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Lyapunov exponents and stability of linear stochastic systems

Feng, Xiangbo
http://rave.ohiolink.edu/etdc/view?acc_num=case1054928844

Abstract Details

1990, Doctor of Philosophy, Case Western Reserve University, Systems and Control Engineering.
The Lyapunov exponents (exponential growth rates) and stability properties of linear stochastic systems are investigated in this thesis. Following the work of Oseledec and Kifer, a nonrandom spectrum theorem for Lyapunov exponents is developed for a class of linear Markovian systems (jump linear systems) and/or products of random matrices with a Markovian structure. The conjecture is that the same result can be proved using a more abstract problem formulation for random bundle maps given by Kifer, this is a topic to be considered in future research studies. Using a stochastic "averaging" method, an analytic series expansion of the (top) Lyapunov exponent of two-dimensional linear stochastic systems with telegraphic noise is obtained. This procedure can be used to compute the general term in the expansion for the exponent and the rotation number. The series expansion for the top exponent is proved to be converging for a range of parameter values. For the random harmonic oscillator, we establish the positivity of the exponent and therefore prove almost sure instability of the system. Lower and upper bound estimates in terms of the system and noise parameters for the exponent are also obtained for the random harmonic oscillator. We also establish the equivalence of three sec ond moment stability properties for jump linear systems, which are of importance in the stabilization and control of the class of stochastic systems. The relationship between moment stability properties and the almost sure sample stability property for jump linear systems is also studied. For scalar systems, we completely characterize the relationship between the domain of moment and almost sure stability in the parameter space of the systems. Extensions of this result to higher dimensional systems is a topic for future research studies.
Kenneth Loparo (Advisor)
160 p.
Lyapunov exponents; stability; linear stochastic systems

Recommended Citations

Citations

  • Feng, X. (1990). Lyapunov exponents and stability of linear stochastic systems [Doctoral dissertation, Case Western Reserve University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=case1054928844

    APA Style (7th edition)

  • Feng, Xiangbo. Lyapunov exponents and stability of linear stochastic systems. 1990. Case Western Reserve University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=case1054928844.

    MLA Style (8th edition)

  • Feng, Xiangbo. "Lyapunov exponents and stability of linear stochastic systems." Doctoral dissertation, Case Western Reserve University, 1990. http://rave.ohiolink.edu/etdc/view?acc_num=case1054928844

    Chicago Manual of Style (17th edition)

case1054928844
961
© 1990, all rights reserved.
This open access ETD is published by Case Western Reserve University School of Graduate Studies and OhioLINK.