Stability criteria for parabolic type partial difference equations

Xie, Shengli; Cheng, Sui Sun

doi:10.1016/s0377-0427(96)00058-1

Cited by 19 publications

(8 citation statements)

References 1 publication

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“…Lemma 1 (Discrete Gronwall's inequality, [2,5]) Let constant sequences {v(x)}, {B(x)}, and {D(x)} be real sequences defined for x ≥ 0, which satisfy…”

Section: Ilc System Descriptionmentioning

confidence: 99%

“…In fact, the state space model of partial difference system can cover many nature laws, such as logistic model with spatial migration, mathematics physical processes (discrete heat equation), engineering technology (image processing, digital signal processing, circuit systems) (see [1,2] and the references therein). There are many excellent results on the partial difference equations/systems, including exist/no exist [3,4], stability [5], oscillation [6], positivity of solutions [7], etc. From the view point of practicality, since the first step, that is, the implementation of control for distributed parameter systems modeled by partial differential equations, requires to discretize the variable of systems, studying the control of partial difference systems has a great value.…”

Section: Introductionmentioning

confidence: 99%

“…The system in this paper is governed by partial difference equations and simultaneously involving three different indices: time, space, and iteration; therefore the convergence analysis is more complex. 3We used the methods of partial difference equations which have been applied in stability analysis from partial difference systems [2,[4][5][6], instead of Lyapunov method or linear matrix inequality (for multi dimensional dynamic systems) [21,22].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Iterative learning control for MIMO parabolic partial difference systems with time delay

Dai

Zhao³

et al. 2018

Adv Differ Equ

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In this paper, the iterative learning control (ILC) technique is extended to multi-input multi-output (MIMO) systems governed by parabolic partial difference equations with time delay. Two types of ILC algorithm are presented for the system with state delay and input delay, respectively. The sufficient conditions for tracking error convergence are established under suitable assumptions. Detailed convergence analysis is given based on discrete Gronwall's inequality and discrete Green's formula for the systems with time-varying uncertainty coefficients. Numerical results show the effectiveness of the proposed ILC algorithms.

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“…Lemma 1 (Discrete Gronwall's inequality, [2,5]) Let constant sequences {v(x)}, {B(x)}, and {D(x)} be real sequences defined for x ≥ 0, which satisfy…”

Section: Ilc System Descriptionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Iterative learning control for MIMO parabolic partial difference systems with time delay

Dai

Zhao³

et al. 2018

Adv Differ Equ

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show abstract

“…Lemma 1 (A analogues discrete of difference type Gronwall inequality) [27] . Let the constant sequences {v(i)}, {B(i)} and {D(i)} be real sequences defined for i ≥ 0 satisfying…”

Section: Preliminarymentioning

confidence: 99%

“…Especially, the parabolic type partial difference equation was considered in [27,28]. In this paper, ILC technique is applied to a class of discrete parabolic distributed parameter systems described by partial difference equations.…”

Section: Introductionmentioning

confidence: 99%

Iterative learning control for discrete parabolic distributed parameter systems

Dai

Tian

Ya-jun

2015

Int. J. Autom. Comput.

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Abstract:In this paper, iterative learning control (ILC) technique is applied to a class of discrete parabolic distributed parameter systems described by partial difference equations. A P-type learning control law is established for the system. The ILC of discrete parabolic distributed parameter systems is more complex as 3D dynamics in the time, spatial and iterative domains are involved. To overcome this difficulty, discrete Green formula and analogues discrete Gronwall inequality as well as some other basic analytic techniques are utilized. With rigorous analysis, the proposed intelligent control scheme guarantees the convergence of the tracking error. A numerical example is given to illustrate the effectiveness of the proposed method.

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Closed‐loop consensus control of partial difference multi‐agent systems via variable gain iterative learning strategy

Wang

Zhou

Liu

2022

Intl J Robust & Nonlinear

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In the research on consensus control of multi-agent systems (MASs), it is the key to ensure that the state quantities (speed, displacement, etc.) of all agents tend to be consensus. Under this premise, how to accelerate the convergence rate of the consensus error is also an important issue. Aiming at the accelerated consensus problem of MASs composed of partial difference equations, based on the network topology and the output form of each follower, a distributed closed-loop accelerated iterative learning control (ILC) protocol with variable gain was proposed, which was designed to improve the speed of consensus error. With the help of basic mathematical tools such as discrete Gronwall inequality and contraction mapping method, the sufficient conditions for the convergence of the consensus error are derived and analyzed. Finally, numerical simulations show that the proposed acceleration control protocol is more effective than the traditional open-loop and closed-loop ILC protocols.

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Stability criteria for parabolic type partial difference equations

Cited by 19 publications

References 1 publication

Iterative learning control for MIMO parabolic partial difference systems with time delay

Iterative learning control for MIMO parabolic partial difference systems with time delay

Iterative learning control for discrete parabolic distributed parameter systems

Closed‐loop consensus control of partial difference multi‐agent systems via variable gain iterative learning strategy

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