is:
is the state input,
xi
ui
is the control output and i is the error, then dynamic
xi ( s ) P( s )ui ( s ) P( s )C ( s ) i ( s )
138
(0)
Now let ij is the communication delay between agent i and j, and this delay affects the
140 141 142 143
dynamic of each neighboring agent and changes its states. As the communication topology is assumed undirected where transmission and reception medium is typically same, that is why communication delay can be taken equal. To assure consensus ability, the control protocol of MAS is given as below:
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( s ) r ( s ) x ( s )e i
ij s
i
( s),
( s) ( s),
i 1, 2,...m i m 1, m 2,...m n
i
(2)
s s Such that ( s ) jN aij [ x j ( s )e xi ( s )e ], is the relative state information of agent i from its ij
ij
i
147
(1)
neighbors.
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From (1) and (2), the vector form of the error signal ( s) can be written as: ij s
149
( s ) R ( s ) I n X ( s )e
150
( s ) R ( s ) ( I n L) X ( s )e
LX ( s)e
ij s
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m
ij s
(3)
151
Remark 1: The error signal ( s) is virtually similar to [28] because the Laplacian matrix
152
L ( I n L ) is responsible for communication and works as an information exchange among
153
two agents where L (l ) ij is the matrix, as if aij wij and (l ) ij has already been defined and wij is
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the weighting function and known as communication strength. The communication strength is the coupling strength between agents of the system and plays a crucial role in MAS consensus. In particular communication, depending on communication rate of each channel, this factor wij can be optimized for better performance and robustness. For example, in digital
158 159 160
communication, sometimes, packets are dropped due to the noise in the communication channel which cause intermittency in the received packet. Different communication channel has different intermittency under that condition, wij can be tuned to assure the convergence of
161
whole MAS.
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The equivalent MIMO model of homogenous MAS is shown in Fig.2. Here T T R ( s ) [ r1 ( s ),..., rn ( s )] are the respected consensus values, Z ( s ) [ z1 ( s ),..., z N ( s )] .“State
164
measurements relative to other agents”, X ( s ) [ x1 ( s ),..., xn ( s )]T “are sensed information, i.e.,
165
the output of n agents”.
m
Din (s)
r1 (s)
u1 (s)
e1 (s)
CC( s ( s) )
eN (s)
x1 ( s)
P(s)
xN (s)
uN (s)
C (s)
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0
Dout (s)
ˆ ( s) P
C (s)
P(s)
z1 ( s)
167 168 169 170 171 172
( L I NM )e
ij s
Fig.2: Closed-loop MIMO MAS with communication delay
Where , , ( s ) [ ( s ), ..., ( s )] is the error of the MAS. For homogenous MAS, all controllers and agents are identical, so we
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z N ( s)
1
can write as: Pˆ ( s ) i 1 Pi ( s ) , Cˆ ( s ) i 1 Ci ( s ) . n
N
Hence the multi-agent system could be: X ( s) Cˆ ( s) Pˆ ( s) S ( s) R( s) .
(4)
6
n
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Where S ( s ) [ I n LCˆ ( s ) Pˆ ( s )]1 is the characteristics equation of the overall MAS with
174
L ( I n L ) . The main purpose is to design a control system (controller) which can be used to
175
minimize the error between the actual output and setpoint of all agents in MAS when the
176
MAS is affected by disturbances and uncertainty in addition of ITD and CTD. Thus for
177
reference tracking and disturbance rejection, H2 performance index has been used [28].
178
Therefore,
m
H 2 performance
is:
179
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173
E
min r ( t ),
0
2
T
(t ) E (t ) dt min E (t )
(5)
2
180
Let L ( I nm L ) . As we know that sensitivity transfer function S(s) is the transfer function of
181
error and reference, i.e. the effect of output disturbance to output state, while complementary
182
transfer function T(s) describes the effects of reference on output states, i.e.,
183
S (s)
184
T (s)
X (s)
Dout ( s )
X (s)
(s)
1 [ I LCˆ ( s ) Pˆ ( s )]
R(s)
1 [ I LCˆ ( s ) Pˆ ( s )] Cˆ ( s ) Pˆ ( s )
R( s)
(6)
(7)
185
Thus, the sensitivity function, S(s) described in equation (6), is comparable to the “weighted
186
sensitivity function”. Therefore, the H2 performance index in frequency domain could be:
187
(8)
2
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min S ( s )W ( s )
188
where W ( s) may be called the stable weighting function and has been used to normalize
189
reference.
190
4. Optimal Robust Controller Design In this section, the following analysis is being presented.
192
Lemma 2: Assume the communication delay free case presented by [33] for Homogenous
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MAS shown in Fig.2, in which some agents (m-subsystems) of MAS are connected to external references and these references are equal for all m-subsystems. Whereas, the some agents (
subsystems) have no access with external reference but have access to relative
states of their neighbors as discussed earlier in section 3. Thus, the MAS will asymptotically achieve:
7
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( I nm L) lim s 0 X ( s ) lim s 0 R ( s ) iff lim s 0 [1 / C ( s) P( s)] 0
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n
199
lim s 0 (1 / [Ci ( s ) Pi ( s )]).sX ( s ) 0.
It follows as:
(9)
i 1
Lemma 2 is very important for the performance tracking of MAS. The obtained result is
201
helpful for the controller to achieve the performance tracking of the overall MAS. Where
202
L ( I nm L) is a Laplacian matrix whose full rank is equal to n.
203
Remark 2: From Lemma 2, we can conclude that consensus performance of each agent
204
depends upon the profile of Laplacian. As the main objective of the MAS (shown in Fig.2) is
205
to design a controller which makes it asymptotically stable, minimize error and rejecting
206
disturbances. Thus equation (9) indicates that the system error i which is the difference of
207
vectors R(s) and all relative statements Z(s) is minimized as LX(s) approaches to zero vector
208
asymptotically. This relation also shows that the performance of MAS depends upon the
209
system dynamics and interconnecting properties. Whereas, sensitivity transfer function S(s) is
210
the effect of output disturbance to output state. Thus the effects of disturbances can be
211
eliminated by considering the condition of sensitivity function during controller design.
212
Usually, the reference tracking of all agents of MAS is studied by putting any specific
213
reference prior and n m 1but the interesting thing of this model is that m=0, R(s)=0 has
214
been chosen as a special case. In accordance with the profile of L matrix, each agent could
215
reach consensus. Thus final states of consensus will be equal to R( s) , but this state will not be
216
a fixed value. The authors do not include its proof in this article. Further details or proof can
217
be seen in [29].
218
Next objective is to find n identical controllers for each agent of whole MAS which can
219
minimize the performance index of the overall system. It will be possible only if each
220
subsystem(agent) is satisfied without loss. Let’s borrow the following Lemma:
221
Lemma 3: As main concern is to find a controller of MAS with undirected and connected
223 224 225
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200
graph in terms of H 2 norms. According to [33], performance criteria of MAS can be calculated from its agent’s norm i.e. square of any norm of the whole MAS is equivalent to the summation of all the n subsystem’s local-performance. Thus, by using (6) and (8), the performance criteria of MAS can be written as under:
8
H2
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n
226
min S ( s )W ( s )
2 2
min
Si ( s)W ( s)
2
(10)
2
lP repro of
i 1
227
Where Si ( s) 1 i P( s)C ( s) . Similarly, if each agent is satisfied without loss, the
228
performance index MAS is minimized. That is why the 2-norm of (6) is: S ( s) 2 K ( s)
229
where K ( s ) is the diagonal matrix, K ( s ) I Pˆ ( s )Cˆ ( s ) , where is the diagonal matrix of i
230
.
231
Remark 3: As we know that the
232
Thus Lemma 3 describes the H performance index of the overall system regarding reference
233
and output disturbance. For a multiagent system with connected graph, square of the H 2
234
norm of the whole MAS is equivalent to the summation of all the n-subsystem’s local
235
performance. Hence it has been concluded that optimization problem might be decomposed
236
into n singles.
237
4.1 Controller parametrization
238
Definition 1: A system is said to be internally stable (IS) iff the transfer function between any
239
two points is stable.
240
Therefore, IS method is more appropriate than input-output mapping to describe the internal
241
states of the system [34]. If RT ( s), DinT ( s) and X T ( s), U T ( s) are the input & output signals
242
for the MIMO system shown in Fig.2 respectively, then its closed loop transfer function is:
1
2
2 2
1
H2
norm of a system is the root mean square of its error,
2
X T ( s), U T ( s) H ( s) RT ( s), DinT ( s) T
243
T
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T
T
(11)
Here H(s) is the transfer function matrix (TFM), Thus if we employed the definition 1 on
245
system (11) then it can be concluded that (11) is stable iff elements of the following TFM are
246
stable:
247
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Pˆ ( s)Cˆ ( s) S ( s) H ( s) ˆ C ( s) S ( s)
Pˆ ( s) S ( s)
LP( S )Cˆ ( s) S ( s)
9
(12)
Journal Pre-proof
where S ( s ) is defined already in earlier section. The following lemma will cover the IS
249
analysis of the closed loop system.
250
Lemma 4: Let’s recall [34] and definition 1, the system (11) is IS if there happens an
251
identical coordination controller such that all the agents are stable asymptotically. Hence in
252
the light of lemma 4, the transfer function of any agent i 1, 2,..., n is:
x ( s), u ( s)
T
253 254
i
i
lP repro of
248
H i ( s ) ri ( s ), d i in ( s )
T
where,
(13)
P ( s )C ( s ) S (s) i H i (s) C (s) S (s) i
255
, i P ( s )C ( s ) Si ( s ) P( s)
Si ( s )
256
and are the eigenvalues of L and Si ( s) 1 i P( s)C ( s) .
257
It is concluded that the closed loop system shown in (11) is composed of n-subsystems. All
258
these n-subsystems are identical and isolated. Moreover (11) is asymptotically stable if and
259
only if all the element in
260
4.2 Controller Design with Multiple Cascaded Time delays
261
As from our proceeding work [34], the identical dynamic of each agent of MAS is shown as:
1
i
of each n subsystems are asymptotically stable.
rna P( s)
e
1 s
p1 s 1
Ke
2 s
p2 s 1
,
(14)
263
where,
264
half-plane (LHP) roots while & are the two time delays. It has been assumed that each
265
agent of MAS network is (14), since each system in MAS consists of multiple time delays
266
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262
H i (s)
267 268
269
is a constant with real values and is known as forward gain, 1
p1
and
p2
are close left
2
that is why it is necessary to reduce it into a rational form with time delay. So the method of controller design for this work is considered with dual time delay. Let if 2 1
p1 p2 , 1 2
then (14) will become: P( s )
1 Ke
s
ps 1
e
1 s
,
(15)
10
and
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Now if we go through the internal model controller (IMC) structure which is constructed by
271
[28]. By using the concept of transfer function matrix H i ( s ) and IMC for overall MAS, we can
272
define the transfer function Qi(s) from (13) as: Qi ( s) C ( s) 1 i C ( s) P( s)
1
273 274
275
lP repro of
270
(16)
and the block matrix H i ( s ) is concluded as:
P ( s )Qi ( s ) P ( s ) 1 i P( s )Qi ( s ) i P ( s )Qi ( s ) Qi ( s )
H i (s)
(17)
276
Thus, overall stability analysis of MAS can be determine from H i ( s ) , the transfer function
277
matrix (17) will be stable if its all elements (quasi-polynomials) are stable. Hence we can
278
conclude the following conditions should be satisfied:
279
1. Q ( s ) is stable.
280
2.
i
Qi ( s ) and 1 i P( s )Qi ( s ) have
zeros whenever P(s) has RHP poles.
281
Theorem 1:
282
For a homogenous MAS shown in (11), in which each-agent (with multiple time delays)
283
described in (15); the topology of MAS is undirected and connected graph achieve consensus
284
asymptotically and
285
form of:
2
is improved, iff, there exist similar controllers C(s) in the
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min W ( s ) S ( s )
C (s)
286
ak ( ps 1)
s s s s 1 K e 1 Ke e
(18)
1
287
Where K 1 is system gain, is the adjustable filter parameter while
288
time delays such that
289
Proof: We know that the optimal controller Q(s) are similar for each agent in the
291
292
2
and
2
1 .
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290
1
are the
homogeneous MAS system. If the design specification treats lemma 3, i.e., to minimize W (s)S (s)
2
, for this Q(s) must be the same as the following plant: P( s )
1 Ke
s
(19)
ps 1
11
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293
For step input, the
294
asymptotic tracking property, can be written as:
Where
297
Q1 ( s ) is
1
the agent i of MAS which satisfies the IS (definition 1) and
1 sQ ( s ) 1
i 1 K
stable and make
W (s)S (s)
300
2
( ps 1)(1 K ) (1 Ke
s ( ps 1)(1 K )
s
zk
ln K 2 k j
)
2
2
1 Ke
2
s
Q1 ( s )
ps 1
and
pk
2
ln K 2 k j
(21)
ps 1
rna
=
s ( ps 1)(1 K ) 1 Ke
K e
s
1 Ke
s 1 Ke
K e
s
s
1 Ke
s 1 Ke
s
s
( ps 1)(1 K ) ( K e s ) K e s Q1 ( s ) s ( ps 1)(1 K )
s
2
ps 1
2
2
2
2
( ps 1)(1 K ) ( K e s ) K e s Q1 ( s )
2
s ( ps 1)(1 K )
ps 1
2
2
Simplifying the right-hand side of the above equation to generate Q1(s), thus the optimal
Jou
309
s
2
2
( ps 1)(1 K ) (1 Ke s ) K e s 1 Ke s Q1 ( s ) s
2 2
W (s)S (s)
305
308
1 sQ ( s ) 1 1 K
s
304
307
2
W (s) 1 P(s)
Basically, the solution (roots) of K e 0 gives poles which are the reflections of zk. so
303
306
2
Let 1 Ke 0 . Then RHP zeros zk and poles pk of P(s) are attained as:
301
302
proper and satisfy the asymptotic tracking.
1 1 Ke 1 sQ1 ( s ) 1 s ps 1 1 K
299
Qi ( s )
(20)
W ( s ) 1 i P( s )Qi ( s )
2
s
298
lP repro of
Qi ( s )
295
296
Q1 ( s ) of
controller Q1 opt ( s ) is:
( ps 1)(1 K ) ( K e s ) K e s Q1 ( s ) 0 s ( ps 1)(1 K )
Q1 opt ( s )
( ps 1)(1 K ) ( K e s ) s (1 K )
ps 1 1
K e
12
s
(22)
Journal Pre-proof
By using the equations (20) and (22) for the optimal controller Qopt ( s ) is Qopt ( s )
311
lP repro of
310
ps 1
(23)
s
i ( K e )
312
This is an H optimal controller for each SISO agent of MAS in which are the eigenvalues
313
of L ( I n L ) . Hence, Eq. (23) shows the importance of communication delay in controller
314
design. Since eigenvalues depend upon communication strength wij which is the weighting
315
function and appears as Laplacian element so we can conclude that input delay and
316
communication delay both influence the performance of controller in MASs. Hence, the
317
feasibility of can be verified by using Lemma 2.
318
From equation(23), it is clear that Qopt ( s ) is an improper function which is unattainable in its
319
physical properties. In order to make it proper , a filter J ( s) should be acquainted with
320
Qopt ( s ). Frankly speaking, J ( s ) should fulfil the following requirements:
2
i
m
i
i
321
1. Qi ( s ) Qopt ( s ) J ( s ) is proper.
322
2. The closed-loop system must follow IS (definition 1).
323
3. The asymptotic tracking should be accomplished.
Let us consider step signal as a system reference; then the filter may be in the form of:
rna
324
J (s)
325
1
(s 1)
ni
, i 1, 2,..., n,
326
Where are the performance degrees. Then, taking
327
J (s)
329 330
ni 1 the
filter in (24) could be: (25)
s 1
Thus the proper form of the optimal controller can be obtained by combining (23) and (25)
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328
1
(24)
as:
Qi ( s )
ps 1 s
i (s 1)( K e )
13
Journal Pre-proof
1
331
If we replace
332
controller could be calculated as follows:
by
ak
, then the unity feedback controller which is the real application
C (s)
333
lP repro of
i
ak ( ps 1)
s s s s 1 K e 1 Ke e
(26)
1
334
Thus the relation " S ( s) LT ( s ) I " has obtained in the form of C(s) which is a physically
335
realizable controller. That is why, a balance between the robustness and nominal performance
336
has been achieved. Ω is the positive adjustable parameter for filter amplitude, and its range is
337
(0, ) .The overall performance of the subsystem or whole system can be achieved by
338
adjusting this weighting parameter Ω. Hence, the H
339
achieved is optimal if all eigenvalues are real and equal. The performance index of a
340
system is always impossible to achieve an optimal solution but can be improved
341
simultaneously. In order to achieve optimal performance, communication topology should be
342
designed for the same non-zero eigenvalues. As a comparison with another structured
343
controller, this analytical algebraic method computed by H index ensures the performance
344
improvement of the whole MAS system with communication delay and multiple input
345
arbitrary delays.
346
Since this is an ideal controller and is challenging to implement practically, so for
347
implementation, the reduction techniques have been used. By using McLaurin expansion
348
series, PID can be generated as:
performance index that has been
i
2
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349
2
1
TI s
C (s) Kc 1
TD s
(27)
Let A( s) sC ( s) be a function of an appropriate order which consists of three parameters; that
351
can be calculated as:
352
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350
K c A '(0),
TI
A '(0) A(0)
and
TD
A ''(0) 2 A '(0)
14
Journal Pre-proof
di out ( s )
di in ( s )
ei ( s )
Ci ( s )
Gi ( s )
Zi (s) 353 354
lP repro of
ri ( s )
xi ( s )
i
Fig 3. Closed loop control structure Remark 4.
356
The designed controller of Eq. (26) will work locally for each agent under which the
357
homogenous agents can achieve consensuses and communicate efficiently. The performance
358
and robustness of MAS under connected and undirected topology have been improved with
359
multiple time delays. For practical application, the PID consensus controller (27) can also be
360
obtained approximately. Now the problem of communication delay will be dealt with the
361
following theorem.
362
Theorem 2:
363
If the fixed directed graph or undirected graph in the MAS (11) has a spanning tree, then the
364
system model shown in figure 2 using a controller (18) guarantees that all agents are
365
uniformly ultimately bounded no matter how large the communication delay is, iff it satisfies
366
the following conditions:
K 1 , ( K 1) 2 21 and ak
( K 1)
(28)
K max
368
Proof: Let us consider that the lemma 4 is true, then the characteristics polynomial of each
369
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367
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355
370 371
372
subsystem in (13) should be Hurwitz stable for consensus, i.e., characteristics poles lie in the left-half of the -plane.
For this, let us consider, det[1 i C ( s) P( s)] 0 . Thus from (19) and (26), we can deduce
det 1
ps 1
i s 1 K e
s
1 Ke e s
15
1 s
1 Ke s 0 , for all i . ps 1
Journal Pre-proof
After rearranging the above equation, we could get K 1 and ak
374
K 2 2 21
( K 1)
lP repro of
373
K max
375
where,
376
Remark 5: Theorem 2 presents a condition for communication delay of multi-agent systems
377
described in figure 2. As is a function of input delays which is a part of system application
378
requirements. we note that ITD is the inherent delay of the system. It is an inner loop control
379
delay and compensates the performance of single agent when considering system as
380
standalone. Whereas, CTD is considered in the case of MAS. CTD is the secondary loop
381
control delay. When MAS is considered, then ITD & CTD both are taken into account. Both
382
of the delays ITD and CTD are treated as input time delays. In addition, states of MAS are
383
affected by both of the delays. ITD and CTD can be addressed with inner-loop and outer-loop
384
control but it will be more optimal if we consider both of the delays simultaneously to
385
stabilize the system and achieve its performance under the circumstances of sufficient
386
conditions given in theorem 2. In this conclusion, the value of depends on the
387
communication topology; and if the eigenvalues are considered to be constant at its
388
maximum level, then the dependency will be between and , . For example, if input
389
delays and K are all considered constant for a special case, then the communication delay can
390
be compensated by under the circumstances of given conditions (28).
391
The proposed analytical controller for the H2 performance of homogenous MAS with
392
multiple ITD and CTD can be obtained through the following steps:
393
STEP-1:
Analyze the network topology and calculate its corresponding eigenvalues.
394
STEP-2:
Choose the dynamics of the sub-system.
395
STEP-3:
Calculate the controller transfer function by using theorem 2.
397 398 399
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396
i
STEP-4:
Compute the Ω for the robustness requirement.
STEP-5:
Determine the value of by using controller parametrization concepts.
STEP-6:
For higher order controller, deduce PID controller by using (27).
5. Numerical Example
16
Journal Pre-proof
This section discusses simulation result by assuming an example of a homogenous network
401
which consists of a stable agent and the topology of MAS is borrowed from [28], as shown in
402
Fig.4.The dynamic of each agent is:
lP repro of
400
P( s)
403
404 405
e
1 s
m1 s 1
Ke
2 s
for K 1
m2 s 1
1
2
4
3
Fig.4: Undirected topology
m1 m2 1, 2 1 such
that
0.1, 2 0.5, c 2 and K
>1. The Laplacian matrix “L”
406
Let set
407
for the information shown in Fig.4 with communication strength
1
wi j 2 is
as follows:
4 2 2 0 2 6 2 2 L 2 2 6 2 0 2 2 4
408
409
Hence
410
time delays, it is aimed at designing an analytical controller which can achieve performance
411
tracking and stability as well as fulfils the consensuses condition of communication delay.
412
According to the proposed method
{0, 4, 8, 8}
Qi opt
413
415 416
ak ( s 1) ( K e
0.4 s
)
If we introduce a filter in series of Qi opt , then according to the theorem 1, the sub-optimal
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414
are the eigenvalues of L. For the given dynamic system with multiple
rna
i
controller is:
Qi ( s )
ak ( s 1) ( K e
17
0.4 s
)(s 1)
Journal Pre-proof
0.1 s
1 0.1s
417
For the simplification of analysis, let us put e
418
approximation of the irrational function by using Taylor series expansion), then the feedback
419
controller of the proposed system can be achieved as follows:
Qi ( s )
(to make an
lP repro of
into
C (s)
420
ak ( s 1)
( K e
0.1 s
)(s 1) (1 Ke
0.4 s
)e
0.1 s
421
Thus it is a typical PI controller and shows its characteristics polynomial. For the verification
422
of communication delay and consensus condition; according to theorem 2 for above
423
considered values, the overall MAS network must be in consensus if K 1 and ak
424
By using proposed analytical controllers for each agent, the output responses are shown in
425
Fig 5(a) and (b) at 10 , when a
426
H2
427
though along with communication delay; if
428
theorem 2.
k
1
0.6
0.8 max
.
the output response is stable and is in consensus i.e.,
performance index of whole system has improved as compared with [28,29,35] even
429
the response will diverge which satisfy the
Effects of K:
432
Now if we put K 1 in our proposed system for all above considered values, the output
433
response of all agents diverge as shown in Fig 6(a) and 6(b).
434
Impact of delay on Stability
435
If the ITD increases, the same algorithm can be used to control the effects of different time
436
delay on performance of MAS. So the output results of MAS with
437
are shown in Fig7(a) and 7(b). So the tuning of performance index can be made in order to
438 439 440 441
rna
431
Jou
430
ak 2
1
1, 2 5, 10, ak 0.1
fulfil the demand of MASs. As CTD increases or decreases, it affects the performance and stability of MAS. According to the demand for eigenvalues, the controller shown in (26) adjusts the values of ak and Ω according to the condition (28) . For varying communication delay, let
c
4 , wi j 4
then the eigenvalues are
i
18
{0, 8,16,16}
, and Laplacian will be:
Journal Pre-proof
8 4 L 4 0
4
12
4
4
4
12
4
4
4
8
0
lP repro of
442
4
443
For this MAS, the same controller achieves consensus with H performance index if
444
for small ITD case and
445
Fig.8(a) and 8(b).
2
ak 0.12
446
Fig.5(a) Converges at
c
for large delay case. The simulation results are shown in
2, 10, ak 1
(b) Divergence at
c
2, 10, ak 2
448 449
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rna
447
ak 0.82
Fig.6 Divergence (a) when =2, 10, a c
k
2, K 1
19
(b)
when = 2 10, a c
k
2, K 0.5
450 451 452
lP repro of
Journal Pre-proof
Fig.7 (a) Consensus Performance when = 2, 10, a c
6. Conclusions
k
0.2
(b) when = 2, 10, a c
k
1
453
An analytical controller has been designed for a homogenous MAS with multiple input
454
delays and communication delay in the frequency domain. The necessary condition
455
guarantees consensus on the overall network with delay. A simple controller parametrization
456
scheme has been used to design an analytical controller for IS and improve H performance
457
index for multiple input delays and communication delay at the same time. The necessary
458
condition has been used to trade off the nominal performance of MAS with communication
459
delay. The simulation results have verified the calculation. Since this method is based on
460
simple algebraic calculations that is why it is much easier to use for the real application.
461
Further studies will focus on designing of consensus controllers for a multi-agent system
462
under attack.
rna
Jou
463
2
20
Journal Pre-proof
c
4, 10, ak 0.12 (b) when c 4, 10, ak 0.5
Fig.8 (a) Consensus Performance when
465
Acknowledgement
466
This paper is partly supported by the National Science Foundation of China
467
(61473183，U1509211，61627810),
468
(SQ2017YFGH001005).
469
References
470
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[34] Zhang, Weidong. Quantitative process control theory. CRC Press, 2011.
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*Conflict of Interest
lP repro of
No conflict of interest exists in the submission of this manuscript, and the manuscript is
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approved by all authors for publication.
PII: DOI: Reference:
S0019-0578(20)30055-0 https://doi.org/10.1016/j.isatra.2020.02.005 ISATRA 3485
To appear in:
ISA Transactions
Received date : 19 November 2018 Revised date : 7 October 2019 Accepted date : 3 February 2020 Please cite this article as: Z. Ahmed, M.M. Khan, M.A. Saeed et al., Consensus control of multi-agent systems with input and communication delay: A frequency domain perspective. ISA Transactions (2020), doi: https://doi.org/10.1016/j.isatra.2020.02.005. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier Ltd on behalf of ISA.
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*Title page showing Author Details
CONSENSUS CONTROL OF MULTI-AGENT SYSTEMS WITH INPUT AND
lP repro of
COMMUNICATION DELAY: A FREQUENCY DOMAIN PERSPECTIVE
Zahoor Ahmed1, Muhammad Mansoor Khan2, Muhammad Abid Saeed1, Zhang Weidong1* 1
Department of Automation, Shanghai Jiaotong University, Shanghai 200240, PRC. Department of Electrical Engineering, Shanghai Jiaotong University, Shanghai, PRC. *Corresponding Author: [email protected]
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2
1
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lP repro of
*Highlights (for review)
Highlights
Consensus control of Multi-agent system (MAS) with input and communication delay in frequency domain.
The time delay is an inherent feature of practical systems.
A sufficient criterion for gain and delay margin.
Homogenous linear dynamic model of each agent involves multiple input delays.
The performance tracking, decomposition performance index and internal stability are
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used to design H2 controller.
Journal Pre-proof
*Blinded Manuscript - without Author Details Click here to view linked References
CONSENSUS CONTROL OF MULTI-AGENT SYSTEMS WITH INPUT AND
2
COMMUNICATION DELAY: A FREQUENCY DOMAIN PERSPECTIVE
3 4
Abstract: This paper describes consensus control of multi-agent systems (MAS) with input
5
and communication delay in the frequency domain. Each agent of MAS is a linear
6
continuous-time system. The considered linear dynamic model of each agent involves
7
multiple input delays. An H2 controller is proposed for optimal performance and robustness
8
of the MAS. The internal stability approach is employed to compute the H2 controller and the
9
performance index of the overall MAS. A sufficient criterion is derived for gain and delay
10
margin to reach convergence. The simulation paradigm shows the effectiveness of the
11
proposed control scheme.
12
Keywords: Consensus, Multi-agent Systems, Communication Delay, Input Delay, Frequency
13
Domain.
14
1. Introduction
lP repro of
1
During the past twenty years of research, it has been determined that the distributed
16
coordination of MASs [1] have comprehensive applications such as flocking [2] autonomous
17
vehicles [3-5], formation control of spacecraft [6], multi-robot [7] and DC microgrid [8]. The
18
key objective of all these studies was to design a control protocol based on local information
19
exchange upon which all agents of MAS agree to certain conditions or interests which is
20
known as a consensus problem.
21
The consensus problems of MAS with time delay have been widely addressed. Two types of
22
delay in MAS can be found: input time delay (ITD) and communication time delay (CTD).
23
ITD is related to the processing time of data [9]. It may occur when actuators, controllers or
24
other components are connected to the network while, CTD is associated with the inter-
25
connecting time between two agents. Each agent may receive delayed information from its
26
neighbouring agents because of CTD [10]. CTD has been taken into account, because it exists
28 29 30 31 32
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27
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15
in most distribution system in real applications. For example, PWM and computational delays in an LCL type grid-connected inverter [11,12], induced delay in controller area networks (CAN) bus of distributed control network (DCN) [13]. Thus coexistence of delay influences the convergence, speed, stability and performance of the overall system. In practice, these “two types of delays” coexist at the same time [9,14]. These types of time delays have been addressed in numerous studies, some of which considered time varying delays and some as 1
multiple ITD [1,15-16]. Besides this, a few work on separate CTD and ITD can be also found
34
in the literature [17,18] among different interacting systems. Thus current study discusses
35
cascaded multiple input time delays in addition to communication time delay.
36
This is a more complex case in which multiple time delays are part of either the nominator or
37
denominator of the system along with CTD. Therefore, we will discuss the consensus
38
problems of continuous time linear homogenous MAS with ITD and CTD. The CTDs are
39
single time delays, and ITD is considered as multiple input time delays. We proposed an
40
analytical H2 controller to solve this problem. The communication topology is typically
41
treated as a directed graph [6,9-10,14] but for the sake of convenience, here it is represented
42
by an undirected graph [12,16]. Generally, the communication graph is directed for one-way
43
communication (half duplex mode) and undirected or bidirectional for two-way
44
communication(full-duplex). When systems are connected through communication wires as
45
the medium in half duplex mode, bidirectional communication remains stable as long the
46
medium (a wire in this case) is intact. Under medium failure both way communication is lost
47
simultaneously. This is also substantially true for the half duplexed communication mode
48
where the same medium is used for transmission and reception. In the literature, some other
49
applications related to an undirected connected MASs can also be seen in [19,20]. One
50
example of such system is distributed battery energy storage systems (BESSs) linked with
51
grid for charge balancing with distributed packets level [21], where each BESS is employed
52
as an agent while ac-dc converters are treated as edges i.e. links between batteries and grid.
53
Thus for the frequency and voltage regulation, all BESSs are connected to each other through
54
a communication topology. A virtual leader connected to atleast one BESSs in the system is
55
presumed for the compatibility of all BESSs to track the reference frequency and voltage. In
56
this study, the authors assumed a generalized system with weighted undirected and fixed
57
communication topology.
58
In search of explicit work which can define the overall performance of all kinds of systems
59
having CTD as well ITD, we found that [22-23] solved a problem of MAS with ITD and
60 61 62 63 64 65
rna
lP repro of
33
Jou
Journal Pre-proof
proposed a PID controller in frequency domain. In [24], a model for heterogeneous systems and homogeneous systems has been presented in the frequency domain. Then in [25], the authors have proved that the previous conditions are insufficient for the stability of MAS and they introduced the new necessary and sufficient conditions. Li et al. [26] worked on H 2 and H performance
regions, and described a comparative analysis between the two.
Furthermore, Wang et al. [27] have designed a state feedback controller for transient 2
Journal Pre-proof
performance and have proved that the H 2 performance of uncertain network is equal to the
67
min H index. The main objective of H2 optimal control is finding a controller that makes
68
the system stable and reduces integral square error (ISE). It can be able to quantify the
69
robustness and performance of the system.
70
Recently, Ye et al. [28-29] worked on a consensus control for MAS in the frequency domain
71
with input delay and disturbance attenuation. Although the problem of time delay in
72
consensus is discussed in detail by using the new technique of controller parametrization in
73
the frequency domain by checking the performance tracking and stability, CTD which often
74
appears in the real distributed system applications and causes of uncertainties such as
75
message losses, message delays and link failure [12], was not considered . In this work, the
76
key purpose is to design a controller that meets the desired H2 performance consensus
77
conditions of MAS with CTD, in which each agent is a general linear dynamical model with
78
ITD. The Multi-agent MIMO model is extended to multiple ITDs and CTDs. Then, based on
79
the newly obtained MIMO model, we design an analytical controller with multiple time delay
80
in the frequency domain by using controller parametrisation. Delay margin criteria is derived
81
by using a quasi-polynomial in order to achieve robustness and high nominal performance. A
82
sufficient condition clue is knowing that there exists a tradeoff between consensus high
83
nominal performance and robustness of MAS under both considered time delays. Hence the
84
main focus of this study is to introduce a general framework for MAS which hopefully
85
reveals many properties that facilitate study of the complete systems.
86
In addition, this work is structured follows as: basic concepts of graph theory and
87
mathematics are described in Section-II, Section-III defines problem formulation, Section IV
88
represents the controller design by using parametrization, its stability, H 2 performance
89
tracking and robustness; Section V describes the simulation results of an example; and
90
Section VI finalizes the paper with conclusion.
92 93 94 95 96
rna
2. Mathematical Preliminaries
Jou
91
lP repro of
66
In this paper, some basic concepts of graph theory have been used to design the MAS model and solve the consensus problem with communication delay. We assumed a MAS network of identical agents, whose communication structure is symbolized by a graph ,
edges,
is the finite vertices set as agents ,i.e. and
, where
is the finite set of
is the adjacency matrix of the graph such that A [ aij ] , 3
Journal Pre-proof
where aij wij , if
98
aij a ji , then it is called the directed graph. The Laplacian L (l ) ij , where (l ) ij aij when
99
i j
and
and aij 0 , otherwise. If aij a ji , the graph is undirected and if
N
(l ) ij
a
i , j 1
ij
lP repro of
97
when i j , where wij between agent i and j , is called weighting
100
function and known as communication strength. The L of a graph is positive semi-definite
101
with a simple zero eigenvalue iff the graph is connected. The corresponding right eigenvector
102
to this simple zero eigenvector is 1n i.e., L1n=0n. The authors refer the readers to” Diesetel
103
(2000)” for further concepts or related information on graph theory.
104
the eigenvalues. Laplacian is an asymmetric matrix for a directed graph and symmetric
105
matrix for an undirected graph. A directed tree of a graph is called a spanning tree which is
106
formed by graph edges such that these edges connect all vertices (nodes) of the graph. If any
107
two nodes are linked through communication, then a directed graph should exist between two
108
nodes. The directed spanning tree is called the leader-following topology under consensus
109
tracking in which the root edge (for example ) is called leader [30-31]. Moreover, ‘tr’ and ‘
110
’ represents the trace of a matrix and direct sum, respectively.
111
Lemma 1.
112
For a directed and an undirected graph, zero is invariably the smallest and simple eigenvalue
113
of L, the laplacian matrix with right eigenvector 1 iff it has a spanning tree or its graph is
114
firmly connected. Specially for an undirected connected graph all eigenvalues of “L” are real
115
and can be arranged as
116
notations in the present paper: R nn is a set of real matrix of order n, I NM diag{1M , 0 N M } is
117
denoted by 1 and 0 with all elements equal to ones and zeros respectively,
118
matrix while 0 is a null matrix. The 2-norm of Y ( s) is defined as:
120 121 122 123 124
rna
( 0 1 2 ... N ) [32].We have used the following further
IN
is identity
N
Y (s)
Jou
119
(for i = 1,2,3….) are
2
1
1 2 tr Y ( jw)Y ( jw) dw 2
3. Problem Formulation
Consider a homogenous MAS described in an n-first order linear dynamic model with both ITD and CTD. The ITD is the inherent delay of the system. Many systems have intrinsic delay which is more deterministic and can be treated as almost constant. While CTD is associated with communication from one agent to another. In this case, each agent gets 4
Journal Pre-proof
delayed state information from another agent. ITD is applicable to single system. While CTD
126
is considered among agents of MAS because agents are interacting with each other through
127
networks. The traditional topology structure of MAS with communication delay is shown in
128
Fig.1. Subsystem 1 r1 (s)
Subsystem 2
r2 (s)
C1 ( s )
P1 (s)
X 1 ( s )e
lP repro of
125
X 2 ( s )e
- 13 s
C2 ( s )
- 12 s
P 2 ( s)
X 2 ( s )e
r3 (s)
-
23 s
rN (s)
C3 ( s )
X N ( s )e
P3 ( s)
-
MN
s
Subsystem 3
129
CN ( s)
PN ( s)
Subsystem N
130
Fig.1: Multi-agent System with ITD and CTD
131
The transfer function for each agent is P ( s) P ( s) ... P ( s) P( s) , while their corresponding controllers are C ( s) C ( s) ... Cn ( s) C ( s) . Some agents have external references (msubsystems), i.e., r ( s) r ( s) ... r ( s) r ( s) , while rest ( n m subsystems) have no external references, i.e., r ( s) r ( s ) ... r ( s ) 0 , but these have access to relative states of their neighbours.
133 134 135
1
1
2
1
2
M 1
2
n
m
M 2
mn
rna
132
136
For agent i, if
137
model of each agent is:
is the state input,
xi
ui
is the control output and i is the error, then dynamic
xi ( s ) P( s )ui ( s ) P( s )C ( s ) i ( s )
138
(0)
Now let ij is the communication delay between agent i and j, and this delay affects the
140 141 142 143
dynamic of each neighboring agent and changes its states. As the communication topology is assumed undirected where transmission and reception medium is typically same, that is why communication delay can be taken equal. To assure consensus ability, the control protocol of MAS is given as below:
144 145 146
Jou
139
( s ) r ( s ) x ( s )e i
ij s
i
( s),
( s) ( s),
i 1, 2,...m i m 1, m 2,...m n
i
(2)
s s Such that ( s ) jN aij [ x j ( s )e xi ( s )e ], is the relative state information of agent i from its ij
ij
i
147
(1)
neighbors.
5
Journal Pre-proof
148
From (1) and (2), the vector form of the error signal ( s) can be written as: ij s
149
( s ) R ( s ) I n X ( s )e
150
( s ) R ( s ) ( I n L) X ( s )e
LX ( s)e
ij s
lP repro of m
m
ij s
(3)
151
Remark 1: The error signal ( s) is virtually similar to [28] because the Laplacian matrix
152
L ( I n L ) is responsible for communication and works as an information exchange among
153
two agents where L (l ) ij is the matrix, as if aij wij and (l ) ij has already been defined and wij is
154 155 156 157
the weighting function and known as communication strength. The communication strength is the coupling strength between agents of the system and plays a crucial role in MAS consensus. In particular communication, depending on communication rate of each channel, this factor wij can be optimized for better performance and robustness. For example, in digital
158 159 160
communication, sometimes, packets are dropped due to the noise in the communication channel which cause intermittency in the received packet. Different communication channel has different intermittency under that condition, wij can be tuned to assure the convergence of
161
whole MAS.
162 163
The equivalent MIMO model of homogenous MAS is shown in Fig.2. Here T T R ( s ) [ r1 ( s ),..., rn ( s )] are the respected consensus values, Z ( s ) [ z1 ( s ),..., z N ( s )] .“State
164
measurements relative to other agents”, X ( s ) [ x1 ( s ),..., xn ( s )]T “are sensed information, i.e.,
165
the output of n agents”.
m
Din (s)
r1 (s)
u1 (s)
e1 (s)
CC( s ( s) )
eN (s)
x1 ( s)
P(s)
xN (s)
uN (s)
C (s)
rna
0
Dout (s)
ˆ ( s) P
C (s)
P(s)
z1 ( s)
167 168 169 170 171 172
( L I NM )e
ij s
Fig.2: Closed-loop MIMO MAS with communication delay
Where , , ( s ) [ ( s ), ..., ( s )] is the error of the MAS. For homogenous MAS, all controllers and agents are identical, so we
Jou
166
z N ( s)
1
can write as: Pˆ ( s ) i 1 Pi ( s ) , Cˆ ( s ) i 1 Ci ( s ) . n
N
Hence the multi-agent system could be: X ( s) Cˆ ( s) Pˆ ( s) S ( s) R( s) .
(4)
6
n
Journal Pre-proof
Where S ( s ) [ I n LCˆ ( s ) Pˆ ( s )]1 is the characteristics equation of the overall MAS with
174
L ( I n L ) . The main purpose is to design a control system (controller) which can be used to
175
minimize the error between the actual output and setpoint of all agents in MAS when the
176
MAS is affected by disturbances and uncertainty in addition of ITD and CTD. Thus for
177
reference tracking and disturbance rejection, H2 performance index has been used [28].
178
Therefore,
m
H 2 performance
is:
179
lP repro of
173
E
min r ( t ),
0
2
T
(t ) E (t ) dt min E (t )
(5)
2
180
Let L ( I nm L ) . As we know that sensitivity transfer function S(s) is the transfer function of
181
error and reference, i.e. the effect of output disturbance to output state, while complementary
182
transfer function T(s) describes the effects of reference on output states, i.e.,
183
S (s)
184
T (s)
X (s)
Dout ( s )
X (s)
(s)
1 [ I LCˆ ( s ) Pˆ ( s )]
R(s)
1 [ I LCˆ ( s ) Pˆ ( s )] Cˆ ( s ) Pˆ ( s )
R( s)
(6)
(7)
185
Thus, the sensitivity function, S(s) described in equation (6), is comparable to the “weighted
186
sensitivity function”. Therefore, the H2 performance index in frequency domain could be:
187
(8)
2
rna
min S ( s )W ( s )
188
where W ( s) may be called the stable weighting function and has been used to normalize
189
reference.
190
4. Optimal Robust Controller Design In this section, the following analysis is being presented.
192
Lemma 2: Assume the communication delay free case presented by [33] for Homogenous
193 194 195 196 197
Jou
191
MAS shown in Fig.2, in which some agents (m-subsystems) of MAS are connected to external references and these references are equal for all m-subsystems. Whereas, the some agents (
subsystems) have no access with external reference but have access to relative
states of their neighbors as discussed earlier in section 3. Thus, the MAS will asymptotically achieve:
7
Journal Pre-proof
( I nm L) lim s 0 X ( s ) lim s 0 R ( s ) iff lim s 0 [1 / C ( s) P( s)] 0
lP repro of
198
n
199
lim s 0 (1 / [Ci ( s ) Pi ( s )]).sX ( s ) 0.
It follows as:
(9)
i 1
Lemma 2 is very important for the performance tracking of MAS. The obtained result is
201
helpful for the controller to achieve the performance tracking of the overall MAS. Where
202
L ( I nm L) is a Laplacian matrix whose full rank is equal to n.
203
Remark 2: From Lemma 2, we can conclude that consensus performance of each agent
204
depends upon the profile of Laplacian. As the main objective of the MAS (shown in Fig.2) is
205
to design a controller which makes it asymptotically stable, minimize error and rejecting
206
disturbances. Thus equation (9) indicates that the system error i which is the difference of
207
vectors R(s) and all relative statements Z(s) is minimized as LX(s) approaches to zero vector
208
asymptotically. This relation also shows that the performance of MAS depends upon the
209
system dynamics and interconnecting properties. Whereas, sensitivity transfer function S(s) is
210
the effect of output disturbance to output state. Thus the effects of disturbances can be
211
eliminated by considering the condition of sensitivity function during controller design.
212
Usually, the reference tracking of all agents of MAS is studied by putting any specific
213
reference prior and n m 1but the interesting thing of this model is that m=0, R(s)=0 has
214
been chosen as a special case. In accordance with the profile of L matrix, each agent could
215
reach consensus. Thus final states of consensus will be equal to R( s) , but this state will not be
216
a fixed value. The authors do not include its proof in this article. Further details or proof can
217
be seen in [29].
218
Next objective is to find n identical controllers for each agent of whole MAS which can
219
minimize the performance index of the overall system. It will be possible only if each
220
subsystem(agent) is satisfied without loss. Let’s borrow the following Lemma:
221
Lemma 3: As main concern is to find a controller of MAS with undirected and connected
223 224 225
Jou
222
rna
200
graph in terms of H 2 norms. According to [33], performance criteria of MAS can be calculated from its agent’s norm i.e. square of any norm of the whole MAS is equivalent to the summation of all the n subsystem’s local-performance. Thus, by using (6) and (8), the performance criteria of MAS can be written as under:
8
H2
Journal Pre-proof
n
226
min S ( s )W ( s )
2 2
min
Si ( s)W ( s)
2
(10)
2
lP repro of
i 1
227
Where Si ( s) 1 i P( s)C ( s) . Similarly, if each agent is satisfied without loss, the
228
performance index MAS is minimized. That is why the 2-norm of (6) is: S ( s) 2 K ( s)
229
where K ( s ) is the diagonal matrix, K ( s ) I Pˆ ( s )Cˆ ( s ) , where is the diagonal matrix of i
230
.
231
Remark 3: As we know that the
232
Thus Lemma 3 describes the H performance index of the overall system regarding reference
233
and output disturbance. For a multiagent system with connected graph, square of the H 2
234
norm of the whole MAS is equivalent to the summation of all the n-subsystem’s local
235
performance. Hence it has been concluded that optimization problem might be decomposed
236
into n singles.
237
4.1 Controller parametrization
238
Definition 1: A system is said to be internally stable (IS) iff the transfer function between any
239
two points is stable.
240
Therefore, IS method is more appropriate than input-output mapping to describe the internal
241
states of the system [34]. If RT ( s), DinT ( s) and X T ( s), U T ( s) are the input & output signals
242
for the MIMO system shown in Fig.2 respectively, then its closed loop transfer function is:
1
2
2 2
1
H2
norm of a system is the root mean square of its error,
2
X T ( s), U T ( s) H ( s) RT ( s), DinT ( s) T
243
T
rna
T
T
(11)
Here H(s) is the transfer function matrix (TFM), Thus if we employed the definition 1 on
245
system (11) then it can be concluded that (11) is stable iff elements of the following TFM are
246
stable:
247
Jou
244
Pˆ ( s)Cˆ ( s) S ( s) H ( s) ˆ C ( s) S ( s)
Pˆ ( s) S ( s)
LP( S )Cˆ ( s) S ( s)
9
(12)
Journal Pre-proof
where S ( s ) is defined already in earlier section. The following lemma will cover the IS
249
analysis of the closed loop system.
250
Lemma 4: Let’s recall [34] and definition 1, the system (11) is IS if there happens an
251
identical coordination controller such that all the agents are stable asymptotically. Hence in
252
the light of lemma 4, the transfer function of any agent i 1, 2,..., n is:
x ( s), u ( s)
T
253 254
i
i
lP repro of
248
H i ( s ) ri ( s ), d i in ( s )
T
where,
(13)
P ( s )C ( s ) S (s) i H i (s) C (s) S (s) i
255
, i P ( s )C ( s ) Si ( s ) P( s)
Si ( s )
256
and are the eigenvalues of L and Si ( s) 1 i P( s)C ( s) .
257
It is concluded that the closed loop system shown in (11) is composed of n-subsystems. All
258
these n-subsystems are identical and isolated. Moreover (11) is asymptotically stable if and
259
only if all the element in
260
4.2 Controller Design with Multiple Cascaded Time delays
261
As from our proceeding work [34], the identical dynamic of each agent of MAS is shown as:
1
i
of each n subsystems are asymptotically stable.
rna P( s)
e
1 s
p1 s 1
Ke
2 s
p2 s 1
,
(14)
263
where,
264
half-plane (LHP) roots while & are the two time delays. It has been assumed that each
265
agent of MAS network is (14), since each system in MAS consists of multiple time delays
266
Jou
262
H i (s)
267 268
269
is a constant with real values and is known as forward gain, 1
p1
and
p2
are close left
2
that is why it is necessary to reduce it into a rational form with time delay. So the method of controller design for this work is considered with dual time delay. Let if 2 1
p1 p2 , 1 2
then (14) will become: P( s )
1 Ke
s
ps 1
e
1 s
,
(15)
10
and
Journal Pre-proof
Now if we go through the internal model controller (IMC) structure which is constructed by
271
[28]. By using the concept of transfer function matrix H i ( s ) and IMC for overall MAS, we can
272
define the transfer function Qi(s) from (13) as: Qi ( s) C ( s) 1 i C ( s) P( s)
1
273 274
275
lP repro of
270
(16)
and the block matrix H i ( s ) is concluded as:
P ( s )Qi ( s ) P ( s ) 1 i P( s )Qi ( s ) i P ( s )Qi ( s ) Qi ( s )
H i (s)
(17)
276
Thus, overall stability analysis of MAS can be determine from H i ( s ) , the transfer function
277
matrix (17) will be stable if its all elements (quasi-polynomials) are stable. Hence we can
278
conclude the following conditions should be satisfied:
279
1. Q ( s ) is stable.
280
2.
i
Qi ( s ) and 1 i P( s )Qi ( s ) have
zeros whenever P(s) has RHP poles.
281
Theorem 1:
282
For a homogenous MAS shown in (11), in which each-agent (with multiple time delays)
283
described in (15); the topology of MAS is undirected and connected graph achieve consensus
284
asymptotically and
285
form of:
2
is improved, iff, there exist similar controllers C(s) in the
rna
min W ( s ) S ( s )
C (s)
286
ak ( ps 1)
s s s s 1 K e 1 Ke e
(18)
1
287
Where K 1 is system gain, is the adjustable filter parameter while
288
time delays such that
289
Proof: We know that the optimal controller Q(s) are similar for each agent in the
291
292
2
and
2
1 .
Jou
290
1
are the
homogeneous MAS system. If the design specification treats lemma 3, i.e., to minimize W (s)S (s)
2
, for this Q(s) must be the same as the following plant: P( s )
1 Ke
s
(19)
ps 1
11
Journal Pre-proof
293
For step input, the
294
asymptotic tracking property, can be written as:
Where
297
Q1 ( s ) is
1
the agent i of MAS which satisfies the IS (definition 1) and
1 sQ ( s ) 1
i 1 K
stable and make
W (s)S (s)
300
2
( ps 1)(1 K ) (1 Ke
s ( ps 1)(1 K )
s
zk
ln K 2 k j
)
2
2
1 Ke
2
s
Q1 ( s )
ps 1
and
pk
2
ln K 2 k j
(21)
ps 1
rna
=
s ( ps 1)(1 K ) 1 Ke
K e
s
1 Ke
s 1 Ke
K e
s
s
1 Ke
s 1 Ke
s
s
( ps 1)(1 K ) ( K e s ) K e s Q1 ( s ) s ( ps 1)(1 K )
s
2
ps 1
2
2
2
2
( ps 1)(1 K ) ( K e s ) K e s Q1 ( s )
2
s ( ps 1)(1 K )
ps 1
2
2
Simplifying the right-hand side of the above equation to generate Q1(s), thus the optimal
Jou
309
s
2
2
( ps 1)(1 K ) (1 Ke s ) K e s 1 Ke s Q1 ( s ) s
2 2
W (s)S (s)
305
308
1 sQ ( s ) 1 1 K
s
304
307
2
W (s) 1 P(s)
Basically, the solution (roots) of K e 0 gives poles which are the reflections of zk. so
303
306
2
Let 1 Ke 0 . Then RHP zeros zk and poles pk of P(s) are attained as:
301
302
proper and satisfy the asymptotic tracking.
1 1 Ke 1 sQ1 ( s ) 1 s ps 1 1 K
299
Qi ( s )
(20)
W ( s ) 1 i P( s )Qi ( s )
2
s
298
lP repro of
Qi ( s )
295
296
Q1 ( s ) of
controller Q1 opt ( s ) is:
( ps 1)(1 K ) ( K e s ) K e s Q1 ( s ) 0 s ( ps 1)(1 K )
Q1 opt ( s )
( ps 1)(1 K ) ( K e s ) s (1 K )
ps 1 1
K e
12
s
(22)
Journal Pre-proof
By using the equations (20) and (22) for the optimal controller Qopt ( s ) is Qopt ( s )
311
lP repro of
310
ps 1
(23)
s
i ( K e )
312
This is an H optimal controller for each SISO agent of MAS in which are the eigenvalues
313
of L ( I n L ) . Hence, Eq. (23) shows the importance of communication delay in controller
314
design. Since eigenvalues depend upon communication strength wij which is the weighting
315
function and appears as Laplacian element so we can conclude that input delay and
316
communication delay both influence the performance of controller in MASs. Hence, the
317
feasibility of can be verified by using Lemma 2.
318
From equation(23), it is clear that Qopt ( s ) is an improper function which is unattainable in its
319
physical properties. In order to make it proper , a filter J ( s) should be acquainted with
320
Qopt ( s ). Frankly speaking, J ( s ) should fulfil the following requirements:
2
i
m
i
i
321
1. Qi ( s ) Qopt ( s ) J ( s ) is proper.
322
2. The closed-loop system must follow IS (definition 1).
323
3. The asymptotic tracking should be accomplished.
Let us consider step signal as a system reference; then the filter may be in the form of:
rna
324
J (s)
325
1
(s 1)
ni
, i 1, 2,..., n,
326
Where are the performance degrees. Then, taking
327
J (s)
329 330
ni 1 the
filter in (24) could be: (25)
s 1
Thus the proper form of the optimal controller can be obtained by combining (23) and (25)
Jou
328
1
(24)
as:
Qi ( s )
ps 1 s
i (s 1)( K e )
13
Journal Pre-proof
1
331
If we replace
332
controller could be calculated as follows:
by
ak
, then the unity feedback controller which is the real application
C (s)
333
lP repro of
i
ak ( ps 1)
s s s s 1 K e 1 Ke e
(26)
1
334
Thus the relation " S ( s) LT ( s ) I " has obtained in the form of C(s) which is a physically
335
realizable controller. That is why, a balance between the robustness and nominal performance
336
has been achieved. Ω is the positive adjustable parameter for filter amplitude, and its range is
337
(0, ) .The overall performance of the subsystem or whole system can be achieved by
338
adjusting this weighting parameter Ω. Hence, the H
339
achieved is optimal if all eigenvalues are real and equal. The performance index of a
340
system is always impossible to achieve an optimal solution but can be improved
341
simultaneously. In order to achieve optimal performance, communication topology should be
342
designed for the same non-zero eigenvalues. As a comparison with another structured
343
controller, this analytical algebraic method computed by H index ensures the performance
344
improvement of the whole MAS system with communication delay and multiple input
345
arbitrary delays.
346
Since this is an ideal controller and is challenging to implement practically, so for
347
implementation, the reduction techniques have been used. By using McLaurin expansion
348
series, PID can be generated as:
performance index that has been
i
2
rna
349
2
1
TI s
C (s) Kc 1
TD s
(27)
Let A( s) sC ( s) be a function of an appropriate order which consists of three parameters; that
351
can be calculated as:
352
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K c A '(0),
TI
A '(0) A(0)
and
TD
A ''(0) 2 A '(0)
14
Journal Pre-proof
di out ( s )
di in ( s )
ei ( s )
Ci ( s )
Gi ( s )
Zi (s) 353 354
lP repro of
ri ( s )
xi ( s )
i
Fig 3. Closed loop control structure Remark 4.
356
The designed controller of Eq. (26) will work locally for each agent under which the
357
homogenous agents can achieve consensuses and communicate efficiently. The performance
358
and robustness of MAS under connected and undirected topology have been improved with
359
multiple time delays. For practical application, the PID consensus controller (27) can also be
360
obtained approximately. Now the problem of communication delay will be dealt with the
361
following theorem.
362
Theorem 2:
363
If the fixed directed graph or undirected graph in the MAS (11) has a spanning tree, then the
364
system model shown in figure 2 using a controller (18) guarantees that all agents are
365
uniformly ultimately bounded no matter how large the communication delay is, iff it satisfies
366
the following conditions:
K 1 , ( K 1) 2 21 and ak
( K 1)
(28)
K max
368
Proof: Let us consider that the lemma 4 is true, then the characteristics polynomial of each
369
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367
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355
370 371
372
subsystem in (13) should be Hurwitz stable for consensus, i.e., characteristics poles lie in the left-half of the -plane.
For this, let us consider, det[1 i C ( s) P( s)] 0 . Thus from (19) and (26), we can deduce
det 1
ps 1
i s 1 K e
s
1 Ke e s
15
1 s
1 Ke s 0 , for all i . ps 1
Journal Pre-proof
After rearranging the above equation, we could get K 1 and ak
374
K 2 2 21
( K 1)
lP repro of
373
K max
375
where,
376
Remark 5: Theorem 2 presents a condition for communication delay of multi-agent systems
377
described in figure 2. As is a function of input delays which is a part of system application
378
requirements. we note that ITD is the inherent delay of the system. It is an inner loop control
379
delay and compensates the performance of single agent when considering system as
380
standalone. Whereas, CTD is considered in the case of MAS. CTD is the secondary loop
381
control delay. When MAS is considered, then ITD & CTD both are taken into account. Both
382
of the delays ITD and CTD are treated as input time delays. In addition, states of MAS are
383
affected by both of the delays. ITD and CTD can be addressed with inner-loop and outer-loop
384
control but it will be more optimal if we consider both of the delays simultaneously to
385
stabilize the system and achieve its performance under the circumstances of sufficient
386
conditions given in theorem 2. In this conclusion, the value of depends on the
387
communication topology; and if the eigenvalues are considered to be constant at its
388
maximum level, then the dependency will be between and , . For example, if input
389
delays and K are all considered constant for a special case, then the communication delay can
390
be compensated by under the circumstances of given conditions (28).
391
The proposed analytical controller for the H2 performance of homogenous MAS with
392
multiple ITD and CTD can be obtained through the following steps:
393
STEP-1:
Analyze the network topology and calculate its corresponding eigenvalues.
394
STEP-2:
Choose the dynamics of the sub-system.
395
STEP-3:
Calculate the controller transfer function by using theorem 2.
397 398 399
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396
i
STEP-4:
Compute the Ω for the robustness requirement.
STEP-5:
Determine the value of by using controller parametrization concepts.
STEP-6:
For higher order controller, deduce PID controller by using (27).
5. Numerical Example
16
Journal Pre-proof
This section discusses simulation result by assuming an example of a homogenous network
401
which consists of a stable agent and the topology of MAS is borrowed from [28], as shown in
402
Fig.4.The dynamic of each agent is:
lP repro of
400
P( s)
403
404 405
e
1 s
m1 s 1
Ke
2 s
for K 1
m2 s 1
1
2
4
3
Fig.4: Undirected topology
m1 m2 1, 2 1 such
that
0.1, 2 0.5, c 2 and K
>1. The Laplacian matrix “L”
406
Let set
407
for the information shown in Fig.4 with communication strength
1
wi j 2 is
as follows:
4 2 2 0 2 6 2 2 L 2 2 6 2 0 2 2 4
408
409
Hence
410
time delays, it is aimed at designing an analytical controller which can achieve performance
411
tracking and stability as well as fulfils the consensuses condition of communication delay.
412
According to the proposed method
{0, 4, 8, 8}
Qi opt
413
415 416
ak ( s 1) ( K e
0.4 s
)
If we introduce a filter in series of Qi opt , then according to the theorem 1, the sub-optimal
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414
are the eigenvalues of L. For the given dynamic system with multiple
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i
controller is:
Qi ( s )
ak ( s 1) ( K e
17
0.4 s
)(s 1)
Journal Pre-proof
0.1 s
1 0.1s
417
For the simplification of analysis, let us put e
418
approximation of the irrational function by using Taylor series expansion), then the feedback
419
controller of the proposed system can be achieved as follows:
Qi ( s )
(to make an
lP repro of
into
C (s)
420
ak ( s 1)
( K e
0.1 s
)(s 1) (1 Ke
0.4 s
)e
0.1 s
421
Thus it is a typical PI controller and shows its characteristics polynomial. For the verification
422
of communication delay and consensus condition; according to theorem 2 for above
423
considered values, the overall MAS network must be in consensus if K 1 and ak
424
By using proposed analytical controllers for each agent, the output responses are shown in
425
Fig 5(a) and (b) at 10 , when a
426
H2
427
though along with communication delay; if
428
theorem 2.
k
1
0.6
0.8 max
.
the output response is stable and is in consensus i.e.,
performance index of whole system has improved as compared with [28,29,35] even
429
the response will diverge which satisfy the
Effects of K:
432
Now if we put K 1 in our proposed system for all above considered values, the output
433
response of all agents diverge as shown in Fig 6(a) and 6(b).
434
Impact of delay on Stability
435
If the ITD increases, the same algorithm can be used to control the effects of different time
436
delay on performance of MAS. So the output results of MAS with
437
are shown in Fig7(a) and 7(b). So the tuning of performance index can be made in order to
438 439 440 441
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431
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430
ak 2
1
1, 2 5, 10, ak 0.1
fulfil the demand of MASs. As CTD increases or decreases, it affects the performance and stability of MAS. According to the demand for eigenvalues, the controller shown in (26) adjusts the values of ak and Ω according to the condition (28) . For varying communication delay, let
c
4 , wi j 4
then the eigenvalues are
i
18
{0, 8,16,16}
, and Laplacian will be:
Journal Pre-proof
8 4 L 4 0
4
12
4
4
4
12
4
4
4
8
0
lP repro of
442
4
443
For this MAS, the same controller achieves consensus with H performance index if
444
for small ITD case and
445
Fig.8(a) and 8(b).
2
ak 0.12
446
Fig.5(a) Converges at
c
for large delay case. The simulation results are shown in
2, 10, ak 1
(b) Divergence at
c
2, 10, ak 2
448 449
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447
ak 0.82
Fig.6 Divergence (a) when =2, 10, a c
k
2, K 1
19
(b)
when = 2 10, a c
k
2, K 0.5
450 451 452
lP repro of
Journal Pre-proof
Fig.7 (a) Consensus Performance when = 2, 10, a c
6. Conclusions
k
0.2
(b) when = 2, 10, a c
k
1
453
An analytical controller has been designed for a homogenous MAS with multiple input
454
delays and communication delay in the frequency domain. The necessary condition
455
guarantees consensus on the overall network with delay. A simple controller parametrization
456
scheme has been used to design an analytical controller for IS and improve H performance
457
index for multiple input delays and communication delay at the same time. The necessary
458
condition has been used to trade off the nominal performance of MAS with communication
459
delay. The simulation results have verified the calculation. Since this method is based on
460
simple algebraic calculations that is why it is much easier to use for the real application.
461
Further studies will focus on designing of consensus controllers for a multi-agent system
462
under attack.
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463
2
20
Journal Pre-proof
c
4, 10, ak 0.12 (b) when c 4, 10, ak 0.5
Fig.8 (a) Consensus Performance when
465
Acknowledgement
466
This paper is partly supported by the National Science Foundation of China
467
(61473183，U1509211，61627810),
468
(SQ2017YFGH001005).
469
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470
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*Conflict of Interest
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No conflict of interest exists in the submission of this manuscript, and the manuscript is
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approved by all authors for publication.