Stability analysis for discrete-time fractional-order LTI state-space systems. Part II: New stability criterion for FD-based systems

Stanisławski, Rafał; Latawiec, Krzysztof J.

doi:10.2478/bpasts-2013-0035

Cited by 42 publications

(47 citation statements)

References 13 publications

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“…Equation (25) with respect to (21)(22)(23)(24) at the time step (k + 1) turns to the following form:…”

Section: Resultsmentioning

confidence: 99%

“…This approach has been applied by many Authors, for example: [10], [13], [14], [23]. To do it let us introduce extended state and control vectors, denoted by x q and u q , respectively:…”

Section: Resultsmentioning

confidence: 99%

“…The stability analysis of discrete time approximations to fractional-order systems is usually based on the analysis of stability/instability areas with respect to eigenvalues of the state matrix of the actual system, or poles of its characteristic equation (see e.g. [20][21][22][23]). In this paper we will present a similar solution.…”

Section: Stability Analysismentioning

confidence: 99%

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A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equation

Oprzędkiewicz¹,

Stanisławski²,

Gawin³

et al. 2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method is simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with any approximator to the operator expressed by a discrete transfer function, e.g. CFE-based Al-Alaoui approximation. A simulation example confirms the usefulness of the method. A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equationbstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with ny approximator to the operator expressed by a discrete transfer function, e.g. CFE-based Al-Alaoui approximation. A simulation example onfirms the usefulness of the method. Many real applications, to mention model-based conrol, model-based fault detection, require to implement a oninteger-order model at a digital platform like PLC or PGA. Known discrete-time state space models of nonintegerrder systems are typically based on the Grünvald-Letnikov GL) definition. An accurate implementation of this model, in articular at the bounded resource platforms, requires a (very) ong-length approximation of the GL-based system [33].The purpose of this paper is to propose a new, discrete-time tate space model of a noninteger-order system, constructed ith the use of the continuous fraction expansion (CFE) imlemented for the Al-Alaoui operator. The use of such an aproximant enables to obtain a much more effective model in hat the memory length is quite low. This recommends its use t industrial digital platforms.The paper is organized as follows. Heaving recalled the ackground of the paper in Section 1, Section 2 outlines the undamentals of fractional-order calculus and introduces a operator is defined aswhere a and t denote time limits for calculation of the operator, α ∈ R denotes the noninteger order of the operation.Next, an idea of the Gamma (Euler) function (see for example [12]) can be given: DEFINITION 2. The Gamma function is defined asThe fractional-order, integro-differential operator (1) can be described by different definitions, given by Grünvald and Letnikov (GL definition), Riemann and Liouville (RL definition) and Caputo (C definition). All these definitions are given below. With respect to particular additional assumptions these definitions can be considered equivalent. A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equationAbstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method is simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with any approximator to the operator expressed by a discrete tra...

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“…Equation (25) with respect to (21)(22)(23)(24) at the time step (k + 1) turns to the following form:…”

Section: Resultsmentioning

confidence: 99%

“…This approach has been applied by many Authors, for example: [10], [13], [14], [23]. To do it let us introduce extended state and control vectors, denoted by x q and u q , respectively:…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equation

Oprzędkiewicz¹,

Stanisławski²,

Gawin³

et al. 2017

Bulletin of the Polish Academy of Sciences Technical Sciences

Self Cite

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show abstract

“…This dynamic element is described by linear time-invariant first-order difference equation; see [11,12]. As a generalization of the classical discrete integrator, we can consider discrete summation of fractional order [3,[13][14][15][16][17][18][19][20]. In this paper, we propose a generalization of the fractional-order discrete integrator and call it the variable-, fractional-order discrete-time integrator.…”

Section: Introductionmentioning

confidence: 99%

Generalized Fractional-Order Discrete-Time Integrator

Mozyrska

Ostalczyk

2017

Complexity

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“…The problems of stability of linear continuous-time and discrete-time fractional order systems, standard and positive, have been investigated in the above mentioned monographs and in many papers, see [5][6][7][8][9][10][11][12] for example, and references therein.…”

Section: Introductionmentioning

confidence: 99%

Stability conditions for linear continuous-time fractional-order state-delayed systems

Busłowicz

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. The stability problem of continuous-time linear fractional order systems with state delay is considered. New simple necessary and sufficient conditions for the asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix and time delay. It is shown that in the complex plane there exists such a region that location in this region of all eigenvalues of the state matrix multiplied by delay in power equal to the fractional order is necessary and sufficient for the asymptotic stability. Parametric description of boundary of this region is derived and simple new analytic necessary and sufficient conditions for the stability are given. Moreover, it is shown that the stability of the fractional order system without delay is necessary for the stability of this system with delay. The considerations are illustrated by a numerical example.

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Stability analysis for discrete-time fractional-order LTI state-space systems. Part II: New stability criterion for FD-based systems

Cited by 42 publications

References 13 publications

A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equation

A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equation

Generalized Fractional-Order Discrete-Time Integrator

Stability conditions for linear continuous-time fractional-order state-delayed systems

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