The Lyapunov functionals method in stability problems for functional differential equations

Andreev, Aleksandr

doi:10.1134/s0005117909090021

Cited by 26 publications

(6 citation statements)

References 35 publications

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“…Let F be the set of all continuous functions f : R × Γ → R p . Define the convergence in F according to a compact-open topology [Andreev, 2009]; namely, the sequence {f n ∈ F } converges to f ∈ F , if for each compact set K ⊂ R×Γ and for each ε > 0 the following estimate holds |f n (t, ϕ) − f (t, ϕ)| < ε for all n ≥ N (ε) and (t, ϕ) ∈ K. This convergence is metrizable.…”

Section: Preliminariesmentioning

confidence: 99%

“…forms a family of limiting equations for (1) [Andreev, 2009]. Note that the functions f * ∈ G(f ) satisfy the conditions (2).…”

Section: Preliminariesmentioning

confidence: 99%

“…Theorem 2.1. [Andreev, 2009] Let x = x(t, α, ϕ) be some solution of (1) bounded for all t ≥ α − h 0 . Then, the set ω + (x(t, α, ϕ)) is quasi-invariant with respect to the family of limiting equations (3).…”

Section: Preliminariesmentioning

confidence: 99%

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On global trajectory tracking control of robot manipulators with a delayed feedback

Andreev

Peregudova

2021

CAP

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In this paper, the trajectory tracking control problem of a robot manipulator with cylindrical joints is considered by means of a nonlinear PD controller taking into account the delayed feedback structure. The conclusion about stability of a closed-loop system is obtained on the basis of the development of the direct Lyapunov method in the study of the stability property for a non-autonomous functional differential equation by constructing a Lyapunov functional with a semi-definite time derivative.

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Section: Preliminariesmentioning

confidence: 99%

“…forms a family of limiting equations for (1) [Andreev, 2009]. Note that the functions f * ∈ G(f ) satisfy the conditions (2).…”

Section: Preliminariesmentioning

confidence: 99%

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On global trajectory tracking control of robot manipulators with a delayed feedback

Andreev

Peregudova

2021

CAP

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“…Lyapunov stability focuses on infinite‐time convergence 1–5 ; however, in many practical systems, the behavior of the system in a finite time is important 6,7 . In other words, the state variables should not exceed a certain threshold in a finite period 8 .…”

Section: Introductionmentioning

confidence: 99%

Observer‐based finite‐time passive control of discrete‐time switched nonlinear systems with time‐delay

Gholami

Shafiei

Safarinejadian

2022

Adaptive Control & Signal

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In this paper, an observer-based controller is designed to control a class of discrete-time switched nonlinear systems with state delay and external disturbances. Based on the passivity concept, the average dwell time method, and auxiliary matrices, less-conservative sufficient conditions are derived to make the closed-loop system finite-time bounded and passive in the presence of disturbances. In contrast to existing papers in this field, this paper deals with discrete-time nonlinear switched systems and especially with an observer-based controller. The developed conditions are presented in the form of linear matrix inequalities. Finally, computer simulations (numerical and practical examples) are provided to illustrate the performance of the proposed controller. K E Y W O R D Saverage dwell time method, discrete-time switched systems, finite-time boundedness, observer, time-delay INTRODUCTIONLyapunov stability focuses on infinite-time convergence 1-5 ; however, in many practical systems, the behavior of the system in a finite time is important. 6,7 In other words, the state variables should not exceed a certain threshold in a finite period. 8 For this purpose, the concept of finite-time stability which focuses on the transient behavior of the system has been introduced. 9 Then, due to the importance of the system performance in the presence of external disturbances, this concept has been generalized to finite-time boundedness. 10 It means that the state variables do not exceed a determined threshold over an interval in the presence of external disturbances with a bounded initial condition. 11 On the other hand, switched systems as a class of hybrid systems have attracted the attention of many researchers. 12-16 Switched systems include multi-subsystems and a switching rule specifying the active subsystem at each time instant. Many researchers have investigated finite-time stability or boundedness of switched systems including discrete-time or continuous-time cases. 1,[17][18][19][20] Practically, discrete-time systems are more important when a digital controller should be implemented. Moreover, finite-time control of discrete-time switched systems can be found in the literature for linear, 21-23 singular, 24 nonlinear, 8,25 stochastic, 26 and markovian jump 27 systems.Passivity plays an important role in the controller design for many practical systems, and also switched systems. [28][29][30] The passivity theory has been presented in various fields such as time-delay systems, 31 signal processing, 32 fuzzy control, 33 neural networks systems, 34 and chemical processes. 30 Generally, passivity-based control allows a better closed-loop performance under transient and steady-state operating conditions. 33 In this method, the model of the system is not necessarily passive; the designed controller will make the closed-loop system passive according to the definition that will be introduced later.

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“…Using the representation of integral terms in the structure of control signals as regulators with unlimited after-effect (Anan'evskii & Kolmanovskii, 1989;Andreev, 2009), the motion of CONTACT Olga Peregudova peregudovaoa@gmail.com mechanical systems with these types of regulators can be modeled by Volterra integro-differential equations (Volterra, 1959). Such equations arise in the mathematical modelling of viscoelastic materials (Dafermos & Nohel 1981;Mac 1977;Sergeev 2007b;Volterra, 1959), population dynamics (Britton 1990;Volterra 1959), agedependent epidemic of a disease (El-Doma, 1987), nuclear reactor dynamics (Kappel & Di, 1972).…”

Section: Introductionmentioning

confidence: 99%

Non-linear PI regulators in control problems for holonomic mechanical systems

Andreev

Peregudova

2017

Systems Science & Control Engineering

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In this paper we propose a solution to the regulation problem of holonomic mechanical systems without velocity measurements by constructing nonlinear PI controllers. An approach based on the Lyapunov functional method is used to analyse the stability of the closed-loop system which has the form of Volterra integro-differential equation. The paper presents the quasi-invariance principle for Volterra nonlinear integral differential equations. Furthermore, we present sufficient conditions that one can use to conclude the asymptotic stability of the closed-loop system via using a Lyapunov functional with semi-definite time derivative. The simulation results of the two-link robot manipulator demonstrate the effectiveness of the proposed PI controller. ARTICLE HISTORY

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The Lyapunov functionals method in stability problems for functional differential equations

Cited by 26 publications

References 35 publications

On global trajectory tracking control of robot manipulators with a delayed feedback

On global trajectory tracking control of robot manipulators with a delayed feedback

Observer‐based finite‐time passive control of discrete‐time switched nonlinear systems with time‐delay

Non-linear PI regulators in control problems for holonomic mechanical systems

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