The Robust Exact Differentiator Toolbox: Improved Discrete-Time Realization

Reichhartinger, Markus; Koch, Stefan; Niederwieser, Helmut; Spurgeon, Sarah K.

doi:10.1109/vss.2018.8460388

Cited by 12 publications

(10 citation statements)

References 18 publications

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“…For example, local stability implies global stability. Moreover, negative homogeneous degree implies finite‐time convergence 29,30,33,34,55,58,59 . Based on the HDs, a differentiation toolbox has been introduced 33 which deals with the implementation of the AO‐STD with high‐degree linear terms 32 .…”

Section: A Summary Of the Continuous‐time Differentiatorsmentioning

confidence: 99%

“…Based on the HDs, a differentiation toolbox has been introduced 33 which deals with the implementation of the AO‐STD with high‐degree linear terms 32 . Subsequently, an improved discretization method of the HDs was introduced 29 which led to a newer version of the differentiation toolbox 30 . Throughout this manuscript, the term HD, refers to the method which was presented in References 29,30.…”

Section: A Summary Of the Continuous‐time Differentiatorsmentioning

confidence: 99%

“…Subsequently, an improved discretization method of the HDs was introduced 29 which led to a newer version of the differentiation toolbox 30 . Throughout this manuscript, the term HD, refers to the method which was presented in References 29,30.…”

Section: A Summary Of the Continuous‐time Differentiatorsmentioning

confidence: 99%

“…In this scheme, the differentiation problem is formulated as the following chain of integrator:

\begin{align} \overset{x}{true} false(t false) & = A x false(t false) + e_{n + 1} f^{false(n + 1 false)} false(t false) \\ y false(t false) & = e_{1}^{T} x false(t false), \end{align}

where, as before,

f (t)

is the signal to be differentiated,

A = [\begin{matrix} 0_{n prefix\times 1} & I_{n prefix\times n} \\ 0 & 0_{1 prefix\times n} \end{matrix}]

, x is the vector of differentiations, and

e_{i}

indicates a column vector with i th element equal to one, and other elements equal to zero. Discrete‐form of (26) has been given in 29,30 as follows:

\begin{array}{l} x_{k + 1} = normalΦ x_{k} + h H_{k} \\ y_{k} = e_{1}^{T} x_{k}, \end{array}

where

Φ = \exp (A h)

, and

H_{k} = [\frac{h^{n}}{(n + 1)!} f_{k}^{(n + 1)}, h^{n}

…”

Section: Explicit Discretization Of the Differentiatorsmentioning

confidence: 99%

“…To alleviate the drawbacks of the explicit discretization, improved explicit discretization methods are introduced. In this context, in a few studies it is tried to improve the accuracy of the explicit AO‐STD through a nonlinear eigenvalue placement using the matching approach 29‐31 . Another attempt to improve the explicit discretization was to add the higher‐degree terms of the Taylor expansions to keep the homogeneity properties of the continuous‐time AO‐STD 32,33 .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Time‐discretizations of differentiators: Design of implicit algorithms and comparative analysis

Mojallizadeh

Brogliato

Acary

2021

Intl J Robust & Nonlinear

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A complete review of the known differentiators and their time‐discretizations have been addressed in this study. To resolve the drawbacks of the explicit (forward Euler) discretization, which is commonly utilized in sliding‐mode‐based differentiators, implicit time discretization methods are proposed to handle the set‐valued functions. The proposed schemes are supported by some analytical results to show their crucial properties, for example, finite‐time convergence, exactness, invariant sliding‐surface, chattering elimination, insensitivity to the gains during the sliding‐phase, and the well‐posedness. The causal implementation of the proposed implicit schemes has been addressed clearly and supported by flowcharts. Semi‐implicit schemes are also presented to provide a compromise between the performance and the ease of implementation. Finally, an exhaustive comparison is made using numerical experiments among 25 different state‐of‐the‐art differentiators to evaluate the behaviors of the differentiators facing the noise and initial error. Realistic conditions are considered in the simulations to provide useful information based on practical conditions. General conclusions are that implicit discretizations can supersede explicit and semi‐implicit ones.

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Section: A Summary Of the Continuous‐time Differentiatorsmentioning

confidence: 99%

Section: A Summary Of the Continuous‐time Differentiatorsmentioning

confidence: 99%

Section: A Summary Of the Continuous‐time Differentiatorsmentioning

confidence: 99%

“…In this scheme, the differentiation problem is formulated as the following chain of integrator:

\begin{align} \overset{x}{true} false(t false) & = A x false(t false) + e_{n + 1} f^{false(n + 1 false)} false(t false) \\ y false(t false) & = e_{1}^{T} x false(t false), \end{align}

where, as before,

f (t)

is the signal to be differentiated,

A = [\begin{matrix} 0_{n prefix\times 1} & I_{n prefix\times n} \\ 0 & 0_{1 prefix\times n} \end{matrix}]

, x is the vector of differentiations, and

e_{i}

indicates a column vector with i th element equal to one, and other elements equal to zero. Discrete‐form of (26) has been given in 29,30 as follows:

\begin{array}{l} x_{k + 1} = normalΦ x_{k} + h H_{k} \\ y_{k} = e_{1}^{T} x_{k}, \end{array}

where

Φ = \exp (A h)

, and

H_{k} = [\frac{h^{n}}{(n + 1)!} f_{k}^{(n + 1)}, h^{n}

…”

Section: Explicit Discretization Of the Differentiatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Time‐discretizations of differentiators: Design of implicit algorithms and comparative analysis

Mojallizadeh

Brogliato

Acary

2021

Intl J Robust & Nonlinear

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On implementing the implicit discrete‐time super‐twisting observer on mechanical systems

Winkler

Grabmair

Reger

2023

Intl J Robust & Nonlinear

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In this paper, an extension of an algorithm for the implicit discretization of a super-twisting sliding mode observer is presented. Implicit and explicit discretization algorithms for homogeneous differentiators, where no physical model information is considered, are investigated in literature. This article studies the behavior when considering models of a rather general class of nonlinear systems. The discrete equations of the super-twisting observer are reformulated as generalized equation and an algorithm for the step-by-step solution is given. The uniqueness of the derived algorithm is investigated with an equivalent variational inequality formulation which is derived for a class of nonlinear systems. Furthermore, a semi-implicit predictor-corrector discretization is presented which is an approximation method for the presented algorithms and allows an explicit implementation in practical applications. Accuracy properties under noise and sampling are given. The algorithm is applied on two mechanical example systems taken from practice.

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State Observer for a Chemical Process Using Modern Discretization Algorithms

Molina-Gómez

Botero-Castro

2020

Communications in Computer and Information Science

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The Robust Exact Differentiator Toolbox: Improved Discrete-Time Realization

Cited by 12 publications

References 18 publications

Time‐discretizations of differentiators: Design of implicit algorithms and comparative analysis

Time‐discretizations of differentiators: Design of implicit algorithms and comparative analysis

On implementing the implicit discrete‐time super‐twisting observer on mechanical systems

State Observer for a Chemical Process Using Modern Discretization Algorithms

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