Almost sure exponential stabilization by discrete-time stochastic feedback control

Research output: Contribution to journalArticle

23 Citations (Scopus)

Abstract

Given an unstable linear scalar differential equation x˙ (t) = αx(t) (α > 0), we will show that the discrete-time stochastic feedback control σx([t/τ ]τ )dB(t) can stabilize it. That is, we will show that the stochastically controlled system dx(t) = αx(t)dt +σx([t/τ ]τ )dB(t) is almost surely exponentially stable when σ2 > 2α and τ > 0 is sufficiently small, where B(t) is a Brownian motion and [t/τ ] is the integer part of t/τ . We will also discuss the nonlinear stabilization problem by a discrete- time stochastic feedback control. The reason why we consider the discrete-time stochastic feedback control is because that the state of the given system is in fact observed only at discrete times, say 0, τ, 2τ, • • • , for example, where τ > 0 is the duration between two consecutive observations. Accordingly, the stochastic feedback control should be designed based on these discrete-time observations, namely the stochastic feedback control should be of the form σx([t/τ ]τ )dB(t). From the point of control cost, it is cheaper if one only needs to observe the state less frequently. It is therefore useful to give a bound on τ from below as larger as better.
LanguageEnglish
Pages1619-1624
Number of pages6
JournalIEEE Transactions on Automatic Control
Volume61
Issue number6
DOIs
Publication statusPublished - 1 Jun 2016

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Feedback control
Stabilization
Brownian movement
Differential equations
Costs

Keywords

  • Brownian motion
  • stochastic differential delay equations
  • difference equations
  • stochastic stabilization
  • discrete-time feedback control

Cite this

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title = "Almost sure exponential stabilization by discrete-time stochastic feedback control",
abstract = "Given an unstable linear scalar differential equation x˙ (t) = αx(t) (α > 0), we will show that the discrete-time stochastic feedback control σx([t/τ ]τ )dB(t) can stabilize it. That is, we will show that the stochastically controlled system dx(t) = αx(t)dt +σx([t/τ ]τ )dB(t) is almost surely exponentially stable when σ2 > 2α and τ > 0 is sufficiently small, where B(t) is a Brownian motion and [t/τ ] is the integer part of t/τ . We will also discuss the nonlinear stabilization problem by a discrete- time stochastic feedback control. The reason why we consider the discrete-time stochastic feedback control is because that the state of the given system is in fact observed only at discrete times, say 0, τ, 2τ, • • • , for example, where τ > 0 is the duration between two consecutive observations. Accordingly, the stochastic feedback control should be designed based on these discrete-time observations, namely the stochastic feedback control should be of the form σx([t/τ ]τ )dB(t). From the point of control cost, it is cheaper if one only needs to observe the state less frequently. It is therefore useful to give a bound on τ from below as larger as better.",
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Almost sure exponential stabilization by discrete-time stochastic feedback control. / Mao, Xuerong.

In: IEEE Transactions on Automatic Control, Vol. 61, No. 6, 01.06.2016, p. 1619-1624.

Research output: Contribution to journalArticle

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