On <i>p</i>th moment stabilization of hybrid systems by discrete-time feedback control

Dong, Ran; Mao, Xuerong

doi:10.1080/07362994.2017.1324798

Cited by 13 publications

(10 citation statements)

References 35 publications

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“…[5, p. 40]) and [5, Theorem 7.1 on p. 39], we obtain that (see e.g. [23])

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ double-struckE | & x (t) - x (δ_{t}) |^{p} \\ \leq & 2^{p - 1} ξ_{t}^{\frac{p - 2}{2}} double-struckE \int_{δ_{t}}^{t} 1.623em1.623em[ξ_{t}^{p / 2} | f (x (s), r (s), s) + u (x (δ_{s}), r (δ_{s}), s) |^{p} \\ + ζ | g (x (s), r (s), s) |^{p} 1.623em1.623em] normald s . \end{matrix}

Let

\overset{false^}{U} false(x false(t false), r false(t false), t false) = e^{\int_{0}^{t} β false(s false) d s} U false(x false(t false), r false(t false), t false) .

We can obtain from the generalised Itô formula that

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ double-struckE \hat{U} & (x (t), r (t), t) \\ = & double-struckE ... \end{matrix}

…”

Section: Stabilisation Problemmentioning

confidence: 95%

“…The highest observation frequency is required for t ∈ [11,12). Moreover, existing theory yields the constant observation interval τ ≤ 0.00026, calculated with the same controller and same Lyapunov function, according to [23] with observation of system mode discretised. Previously frequent observations were required for all times.…”

Section: Examplementioning

confidence: 99%

“…hence has to require the unnecessary highest observation frequency for all times. Calculated with the same form of the controller and same Lyapunov function, existing theory yields the constant observation interval

τ \leq 0.00026

, according to Chapter 3 of [40], which discretised the observation of system mode based on [23].…”

Section: Examplementioning

confidence: 99%

See 2 more Smart Citations

Exponential stabilisation of continuous‐time periodic stochastic systems by feedback control based on periodic discrete‐time observations

Dong

Mao

Birrell

2020

IET Control Theory & Applications

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Since Mao in 2013 discretised the system observations for stabilisation problem of hybrid SDEs (stochastic differential equations with Markovian switching) by feedback control, the study of this topic using a constant observation frequency has been further developed. However, the time-varying observation frequencies have not been considered yet. Particularly, an observational more efficient way is to consider the time-varying property of the system and observe a periodic SDE system at the periodic time-varying frequencies. This paper investigates how to stabilise a periodic hybrid SDE by a periodic feedback control, based on periodic discrete-time observations. This paper provides sufficient conditions under which the controlled system can achieve pth moment exponential stability for p > 1 and almost sure exponential stability. The Lyapunov method and inequalities are main tools of our derivation and analysis. The existence of observation interval sequence is verified and one way of its calculation is provided. Finally, an example is given for illustration. Our new techniques not only reduce the observational cost by reducing observation frequency dramatically, but also offer the flexibility on system observation settings. This paper allows readers to set observation frequencies for some time intervals according to their needs to some extent.

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“…[5, p. 40]) and [5, Theorem 7.1 on p. 39], we obtain that (see e.g. [23])

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ double-struckE | & x (t) - x (δ_{t}) |^{p} \\ \leq & 2^{p - 1} ξ_{t}^{\frac{p - 2}{2}} double-struckE \int_{δ_{t}}^{t} 1.623em1.623em[ξ_{t}^{p / 2} | f (x (s), r (s), s) + u (x (δ_{s}), r (δ_{s}), s) |^{p} \\ + ζ | g (x (s), r (s), s) |^{p} 1.623em1.623em] normald s . \end{matrix}

Let

\overset{false^}{U} false(x false(t false), r false(t false), t false) = e^{\int_{0}^{t} β false(s false) d s} U false(x false(t false), r false(t false), t false) .

We can obtain from the generalised Itô formula that

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ double-struckE \hat{U} & (x (t), r (t), t) \\ = & double-struckE ... \end{matrix}

…”

Section: Stabilisation Problemmentioning

confidence: 95%

Section: Examplementioning

confidence: 99%

See 1 more Smart Citation

Exponential stabilisation of continuous‐time periodic stochastic systems by feedback control based on periodic discrete‐time observations

Dong

Mao

Birrell

2020

IET Control Theory & Applications

Self Cite

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show abstract

“…., where τ is the duration time of every observation. Hence, many discrete time feedback control techniques have been proposed and discussed by several authors [20][21][22][23][24]. It is worth pointing out that Mao and his collaborators [25,26] combined stochastic stabilization technology with discrete-time feedback control or time-delay feedback control techniques respectively, and they have stabilized an unstable system via discrete-time observation noise or delay time observation noise.…”

Section: Introductionmentioning

confidence: 99%

Aperiodically intermittent stochastic stabilization via discrete time or delay feedback control

Liu

Perc

Cao

2019

Sci. China Inf. Sci.

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In this paper, we present stochastic intermittent stabilization based on the feedback of the discrete time or the delay time. By using the stochastic comparison principle, the Itô formula, and the Borel-Cantelli lemma, we obtain two sufficient criteria for stochastic intermittent stabilization. The established criteria show that an unstable system can be stabilized by means of a stochastic intermittent noise via a discrete time feedback if the duration time τ is bounded by τ *. Similarly, stabilization via delay time feedback is equally possible if the lag time τ is bounded by τ * *. The upper bound τ * and τ * * can be computed numerically by solving corresponding equation.

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“…As a result, Mao [16] initiated the study of stabilization of (1) by feedback control based on discrete-time state observations. Later, the observation interval was increased, more stabilities and uncertain systems were also investigated ( [6,20,33,34]). Recently, observations of system mode have also been discretized ( [10,12,25]) and the controlled system regarding to (1) becomes dx(t) =[f (x(t), r(t), t) + u(x([t/τ ]τ ), r([t/τ ]τ ), t)]dt + g(x(t), r(t), t)dB(t), (2) where τ is a positive number representing observation interval, [t/τ ] denotes the integer part of t/τ .…”

mentioning

confidence: 99%

Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations

Dong¹,

Mao²

2020

Mathematical Control &Amp; Related Fields

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In 2013, Mao initiated the study of stabilization of continuoustime hybrid stochastic differential equations (SDEs) by feedback control based on discrete-time state observations. In recent years, this study has been further developed while using a constant observation interval. However, time-varying observation frequencies have not been discussed for this study. Particularly for non-autonomous periodic systems, it's more sensible to consider the timevarying property and observe the system at periodic time-varying frequencies, in terms of control efficiency. This paper introduces a periodic observation interval sequence, and investigates how to stabilize a periodic SDE by feedback control based on periodic observations, in the sense that, the controlled system achieves L p-stability for p > 1, almost sure asymptotic stability and pth moment asymptotic stability for p ≥ 2. This paper uses the Lyapunov method and inequalities to derive the theory. We also verify the existence of the observation interval sequence and explain how to calculate it. Finally, an illustrative example is given after a useful corollary. By considering the time-varying property of the system, we reduce the observation frequency dramatically and hence reduce the observational cost for control.

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On pth moment stabilization of hybrid systems by discrete-time feedback control

Cited by 13 publications

References 35 publications

Exponential stabilisation of continuous‐time periodic stochastic systems by feedback control based on periodic discrete‐time observations

Exponential stabilisation of continuous‐time periodic stochastic systems by feedback control based on periodic discrete‐time observations

Aperiodically intermittent stochastic stabilization via discrete time or delay feedback control

Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations

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