Abstract

In this paper, the non-zero finite positive state consensus problem of leader-following multi-agent systems (MASs) connected over a directed network is investigated. The multi-agent network consists of homogeneous linear time-invariant (LTI) positive agents whose minimal positive state-space realization is assumed to be known. In this paper, a state-feedback hierarchical control protocol is proposed where the local controller synthesizes a singular, Lyapunov stable, and Metzler system matrix with a simple dominant eigenvalue at origin. The obtained consensus vector is the positive eigenvector associated with the zero eigenvalue of the synthesized system matrix. The controller gain matrices are obtained by solving a set of necessary and sufficient conditions derived in linear programing framework. A numerical example is given to elucidate the usefulness of the proposed results.

References

1.
Jadbabaie
,
A.
,
Lin
,
J.
, and
Morse
,
A. S.
,
2003
, “
Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules
,”
IEEE Trans. Autom. Control
,
48
(
6
), pp.
988
1001
.10.1109/TAC.2003.812781
2.
Olfati-Saber
,
R.
, and
Murray
,
R. M.
,
2004
, “
Consensus Problems in Networks of Agents With Switching Topology and Time-Delays
,”
IEEE Trans. Autom. Control
,
49
(
9
), pp.
1520
1533
.10.1109/TAC.2004.834113
3.
Moreau
,
L.
,
2005
, “
Stability of Multiagent Systems With Time-Dependent Communication Links
,”
IEEE Trans. Autom. Control
,
50
(
2
), pp.
169
182
.10.1109/TAC.2004.841888
4.
Ren
,
W.
,
Beard
,
R. W.
, and
Atkins
,
E. M.
,
2007
, “
Information Consensus in Multivehicle Cooperative Control
,”
IEEE Control Syst. Mag.
,
27
(
2
), pp.
71
82
.10.1109/MCS.2007.338264
5.
Wieland
,
P.
,
Kim
,
J.-S.
, and
Allgöwer
,
F.
,
2011
, “
On Topology and Dynamics of Consensus Among Linear High-Order Agents
,”
Int. J. Syst. Sci.
,
42
(
10
), pp.
1831
1842
.10.1080/00207721003658202
6.
Lin
,
Z.
,
Francis
,
B.
, and
Maggiore
,
M.
,
2007
, “
State Agreement for Continuous-Time Coupled Nonlinear Systems
,”
SIAM J. Control Optim.
,
46
(
1
), pp.
288
307
.10.1137/050626405
7.
Chen
,
Y.
,
Lu
,
J.
,
Yu
,
X.
, and
Hill
,
D. J.
,
2013
, “
Multi-Agent Systems With Dynamical Topologies: Consensus and Applications
,”
IEEE Circuits Syst. Mag.
,
13
(
3
), pp.
21
34
.10.1109/MCAS.2013.2271443
8.
Xu
,
Y.
, and
Tian
,
Y.-P.
,
2013
, “
Design of a Class of Nonlinear Consensus Protocols for Multi-Agent Systems
,”
Int. J. Robust Nonlinear Control
,
23
(
13
), pp.
1524
1536
.10.1002/rnc.2838
9.
Xing
,
W.
,
Shi
,
P.
,
Agarwal
,
R. K.
, and
Zhao
,
Y.
,
2019
, “
A Survey on Global Pinning Synchronization of Complex Networks
,”
J. Franklin Inst.
,
356
(
6
), pp.
3590
3611
.10.1016/j.jfranklin.2019.02.021
10.
He
,
C.
, and
Huang
,
J.
,
2021
, “
Leader-Following Consensus Over Acyclic Switching Digraphs
,”
ASME J. Dyn. Syst., Meas., Control
,
143
(
8
), p.
081008
.10.1115/1.4050507
11.
Sakaguchi
,
A.
, and
Ushio
,
T.
,
2017
, “
Dynamic Pinning Consensus Control of Multi-Agent Systems
,”
IEEE Control Syst. Lett.
,
1
(
2
), pp.
340
345
.10.1109/LCSYS.2017.2717854
12.
Farina
,
L.
, and
Rinaldi
,
S.
,
2000
,
Positive Linear Systems: Theory and Applications
,
Wiley
,
New York
.
13.
De Leenheer
,
P.
, and
Aeyels
,
D.
,
2001
, “
Stability Properties of Equilibria of Classes of Cooperative Systems
,”
IEEE Trans. Autom. Control
,
46
(
12
), pp.
1996
2001
.10.1109/9.975508
14.
Briat
,
C.
,
2018
, “
Stability and Performance Analysis of Linear Positive Systems With Delays Using Input–Output Methods
,”
Int. J. Control
,
91
(
7
), pp.
1669
1692
.10.1080/00207179.2017.1326628
15.
Bhattacharyya
,
S.
, and
Patra
,
S.
,
2018
, “
Static Output-Feedback Stabilization for MIMO LTI Positive Systems Using LMI-Based Iterative Algorithms
,”
IEEE Control Syst. Lett.
,
2
(
2
), pp.
242
247
.10.1109/LCSYS.2018.2816969
16.
Haddad
,
W. M.
,
Chellaboina
,
V. S.
, and
August
,
E.
,
2003
, “
Stability and Dissipativity Theory for Discrete-Time Non-Negative and Compartmental Dynamical Systems
,”
Int. J. Control
,
76
(
18
), pp.
1845
1861
.10.1080/00207170310001635400
17.
Knorn
,
F.
,
Corless
,
M. J.
, and
Shorten
,
R. N.
,
2011
, “
A Result on Implicit Consensus With Application to Emissions Control
,”
50th IEEE Conference on Decision and Control and European Control Conference
, Orlando, FL, Dec. 12–15, pp.
1299
1304
.10.1109/CDC.2011.6160599
18.
Pahuja
,
R.
,
Verma
,
H.
, and
Uddin
,
M.
,
2013
, “
A Wireless Sensor Network for Greenhouse Climate Control
,”
IEEE Pervasive Comput.
,
12
(
2
), pp.
49
58
.10.1109/MPRV.2013.26
19.
Valcher
,
M. E.
, and
Misra
,
P.
,
2014
, “
On the Stabilizability and Consensus of Positive Homogeneous Multi-Agent Dynamical Systems
,”
IEEE Trans. Autom. Control
,
59
(
7
), pp.
1936
1941
.10.1109/TAC.2013.2294621
20.
Valcher
,
M. E.
, and
Zorzan
,
I.
,
2016
, “
On the Consensus Problem With Positivity Constraints
,”
American Control Conference (ACC)
, Boston, MA, July 6–8, pp.
2846
2851
.10.1109/ACC.2016.7525350
21.
Valcher
,
M. E.
, and
Zorzan
,
I.
,
2016
, “
New Results on the Solution of the Positive Consensus Problem
,”
IEEE 55th Conference on Decision and Control (CDC)
, Las Vegas, NV, Dec. 12–14, pp.
5251
5256
.10.1109/CDC.2016.7799073
22.
Valcher
,
M. E.
, and
Zorzan
,
I.
,
2017
, “
On the Consensus of Homogeneous Multiagent Systems With Positivity Constraints
,”
IEEE Trans. Autom. Control
,
62
(
10
), pp.
5096
5110
.10.1109/TAC.2017.2691305
23.
Liu
,
J. J.
,
Lam
,
J.
, and
Shu
,
Z.
,
2020
, “
Positivity-Preserving Consensus of Homogeneous Multiagent Systems
,”
IEEE Trans. Autom. Control
,
65
(
6
), pp.
2724
2729
.10.1109/TAC.2019.2946205
24.
Wu
,
H.
, and
Su
,
H.
,
2019
, “
Observer-Based Consensus for Positive Multiagent Systems With Directed Topology and Nonlinear Control Input
,”
IEEE Trans. Syst., Man, Cybern.
,
49
(
7
), pp.
1459
1469
.10.1109/TSMC.2018.2852704
25.
Yang
,
N.
,
Yin
,
Y.
, and
Liu
,
J.
,
2019
, “
Positive Consensus of Directed Multi-Agent Systems Using Dynamic Output-Feedback Control
,”
IEEE 58th Conference on Decision and Control (CDC)
, Nice, France, Dec. 11–13, pp.
897
902
.10.1109/CDC40024.2019.9029331
26.
Cao
,
X.
, and
Li
,
Y.
,
2021
, “
Positive Consensus for Multi-Agent Systems With Average Dwell Time Switching
,”
J. Franklin Inst.
,
358
(
16
), pp.
8308
8329
.10.1016/j.jfranklin.2021.08.024
27.
Deplano
,
D.
,
Franceschelli
,
M.
, and
Giua
,
A.
,
2020
, “
A Nonlinear Perron–Frobenius Approach for Stability and Consensus of Discrete-Time Multi-Agent Systems
,”
Automatica
,
118
, p.
109025
.10.1016/j.automatica.2020.109025
28.
Gong
,
X.
,
Liu
,
J.
,
Wang
,
Y.
, and
Lam
,
J.
,
2019
, “
Consensus of Discrete-Time Positive Multi-Agent Systems With Observer-Type Protocols
,”
IEEE 15th International Conference on Control and Automation (ICCA)
, Edinburgh, Scotland, July 16–19, pp.
846
850
.10.1109/ICCA.2019.8899988
29.
Horn
,
R. A.
, and
Johnson
,
C. R.
,
1990
,
Matrix Analysis
,
Cambridge University Press
,
Cambridge, UK
.
30.
Wu
,
C. W.
,
2007
,
Synchronization in Complex Networks of Nonlinear Dynamical Systems
,
World Scientific
,
Singapore
.
31.
Zhou
,
K.
,
Doyle
,
J. C.
, and
Glover
,
K.
,
1996
,
Robust and Optimal Control
,
Prentice Hall
,
Upper Saddle River, NJ
.
32.
Beauwens
,
R.
,
1976
, “
Semistrict Diagonal Dominance
,”
SIAM J. Numer. Anal.
,
13
(
1
), pp.
109
112
.10.1137/0713013
33.
Ebihara
,
Y.
,
Peaucelle
,
D.
, and
Arzelier
,
D.
,
2014
, “
LMI Approach to Linear Positive System Analysis and Synthesis
,”
Syst. Control Lett.
,
63
, pp.
50
56
.10.1016/j.sysconle.2013.11.001
34.
Ait Rami
,
M.
,
2011
, “
Solvability of Static Output-Feedback Stabilization for LTI Positive Systems
,”
Syst. Control Lett.
,
60
(
9
), pp.
704
708
.10.1016/j.sysconle.2011.05.007
35.
Bhattacharyya
,
S.
, and
Patra
,
S.
,
2022
, “
Positive Consensus of Multi-Agent Systems With Hierarchical Control Protocol
,”
Automatica
,
139
, p.
110191
.10.1016/j.automatica.2022.110191
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