In this paper, linear quadratic regulator (LQR) theory is applied to solve the inverse optimal consensus problem for a second-order linear multi-agent systems (MAS) under independent position and velocity topology. The optimal Laplacian matrices related to the topologies of position and velocity are derived by solving the algebraic Riccati equation (ARE). Theoretically, we obtain the optimal Laplacian matrices, which correspond to the directed strongly connected graphs, for the second-order multi-agent systems. Finally, two simulation examples are provided to verify the theoretical analysis of this paper.

References

References
1.
Olfatisaber
,
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
.
2.
Ren
,
W.
,
Beard
,
R. W.
, and
Atkins
,
E. M.
,
2005
, “
A Survey of Consensus Problems in Multi-Agent Coordination
,”
American Control Conference
(
ACC
), Portland, OR, June 8–10, Vol.
3
, pp.
1859
1864
.
3.
Cao
,
Y.
, and
Ren
,
W.
,
2010
, “
Optimal Linear-Consensus Algorithms: An LQR Perspective
,”
IEEE Trans. Syst. Man Cybern., Part B
,
40
(
3
), pp.
819
830
.
4.
Semsar-Kazerooni
,
E.
, and
Khorasani
,
K.
,
2008
, “
Optimal Consensus Algorithms for Cooperative Team of Agents Subject to Partial Information
,”
Automatica
,
44
(
11
), pp.
2766
2777
.
5.
Shi
,
G.
,
Johansson
,
K. H.
, and
Hong
,
Y.
,
2013
, “
Reaching an Optimal Consensus Dynamical Systems That Compute Intersections of Convex Sets
,”
IEEE Trans. Autom. Control
,
58
(
3
), pp.
610
622
.
6.
Bauso
,
D.
,
Giarre
,
L.
, and
Pesenti
,
R.
,
2006
, “
Non-Linear Protocols for Optimal Distributed Consensus in Networks of Dynamic Agents
,”
Syst. Control Lett.
,
55
(
11
), pp.
918
928
.
7.
Zhang
,
H.
,
Zhang
,
J.
,
Yang
,
G. H.
, and
Luo
,
Y.
,
2015
, “
Leader-Based Optimal Coordination Control for the Consensus Problem of Multiagent Differential Games Via Fuzzy Adaptive Dynamic Programming
,”
IEEE Trans. Fuzzy Syst.
,
23
(
1
), pp.
152
163
.
8.
Vamvoudakis
,
K. G.
,
Lewis
,
F. L.
, and
Hudas
,
G. R.
,
2012
, “
Multi-Agent Differential Graphical Games: Online Adaptive Learning Solution for Synchronization With Optimality
,”
Automatica
,
48
(
8
), pp.
1598
1611
.
9.
Semsar-Kazerooni
,
E.
, and
Khorasani
,
K.
,
2009
, “
Multi-Agent Team Cooperation: A Game Theory Approach
,”
Automatica
,
45
(
10
), pp.
2205
2213
.
10.
Wei
,
Q.
,
Liu
,
D.
, and
Lewis
,
F. L.
,
2015
, “
Optimal Distributed Synchronization Control for Continuous-Time Heterogeneous Multi-Agent Differential Graphical Games
,”
Inf. Sci.
,
317
, pp.
96
113
.
11.
Ma
,
J.
,
Zheng
,
Y.
, and
Wang
,
L.
,
2015
, “
LQR-Based Optimal Topology of Leader-Following Consensus
,”
Int. J. Robust Nonlinear Control
,
25
(
17
), pp.
3404
3421
.
12.
Dong
,
W.
,
2010
, “
Distributed Optimal Control of Multiple Systems
,”
Int. J. Control
,
83
(
10
), pp.
2067
2079
.
13.
Dunbar
,
W. B.
, and
Murray
,
R. M.
,
2006
, “
Distributed Receding Horizon Control for Multi-Vehicle Formation Stabilization
,”
Automatica
,
42
(
4
), pp.
549
558
.
14.
Zhang
,
H.
,
Lewis
,
F. L.
, and
Das
,
A.
,
2011
, “
Optimal Design for Synchronization of Cooperative Systems: State Feedback, Observer and Output Feedback
,”
IEEE Trans. Autom. Control
,
56
(
8
), pp.
1948
1952
.
15.
Movric
,
K. H.
, and
Lewis
,
F. L.
,
2014
, “
Cooperative Optimal Control for Multi-Agent Systems on Directed Graph Topologies
,”
IEEE Trans. Autom. Control
,
59
(
3
), pp.
769
774
.
16.
Zhang
,
Y.
, and
Hong
,
Y.
,
2015
, “
Distributed Optimization Design for High-Order Multi-Agent Systems
,”
34th Chinese Control Conference
(
CCC
), Hangzhou, China, July 28–30, pp.
7251
7256
.
17.
Qin
,
J.
, and
Yu
,
C.
,
2013
, “
Coordination of MultiAgents Interacting Under Independent Position and Velocity Topologies
,”
IEEE Trans. Neural Networks Learn. Syst.
,
24
(
10
), pp.
1588
1597
.
18.
Alefeld
,
G.
, and
Schneider
,
N.
,
1982
, “
On Square Roots of M-Matrices
,”
Linear Algebra Appl.
,
42
, pp.
119
132
.
19.
Chung
,
F. R. K.
,
1992
,
Spectral Graph Theory
,
American Mathematical Society
,
Providence, RI
, Chap. 1.
20.
Ren
,
W.
, and
Atkins
,
E.
,
2005
, “
Second-Order Consensus Protocols in Multiple Vehicle Systems With Local Interactions
,”
AIAA
Paper No. 2005-6238.
21.
Ren
,
W.
,
2007
, “
Second-Order Consensus Algorithm With Extensions to Switching Topologies and Reference Models
,”
American Control Conference
(
ACC
), New York, July 9–13, pp.
1431
1436
.
22.
Ren
,
W.
,
2008
, “
On Consensus Algorithms for Double-Integrator Dynamics
,”
IEEE Trans. Autom. Control
,
53
(
6
), pp.
1503
1509
.
23.
Cheng
,
F.
,
Yu
,
W.
,
Wang
,
H.
, and
Li
,
Y.
,
2013
, “
Second-Order Consensus Protocol Design in Multi-Agent Systems: A General Framework
,”
32nd Chinese Control Conference
(
CCC
), Xi’an, China, July 26–28, pp.
7246
7251
.http://ieeexplore.ieee.org/document/6640712/
24.
Xie
,
G.
, and
Wang
,
L.
,
2007
, “
Consensus Control for a Class of Networks of Dynamic Agents
,”
Int. J. Robust Nonlinear Control
,
17
(
10–11
), pp.
941
959
.
25.
Ren
,
W.
, and
Beard
,
R. W.
,
2005
, “
Consensus Seeking in Multiagent Systems Under Dynamically Changing Interaction Topologies
,”
IEEE Trans. Autom. Control
,
50
(
5
), pp.
655
661
.
26.
Liberzon
,
D.
,
2012
,
Calculus of Variations and Optimal Control Theory: A Concise Introduction
,
Princeton University Press
,
Princeton, NJ
, Chap. 6.
27.
Lewis
,
F. L.
,
Vrabie
,
D. L.
, and
Syrmos
,
V. L.
,
2012
,
Optimal Control
,
3rd ed.
,
Wiley
,
NJ
, Chaps. 3 and 4.
You do not currently have access to this content.