Fractional Linear Systems and Electrical Circuits

A new, state space, non-integer order model for the heat transfer process is presented. The proposed model is based on a Feller semigroup one, the derivative with respect to time is expressed by the non-integer order Caputo operator, and the derivative with respect to length is described by the non-integer order Riesz operator. Elementary properties of the state operator are proven and a formula for the step response of the system is also given. The proposed model is applied to the modeling of temperature distribution in a one dimensional plant. Results of experiments show that the proposed model is more accurate than the analogical integer order model in the sense of the MSE cost function.

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“…We will start the set of elementary ideas with recalling the idea of the gamma Euler function (see, e.g., Kaczorek and Rogowski, 2014).…”

Section: Preliminariesmentioning

confidence: 99%

“…Consequently, the inverse Laplace transform for a non-integer order function is expressed as follows (see, e.g., Kaczorek and Rogowski, 2014):…”

Section: Definition 7 the Riesz Definition Of The Fo Spatial Derivatmentioning

confidence: 99%

Modeling heat distribution with the use of a non-integer order, state space model

Oprzźdkiewicz

Gawin

Mitkowski

2016

International Journal of Applied Mathematics and Computer Science

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“…The first one involves the study of the frequency properties and characteristics of circuits of the class RL β C α [21,25,26,35,36] and circuits of the class RLC α with supercapacitors [9,30,33,34], analysis of transient states in such systems [3,7,9,10,12] and analysis of their stability [11,22].…”

Section: Introductionmentioning

confidence: 99%

“…There are a few works concerning this issue [3,7,9,10,12]. The paper [9] presents an analysis of an electrical circuit of the class RC α , where the voltage excitation is a unit step function and it analyzes the delay, rise and settling times of the fractionalorder circuit.…”

Section: Introductionmentioning

confidence: 99%

“…For the specific case, α = β, an analytical solution has been found, but for the general case, α = β, solutions have been found numerically. Generalized method of solving transient states in circuits of the class RL β C α , by defining state equations of the concerned circuit, has been described in the work [12]. In contrast to previous works, this paper describes a method of analysis of transient states in a series circuit of the class RL β C α for any kind of the voltage excitation and arbitrary values of parameters α, β ∈ R + , resulting in closed form solutions.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of the Transient State in a Series Circuit of the Class $$RL_{\beta }C_{\alpha }$$ R L β C α

Jakubowska¹,

Walczak²

2016

Circuits Syst Signal Process

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Results of analysis of transient states in a series circuit of the class RL β C α , supplied by an ideal voltage source, have been described in the paper. This circuit consists of a coil L β and a supercapacitor C α described by fractional-order differential equations. A method for determining the current and voltage waveforms in the analyzed circuit, based on the decomposition of rational functions into partial fractions, has been described. This method allows to determine transient waveform shapes in the system for any kind of voltage excitation. Two cases of the problem solutions have been considered. The first case concerns a situation where poles of rational functions are real, and the second where rational functions have complex poles. Effective relations enabling the determination of transient waveforms in a closed form have been given. Analytical formulae describing transient state waveforms in the system for different types of voltage excitations: constant, monoharmonic, periodic and arbitrary being an element of a Hilbert space, have been determined, too. The obtained results have been illustrated by an example.

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Interval observers for linear functions of state vectors of linear fractional‐order systems with delayed input and delayed output

Huong

2018

Adaptive Control & Signal

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This paper addresses the problem of interval observer design for linear functions of state vectors of linear fractional-order systems, which are subjected to time delays in the measured output as well as the control input. By using the information of both the delayed output and input, we design two linear functional state observers to compute two estimates, an upper one and a lower one, which bound the unmeasured linear functions of state vectors. As a particular case with output delay only, we design a linear functional state observer to estimate (asymptotically) the unmeasured linear functions of state vectors. Existence conditions of such observers are provided, and some of them are translated into a linear programming problem, in which the observers' matrices can be effectively computed. Constructive design algorithms are introduced. Numerical examples are provided to illustrate the design procedure, practicality, and effectiveness of the proposed design method. KEYWORDS fractional-order systems, linear functional state observers, linear programming, positive systems, time-delay systems Int J Adapt Control Signal Process. 2019;33:445-461.wileyonlinelibrary.com/journal/acs

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Fractional Linear Systems and Electrical Circuits

Cited by 260 publications

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Modeling heat distribution with the use of a non-integer order, state space model

Modeling heat distribution with the use of a non-integer order, state space model

Analysis of the Transient State in a Series Circuit of the Class $$RL_{\beta }C_{\alpha }$$ R L β C α

Interval observers for linear functions of state vectors of linear fractional‐order systems with delayed input and delayed output

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