Qualitative differences of two classes of multivariable super-twisting algorithms

Intl J Robust & Nonlinear

2018

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Summary The purpose of this paper is to present an extension of the generalised supertwisting algorithm (STA) to the multivariable framework. We begin by introducing an algorithm that may be deemed as a linear, quasicontinuous, or discontinuous multivariable system, depending on the functions that define them. For the class represented by such an algorithm we prove the robust, Lyapunov stability of the origin and characterise the perturbations that preserve its stability. In particular, when its vector field is discontinuous or quasicontinuous our algorithm is endowed with finite‐time stability. Due to its resemblance to the scalar case, we denote such finite‐time stable systems as generalised multivariable STA. Furthermore, the class of finite‐time stable systems comprise the currently available versions of STAs. To finalise, by means of simulation examples, we show that our proposed finite‐time stable algorithms are well suited for signals online differentiation and highlight their dynamical traits.

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\frac{d}{normald t} z^{1} = boldBu false(t false) + ρ^{1} false(t, boldx false)

\frac{d}{normald t} \overset{θ}{true} = - K^{1} ϕ^{1} false(\overset{θ}{true} - bold-italicθ false) + \overset{ω}{true}

\frac{d}{normald t} \overset{ω}{true} = - K^{2} ϕ^{2} false(\overset{θ}{true} - bold-italicθ false) .

By letting

z^{1} : = \overset{θ}{true} - bold-italicθ

and

z^{2} : = \overset{ω}{true} - bold-italicω

, the differentiation error becomes

\begin{matrix} aligncenter \\ align-1 \frac{normald}{d t} {boldz}^{1} & align-2 = - {boldK}^{bold1} {bold-italicϕ}^{1} ({boldz}^{1}) + {boldz}^{2} \\ align-1 \frac{normald}{d t} {boldz}^{2} & align-2 = - {boldK}^{bold2} {bold-italicϕ}^{2} ({boldz}^{1}) - \end{matrix}

…”

Section: Multivariable Algorithmmentioning

confidence: 99%

“…To conclude these remarks, please notice that, in this case, the switching manifold is described by

double-struckS = false{\cup_{i = 1}^{n} z_{i}^{1} = 0 false}

; therefore, we foresee the appearance of chattering whenever any of the states

z_{i}^{1}

converge to zero. We refer the interested reader to the work of López‐Caamal and Moreno for a more detailed discussion.…”

Section: Generalised Multivariable Stamentioning

confidence: 99%

Section: Generalised Multivariable Stamentioning

confidence: 99%

Section: Multivariable Algorithmmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Generalised multivariable supertwisting algorithm

Intl J Robust & Nonlinear

2018

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A quasicontinuous multivariable super‐twisting observer for 2n states systems with Lipschitz nonlinearities

Intl J Robust & Nonlinear

2022

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In this article we present a global, finite‐time observer for continuous‐time nonlinear systems with states whose nonlinearities are characterized by a sufficiently small, globally Lipschitz constant. We consider the measurement of independent linear combinations of the system's states and assume that the system is subject to a class of external perturbations and/or an unknown input with a bounded norm for every time. To make use of this observer one requires the linear part of the plant to be observable. Furthermore, our design takes into account the linear part of the plant, instead of considering it as a perturbation in the observer's error dynamics. The observer is endowed with two correction terms: a Luenberger‐like gain and the nonlinearities of the generalized multivariable super‐twisting algorithm; and it is designed in the original plant's coordinates rather that designing it in a form obtained via the transformation of the state space. To finalize, we assess the observer performance via numerical simulations.

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Unmeasured Concentrations and Reaction Rates Estimation in CSTRs

2016

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