Journal of Function Spaces

Volume 2018, Article ID 4980434, 8 pages

https://doi.org/10.1155/2018/4980434

## A New Sufficient Condition for Checking the Robust Stabilization of Uncertain Descriptor Fractional-Order Systems

^{1}School of Mathematics, Southeast University, Nanjing 210096, China^{2}School of Science, Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, Nanning 530006, China^{3}School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

Correspondence should be addressed to Hongxing Wang; moc.361@2090gnixgnohgniw

Received 14 April 2018; Accepted 11 June 2018; Published 18 July 2018

Academic Editor: Xinguang Zhang

Copyright © 2018 Hongxing Wang and Aijing Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider the robust asymptotical stabilization of uncertain a class of descriptor fractional-order systems. In the state matrix, we require that the parameter uncertainties are time-invariant and norm-bounded. We derive a sufficient condition for the system with the fractional-order satisfying in terms of linear matrix inequalities (LMIs). The condition of the proposed stability criterion for fractional-order system is easy to be verified. An illustrative example is given to show that our result is effective.

#### 1. Introduction

Descriptor systems arise naturally in many applications such as aerospace engineering, social economic systems, and network analysis. Sometimes we also call descriptor systems singular systems. Descriptor system theory is an important part in control systems theory. Since 1970s, descriptor systems have been wildly studied, for example, descriptor linear systems [1], descriptor nonlinear systems [2–4], and discrete descriptor systems [5–7]. In particular, Dai has systematically introduced the theoretical basis of descriptor systems in [8], which is the first monograph on this subject. A detailed discussion of descriptor systems and their applications can be found in [9, 10].

It is well known that fractional-order systems have been studied extensively in the last 20 years, since the fractional calculus has been found many applications in viscoelastic systems [11–14], robotics [15–18], finance system [19–21], and many others [22–26]. Studying on fractional-order calculus has become an active research field. To the best of our knowledge, although stability analysis is a basic problem in control theory, very few works existed for the stability analysis for descriptor fractional-order systems.

Many problems related to stability of descriptor fractional-order control systems are still challenging and unsolved. For the nominal stabilization case, N’Doye et al. [27] study the stabilization of one descriptor fractional-order system with the fractional-order , , in terms of LMIs. N’Doye et al. [28] derive some sufficient conditions for the robust asymptotical stabilization of uncertain descriptor fractional-order systems with the fractional-order satisfying . Furthermore, Ma et al. [29] study the robust stability and stabilization of fractional-order linear systems with positive real uncertainty. Note that, in Example 1, by applying Theorem 2 [27], it is harder to determine whether the uncertain descriptor fractional-order system (6) is asymptotically stable. Therefore, it is valuable to seek sufficient conditions, for checking the robust asymptotical stabilization of uncertain descriptor fractional-order systems.

In this paper, we study the stabilization of a class of descriptor fractional-order systems with the fractional-order , , in terms of LMIs. We derive a new sufficient condition for checking the robust asymptotical stabilization of uncertain descriptor fractional-order systems with the fractional-order satisfying , in terms of LMIs. It should be mentioned that, compared with some prior works, our main contributions consist in the following: we assume that the matrix of uncertain parameters in the uncertain descriptor fractional-order system is diagonal. Thus, compared with the results in [28], our conclusion, Theorem 8, is more feasible and effective and has wider applications; compared with some stability criteria of fractional-order nonlinear systems, for example, in [9, 22], our method is easier to be used.

Notations: throughout this paper, stands for the set of by matrices with real entries, stands for the transpose of , denotes the expression , denotes the identity matrix of order , denotes the diagonal matrix, and will be used in some matrix expressions to indicate a symmetric structure; i.e., if given matrices and , then

#### 2. Preliminary Results

Consider the following class of linear fractional-order systems:where is the fractional-order, is the state vector, is a constant matrix, and represent the fractional-order derivative, which can be expressed as where is the Euler Gamma function. For convenience, we use to replace in the rest of this paper. It is well known that system (2) is stable if [30–32]where and is the spectrum of all eigenvalues of .

The next lemma, given by Chilali et al. [33], contains the necessary and sufficient conditions of (4) in terms of LMI, when the fractional-order belongs to .

Lemma 1 (see [33]). *Let be a real matrix and . Then if and only if there exists such that*

Consider the following uncertain descriptor fractional-order systems:where , is the semistate vector, is the control input, is singular, and are constant matrices, and the time-invariant matrix corresponds to a norm-bounded parameter uncertainty, which is the following form: where and are real constant matrices of appropriate sizes, and the uncertain matrix satisfies

*Remark 2. *Condition is rational because a lot of system uncertainties satisfy this inequality. Besides, this condition can also be used in many literatures, for example, in [9, 34–39].

It is well known that the following systemis normalizable if and only if Further we have that the uncertain descriptor fractional-order systems (6) is normalizable if and only if the nominal descriptor fractional-order system (9) is normalizable.

Lemma 3 (see [28], Theorem 1). *System (6) is normalizable if and only if there exist a nonsingular matrix and a matrix such that the following LMIis satisfied. In this case, the gain matrix is given by *

Assume that (6) is normalizable; by applying LMI (11), we obtain such that . Consider the feedback control for (6) in the following form:where is one gain matrix such that the obtained normalized system is asymptotically stable. Then we have the closed-loop system: that is,where

To facilitate the description of our main results, we need the following results.

In [28], N’Doye et al. derive a sufficient condition for the robust asymptotical stabilization of uncertain descriptor fractional-order systems with the fractional-order satisfying in terms of LMIs.

Lemma 4 (see [28], Theorem 2). *Assume that (6) is normalizable; then there exists gain matrix such that the uncertain descriptor fractional-order system (6) with fractional-order controlled by the control (13) is asymptotically stable, if there exist matrices , and a real scalar , such thatwhere with and matrices and are given by LMI (11).**Moreover, the gain matrix is given by *

Lemma 5 (see [40]). *For any matrices and with appropriate sizes, we have for any .*

Lemma 6 (see [41]). *Let , , and be real matrices of appropriate sizes. Then, for any ,*

#### 3. Main Result

In this section, we present a new sufficient condition to design the gain matrix . In the following theorem, and are given nonsingular matrices, such that

From now on, we denote , , and . It is obvious that . Thus, for any and , by using Lemmas 5 and 6 and , we have andthat is,

*Remark 7. *Note that, when , we have and That is, for any real scalar , and two matrices and , we cannot obtain real scalars and such that where

Theorem 8. *Assume that (6) is normalizable; then there exists a gain matrix such that the uncertain descriptor fractional-order system (6) with fractional-order controlled by the controller (13) is asymptotically stable, if there exist matrices , and two real scalars and , such thatwhere with and matrices and are given by LMI (11).**Moreover, the gain matrix is given by *

*Proof. *Suppose that there exist matrices , and two real scalars and such that (29) holds. It is easy to derive thatBy using the Schur complement of (29), one obtainsWrite . It follows from applying (25) thatBy using the above inequality (34) and Lemma 1, we obtain Therefore, system (6) is asymptotically stable. This ends the proof.

*Remark 9. *Write Note that if we choose and in LMI (29), It is easy to see the following: (1)For given , when , it is always true that ; that is, there do not exist and such that . Therefore, Theorem 8 is not a special case of Lemma 4 [28, Theorem 2], when and .(2)For given and , when , there exists such that is positive definite; that is, there exists such that . Since conditions in Lemma 4 and Theorem 8 are both sufficient, we cannot derive Lemma 4 by applying Theorem 8; that is, Theorem 8 is not a generalization of Lemma 4 [28, Theorem 2].

#### 4. A Numerical Example

In this section, we assume that the matrix of uncertain parameters in the uncertain descriptor fractional-order system (6) is diagonal. We provide a numerical example to illustrate that Theorem 8 is feasible and effective with wider applications.

*Example 1. *Consider the uncertain descriptor fractional-order system described in (6) with parameters as follows: where .

It is easy to check that ; that is, is singular. By applying the LMI (11), we obtain and the gain matrix It follows from (16) that Firstly, we compute , , , and by using Lemma 4 [28, Theorem 2]. A feasible solution of LMI (11) is as follows: We choose It follows from (15) that and the arguments of all eigenvalues of are Based on those results, it is debatable whether or not system (6) is stable.

In the second way, we compute , , , , and by using Theorem 8; we choose It is easy to check that It follows that a feasible solution of LMI (11) is asymptotically stabilizing state-feedback gain is and the arguments of all eigenvalues of are Therefore, system (6) is stable.

#### 5. Conclusion

In this paper, the robust asymptotical stability of uncertain descriptor fractional-order systems (6) with the fractional-order belonging to has been studied. We derive a new sufficient condition for checking the robust asymptotical stabilization of (6) in terms of LMIs. Out results can be seen as a generalization of [28, Theorem 2]. By adding appropriate parameters into LMIs, our result has wider applications. One special numerical example has shown that our results are feasible and easy to be used.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The first author was supported partially by China Postdoctoral Science Foundation [Grant no. 2015M581690], the National Natural Science Foundation of China [Grant no. 11361009], High level innovation teams and distinguished scholars in Guangxi Universities, the Special Fund for Scientific and Technological Bases and Talents of Guangxi [Grant no. 2016AD05050], and the Special Fund for Bagui Scholars of Guangxi. The second author was supported partially by the National Natural Science Foundation of China [Grant no. 11701320] and the Shandong Provincial Natural Science Foundation [Grant no. ZR2016AM04].

#### References

- D. G. Luenberger, “Boundary recursion for descriptor variable systems,”
*Institute of Electrical and Electronics Engineers Transactions on Automatic Control*, vol. 34, no. 3, pp. 287–292, 1989. View at Publisher · View at Google Scholar · View at MathSciNet - D. G. Luenberger, “A double look at duality,”
*Institute of Electrical and Electronics Engineers Transactions on Automatic Control*, vol. 37, no. 10, pp. 1474–1482, 1992. View at Publisher · View at Google Scholar · View at MathSciNet - C. Liu and Y.-J. Peng, “Stability of periodic steady-state solutions to a non-isentropic Euler-Maxwell system,”
*Zeitschrift für Angewandte Mathematik und Physik*, vol. 68, no. 5, Art. 105, 17 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - Y.-H. Feng and C.-M. Liu, “Stability of steady-state solutions to Navier-Stokes-Poisson systems,”
*Journal of Mathematical Analysis and Applications*, vol. 462, no. 2, pp. 1679–1694, 2018. View at Publisher · View at Google Scholar · View at MathSciNet - F. Li and Y. Bao, “Uniform stability of the solution for a memory-type elasticity system with nonhomogeneous boundary control condition,”
*Journal of Dynamical and Control Systems*, vol. 23, no. 2, pp. 301–315, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Zheng, Y. Shang, and X. Peng, “Orbital stability of solitary waves of the coupled Klein-Gordon-Zakharov equations,”
*Mathematical Methods in the Applied Sciences*, vol. 40, no. 7, pp. 2623–2633, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Zheng, Y. Shang, and X. Peng, “Orbital stability of periodic traveling wave solutions to the generalized Zakharov equations,”
*Acta Mathematica Scientia*, vol. 37, no. 4, pp. 998–1018, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. L. Debeljkovic, “Singular control systems,”
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis*, vol. 11, no. 5-6, pp. 691–705, 2004. View at Google Scholar · View at MathSciNet - H. Liu, Y. Pan, S. Li, and Y. Chen, “Synchronization for fractional-order neural networks with full/under-actuation using fractional-order sliding mode control,”
*International Journal of Machine Learning and Cybernetics*, pp. 1–14, 2017. View at Publisher · View at Google Scholar - G.-R. Duan,
*Analysis and Design of Descriptor Linear Systems*, vol. 23 of*Advances in Mechanics and Mathematics*, Springer, New York, NY, USA, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - X. Du and A. Mao, “Existence and Multiplicity of Nontrivial Solutions for a Class of Semilinear Fractional Schrödinger Equations,”
*Journal of Function Spaces*, vol. 2017, pp. 1–7, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - H. Qin, X. Zuo, J. Liu, and L. Liu, “Approximate controllability and optimal controls of fractional dynamical systems of order (Formula presented.) in Banach spaces,”
*Advances in Difference Equations*, vol. 2015, no. 1, pp. 1–17, 2015. View at Google Scholar · View at Scopus - S. Dadras and H. R. Momeni, “Control of a fractional-order economical system via sliding mode,”
*Physica A: Statistical Mechanics and its Applications*, vol. 389, no. 12, pp. 2434–2442, 2010. View at Publisher · View at Google Scholar · View at Scopus - X. Zhang, L. Liu, Y. Wu, and Y. Cui, “New Result on the Critical Exponent for Solution of an Ordinary Fractional Differential Problem,”
*Journal of Function Spaces*, vol. 2017, pp. 1–4, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - M. Wang and G.-L. Tian, “Robust group non-convex estimations for high-dimensional partially linear models,”
*Journal of Nonparametric Statistics*, vol. 28, no. 1, pp. 49–67, 2016. View at Publisher · View at Google Scholar · View at Scopus - V.-T. Pham, S. T. Kingni, C. Volos, S. Jafari, and T. Kapitaniak, “A simple three-dimensional fractional-order chaotic system without equilibrium: Dynamics, circuitry implementation, chaos control and synchronization,”
*AEÜ - International Journal of Electronics and Communications*, vol. 78, pp. 220–227, 2017. View at Publisher · View at Google Scholar · View at Scopus - K. Rajagopal, L. Guessas, A. Karthikeyan, A. Srinivasan, and G. Adam, “Fractional Order Memristor No Equilibrium Chaotic System with Its Adaptive Sliding Mode Synchronization and Genetically Optimized Fractional Order PID Synchronization,”
*Complexity*, vol. 2017, pp. 1–19, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - X. Zhang, C. Mao, L. Liu, and Y. Wu, “Exact iterative solution for an abstract fractional dynamic system model for bioprocess,”
*Qualitative Theory of Dynamical Systems*, vol. 16, no. 1, pp. 205–222, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - J. Wang, Y. Yuan, and S. Zhao, “Fractional factorial split-plot designs with two- and four-level factors containing clear effects,”
*Communications in Statistics—Theory and Methods*, vol. 44, no. 4, pp. 671–682, 2015. View at Publisher · View at Google Scholar · View at MathSciNet - S. Zhao and X. Chen, “Mixed two- and four-level fractional factorial split-plot designs with clear effects,”
*Journal of Statistical Planning and Inference*, vol. 142, no. 7, pp. 1789–1793, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Zhao and X. Chen, “Mixed-level fractional factorial split-plot designs containing clear effects,”
*Metrika. International Journal for Theoretical and Applied Statistics*, vol. 75, no. 7, pp. 953–962, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - H. Liu, S. Li, H. Wang, and Y. Sun, “Adaptive fuzzy control for a class of unknown fractional-order neural networks subject to input nonlinearities and dead-zones,”
*Information Sciences*, vol. 454-455, pp. 30–45, 2018. View at Publisher · View at Google Scholar - H. Liu, Y. Pan, S. Li, and Y. Chen, “Adaptive Fuzzy Backstepping Control of Fractional-Order Nonlinear Systems,”
*IEEE Transactions on Systems, Man, and Cybernetics: Systems*, vol. 47, no. 8, pp. 2209–2217, 2017. View at Publisher · View at Google Scholar - W. Yu, Y. Li, G. Wen, X. Yu, and J. Cao, “Observer design for tracking consensus in second-order multi-agent systems: fractional order less than two,”
*Institute of Electrical and Electronics Engineers Transactions on Automatic Control*, vol. 62, no. 2, pp. 894–900, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - Q. Feng and F. Meng, “Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method,”
*Mathematical Methods in the Applied Sciences*, vol. 40, no. 10, pp. 3676–3686, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. Xu and Y. Zhang, “Generalized Gronwall fractional summation inequalities and their applications,”
*Journal of Inequalities and Applications*, vol. 2015, no. 42, 2015. View at Publisher · View at Google Scholar · View at MathSciNet - I. N'Doye, M. Zasadzinski, M. Darouach, and N.-E. Radhy, “Stabilization of singular fractional-order systems: an lmi approach,” in
*Proceedings of the in Control Automation (MED, 2010 18th Mediterranean Conference on. 1em plus 0.5em minus 0.4em IEEE*, pp. 209–213, 2010. - I. N'Doye, M. Darouach, M. Zasadzinski, and N.-E. Radhy, “Robust stabilization of uncertain descriptor fractional-order systems,”
*Automatica*, vol. 49, no. 6, pp. 1907–1913, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. D. Ma, J. G. Lu, and W. D. Chen, “Robust stability and stabilization of fractional order linear systems with positive real uncertainty,”
*ISA Transactions®*, vol. 53, no. 2, pp. 199–209, 2014. View at Publisher · View at Google Scholar · View at Scopus - D. Matignon, “Stability result for fractional differential equations with applications to control processing,” in
*Computational Engineering in Systems Applications*, pp. 963–968, 1997. View at Google Scholar - M. Moze, J. Sabatier, and A. Oustaloup, “LMI tools for stability analysis of fractional systems,” in
*Proceedings of the ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE '05)*, pp. 1611–1619, Long Beach, Calif, USA, 2005. View at Publisher · View at Google Scholar - J. Sabatier, M. Moze, and C. Farges, “On stability of fractional order systems,”
*in Third IFAC workshop on fractional differentiation and its applications FDA08*, 2008. View at Google Scholar - M. Chilali, P. Gahinet, and P. Apkarian, “Robust pole placement in LMI regions,”
*IEEE Transactions on Automatic Control*, vol. 44, no. 12, pp. 2257–2270, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - O. Nelles,
*Nonlinear System Identification from Classical Approaches to Neural Networks and Fuzzy Models*, Springer, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - L. Liu, F. Sun, X. Zhang, and Y. Wu, “Bifurcation analysis for a singular differential system with two parameters via to topological degree theory,”
*Nonlinear Analysis: Modelling and Control*, vol. 22, no. 1, pp. 31–50, 2017. View at Publisher · View at Google Scholar · View at Scopus - Y. Sun, L. Liu, and Y. Wu, “The existence and uniqueness of positive monotone solutions for a class of nonlinear Schrödinger equations on infinite domains,”
*Journal of Computational and Applied Mathematics*, vol. 321, pp. 478–486, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Pan, C. Yang, L. Pan, and H. Yu, “Integral Sliding Mode Control: Performance, Modification and Improvement,”
*IEEE Transactions on Industrial Informatics*, 2017. View at Google Scholar · View at Scopus - A. Maghareh, C. E. Silva, and S. J. Dyke, “Parametric model of servo-hydraulic actuator coupled with a nonlinear system: Experimental validation,”
*Mechanical Systems and Signal Processing*, vol. 104, pp. 663–672, 2018. View at Publisher · View at Google Scholar · View at Scopus - H. Wu, “Liouville-type theorem for a nonlinear degenerate parabolic system of inequalities,”
*Mathematical Notes*, vol. 103, no. 1-2, pp. 155–163, 2018. View at Publisher · View at Google Scholar · View at MathSciNet - P. P. Khargonekar, I. R. Petersen, and K. Zhou, “Robust stabilization of uncertain linear systems: quadratic stabilizability and control theory,”
*IEEE Transactions on Automatic Control*, vol. 35, no. 3, pp. 356–361, 1990. View at Publisher · View at Google Scholar · View at MathSciNet - I. R. Petersen, “A stabilization algorithm for a class of uncertain linear systems,”
*Systems & Control Letters*, vol. 8, no. 4, pp. 351–357, 1987. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus