Smooth Lyapunov function and gain design for a Second Order Differentiator

Ortiz-Ricardez, Fernando A.; Sánchez, Tonámetl; Moreno, Jaime A.

doi:10.1109/cdc.2015.7403065

Cited by 26 publications

(25 citation statements)

References 12 publications

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“…This Lemma says V is a ISS Lyapunov function for the auxiliary system (19 (17) give the error system (14) and ri+1 rj ≤ 1, we obtain, for all L ≥ 1…”

Section: Homogeneous Observermentioning

confidence: 81%

Observers for a non-Lipschitz triangular form

2017

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We address the problem of designing an observer for triangular non locally Lipschitz dynamical systems. We show the convergence with an arbitrary small error of the classical high gain observer in presence of nonlinearities verifying some Hölder-like condition. Also, for the case when this Hölder condition is not verified, we propose a novel cascaded high gain observer. Under slightly more restrictive assumptions, we prove the convergence of an homogeneous observer and of its cascaded version with the help of an explicit Lyapunov function.

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“…This Lemma says V is a ISS Lyapunov function for the auxiliary system (19 (17) give the error system (14) and ri+1 rj ≤ 1, we obtain, for all L ≥ 1…”

Section: Homogeneous Observermentioning

confidence: 81%

Observers for a non-Lipschitz triangular form

2017

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“…This LF constitutes a GF, and remarkably, it can be written as a quadratic form. In the case of the second‐order differentiator, the error dynamics is

ė_{1} = - λ_{1} {⌈ e_{1} ⌋}^{\frac{2}{3}} + e_{2}, ė_{2} = - λ_{2} {⌈ e_{1} ⌋}^{\frac{1}{3}} + e_{3}, ė_{3} = - λ_{3} {⌈ e_{1} ⌋}^{0} .

In other works, some GFs have been provided as LFs for the differential inclusion associated to , for example, the GF

V : R^{3} \to double-struckR

given by

V false(e false) = β_{1} {false| e_{1} false|}^{\frac{5}{3}} - β_{12} e_{1} e_{2} + β_{2} {false| e_{2} false|}^{\frac{5}{2}} - β_{23} e_{2} e_{3}^{3} + β_{3} {false| e_{3} false|}^{5}

.…”

Section: Motivational Examplesmentioning

confidence: 99%

“…In other works, 32,[34][35][36] some GFs have been provided as LFs for the differential inclusion associated to (6), for example, the GF V ∶ ℝ 3 → ℝ given by V(e) = 1 |e 1 | 5 3 − 12 e 1 e 2 + 2 |e 2 | 5 2 − 23 e 2 e 3 3 + 3 |e 3 | 5 .…”

Section: Examplementioning

confidence: 99%

Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach

Sánchez

Moreno

2018

Intl J Robust & Nonlinear

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Summary In this paper, we provide a method to design Lyapunov functions (LFs) for a class of homogeneous systems described by functions that we call generalized forms (GFs). Homogeneous polynomial systems and several high‐order sliding modes are included in the class. The LF candidate is chosen from the same class of functions and it is parameterized in its coefficients. Since the derivative of the LF candidate along the system's trajectories is also a GF, the problem is reduced to verify positive definiteness of two GFs. We establish a procedure to represent a GF with a finite set of polynomials. Thus, the problem is changed to determine positive definiteness of a set of polynomials. Such a problem can be solved by means of Pólya's theorem or the sum of squares representation.

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“…In real world applications the setting of the differentiator, however, needs to be adjusted according to the characteristics of the signal to be differentiated. One option to generate a convergent parameter setting is to evaluate recently proposed Lyapunov functions either for certain orders of the differentiator, see [6]- [9], or for the arbitrary order case, see [10]. In order to avoid this cumbersome procedure the toolbox [1] may be used which is an implementation of the tuning paradigm presented in [11].…”

Section: Introductionmentioning

confidence: 99%

The Robust Exact Differentiator Toolbox: Improved Discrete-Time Realization

Reichhartinger

Koch

Niederwieser

et al. 2018

2018 15th International Workshop on Variable Structure Systems (VSS)

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This paper presents a new release of A Robust Exact Differentiator Toolbox for Matlab/Simulink proposed in [1]. This release features a new discrete-time realization of the continuoustime robust exact differentiator. The implemented discretization scheme is less sensitive to gain overestimation and does not suffer from the discretization chattering effect. Hence, the single tuning parameter of the new version of the implemented differentiator is more intuitive to tune. Furthermore, it shows superior estimation performance in the case of large sampling times in comparison to the previous release. This is confirmed by the presented results obtained by numerical simulations and a real world application.

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Smooth Lyapunov function and gain design for a Second Order Differentiator

Cited by 26 publications

References 12 publications

Observers for a non-Lipschitz triangular form

Observers for a non-Lipschitz triangular form

Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach

The Robust Exact Differentiator Toolbox: Improved Discrete-Time Realization

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