Elsevier

Automatica

Volume 49, Issue 6, June 2013, Pages 1907-1913
Automatica

Brief paper
Robust stabilization of uncertain descriptor fractional-order systems

https://doi.org/10.1016/j.automatica.2013.02.066Get rights and content

Abstract

This paper presents sufficient conditions for the robust asymptotical stabilization of uncertain descriptor fractional-order systems with the fractional order α satisfying 0<α<2. The results are obtained in terms of linear matrix inequalities. The parameter uncertainties are assumed to be time-invariant and norm-bounded appearing in the state matrix. A necessary and sufficient condition for the normalization of uncertain descriptor fractional-order systems is given via linear matrix inequality (LMI) formulation. The state feedback control to robustly stabilize such uncertain descriptor fractional-order systems with the fractional order α belonging to 0<α<2 is derived. Two numerical examples are given to demonstrate the applicability of the proposed approach.

Introduction

Recently, fractional-order systems have been studied by many authors in engineering science from an application point of view (see Podlubny, 1999, Hilfer, 2001, Kilbas, Srivastava, & Trujillo, 2006 and references therein). Many systems can be described with the help of fractional derivatives: electromagnetic systems (Engheta, 1996), dielectric polarization (Sun, Abdelwahad, & Onaral, 1984), viscoelastic systems (Bagley and Calico, 1991, Rossikhin and Shitikova, 1997).

Moreover, descriptor systems have been of great interest in the specialized literature because they have many applications (Dai, 1989) in electrical circuit networks, robotics and economics. Intuitively, a descriptor state space description of linear systems is more general than a conventional state space description. Descriptor systems are described by a mixture of differential equations and algebraic equations. The method of transforming these systems into normal ones is called a normalization or regularization (Dai, 1988). The question of stability is very important in control theory. For descriptor fractional-order control systems, there are many challenging and unsolved problems related to stability theory such as robust stability, bounded-input bounded-output stability, internal stability, etc. So far, very few works exist for the stability and stabilization issue for descriptor fractional-order systems. In N’Doye, Zasadzinski, Darouach, and Radhy (2010), for the stabilization of such descriptor fractional-order systems with fractional commensurate orders α, 1<α<2, an LMI formulation is used for, and only for the nominal stabilization case.

In this paper, we consider the problem of the robust asymptotical stabilization for uncertain descriptor fractional-order systems. The structure of the uncertainty has been extensively used in many papers when dealing with the problem of robust stabilization for uncertain standard state-space systems in both continuous and discrete time frameworks (Xie and de Souza, 1990, Xu et al., 2001). However, concerning robust stability and stabilization for uncertain descriptor fractional-order systems, such a structure of the uncertainty has not yet been introduced. The descriptor fractional-order system under consideration is subject to unstructured time-invariant parameter uncertainties in the state matrix. A necessary and sufficient condition for the normalization of descriptor fractional-order systems in an LMI formulation is given. A novel form of normalizing stabilizing controllers that involve the action of derivative and a state feedbacks is introduced. The use of a proportional and derivative feedback has a strong engineering motivation and so far there have been many research papers published to address the importance and the engineering motivation of proportional and derivative state feedback and proportional and derivative output feedback (Bunse-Gerstner et al., 1999, Lin et al., 2005). With the above motivation, the important issues of stability check and stabilization of uncertain descriptor fractional-order systems are further investigated in this paper.

First, we normalize the descriptor fractional-order systems by applying a derivative controller and second, a state feedback controller is given to achieve the robust asymptotical stabilization of the obtained normalized fractional-order systems. The problem of asymptotical stability and stabilization problems for descriptor fractional-order are still open problems to our best knowledge.

This paper is organized as follows. In Section 2, we provide some preliminaries results on the fractional derivative, the stability conditions of the fractional-order systems, the linear algebra and the matrix theory. In Section 3, the problem of normalization by full state derivative controller is firstly studied. The condition of existence of this controller is formulated in terms of linear matrix inequalities (LMIs). Then sufficient robust stabilization of uncertain descriptor fractional-order systems conditions are derived in terms of linear matrix inequalities. Finally, two illustrative examples are presented to illustrate of our proposed results.

Notations

MT is the transpose of M, Sym{X} is used to denote XT+X, stands for the Kronecker product and Dα represents initialized αth order differintegration. Let ACn×n be a complex matrix, then A=A¯T and A¯ denotes the complex conjugate of A. The null space of matrix A is defined by N(A)={xFn such that Ax=0}, where F represents either R or C.

Section snippets

Preliminary results

In this section, we present some preliminaries results on the fractional derivative systems, the linear algebra and matrix theory which will be used in the sequel of this paper. Formulations of noninteger-order derivatives fall into two main classes: the Riemann–Liouville derivative and the Grûnward–Letnikov derivative, on one hand, defined as (Podlubny, 1999) Dαf(t)=1Γ(nα)dndtnatf(τ)(tτ)αn+1dτ,n1α<n or the Caputo derivative on the other, defined as (Podlubny, 2002), Dαf(t)=1Γ(αn)atdnf(τ

Robust stabilization of uncertain descriptor fractional-order systems

Consider the following uncertain descriptor fractional-order systems {EDαx(t)=(A+ΔA)x(t)+Bu(t)x(0)=x00<α<2 where x(t)Rn is the semi-state vector, u(t)Rm is the control input. Matrix ERn×n is a singular square matrix and ARn×n, BRn×m are constant matrices. ΔA is time-invariant matrix representing a norm-bounded parameter uncertainty, and is assumed to be of the following formΔA=MAΔNA where MA and NA are known real constant matrices of appropriate dimensions, and the uncertain matrix Δ

Numerical examples

In this section, we provide two numerical examples to illustrate the applicability of the proposed method.

Conclusion

In this paper, the robust stabilization of uncertain descriptor fractional-order systems for the fractional-order α belonging to 0<α<2 with parameter uncertainties in the state matrix have been studied. The problem of normalization of descriptor fractional-order systems by derivative controller has been proposed. A necessary and sufficient condition for the existence of such normalizing feedback is given in an LMI formulation. Based on this, the robust asymptotical stabilization of uncertain

Ibrahima N’Doye received his Ph.D. degree in Automatic Control from the University Henri Poincaré of Nancy at the Research Center of Automatic Control (CRAN-CNRS, Université de Lorraine), France and the University Hassan II Ain Chock, Casablanca, Morocco, in 2011. He is a Postdoc in the Research Unit in Engineering Science (RUES) at the University of Luxembourg. His research interests are in estimation and control of fractional-order systems and nonlinear dynamic systems.

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    Ibrahima N’Doye received his Ph.D. degree in Automatic Control from the University Henri Poincaré of Nancy at the Research Center of Automatic Control (CRAN-CNRS, Université de Lorraine), France and the University Hassan II Ain Chock, Casablanca, Morocco, in 2011. He is a Postdoc in the Research Unit in Engineering Science (RUES) at the University of Luxembourg. His research interests are in estimation and control of fractional-order systems and nonlinear dynamic systems.

    Mohamed Darouach graduated from “Ecole Mohammadia d’lngénieurs”, Rabat, Morocco, in 1978, and received the Docteur Ingénieur and Doctor of Sciences degrees from Nancy University, France, in 1983 and 1986, respectively. From 1978 to 1986 he was Associate Professor and Professor of automatic control at Ecole Hassania des Travaux Publics, Casablanca, Morocco. Since 1987 he has been a Professor at Université de Lorraine. He has been a Vice Director of the Research Center in Automatic Control of Nancy (CRAN UMR 7039, Nancy-University, CNRS) from 2005 to 2013. He obtained a degree Honoris Causa from the Technical University of IASI and since 2010 he has been a member of the Scientific council of Luxembourg University. He held invited positions at the University of Alberta, Edmonton. His research interests span theoretical control, observers design, and control of large-scale uncertain systems with applications.

    Michel Zasadzinski received his Ph.D. degree in Automatic Control from the Nancy-Université, France, in 1990. He was Assistant Professor at the Université Henri Poincaré and, from 1992 to 2000, he has been a CNRS Researcher in the Centre de Recherche en Automatique de Nancy (CRAN, CNRS). Michel Zasadzinski is now Professor at the Institut Universitaire de Technologie (Longwy, Université de Lorraine (former Nancy-Université)). His research interests encompass the theory and application of robust control and filtering for linear and nonlinear systems, and for stochastic differential equations.

    Nour-Eddine Radhy received a French thesis in Microwaves Engineering from the University of Lille (France) in 1985 and Ph.D. in Automatic Control from University Hassan II of Casablanca (Morocco) in 1992, where he is now a Professor in the Department of Physics (Laboratory of Automatic Control). His current research interests are in model reduction, stability analysis, robust control, identification, control of constrained systems, invariant sets and fuzzy systems.

    The material in this paper was partially presented at the 18th IFAC World Congress, August 28–September 2, 2011, Milano, Italy. This paper was recommended for publication in revised form by Associate Editor Delin Chu under the direction of Editor Ian R. Petersen.

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