Implementing Discrete-time Fractional-order Controllers

The modelling of fractional linear systems through ARMA models is addressed. To perform this study, a new recursive algorithm for impulse response ARMA modelling is presented. This is a general algorithm that allows the recursive construction of ARMA models from the impulse response sequence. This algorithm does not need an exact order specification, as it gives some insights into the correct orders. It is applied to modelling fractional linear systems described by fractional powers of the backward difference and the bilinear transformations. The analysis of the results leads to propose suitable models for those systems. IntroductionPseudo-fractional auto-regressive moving average (ARMA) modelling is a pole-zero modelling of fractional linear systems. These are described by fractional differential equations in the continuous-time case or auto-regressive integrated moving average (ARIMA) models in the discrete-time case. The first case is based on the definition of fractional differintegration, whereas the second deals with the fractional differencing that is a fractional version of the well-known finite differences. These systems are characterised by having a long memory that cannot be explained by the usual linear systems that have short memory (exponential). The desire of finding a theoretical base for such systems led to the fractional calculus that has recently received a great deal of attention in the scientific literature, through the publication of books, special issues of journals, review articles, as well as a very large number of research papers. The interest in fractional calculus comes from the fact that it provides foundations for the understanding of several natural phenomena and the basic theory for building models for the systems underlying them. However, adoption of the fractional calculus by the physicists and engineering community was inhibited historically by the lack of clear experimental evidence for its need and by the difficulty in constructing simple models for simulation or even implementation of simple fractional systems. Fractional calculus is almost as old as the common calculus, but only since 30 years ago it has been a subject of specialised publications and conferences.The basic building block of this kind of systems is the non-integer order derivative and integral that have been approximated by fractional powers of the backward difference or the bilinear transformations -the former is exactly the building block of the fractional differencing, as said earlier. However, these approximations are described infinite impulse response (IIR) systems with nonrational transfer functions. For these, ARMA models are only approximations. However, the usefulness of ARMA models makes them very interesting when constructing discrete-time approximating models for fractional systems. In the last few years, a lot of attempts to obtain such models have been done [1 -5]. However, it is not clear how to perform such modelling and how to choose the most suitable orders, although there are a l...

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“…This result is stated in the following theorem (for proof refer to Machado [2]). As consequence of this theorem, we obtain…”

Section: Algorithmmentioning

confidence: 87%

Pseudo-fractional ARMA modelling using a double Levinson recursion

Ortigueira

Serralheiro

2007

IET Control Theory &Amp; Applications

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“…Petráš [64] presented methods of tuning and implementation of Fractional-Order Controllers. Machado discussed the design of fractional-order discrete-time controllers [86] and proposed a new method for optimal tuning of fractional controllers by using genetic algorithms [87]. However, publications on the FOCP of stochastic systems are very limited.…”

Section: Stochastic Fractional Optimal Controlmentioning

confidence: 99%

Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping

Chen

Zhu

2013

fcaa

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In the present survey, some progress in the stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping is reviewed. First, the stochastic averaging method for quasi integrable Hamiltonian systems with fractional derivative damping under various random excitations is briefly introduced. Then, the stochastic stability, stochastic bifurcation, first passage time and reliability, and stochastic fractional optimal control of the systems studied by using the stochastic averaging method are summarized. The focus is placed on the effects of fractional derivative order on the dynamics and control of the systems. Finally, some possible extensions are pointed out.MSC 2010 : Primary 26A33; Secondary 33E12, 34A08, 34K37, 35R11, 60G22

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“…However, we intend to find approximations using finite orders ARMA models. There have been a lot of attempts to do it [4][5][6]8]. Here we propose a new LS identification algorithm different from that one proposed in [4].…”

Section: The Algorithmmentioning

confidence: 99%

“…For them, ARMA models are only approximations that we call pseudo-fractional ARMA models [3]. In the last few years, a lot of attempts to obtain such models have been done (see [4][5][6][7][8][9]). However, it remains to clarify two important questions: (a) how to perform such modeling and (b) how to choose the most suitable orders.…”

Section: Introductionmentioning

confidence: 99%

A new least-squares approach to differintegration modeling

Ortigueira

Serralheiro

2006

Signal Processing

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In this paper a new least-squares (LS) approach is used to model the discrete-time fractional differintegrator. This approach is based on a mismatch error between the required response and the one obtained by the difference equation defining the auto-regressive, moving-average (ARMA) model. In minimizing the error power we obtain a set of suitable normal equations that allow us to obtain the ARMA parameters. This new LS is then applied to the same examples as in

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Implementing Discrete-time Fractional-order Controllers

Cited by 83 publications

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Pseudo-fractional ARMA modelling using a double Levinson recursion

Pseudo-fractional ARMA modelling using a double Levinson recursion

Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping

A new least-squares approach to differintegration modeling

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