Homogeneity approach to high-order sliding mode design

Levant, Arie

doi:10.1016/j.automatica.2004.11.029

Cited by 851 publications

(785 citation statements)

References 25 publications

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“…This is the socalled 'super-twisting' approach [11]. Until very recently stability, robustness and convergence rates in higher order sliding mode methods have been analyzed in terms of homogeneity or geometric arguments [5]. However in a succession of papers [6,16,14], Lyapunov methods were employed successfully for the first time to analyze the properties of the super-twisting algorithm for uncertain systems.…”

Section: Introductionmentioning

confidence: 99%

A multivariable super-twisting sliding mode approach

2014

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Section: Introductionmentioning

confidence: 99%

A multivariable super-twisting sliding mode approach

2014

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“…(27) The convergence of the above observer has already been demonstrated in [23] and [24]. It has been shown that, if the rth derivative of the output has a Lipschitz constant L > 0 and by choosing λ > L, then we haveẑ = z after a finite time T .…”

Section: Observer Designmentioning

confidence: 99%

“…is the estimation of y (ρ i ) i which however might be realized by the finite-time differentiators, such as algebraic one and High-Order Sliding Mode differentiator [1], [10], [22], [23], [24]. Sincex and y (ρ i ) i can be estimated in finite time T by the higher order sliding mode observer and following Definition 1, we can conclude that the unknown input u is causally finite-time estimated withû is given from (12).…”

Section: Left Inversion Without Internal Dynamicsmentioning

confidence: 99%

Finite-time and asymptotic left inversion of nonlinear time-delay systems

2018

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In this paper we investigate the left invertibility problem for a class of nonlinear time-delay systems. In both cases of time delay systems with and without internal dynamics the invertibility conditions are given. A new approach based on the use of higher order sliding mode observer is developed for finite-time left invertibility and for asymptotic left inversion. Causal and non-causal estimation of the unknown inputs are respectively discussed. The results are illustrated by numerical examples in order to show the efficiency of the method and its limits.Key words: Left inversion, nonlinear system, nonlinear time-delay system, asymptotic left inversion, internal dynamics. IntroductionTime delay systems represent one of the most studied class of systems in control theory. Since the 60', many different problems are studied such as stability and stabilization of time delay systems, observation and observer design, parameter identification, etc. In the present work we are interested in the left invertibility problem of time delay systems with internal dynamics. The problem of unknown inputs recovering from the outputs is crucial. Such a problem has attracted the interest of the control community since it has direct applications in many domains, such as data secure transmission where the unknown input is the message, and fault detection and isolation where the fault is the unknown input. In fact, left invertibility problem has been studied since at least forty five years ago in linear control theory [33] In the literature, an important tool based on non-commutative ring, proposed in [36], is used to analyze nonlinear time delay systems in algebraic framework. Using this framework, many notions are extended to the case of nonlinear time * Corresponding author.Email address: zohra.kader@inria.fr (Zohra Kader). delay systems and many results have already been obtained and published [40], [37]. In the context of constant time delay, the notions of Lie derivatives and relative degree are defined, and the differences between causal and non-causal invertibility are clarified in [38]. The canonical form of invertibility is also given in [39], and in [42] a method for estimating the unknown inputs is proposed. However, the algorithm for left invertibility proposed in [39] was only for system without internal dynamics (see also [4] and [13] and their references). For system with or without time-delay, the main difficulty when the internal dynamics appears is to estimate the state of such dynamics. One interesting solution in order to overcome such a difficulty is to allow the derivative of the unknown inputs [31] with a geometrical approach and with an algebraic one. If, however, the input derivatives are not possible, then it is necessary to compute and analyze the internal dynamics.For nonlinear systems without delay, if the vector fields associated to the inputs verify involutivity property, then the internal dynamics does not depend on the unknown input. However, this rule is not valid for nonlinear systems with...

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“…Some results in this context can be discovered in the literature. In particular, fast stability of ODEs is represented by the notions of finite-time and fixed-time stabilities [50], [43], [20], [11], [6], [29], [26], [2], [14], [33], [39], but hyper exponential transitions are studied in [38] as fast behavior of time delay systems. Fast models described by partial differential equations may demonstrate the so-called finite-time extinction property [46], [19], [31], [10] also known as super stability [5], [13].…”

Section: State Of the Artmentioning

confidence: 99%