Abstract
Stability and stabilization of fractional-order interval system is investigated. By adding parameters to linear matrix inequalities, necessary and sufficient conditions for stability and stabilization of the system are obtained. The results on stability check for uncertain FO-LTI systems with interval coefficients of dimension n only need to solve one 4n-by-4n LMI. Numerical examples are presented to shown the effectiveness of our results.
1. Introduction
During the last two decades, the study of fractional-order control systems has received more and more attention. As a generalization of the traditional calculus, the fractional calculus have found many applications in viscoelastic systems, robotics, finance, and so on ([1ā4], etc.). Studying on fractional-order calculus has become an active research field.
Stability analysis is a basic problem in control theory. For Caputo fractional derivative-based linear systems, the stability results were formulated with the fractional-order belonging to and , in [5ā7] and so on. In [8], the stability issue of interval fractional-order linear time-invariant (FO-LTI) systems was first presented and discussed. The stability of single-input single-output FO-LTI systems was further discussed based on an experimentally verified Kharitonov-like procedure [9].
Robust stability analysis was carried out for FO-LTI interval system with fractional commensurate order based on the maximum eigenvalue of a Hermitian matrix by applying Lyapunov inequality in [10] and then further discussed in [7].
The uncertain FO-LTI systems have been wildly studied. In [11], the robust stability problem was discussed based on the ranges of the corresponding interval eigenvalues by applying the matrix perturbation theory. In [12], a new and effectively robust stability checking method was first proposed for FO-LTI interval uncertain systems in terms of LMIs, and an analytical design of the stabilizing controllers for fractional-order dynamic interval systems was given. Note that the above-mentioned results on stability check for uncertain FO-LTI systems with interval coefficients of dimension need to solve one LMI. Therefore, it is valuable to seek some simple necessary and sufficient conditions for checking robust stability of uncertain FO-LTI systems with interval coefficients. With the above motivation, based on the results of [12], the robust stability and stabilization problems of uncertain FO-LTI interval systems with the fractional-order belonging to are further investigated in this paper.
This paper is organized as follows: in Section 2, we present some preliminaries results on the fractional derivative, the linear algebra and the matrix theory. In Section 3, we study the problems of the stability and stabilization of uncertain FO-LTI systems with interval coefficients in terms of LMIs. In Section 4, numerical examples are presented to illustrate our proposed results. Finally, Section 5 concludes this work.
Notations. Throughout this paper, stands for the set of by matrices with real entries. The symbols , , and stand for the transpose of , the expression , and the identity matrix of order , respectively. The symbol is used to denote the row vector with the th element being , ; that is, The symbol is the Kronecker product of two matrices and The symbol will be used in some matrix expressions to indicate a symmetric structure; that is, if matrices and were given, thenLet ; consider the symbol in which , .
2. Preliminaries
Throughout the paper, only the Caputo definition is used. The following Caputo definition is adopted for fractional derivatives of order of function [13]: with , , where is the Gamma function:
Consider the following FO-LTI interval system: where is the fractional commensurate order, and stand for the state vector and control input, respectively, and the system matrices and are interval uncertain in the sense that where , , , and are given matrices.
To take into account the stability [14], we introduce the following definition.
Definition 1. The fractional-order interval system (7) is said to be asymptotically stabilizable via linear state-feedback control if there exists a state-feedback controller such that the closed-loop system is asymptotically stable.
Denote To handle the interval uncertain, the following notations are introduced:
Lemma 2. Let ,: Then , .
Proof. Since , , , and are all diagonal, it is easy to check that It follows that Thus, .
Noting that the above proof is reversible, it is easy to know that . Therefore, .
In the same way, we have .
Lemma 3 (see [7, 15]). Let be a deterministic real matrix without uncertainty. Then, a necessary and sufficient condition for the asymptotical stability of is where is the spectrum of all eigenvalues of .
Lemma 4 (see [16]). Let be a real matrix. Then where , if and only if there exists such that where .
Lemma 5 (see [17]). For any matrices and with appropriate dimensions, we have
Lemma 6 (see [18]). Let , , and be real matrices of suitable dimensions. Then, for any ,
Lemma 7 (see [18]). Let , , and be symmetric matrices such that, , and . Furthermore, assume that for all nonzero . Then, there exists a constant such that
3. Main Results
In this section, by adding parameters into linear matrix inequalities, necessary and sufficient conditions for stability and stabilization of the system are obtained. Those results are generalization of the main theorems in [12].
Theorem 8. Let . The uncertain FO-LTI interval system (7) with controller is asymptotically stable if and only if there exist some symmetric positive definite matrix and a real scalar constant such that where
Proof. It is easy to check that Denote ; then we see that where is an arbitrary positive define matrix.
By applying (13), we havein which Sufficiency. Suppose that there exists a symmetric positive definite matrix such that (22) holds. By applying (26), (27), and Lemma 5, we have By using the Schur complement of (22), one obtains It follows from Lemma 4 that . Therefore, by Lemma 3, the uncertain FO-LTI interval system (7) is asymptotically stable.
Necessity. Suppose that the uncertain FO-LTI interval system (7) is asymptotically stable. Then, . It follows from Lemma 4 that there exists a symmetric positive definite matrix such that By using Lemma 2 and after some calculations, one can obtain from (27) that Therefore, for all , , that is, Consequently, given any and , we have Applying Lemma 6, we obtain It follows by Lemma 7 that there exists a constant such that So we derive that Applying the well-known Schur complement yields (22).
This ends the proof.
Next, let us establish a stabilization result.
Theorem 9. Let . The uncertain FO-LTI interval system (7) is asymptotically stable if and only if there are a matrix , a symmetric positive definite matrix , and two real scalars , such that where Moreover, the robustly asymptotically stabilizing state-feedback gain matrix is given by
Remark 10. When one takes and , it is easy to obtain equivalence of Theorem 8 and [12, Theorem 1] and Theorem 9 and [12, Theorem 2], respectively.
4. Numerical Examples
Example 1 (see [12]). Consider the robust stability of the following uncertain FO-LTI interval system: where and : Taking , a feasible solution of(22) is as follows:
Letand let the initial conditions be , , and . Time response of the state variables is depicted in Figure 1.
In the following example, we have shown the effectiveness of our results by choosing different parameters.
Example 2 (see [12]). Consider the robust stability of the following uncertain FO-LTI interval system: where and , with (I)Taking and , a feasible solution of (39) is as follows: Finally, the asymptotically stabilizing state-feedback gain matrix is obtained as (II)Taking and , we have (III)Taking and , we have
Let and let the initial conditions be , , and , and let be as in (49), (51), and (53), respectively. Time response of the state variables is depicted in Figures 2ā4, respectively.
Remark 11. Applying [12, Theorem 2 (28)] to Example 2 gives the same results as in Example 2 (II).
5. Conclusion
In this paper, the robust asymptotical stability of fractional-order interval systems with the fractional-order belonging to has been studied. The results on stability check for uncertain FO-LTI systems with interval coefficients of dimension only need to solve one -by- LMI. LMI stability conditions for fractional systems are proposed. Numerical examples have shown the effectiveness of our results. To the best of our knowledge, the idea of introducing free parameters is used for the first time to derive an analytical design of the stabilizing controllers for fractional-order dynamic interval systems. Working towards relaxing the requirements for the knowledge of system uncertainties and applying the proposed control methods to fractional-order nonlinear systems while maintaining the simplicity of the controller design are our further investigation directions.
Conflict of Interests
The authors declare that they have no competing interests.
Acknowledgments
The authors would like to thank the referees for their helpful comments and suggestions. The work of authors was supported in part by the Natural Science Foundation of China (Grant nos. 11401243 and 61403157) and the Science Foundation of Huainan Normal University (no. 2014xj45).