Abstract

This paper generalizes our previous results on semistability and stochastic semistability for switched nonlinear systems published in the Proceedings of 2021 Modeling, Estimation and Control Conference. The paper also provides the results on semistability in mean square for switched nonlinear discrete-time systems. The theoretical result involves generalized sufficient conditions for (stochastic) semistability and semistability in mean square of discrete-time nonlinear dynamical systems under time-varying or random (arbitrary) switching by means of Fixed Point Theory. An advantage of these results is to overcome fundamental challenges arising from using existing methods such as Lyapunov and LaSalle methods. As an application of the theoretical results presented, a constrained distributed consensus problem over random multi-agent networks is considered for which a generalized asynchronous and totally asynchronous iterative algorithm is derived. The algorithm is able to converge even if the weighted matrix of the graph is periodic and irreducible under synchronous protocol. Finally, a numerical example is given in which there is a distribution dependency among communication graphs to demonstrate the results.

References

1.
Liberzon
,
D.
,
2003
,
Switching in Systems and Control
,
Birkhäuser
,
Cambridge
.
2.
Decarlo
,
R. A.
,
Branicky
,
M. S.
,
Pettersson
,
S.
,
Lennartson
,
B.
, and
Antsaklis
,
P. J.
,
2000
, “
Perspectives and Results on the Stability and Stabilizability of Hybrid Systems
,”
Proc. IEEE: Special Issue Hybrid Syst.
,
88
, pp.
1069
1082
.
3.
Liberzon
,
D.
,
Hespanha
,
J.
, and
Morse
,
A. S.
,
1999
, “
Stability of Switched Linear Systems: A Lie-Algebraic Condition
,”
Syst. Cont. Lett.
,
37
, pp.
117
122
.
4.
Arnold
,
L.
,
1974
,
Stochastic Differential Equations: Theory and Applications
,
John Wiley & Sons Inc
.,
New York
.
5.
Has’minskiĭ
,
R. Z.
,
1980
,
Stochastic Stability of Differential Equations
,
Sijthoff & Noordhoff
,
Germantown, MD
.
6.
Liu
,
K.
,
2006
,
Stability of Infinite Dimensional Stochastic Differential Equations With Applications
,
Monographs and Surveys in pure and Applied Mathematics
, Vol. 134,
Chapman & Hall/CRC
,
New York
.
7.
Alaviani
,
S. S.
,
2019
, “
Applications of Fixed Point Theory to Distributed Optimization, Robust Convex Optimization, and Stability of Stochastic Systems
,” Ph.D. thesis, Departemnt of Electrical and Computer Enginering,
Iowa State University
,
Ames, IA
.
8.
Alaviani
,
S. S.
, and
Kelkar
,
A. G.
,
2021
, “
Semistability and Stochastic Semistability for Switched Nonlinear Systems by Means of Fixed Point Theory: Application to Constrained Distributed Consensus Over Random Networks
,”
Proceedings of Modeling, Estimation and Control Conference, Oct. 24–27
,
Austin, TX
, pp.
560
565
(Best Control Paper Award).
9.
Hui
,
Q.
,
2011
, “
Semistability and Robustness Analysis for Switched Systems
,”
Eur. J. Control
,
1
, pp.
73
88
.
10.
Hui
,
Q.
,
2013
, “
Convergence and Stability Analysis for Iterative Dynamics With Application to Compartmental Networks: A Trajectory Distance Based Lyapunov Approach
,”
J. Franklin Inst.
,
350
, pp.
679
697
.
11.
Zhou
,
J.
, and
Wang
,
Q.
,
2010
, “
Stochastic Semistability With Application to Agreement Problems Over Random Networks
,”
Proceedings of the 2010 American Control Conference
,
Baltimore, MD
,
June 30–July 2
,
IEEE
, pp.
568
573
.
12.
Rajpurohit
,
T.
, and
Haddad
,
W. M.
,
2017
, “
Stochastic Thermodynamics: A Dynamical Systems Approach
,”
Entropy
,
19
, pp.
1
48
.
13.
Haddad
,
W. M.
,
Rajpurohit
,
T.
, and
Jin
,
X.
,
2020
, “
Stochastic Semistability for Nonlinear Dynamical Systems With Application to Consensus on Networks With Communication Uncertainty
,”
IEEE Trans. Autom. Control
,
65
, pp.
2826
2841
.
14.
Haddad
,
W. M.
, and
Lee
,
J.
,
2022
, “
Lyapunov Theorems for Stability and Semistability of Discrete-Time Stochastic Systems
,”
Automatica
,
142
, p.
110393
.
15.
Shen
,
J.
,
Hu
,
J.
, and
Hui
,
Q.
,
2011
, “
Semistability of Switched Linear Systems With Applications to Distributed Sensor Networks: A Generating Function Approach
,”
50th IEEE Conference on Decision and Control and European Control Conference
,
Orlando, FL
,
Dec. 12–1
5,
IEEE
, pp.
8044
8049
.
16.
Shen
,
J.
,
Hu
,
J.
, and
Hui
,
Q.
,
2014
, “
Semistability of Switched Linear Systems with Application to PageRank Algorithms
,”
Eur. J. Control
,
20
, pp.
132
140
.
17.
Zhang
,
W.
,
Lin
,
X.
, and
Chen
,
B.-S.
,
2017
, “
LaSalle-Type Theorem and Its Applications to Infinite Horizon Optimal Control of Discrete-Time Nonlinear Stochastic Systems
,”
IEEE Trans. Autom. Control
,
62
, pp.
250
261
.
18.
Nedić
,
A.
,
2014
, “Distributed Optimization,”
Encyclopedia of Systems and Control
, pp.
1
12
.
19.
Jakovetić
,
D.
,
Bajović
,
D.
,
Xavier
,
J.
, and
Noura
,
J. M. F.
,
2020
, “
Primal-Dual Methods for Large-Scale and Distributed Convex Optimization and Data Analytics
,”
Proc. IEEE
,
108
, pp.
1923
1938
.
20.
Yang
,
T.
,
Yi
,
X.
,
Wu
,
J.
,
Yuan
,
Y.
,
Wu
,
D.
,
Meng
,
Z.
,
Hong
,
Y.
,
Wang
,
H.
,
Lin
,
Z.
, and
Johansson
,
K. H.
,
2019
, “
A Survey of Distributed Optimization
,”
Ann. Rev. Control
,
47
, pp.
278
305
.
21.
Ishikawa
,
S.
,
1974
, “
Fixed Points by a New Iteration Method
,”
Proc. Am. Math. Soc.
,
44
, pp.
147
150
.
22.
Alaviani
,
S. S.
, and
Elia
,
N.
,
2020
, “
Distributed Multi-Agent Convex Optimization Over Random Digraphs
,”
IEEE Trans. Autom. Control
,
65
, pp.
986
998
.
23.
Matsushita
,
S.-Y.
,
2019
, “
On the Convergence Rate of the Krasnoselskii–Mann Iteration
,”
Bull. Aust. Math. Soc.
,
96
, pp.
162
170
.
24.
Alaviani
,
S. S.
, and
Elia
,
N.
,
2019
, “
Distributed Average Consensus Over Random Networks
,”
Proceedings of American Control Conference (ACC)
,
Philadelphia, PA
,
July 10–12
,
IEEE
, pp.
1854
1859
.
25.
Alaviani
,
S. S.
, and
Elia
,
N.
,
2021
, “
A Distributed Algorithm for Solving Linear Algebraic Equations Over Random Networks
,”
IEEE Trans. Autom. Control
,
66
, pp.
2399
2406
.
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