In this paper, we develop a hybrid control framework for addressing multiagent formation control protocols for general nonlinear dynamical systems using hybrid stabilization of sets. The proposed framework develops a novel class of fixed-order, energy-based hybrid controllers as a means for achieving cooperative control formations, which can include flocking, cyclic pursuit, rendezvous, and consensus control of multiagent systems. These dynamic controllers combine a logical switching architecture with the continuous system dynamics to guarantee that a system generalized energy function whose zero level set characterizes a specified system formation is strictly decreasing across switchings. The proposed approach addresses general nonlinear dynamical systems and is not limited to systems involving single and double integrator dynamics for consensus and formation control or unicycle models for cyclic pursuit. Finally, several numerical examples involving flocking, rendezvous, consensus, and circular formation protocols for standard system formation models are provided to demonstrate the efficacy of the proposed approach.

References

References
1.
Haddad
,
W. M.
,
Chellaboina
,
V.
,
Hui
,
Q.
, and
Nersesov
,
S. G.
,
2007
, “
Energy- and Entropy-Based Stabilization for Lossless Dynamical Systems Via Hybrid Controllers
,”
IEEE Trans. Autom. Control
,
52
(
9
), pp.
1604
1614
.10.1109/TAC.2007.904452
2.
Haddad
,
W. M.
,
Chellaboina
,
V.
, and
Nersesov
,
S. G.
,
2006
,
Impulsive and Hybrid Dynamical Systems. Stability, Dissipativity, and Control
,
Princeton University
Press,
Princeton, NJ
.
3.
Mesbahi
,
M.
, and
Egerstedt
,
M.
,
2010
,
Graph Theoretic Methods in Multiagent Networks
,
Princeton University
Press,
Princeton, NJ
.
4.
Hui
,
Q.
, and
Haddad
,
W. M.
,
2008
, “
Distributed Nonlinear Control Algorithms for Network Consensus
,”
Automatica
,
44
(9), pp.
2375
2381
.10.1016/j.automatica.2008.01.011
5.
Chellaboina
,
V.
,
Haddad
,
W. M.
,
Hui
,
Q.
, and
Ramakrishnan
,
J.
,
2008
, “
On System State Equipartitioning and Semistability in Network Dynamical Systems With Arbitrary Time-Delays
,”
Syst. Control Lett.
,
57
(8), pp.
670
679
.10.1016/j.sysconle.2008.01.008
6.
Hui
,
Q.
,
Haddad
,
W. M.
, and
Bhat
,
S. P.
,
2008
, “
Finite-Time Semistability and Consensus for Nonlinear Dynamical Networks
,”
IEEE Trans. Autom. Control
,
53
(8), pp.
1887
1900
.10.1109/TAC.2008.929392
7.
Bhat
,
S. P.
, and
Bernstein
,
D. S.
,
2003
, “
Nontangency-Based Lyapunov Tests for Convergence and Stability in Systems Having a Continuum of Equilibria
,”
SIAM J. Control Optim.
,
42
(5), pp.
1745
1775
.10.1137/S0363012902407119
8.
Bhat
,
S. P.
, and
Bernstein
,
D. S.
,
2010
, “
Arc-Length-Based Lyapunov Tests for Convergence and Stability With Applications to Systems Having a Continuum of Equilibria
,”
Math. Control Signals Syst.
,
22
(2), pp.
155
184
.10.1007/s00498-010-0054-3
9.
Haddad
,
W. M.
, and
Chellaboina
,
V.
,
2008
,
Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach
,
Princeton University Press,
Princeton, NJ
.
10.
Haddad
,
W. M.
,
Chellaboina
,
V.
, and
Nersesov
,
S. G.
,
2005
,
Thermodynamics: A Dynamical Systems Approach
,
Princeton University
Press,
Princeton, NJ
.
11.
Strogatz
,
S. H.
,
2000
, “
From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators
,”
Physica D
,
143
(1–4), pp.
1
20
.10.1016/S0167-2789(00)00094-4
12.
Brown
,
E.
,
Moehlis
,
J.
, and
Holmes
,
P.
,
2004
, “
On the Phase Reduction and Response Dynamics of Neural Oscillator Populations
,”
Neural Comput.
,
16
(4), pp.
673
715
.10.1162/089976604322860668
13.
Hui
,
Q.
,
Haddad
,
W. M.
, and
Bailey
,
J. M.
,
2011
, “
Multistability, Bifurcations, and Biological Neural Networks: A Synaptic Drive Firing Model for Cerebral Cortex Transition in the Induction of General Anesthesia
,”
Nonlinear Anal. Hybrid Syst.
,
5
(3), pp.
554
573
.10.1016/j.nahs.2010.12.002
14.
Tanner
,
H. G.
,
Jadbabaie
,
A.
, and
Pappas
,
G. J.
,
2003
, “
Stable Flocking of Mobile Agents, Part I: Fixed Topology
,”
42nd IEEE Conference on Decision and Control
, Maui, HI, December 9–12, pp.
2010
2015
.10.1109/CDC.2003.1272910
15.
Olfati-Saber
,
R.
,
2006
, “
Flocking for Multi-Agent Dynamic Systems: Algorithms and Theory
,”
IEEE Trans. Autom. Control
,
51
(3), pp.
401
420
.10.1109/TAC.2005.864190
16.
Marshall
,
J. A.
,
Broucke
,
M. E.
, and
Francis
,
B. A.
,
2004
, “
Formations of Vehicles in Cyclic Pursuit
,”
IEEE Trans. Autom. Control
,
49
(11), pp.
1963
1974
.10.1109/TAC.2004.837589
17.
Lakshmikantham
,
V.
,
Bainov
,
D. D.
, and
Simeonov
,
P. S.
,
1989
,
Theory of Impulsive Differential Equations
,
World Scientific
,
Singapore
.
18.
Bainov
,
D. D.
, and
Simeonov
,
P. S.
,
1989
,
Systems With Impulse Effect: Stability, Theory and Applications
,
Ellis Horwood Limited
,
UK
.
19.
Bainov
,
D. D.
, and
Simeonov
,
P. S.
,
1995
,
Impulsive Differential Equations: Asymptotic Properties of the Solutions
,
World Scientific
,
Singapore
.
20.
Samoilenko
,
A. M.
, and
Perestyuk
,
N. A.
,
1995
,
Impulsive Differential Equations
,
World Scientific
,
Singapore
.
21.
Chellaboina
,
V.
,
Bhat
,
S. P.
, and
Haddad
,
W. M.
,
2003
, “
An Invariance Principle for Nonlinear Hybrid and Impulsive Dynamical Systems
,”
Nonlinear Anal. Theory Methods Appl.
,
53
(3–4), pp.
527
550
.10.1016/S0362-546X(02)00316-4
22.
Michel
,
A. N.
,
Wang
,
K.
, and
Hu
,
B.
,
2001
,
Qualitative Theory of Dynamical Systems: The Role of Stability Preserving Mappings
,
Marcel Dekker, Inc.
,
New York
.
23.
Grizzle
,
J. W.
,
Abba
,
G.
, and
Plestan
,
F.
,
2001
, “
Asymptotically Stable Walking for Biped Robots: Analysis Via Systems With Impulse Effects
,”
IEEE Trans. Autom. Control
,
46
(1), pp.
51
64
.10.1109/9.898695
24.
Guillemin
,
V.
, and
Pollack
,
A.
,
1974
,
Differential Topology
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
25.
Dubrovin
,
B. A.
,
Fomenko
,
A. T.
, and
Novikov
,
S. P.
,
1985
,
Modern Geometry—Methods and Applications: Part II: The Geometry and Topology of Manifolds
,
Springer
,
New York
.
26.
Goebel
,
R.
, and
Teel
,
A. R.
,
2006
, “
Solutions to Hybrid Inclusions Via Set and Graphical Convergence With Stability Theory Applications
,”
Automatica
,
42
(4), pp.
573
587
.10.1016/j.automatica.2005.12.019
27.
Bernstein
,
D. S.
,
2005
,
Matrix Mathematics
,
Princeton University
Press,
Princeton, NJ
.
28.
Justh
,
E. W.
, and
Krishnaprasad
,
P. S.
,
2004
, “
Equilibria and Steering Laws for Planar Formations
,”
Syst. Control Lett.
,
52
(1), pp.
25
38
.10.1016/j.sysconle.2003.10.004
29.
Hui
,
Q.
, and
Haddad
,
W. M.
,
2007
, “
Continuous and Hybrid Distributed Control for Multiagent Coordination: Consensus, Flocking, and Cyclic Pursuit
,”
American Control Conference
(
ACC '07
), New York, July 9–13, pp.
2576
2581
.10.1109/ACC.2007.4282465
30.
El-Hawwary
,
M. I.
, and
Maggiore
,
M.
,
2013
, “
Distributed Circular Formation Stabilization for Dynamic Unicycles
,”
IEEE Trans. Autom. Control
,
58
(1), pp.
149
162
.10.1109/TAC.2012.2206720
31.
Kasdin
,
N. J.
, and
Paley
,
D. A.
,
2011
,
Engineering Dynamics
,
Princeton University
Press,
Princeton, NJ
.
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