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Research Papers

Asymptotic Stability With Probability One of Random-Time-Delay-Controlled Quasi-Integrable Hamiltonian Systems

[+] Author and Article Information
R. H. Huan, R. C. Hu, Z. G. Ying

Department of Mechanics,
Zhejiang University,
Hangzhou 310027, China;
State Key Laboratory of Fluid Power and
Mechatronic Systems,
Zhejiang University,
Hangzhou 310027, China;
Key Laboratory of Soft Machines and Smart
Devices of Zhejiang Province,
Zhejiang University,
Hangzhou 310027, China

W. Q. Zhu

Department of Mechanics,
Zhejiang University,
Hangzhou 310027, China;
State Key Laboratory of Fluid Power and
Mechatronic Systems,
Zhejiang University,
Hangzhou 310027, China;
Key Laboratory of Soft Machines and Smart
Devices of Zhejiang Province,
Zhejiang University,
Hangzhou 310027, China
e-mail: wqzhu@zju.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 21, 2016; final manuscript received June 16, 2016; published online July 4, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(9), 091009 (Jul 04, 2016) (8 pages) Paper No: JAM-16-1196; doi: 10.1115/1.4033944 History: Received April 21, 2016; Revised June 16, 2016

A new procedure for determining the asymptotic stability with probability one of random-time-delay-controlled quasi-integrable Hamiltonian systems is proposed. Such a system is formulated as continuous–discrete hybrid system and the random time delay is modeled as a Markov jump process. A three-step approximation is taken to simplify such hybrid system: (i) the randomly periodic approximate solution property of the system is used to convert the random time delay control into the control without time delay but with delay time as parameter; (ii) a limit theorem is used to transform the hybrid system with Markov jump parameter into one without jump parameter; and (iii) the stochastic averaging method for quasi-integrable Hamiltonian systems is applied to reduce the system into a set of averaged Itô stochastic differential equations. An approximate expression for the largest Lyapunov exponent of the system is derived from the linearized averaged Itô equations and the necessary and sufficient condition for the asymptotic stability with probability one of the system is obtained. The application and effectiveness of the proposed procedure are demonstrated by using an example of stochastically driven two-degrees-of-freedom networked control system (NCS) with random time delay.

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Figures

Grahic Jump Location
Fig. 1

The largest Lyapunov exponent Λmax as function of transition rates v1 (or v2). — Analytical result. ▼ Direct simulation.

Grahic Jump Location
Fig. 2

The stability boundaries in (v1,v2) plane for different time delay τ(2)

Grahic Jump Location
Fig. 3

The largest Lyapunov exponent Λmax as functions of control gains f1 (f11=f21=f1) for different jump rules. — Analytical result. ▼ Direct simulation.

Grahic Jump Location
Fig. 4

The largest Lyapunov exponent Λmax as functions of control gains f2 (f12=f22=f2) for different jump rules. — Analytical result. ▼ Direct simulation.

Grahic Jump Location
Fig. 5

Samples of displacement q1 obtained by direct simulations of system represented by points A, B, and C in Fig. 3: (a) for system A ; (b) for system B ; and (c) for system C

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