Tractable Razumikhin-type conditions for input-to-state stability analysis of delay difference inclusions

Gielen, Rob H.; Teel, Andrew R.; Lazăr, Mircea

doi:10.1016/j.automatica.2012.11.048

Cited by 21 publications

(19 citation statements)

References 16 publications

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“…Example Consider a DDS as

x false(k + 1 false) = ((), \begin{matrix} array 0.8 & array a k \\ array 0 & array 0 \end{matrix}) x false(k false) + ((), \begin{matrix} array 0 & array 0 \\ array 0 & array 0.8 \end{matrix}) x false(k - τ false(k false) false) + w false(k false), k \in double-struckN,

where

x false(k false) \in R^{2}

, a ⩾0, and

w false(k false) \in R^{2}

is the external input. When a =0 and τ ( k )=1, DDS is time‐invariant which was studied in Gielen et al and shown to be ISS. Here, since ρ 0 = ρ (| A |+| B |)=0.8<1, thus, by Corollary , even τ ( k ) is infinite, ie,

τ false(k false) \to + \infty

k \to + \infty

, DDS has IS‐ e −1 , ie, exponential ISS, see Figure , where

τ false(k false) = false[\frac{k}{2} false]

, which is the minimal integer no less than

\frac{k}{2}

.…”

Section: Input‐to‐state Negative Exponent and Related Iss For Ddssmentioning

confidence: 99%

“…However, for a time‐varying DDS, by Example , the condition

ρ_{0} = \sup_{k \in double-struckN} false{ρ false(normalΨ false(k false) false) false} = \sup_{k \in double-struckN} false{ρ false(false| A false(k false) false| + false| B false(k false) false| false) false} < 1

cannot guarantee the DDS having ISS property. From Example , for a time‐varying DDS, the exponential stability of DDS with zero external input cannot guarantee ISS of the DDS. Hence, the equivalence between ISS and uniform asymptotic stability under zero external input is no longer true for time‐varying DDSs. In the literature, Razumikhin‐type ISS conditions are proposed for DDSs and delay difference inclusions (DDIs), see Gielen et al and Liu and Hill . When a DDS/DDI is linear and time‐invariant and has finite delay, the trackable Razumikhin‐type ISS conditions via expressions of LMI in Gielen et al can also be used to test the ISS.…”

Section: Input‐to‐state Negative Exponent and Related Iss For Ddssmentioning

confidence: 99%

“…Hence, the equivalence between ISS and uniform asymptotic stability under zero external input is no longer true for time‐varying DDSs. In the literature, Razumikhin‐type ISS conditions are proposed for DDSs and delay difference inclusions (DDIs), see Gielen et al and Liu and Hill . When a DDS/DDI is linear and time‐invariant and has finite delay, the trackable Razumikhin‐type ISS conditions via expressions of LMI in Gielen et al can also be used to test the ISS. But in the case of time‐varying systems and

τ = + \infty

, the ISS conditions via LMI in Gielen et al cannot be used to check the ISS. In the following, we investigate IS‐ e of DDS from IS‐ e properties of the nominal system .…”

Section: Input‐to‐state Negative Exponent and Related Iss For Ddssmentioning

confidence: 99%

“…In the case of time‐delay, the Lyapunov‐Krasovskii approach is used to derive the ISS conditions, eg, Pepe and Jiang, Mazenc et al, and Yeganefar et al However, the functional constructed by this approach is required to decrease along all trajectories of the system and the ISS conditions are often hard to be tested. To overcome these shortages, recently, the Razumikhin approach is used to analyze stability and ISS of delayed discrete‐time systems (DDSs) . The Razumikhin‐type ISS conditions have been established for time‐varying DDSs in Liu and Hill .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Input‐to‐state exponents and related ISS for delayed discrete‐time systems with application to impulsive effects

Liu

Hill

Sun

2017

Intl J Robust & Nonlinear

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Summary This paper aims to investigate the input‐to‐state exponents (IS‐e) and the related input‐to‐state stability (ISS) for delayed discrete‐time systems (DDSs). By using the method of variation of parameters and introducing notions of uniform and weak uniform M‐matrix, the estimates for 3 kinds of IS‐e are derived for time‐varying DDSs. The exponential ISS conditions with parts suitable for infinite delays are thus established, by which the difference from the time‐invariant case is shown. The exponential stability of a time‐varying DDS with zero external input cannot guarantee its ISS. Moreover, based on the IS‐e estimates for DDSs, the exponential ISS under events criteria for DDSs with impulsive effects are obtained. The results are then applied in 1 example to test synchronization in the sense of ISS for a delayed discrete‐time network, where the impulsive control is designed to stabilize such an asynchronous network to the synchronization.

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“…Example Consider a DDS as

x false(k + 1 false) = ((), \begin{matrix} array 0.8 & array a k \\ array 0 & array 0 \end{matrix}) x false(k false) + ((), \begin{matrix} array 0 & array 0 \\ array 0 & array 0.8 \end{matrix}) x false(k - τ false(k false) false) + w false(k false), k \in double-struckN,

where

x false(k false) \in R^{2}

, a ⩾0, and

w false(k false) \in R^{2}

τ false(k false) \to + \infty

k \to + \infty

, DDS has IS‐ e −1 , ie, exponential ISS, see Figure , where

τ false(k false) = false[\frac{k}{2} false]

, which is the minimal integer no less than

\frac{k}{2}

.…”

Section: Input‐to‐state Negative Exponent and Related Iss For Ddssmentioning

confidence: 99%

“…However, for a time‐varying DDS, by Example , the condition

ρ_{0} = \sup_{k \in double-struckN} false{ρ false(normalΨ false(k false) false) false} = \sup_{k \in double-struckN} false{ρ false(false| A false(k false) false| + false| B false(k false) false| false) false} < 1

Section: Input‐to‐state Negative Exponent and Related Iss For Ddssmentioning

confidence: 99%

τ = + \infty

, the ISS conditions via LMI in Gielen et al cannot be used to check the ISS. In the following, we investigate IS‐ e of DDS from IS‐ e properties of the nominal system .…”

Section: Input‐to‐state Negative Exponent and Related Iss For Ddssmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Input‐to‐state exponents and related ISS for delayed discrete‐time systems with application to impulsive effects

Liu

Hill

Sun

2017

Intl J Robust & Nonlinear

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“…Time-varying delays problems in control systems have received much attention and currently remain as active research areas in control engineering field (Bekiaris-Liberis, Jankovic, & Krstic, 2012;Bekiaris-Liberis & Krstic, 2010Bresch-Pietri, Chauvbin, & Petit, 2012;Choi & Lim, 2006Gielen, Teel, & Lazar, 2013;Jankovic, 2010;Karafyllis, 2006;Koo, Choi, & Lim, 2012;Krstic, 2010;Lei & Lin, 2007;Lin & Fang, 2007;Polyakov, Efimov, Perruquetti, & Richard, 2013;Richard, 2003;Yakoubi & Chitour, 2007;Ye, 2011;Zhang, Liu, & Zhang, 2013;Zhou, 2014;Zhou, Duan, & Lin, 2010;Zhou, Li, Zheng, & Duan, 2012). In this paper, we consider a chain of integrators with unknown timevarying delays in both states and input aṡ…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

Output feedback regulation of a chain of integrators with unknown time-varying delays in states and input

Koo

Choi

2015

Automatica

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Event‐triggered control via impulses for exponential stabilization of discrete‐time delayed systems and networks

Liu

Hill

Sun

et al. 2019

Intl J Robust & Nonlinear

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Summary This paper investigates the stabilization issue via event‐triggered controls (ETCs) for discrete‐time delayed systems (DDSs) and networks. Based on the recently proposed ETC scheme for discrete‐time systems without time delays, improved ETC (I‐ETC) and event‐triggered impulsive control (ETIC) are proposed for DDS. The algorithms for ETC, I‐ETC, and ETIC are given respectively to derive criteria of exponential stabilization of DDS. Moreover, the exponential stabilization and stabilization to ISS for discrete‐time delayed networks is achieved by employing the algorithms of ETC and ETIC. The issue of stabilization via ETCs for dynamical networks where different subsystems have different sequences of event instants is solved by introducing the check‐period into ETCs and establishing general ISS estimate of discrete‐time delayed inequality. In order to assess the performances of the control schemes, discussions on nontriviality are given by proposing the concept of rate of control and the function of control cost. Finally, two examples with numerical simulations are presented to demonstrate the effectiveness of theoretical results. From the obtained results on stabilization and the simulations, the ETIC is shown to have clear advantages and well performances than the classical state feedback control, the ETC recently proposed, I‐ETC, and the time‐based impulsive control on aspects of nontriviality, lower rate of control, lower cost of control, and robustness w.r.t. external disturbances.

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Tractable Razumikhin-type conditions for input-to-state stability analysis of delay difference inclusions

Cited by 21 publications

References 16 publications

Input‐to‐state exponents and related ISS for delayed discrete‐time systems with application to impulsive effects

Input‐to‐state exponents and related ISS for delayed discrete‐time systems with application to impulsive effects

Output feedback regulation of a chain of integrators with unknown time-varying delays in states and input

Event‐triggered control via impulses for exponential stabilization of discrete‐time delayed systems and networks

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