Stabilization of MISO fractional systems with delays☆
Introduction
Fractional systems are systems described by differential equations involving non-integer order derivatives and/or integrals. Consequently, in the frequency domain, their transfer functions contain non-integer powers of the Laplace variable . This kind of models has become more popular in many fields in the past two decades since it provides a better fit to data being then more succinct than a standard model. Refer, for example, to Miller and Ross (1993) for basic backgrounds on fractional calculus and to Freeborn (2013) and Sabatier, Agrawal, and Machado (2007) for its recent applications on modeling.
Delays are encountered almost everywhere due, for example, to distance of transmission and it is well-known that they have important effects on the stability of the systems (Richard, 2003).
While integer-order systems with delays have been intensively studied (Richard, 2003), the literature on fractional systems with delays is still quite small. Particularly, the stabilization problem of fractional systems with delays has rarely been addressed. Some available studies are classical (Özbay, Bonnet, & Fioravanti, 2012) and fractional PID controller design (Hamamci, 2007), fractional sliding mode control (Si-Ammour, Djennoune, & Bettayeb, 2009), factorization approach to control synthesis Bonnet and Partington (2002), Bonnet and Partington (2007).
In the framework of fractional representation approach to synthesis problems (Vidyasagar, 1985), SISO fractional delay systems was considered in Bonnet and Partington (2002), Bonnet and Partington (2007) and coprime factorizations together with the corresponding Bézout factors of the transfer function of these systems have been derived. In Curtain, Weiss, and Weiss (1996), coprime factors were presented for a large class of MIMO infinite-dimensional systems which include delay systems. The factors were determined from a state-space realization of the (regular) system which was given in terms of the semigroup of the system. Such realizations are not much considered for fractional systems.
For the particular class of MIMO (integer-order) systems with I/O delays, the problem of parametrization of stabilizing controllers was solved in Mirkin and Raskin (1999) and Moelja and Meinsma (2003). The idea was to reduce the problem to an equivalent finite-dimensional stabilization problem by involving an unstable finite-dimensional system and a stable infinite-dimensional system (FIR filter). In Mondié and Loiseau (2004), a procedure to compute right coprime factorizations over a Bézout domain was proposed for spectrally controllable MIMO (integer-order) systems with input delays. For MISO structure, a class of (integer-order) systems with multiple transmission delays was studied in Bonnet and Partington (2004) and coprime factorizations and associated Bézout factors over were derived.
In this paper, we are interested in the stabilization problem of MISO fractional systems with different I/O delays which are not necessarily commensurate. This MISO structure appeared, for example, in communication systems (Quet, Ataşlar, İftar, Özbay, Kalyanaraman, & Kang, 2002). We would like to obtain the set of all stabilizing controllers by determining a doubly coprime factorization over of the transfer matrix and the associated Bézout factors, which allow the construction of the Youla–Kučera parametrization (Vidyasagar, 1985). As in the finite-dimensional case, the Youla–Kučera parametrization gives the set of all -stabilizing controllers in terms of one free parameter. Note that in Quadrat (2006), a parametrization of the set of all stabilizing controllers is given in terms of two free parameters for MIMO systems once we already know a particular stabilizing controller. Our strategy here is to work directly on the Bézout identity in order to get explicit expressions of Bézout factors in terms of the matrix transfer function. Such explicit expressions could not be easily derived in Mirkin and Raskin (1999), Moelja and Meinsma (2003) and Mondié and Loiseau (2004) even in the case of standard delay systems. We hope that the explicit form will facilitate the use of these factors in controllers design while the use of the frequency domain representation of the systems agrees well with the modeling practice of fractional systems (Sabatier et al., 2007).
The paper is organized as follows. In Section 2, the class of systems of interest and some background are presented. The results are stated in Sections 3 Left coprime factorizations and Bézout factors, 4 Right coprime factorizations and Bézout factors. We give in Section 3 explicit expressions of left coprime factorizations and associated Bézout factors over of the transfer function of the systems under study. Right coprime factorizations and right Bézout factors are given in Section 4 for a large subclass of the class of systems considered. Examples are provided to illustrate the results. Finally, Section 5 gives conclusions and perspectives.
Section snippets
A class of MISO fractional time-delay systems
We consider systems described by transfer matrices of the form where for are the delays; , ; , where and are polynomials of integer degree in , and have no common roots, and for ; is the degree in of ; is in the principle branch , that is , in order to guarantee a unique value of the transfer function involving with .
Some notations used
Left coprime factorizations and Bézout factors
In this section, we present l.c.f.’s and Bézout factors for the transfer matrix (1).
Right coprime factorizations and Bézout factors
The previous section showed that the systems under study admit l.c.f.’s over . Since is a Hermite ring, then there exist r.c.f.’s for (Smith, 1989).
While l.c.f.’s and left Bézout factors of our transfer matrices are vectors and scalars, the right counterparts are vectors and square matrices with a lot more entries to be determined. Regarding calculation complexity, we separate the study into two classes of systems. The first class consists of systems with distinct unstable poles,
Conclusion
In this paper, we have considered MISO fractional systems with input or output delays. Explicit expressions of an l.c.f. over of the transfer matrices as well as the corresponding Bézout factors are given. Right coprime factorizations and right Bézout factors are also found for systems with entries of the transfer matrix containing different poles. In the case of identical poles, the right factors are primarily found for some simple classes of systems. Hence, in conclusion, we can have
Le Ha Vy Nguyen was born in Khanh Hoa, Vietnam, in 1987. She received the M.S. degree in Electrical Engineering from Grenoble Institute of Technology, France, in 2011 and obtained a Ph.D. degree in Automatic Control from the University of Paris XI in 2014. From 2015 to 2016, she was a postdoctoral fellow at the University of Namur, Belgium. She is currently a postdoctoral fellow at Inria Saclay - Île-de-France. Her research interests include analysis and control of linear time-delay and
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Le Ha Vy Nguyen was born in Khanh Hoa, Vietnam, in 1987. She received the M.S. degree in Electrical Engineering from Grenoble Institute of Technology, France, in 2011 and obtained a Ph.D. degree in Automatic Control from the University of Paris XI in 2014. From 2015 to 2016, she was a postdoctoral fellow at the University of Namur, Belgium. She is currently a postdoctoral fellow at Inria Saclay - Île-de-France. Her research interests include analysis and control of linear time-delay and fractional systems as well as nonlinear systems.
Catherine Bonnet obtained her Ph.D. degree in Mathematics at the University of Provence, Marseille, France in 1991 and her French Habilitation (HDR) in Mathematics at the University Paris 6, France in 2008.
From 1986 to 1989 she worked as a research fellow with the Aerospatiale company. She was a temporary lecturer in Aix-Marseille Universities, France from 1990 to 1993, and then a postdoc at the University of Leeds, U.K. and a postdoc at Inria Rocquencourt, France where she became a researcher in 1994.
In 2010, she joined Inria Saclay-Île-de-France where she is currently leading the DISCO team.
Her research interests include approximation, robust control, delay and fractional systems as well as modeling in medicine.
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The material in this paper was partially presented at the 20th International Symposium on Mathematical Theory of Networks and Systems, July 9–13, 2012, Melbourne, Australia and at the 1st IFAC Workshop on Control of Systems Modeled by Partial Differential Equations, September 25–27, 2013, Paris, France. This paper was recommended for publication in revised form by Associate Editor Hitay Ozbay under the direction of Editor Miroslav Krstic.