Robust stability and stabilization of fractional order linear systems with positive real uncertainty

ISA Transactions 53 (2014) 199–209

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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research Article

Robust stability and stabilization of fractional order linear systems with positive real uncertainty Yingdong Ma, Junguo Lu n, Weidong Chen Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, No. 800 Dong Chuan Road, Min Hang, Shanghai 200240, P.R. China

art ic l e i nf o

a b s t r a c t

Article history: Received 6 June 2013 Received in revised form 26 September 2013 Accepted 12 November 2013 Available online 14 December 2013 This paper was recommended for publication by Prof. Y. Chen

This paper investigates the robust stability and stabilization of fractional order linear systems with positive real uncertainty. Firstly, sufficient conditions for the asymptotical stability of such uncertain fractional order systems are presented. Secondly, the existence conditions and design methods of the state feedback controller, static output feedback controller and observer-based controller for asymptotically stabilizing such uncertain fractional order systems are derived. The results are obtained in terms of linear matrix inequalities. Finally, some numerical examples are given to validate the proposed theoretical results. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Fractional order system Positive real uncertainty Robust stability Robust stabilization Linear matrix inequality (LMI)

1. Introduction Fractional order control systems have attracted growing attention and interest of physicists and engineers from an application point of view recently (see [1–7, and the references therein]). On the one hand, this is mainly due to the fact that many real-world physical systems in interdisciplinary fields can be well characterized by fractional order differential equations involving the so-called fractional derivatives and integrals (for an introduction to this theory, see [1]). In particular, it has been shown that viscoelastic materials having memory and hereditary effects [8], biomedical systems [9–11], dynamical processes such as semi-infinite lossy RC transmission [12], mass diffusion and heat conduction [13] can be more adequately modeled by fractional order models than the traditional integer order models. In addition, fractional order derivatives and integrals also provide a powerful instrument for modeling dynamical processes in fractal media [14]. This is a significant advantage of the fractional order models in comparison with integer order models, where such effects or geometry have been neglected [14]. On the other hand, with the success in the synthesis of real noninteger differentiator and the emergence of new electrical circuit element called “fractance” [15,16], fractional order controllers [14,17–22] have been designed and applied to

n

Corresponding author. Tel.: þ 86 21 34205004; fax: þ86 21 34204302. E-mail addresses: [email protected], [email protected] (J. Lu).

control a variety of dynamical processes, including integer order and fractional order systems, so as to enhance the robustness and performance of the control systems. Stability and stabilization is fundamental to fractional order control systems [23–26]. In practice, there exist some uncertainties in the model due, for example, to some uncertain physical parameters, parametrical variations in time, neglected dynamics and so on. These uncertainties, which have to be considered for modeling and analyzing the system, can be introduced through various forms. There have been some stability results about the fractional order systems with interval uncertainties [27–32], polytopic uncertainties [33,34], or norm-bounded uncertainties [35]. For example, the robust stability problem of fractional order linear time-invariant (FO-LTI) interval systems described in the transfer function form was investigated in [30]. The robust stability problem of FO-LTI interval systems described in the state-space form was first considered in [29] by using the matrix perturbation theory. In [28], based on Lyapunov inequality, a new robust stability checking method was proposed for FO-LTI interval uncertain systems. However the results in [28,29] only are sufficient conditions. In [27], the necessary and sufficient condition for the robust stability of FO-LTI interval systems with fractional orders α, 1 r α o2, was presented. In [31,32], robust stability and stabilization problems of FO-LTI interval systems were investigated by using the linear matrix inequality method. In [36,37], sufficient conditions for the robust stability and stabilization of a class of FO-LTI interval systems with linear coupling relationships among the fractional order, the system

0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.11.013

200

Y. Ma et al. / ISA Transactions 53 (2014) 199–209

matrix and the input matrix were derived. In [35], observer-based controller and static output feedback controller for uncertain fractional order systems with norm-bounded uncertainty via linear matrix inequality approach were designed. In [38], synchronization of uncertain chaotic fractional order Duffing–Holmes systems was achieved by using the sliding mode control. In [39], an adaptive fuzzy sliding mode control for synchronizing two different uncertain fractional order time-delay chaotic systems was investigated. In [40], the sliding mode controller for an uncertain chaotic fractional order economic system was designed. It is well known that the interval uncertainty description, polytopic uncertainty description, and norm-bounded uncertainty description only can capture gain uncertainty [41,42]. When uncertainty phase information is available, these uncertainty description may lead to conservative results [41,42]. A way for accounting phase information is to apply the positivity theorem and more precisely to model the uncertainty through a positive real uncertainty matrix as in [41–44]. Note that positive real uncertainty exists in many real systems, and the robust stability and stabilization problem of integer order systems with positive real uncertainty has been studied in [41–44]. To the best of our knowledge, there are few results concerning robust stability and stabilization of fractional order linear systems with positive real uncertainty. Moreover, in most practical applications, the system state vector is not always accessible and only the partial information is available via measured output. In this case, the output feedback control or observer-based control is often needed. With the above motivation, the robust stability and stabilization of fractional order linear systems with positive real uncertainty will be investigated. Firstly, sufficient conditions for the asymptotical stability of such uncertain fractional order systems are presented. Secondly, the existence conditions and design methods of the state feedback controller, static output feedback controller and observerbased controller for asymptotically stabilizing such uncertain fractional order systems are derived. The results are obtained in terms of linear matrix inequalities. Finally, some numerical examples are given to validate the proposed theoretical results. The rest of this paper is organized as follows: in Section 2, the problem formulation and some necessary preliminaries are presented. In Section 3, robust stability conditions of uncertain fractional order systems with positive real uncertainty are derived. In Section 4, robust stabilizable conditions of such uncertain fractional order systems via state feedback control, static output feedback control and observer-based output feedback control and the design methods of the corresponding controllers are derived. For illustration of the effectiveness of the proposed theoretical results, numerical examples are presented in Section 5. Finally, some conclusions are given in Section 6. Notations: We denote by MT the transpose of M, by M the conjugate of M, by Mn the transpose conjugate of M, by z the conjugate of the scalar number z, by ReðzÞ its real part and by ImðzÞ its imaginary part. In is the identity matrix of order n. Matrices, if not explicitly stated, are assumed to have appropriate dimensions.  is the Kronecker product of two matrices and ðA  BÞðC  DÞ ¼ ðACÞ  ðBDÞ. i denotes the imaginary unit. SymfXg denotes the expression X n þ X. The notation  stands for the symmetric component in matrix.

2. Problem formulation and preliminaries Consider the following uncertain fractional order linear system: 8 α > < d xðtÞ ¼ ðA þ ΔAÞxðtÞ þ ðB þ ΔBÞuðtÞ; α ð1Þ dt > : yðtÞ ¼ CxðtÞ;

where α is the fractional commensurate order, xðtÞ A Rn denotes the state vector, uðtÞ A Rl is the control input, yðtÞ A Rm is the output vector, A A Rnn , B A Rnl and C A Rmn are constant matrices. ΔA and ΔB are time-invariant matrices with parametric uncertainties, and are assumed to be of the form (see for example [42–44]) ½ΔA ΔB ¼ M Δðζ Þ½N 1 N2 ;

ð2Þ

Δðζ Þ ¼ Fðζ Þ½I þ JFðζ Þ  1 ;

ð3Þ

SymfJg 4 0;

ð4Þ m0 n

nm0

m0 l

m0 m0

where M A R , N1 A R , N2 A R , and J A R are known real constant matrices. The uncertain matrix Fðζ Þ A Rm0 m0 satisfies SymfFðζ Þg Z 0; where ζ A Ω with

ð5Þ

Ω being a compact set.

Remark 1. It can be verified that the condition (4) guarantees that I JFðζ Þ is invertible for all Fðζ Þ satisfying (5). Therefore, Δðζ Þ in (3) is well defined [42–44]. In this paper, the following Caputo definition is adopted for fractional derivatives of order α of function f(t), since the Laplace transform of the Caputo derivative allows utilization of initial values of classical integer-order derivatives with clear physical interpretations [1]: Z t α ðmÞ d f ðtÞ 1 f ðτ Þ Dα f ðtÞ ¼ ; ð6Þ α ¼ Γ ðα  mÞ 0 ðt  τÞα þ 1  m dt where m is an integer satisfying m  1 o α rm and Γ ðÞ is the wellR1 known Euler Gamma function Γ ðzÞ ¼ 0 t z  1 e  t dt. From a mathematical point of view, the fractional order can be any real even complex number. In engineering applications, α often lies in (0, 2), and is a real number related to physical parameters. Therefore, this paper focuses on the robust stability and stabilization problem of uncertain fractional order systems (1) where α is a real number in (0, 2). To proceed, we need the following assumption and lemmas. Lemma 1 (Matignon [25], Sabatier et al. [45]). Let A A Rnn and 0 o α o 2. Then, a necessary and sufficient condition for the asympα α totical stability of d xðtÞ=dt ¼ AxðtÞ is   argðspecðAÞÞ 4 απ ; 2 where spec(A) is the spectrum of all eigenvalues of A. Lemma 2 (Farges et al. [33]). Let A A Rnn , 0 o α o 1 and θ ¼ ð1  αÞπ =2. The fractional order system dα xðtÞ=dt α ¼ AxðtÞ is asymptotically stable if and only if there exists a positive definite Hermitian matrix X ¼ X n 40, X A Cnn such that ðrX þ rX ÞT AT þ AðrX þ r X Þ o 0;

ð7Þ

where r ¼ eθi . Lemma 3 (Sabatier et al. [45]). Let A A Rnn , 1 r α o 2 and θ ¼ π  απ =2. The fractional order system dα xðtÞ=dt α ¼ AxðtÞ is asymptotically stable if and only if there exists a positive definite matrix X ¼ X T 4 0, X A Rnn such that " T # ðA X þ XAÞ sin θ ðXA AT XÞ cos θ o 0: ð8Þ  ðAT X þ XAÞ sin θ Remark 2. Defining " # sin θ  cos θ Θ¼ ; cos θ sin θ

ð9Þ

Y. Ma et al. / ISA Transactions 53 (2014) 199–209

201

and taking into consideration that A is similar to AT, condition (8) is equivalent to

Inequality (16) is equivalent to that there exist ɛ 4 0 and μ 40 such that

SymfΘ  ðAXÞg o 0:

SymfAðrY þ rY Þ þ MR  1 N 1 ðrY þ rY Þg

ð10Þ

Lemma 4 (Haddad and Bernstein [42]). Let Ω ¼ fΔ A Rm0 m0 jΔ is subjected to (3)–(5)}. Then

Assumption 1 (Ho and Lu [46]). The output matrix C is of full-row rank, i.e., rankðCÞ ¼ m, and then there exist two orthogonal matrices U A Rmm ; V A Rnn , such that ð11Þ

where S ¼ diagfs1 ; s2 ; …; sm g; s1 Z s2 Z ⋯ Z sm Z 0 are singular values of C. Lemma 5 (Ho and Lu [46]). For a given C A Rmn with rankðCÞ ¼ m, assume that Q A Rnn is a symmetric matrix, then there exists a matrix Z A Rmm such that CQ¼ ZC if and only if ! Q1 0 Q ¼V ð12Þ VT 0 Q2 where Q 1 A Rmm , Q 2 A Rðn  mÞðn  mÞ and V A Rnn satisfies (11).

þ ½ɛ

1

The objective of this section is to establish sufficient conditions for the asymptotical stability of system (1) with uðtÞ  0. Theorem 1. Let 0 o α o 1, θ ¼ ð1  αÞπ =2, and r ¼ eθi . If there exists a positive definite Hermitian matrix X ¼ X n 4 0; X A Cnn and a real scalar constant μ 4 0 such that 2 3 M ðrX þr X ÞT N T1 SymfAðrX þ r X Þg 6 7   μI μI ð13Þ 4 5 o 0;  SymfJg  μI

then the uncertain fractional order linear system (1) with uðtÞ  0 is asymptotically stable. Proof. Let AΔ ¼ A þ ΔA ¼ A þ M Δðζ ÞN 1 . It follows from Lemma 2 that the uncertain fractional order system (1) is asymptotically stable if there exists a positive definite Hermitian matrix Y ¼ Y n 4 0; Y A Cnn such that SymfAΔ ðrY þr Y Þg o 0

3 SymfðA þ ΔAÞðrY þ rY Þg o0

3 SymfAðrY þr Y Þg þ SymfM Δðζ ÞN 1 ðrY þ r Y Þg o 0:

ð14Þ

Define T R ¼ SymfJg; Q ¼ R  1=2 ðɛ  1 M T þ ɛN1 ðrY þr Y ÞÞ  R1=2 Δ ðζ Þɛ  1 M T . It T follows from Lemma 4 that SymfΔðζ Þg  Δðζ ÞRΔ ðζ Þ 4 0 and then for any ɛ 40, the following inequality holds T

Q Q r 0 3  SymfMR  1 N 1 ðrY þ rY Þg  ɛ  2 MR  1 M T  ɛ2 ðrY þ rY ÞT NT1 R  1 N 1 ðrY þ rY Þ þSymfM Δðζ ÞN 1 ðrY þ rY Þg T

)  SymfMR  1 N 1 ðrY þ rY Þg ɛ  2 MR  1 M T  ɛ 2 ðrY þ rY ÞT NT1 R  1 N 1 ðrY þ rY Þ þ SymfM Δðζ ÞN 1 ðrY þ rY Þg r 0: ð15Þ It follows from inequality (15) that inequality (14) holds if 2

þ ɛ ðrY þ rY Þ

T

þ ɛ  2 MR  1 M T

N T1 R  1 N 1 ðrY þ rY Þ o 0:

T

þ ɛ  2 MðR  1

N T1 R  1 N 1 ðrY þ rY Þ o 0;

ð17Þ

" T

N T1 

¼ SymfAðrY þ rY Þg

"

þ ½ɛ  1 M ɛðrY þr Y ÞT N T1 

R  1 þ μ  1I

R1

R1

R1

μI  μI  μI R þ μI

#"

#

ɛ  1 MT ɛN 1 ðrY þ rY Þ

#  1"

#

ɛ  1 MT ɛN 1 ðrY þ rY Þ

o 0:

ð18Þ

By the Schur complement, inequality (18) is equivalent to 2 3 SymfAðrY þr Y Þg ɛ  1 M ɛðrY þ rY ÞT NT1 6 7   μI μI 4 5o0    ðR þ μIÞ 3 2 3 2 SymfAðrY þ rY Þg ɛ  1 M ɛðrY þr Y ÞT N T1 ɛI 7 6 7 6   μI μI I 34 5 54    ðRþ μIÞ I 6 4 2 6 34 2 6 36 4

3

ɛI

7 5 o0

I I

ɛ SymfAðrY þ rY Þg

M



 μI

2

 SymfA½rðɛ

ɛ 2 ðrY þ rY ÞT N T1

μI

 ðR þ μIÞ

 2

YÞ þr ɛ2 Y g

M



 μI





3 7 5o0

½rðɛ 2 YÞ þ rɛ 2 Y T NT1

μI

 SymfJg  μI

o 0:

3 7 7 5 ð19Þ

Defining X ¼ ɛ 2 Y, the inequality (19) is equivalent to (13). This ends the proof. □ Theorem 2. Let 1 r α o 2, θ ¼ π  απ =2 and Θ satisfy (9). If there exists a positive definite matrix X ¼ X T 4 0; X A Rnn and a real scalar constant μ 40 such that 2 3 SymfΘ  ðAXÞg Θ  M I 2  ðXN T1 Þ 6 7 μI   μI ð20Þ 4 5 o0;   SymfI 2  Jg  μI then the uncertain fractional order linear system (1) with uðtÞ  0 is asymptotically stable. Proof. Let us denote AΔ ¼ A þ ΔA ¼ A þ M Δðζ ÞN1 . It follows from Lemma 3 that the uncertain fractional order system (1) is asymptotically stable if there exists a positive definite matrix Y ¼ Y T 4 0; Y A Rnn such that SymfΘ  ðAΔ YÞg o0 3 SymfΘ  ½ðA þ ΔAÞYg o0

þ ɛ  2 MðSymfΔðζ Þg  Δðζ ÞRΔ ðζ ÞÞM T r 0

SymfAðrY þr Y Þ þ MR  1 N1 ðrY þr Y Þg

IÞM þ ɛ ðrY þ rY Þ

M ɛðrY þr Y Þ

2



2

which is equivalent to that there exist ɛ 4 0 and μ 40 such that

3. Robust stability



T

SymfAðrY þ rY Þg

Ω ¼ fΔ A Rm0 m0 jdetðI  ΔJÞ a 0 and ΔSymfJgΔ rSymfΔgg: T

C ¼ U½S 0V T ;

þμ

1

ð16Þ

3 SymfΘ  ðAYÞg þSymfΘ  ðM Δðζ ÞN 1 YÞg o 0 ~ ðζ ÞN~ Y~ g o 0; ~Δ 3 Ψ þSymfM 1

ð21Þ

~ ¼ Θ  M; where Ψ ¼ SymfΘ  ðAYÞg; M N~ 1 ¼ I 2  N 1 ; F~ ~ ðζ Þ ¼ I  Δðζ Þ ¼ F~ ðζ Þ½I  J~ F~ ðζ Þ  1 ; ðζ Þ ¼ I 2  Fðζ Þ; J~ ¼ I 2  J; Δ 2 Y~ ¼ I 2  Y.  1=2  1 ~ T þ ɛ N~1 Y~ Þ  R~ 1=2 ΔT ðζ Þɛ  1 M ~ T. Define R~ ¼ SymfJ~ g; Q~ ¼ R~ ðɛ M T ~ ~ ~ ~ It follows from Lemma 4 that SymfΔ ðζ Þg  Δ ðζ ÞR Δ ðζ Þ 4 0 and then

202

Y. Ma et al. / ISA Transactions 53 (2014) 199–209

for any ɛ 4 0, the following inequality holds

Definition 1. The uncertain fractional order linear system (1) with uðtÞ ¼ KxðtÞ is said to be asymptotically stabilizable if there exists a control gain matrix K A Rln such that the closed-loop system

 Q~ Q~ r 0 T

~ R~  1 N~ 1 Y~ g  ɛ  2 M ~ R~  1 M ~T 3  SymfM

α

d xðtÞ α ¼ ðA þ BK þ ΔA þ ΔBKÞxðtÞ dt

T T 1 ~ ðζ ÞN~ Y~ g ~Δ  ɛ Y~ N~ 1 R~ N~ 1 Y~ þSymfM 1   T T 2 ~ ~ ~ ~ ~ ~ o0 þ ɛ M SymfΔ ðζ Þg  Δ ðζ ÞR Δ ðζ Þ M 2

~ R~ )  SymfM

1

is asymptotically stable.

~ R~  1 M ~T N~ 1 Y~ g  ɛ  2 M

Next, stabilization results are established.

T T 1 ~ Δðζ ÞN~ 1 Y~ g  ɛ 2 Y~ N~ 1 R~ N~ 1 Y~ þ SymfM r 0:

ð22Þ

It follows from inequality (22) that inequality (21) holds if

Ψ þ SymfM~ R~

1

~ R~  1 M ~ T þ ɛ 2 Y~ T N~ T R~  1 N~ 1 Y~ o 0: N~ 1 Y~ g þ ɛ  2 M 1

ð23Þ

Inequality (23) is equivalent to that there exist ɛ 4 0 and μ 4 0 such that

Ψ þ SymfM~ R~

1

~ T þ ɛ2 Y~ T N~ T R~  1 N~ 1 Y~ o 0; ~ R~  1 þ μ  1 IÞM N~ 1 Y~ g þ ɛ  2 Mð 1 ð24Þ

which is equivalent to that there exist ɛ 4 0 and μ 4 0 such that 2 3" # ~  1 þ μ  1 I R~  1 1 ~ T R T T 1 ~ 5 ɛ M Ψ þ ½ɛ M ɛY~ N~ 1 4 1 1 ɛ N~ 1 Y~ R~ R~ " #  1" # ~T μI  μI T T ɛ  1M 1 ~ ~ ~ ¼ Ψ þ ½ɛ M ɛ Y N 1  o0: ð25Þ  μI R~ þ μI ɛN~ 1 Y~ By the Schur complement, inequality (25) is equivalent to 2 3 T T ~ Ψ ɛ  1M ɛ Y~ N~ 1 6 7 6 7o0  μI μI 4 5    ðR~ þ μIÞ 32 2 32 3 T T ~ ɛI ɛI Ψ ɛ  1M ɛY~ N~ 1 76 6 76 7 74 I I 34 56 5o0  μI μI 4  5 ~ I I    ðR þ μIÞ 2 3 2 2 ~T ~T ~ M ɛ Y N1 ɛ Ψ 6 7 7o0 36  μI μI 4  5 ~    ðR þ μIÞ 2

6 34

SymfΘ  ðAɛ 2 YÞg 

ΘM  μI





I 2  ðɛ 2 YN T1 Þ

μI

 SymfI 2  Jg  μI

then the uncertain fractional order linear system (1) with uðtÞ ¼ KxðtÞ is asymptotically stabilizable. Moreover, a asymptotically stabilizing state feedback gain matrix is given by b ðrX þ rX Þ  1 : K ¼K Proof. The uncertain fractional order linear system (1) with uðtÞ ¼ KxðtÞ is asymptotically stabilizable if the closed-loop system α

d xðtÞ α ¼ ðA þ BK þ ΔA þ ΔBKÞxðtÞ dt ¼ ðA þ BK þM Δðζ ÞðN 1 þ N2 KÞÞxðtÞ

ð30Þ

b ¼ KðrX þ r X Þ, inequality (31) is equivalent to (29). This Set K completes the proof. □

3

7 5 o 0: ð26Þ

Defining X ¼ ɛ Y, the inequality (26) is equivalent to (20). This ends the proof. □ Remark 3. The conditions given in Theorems 1 and 2 are actually LMIs. A powerful LMI toolbox has been provided by Matlab to check the existence of a solution of LMIs. Thus, by Theorems 1 and 2, one can easily determine whether an uncertain fractional order system (1) with the input uðtÞ  0 is asymptotically stable.

Theorem 4. Let 1 r α o2, θ ¼ π  απ =2 and Θ satisfy (9). If there b A Rln and a real scalar conexist matrices X ¼ X T 4 0; X A Rnn ; K stant μ 4 0 such that 2 3 b Þg Θ  M I 2  ðXN T þ K b T NT Þ SymfΘ  ðAX þ BK 1 2 6 7 6 7 o 0; ð32Þ μI   μI 4 5    SymfI 2  Jg  μI then the uncertain fractional order linear system (1) with uðtÞ ¼ KxðtÞ is asymptotically stabilizable. Moreover, a asymptotically stabilizing state feedback gain matrix is given by b X  1: K ¼K Proof. The uncertain fractional order linear system (1) with uðtÞ ¼ KxðtÞ is asymptotically stabilizable if the closed-loop system α

d xðtÞ α ¼ ðA þ BK þ ΔA þ ΔBKÞxðtÞ dt ¼ ðA þ BK þM Δðζ ÞðN 1 þ N2 KÞÞxðtÞ

4. Robust stabilization 4.1. State feedback stabilization In this subsection, we assume that the state of the uncertain fractional order system (1) is perfectly available and consider linear state feedback control, that is, ð27Þ ln

Theorem 3. Let 0 o α o1, θ ¼ ð1  αÞπ =2, and r ¼ eθi . If there exist b A Rln and a real scalar constant matrices X ¼ X n 4 0; X A Cnn ; K μ 40 such that 2 3 bg b T NT SymfAðrX þ rX Þ þ BK M ðrX þ rX ÞT N T1 þ K 2 6 7 6 7 o0; ð29Þ   μI μI 4 5    SymfJg  μI

is asymptotically stable. It follows from Theorem 1 that the closedloop system (30) is asymptotically stable if there exist a Hermitian positive definite matrix X A Cnn and a real scalar μ 4 0 such that 2 3 SymfðA þ BKÞðrX þ rX Þg M ðrX þ rX ÞT ðN 1 þ N2 KÞT 6 7   μI μI 4 5 o0: ð31Þ    SymfJg  μI

2

uðtÞ ¼ KxðtÞ;

ð28Þ

where K A R is the control gain matrix to be designed. We introduce the following definition.

ð33Þ

is asymptotically stable. It follows from Theorem 2 that the closedloop system (33) is asymptotically stable if there exist a symmetric positive-definite matrix X A Rnn and a real scalar μ 4 0 such that 2 3 SymfΘ  ððA þ BKÞXÞg Θ  M I 2  ðXðN 1 þ N2 KÞT Þ 6 7 μI   μI 4 5 o 0: ð34Þ    SymfI 2  Jg  μI

Y. Ma et al. / ISA Transactions 53 (2014) 199–209

b ¼ KðrX þ rX Þ, inequality Set K This completes the proof. □

(34)

is

equivalent

to

(32).

From (38) and (41), we obtain 2 6 4

4.2. Output feedback stabilization In most practical applications, the system state vector is not always accessible and only the partial information is available via measured output. In this case, to stabilize the uncertain fractional order linear system (1) via the measure output y(t), we use static output feedback control and observer-based output feedback control.

203

SymfAðrX þ rX Þ þ BFCðrX þrX Þg  

T T T T T 3 M ðrX þ rX Þ N 1 þðCðrX þ rX ÞÞ F N 2 7 μI  μI 5 o 0: SymfJg  μI 

ð42Þ Similar to the proof of Theorem 1, from (42), we have SymfAðrX þ rX Þ þ BFCðrX þr X Þg   þ SymfM Δðζ Þ N 1 ðrX þ r X Þ þ N2 FCðrX þ rX Þ g o 0;

ð43Þ

which implies 4.2.1. Static output feedback stabilization Here, we consider the following static output feedback control:

SymfðA þ BFC þM Δðζ ÞðN 1 þ N 2 FCÞÞðrX þ rX Þg o 0:

uðtÞ ¼ FyðtÞ;

From Lemma 2 and (44), we can conclude that the closed-loop system

ð35Þ

where yðtÞ A Rm is the output vector and F A Rlm is the control gain matrix to be designed. We introduce the following definition. Definition 2. The uncertain fractional order linear system (1) with uðtÞ ¼ FyðtÞ is said to be asymptotically stabilizable if there exists a control gain matrix F A Rlm such that the closed-loop system α

d xðtÞ α ¼ ðA þ BFC þ ΔA þ ΔBFCÞxðtÞ dt

ð36Þ

is asymptotically stable. Next, static output feedback stabilization results are established. Theorem 5. Let 0 o α o 1; θ ¼ ð1  αÞπ =2, r ¼ eθi and the output matrix C satisfy Assumption 1. If there exist matrices X ¼ X n 4 0; X A Cnn ; Fb A Rlm , Z A Rmm and a real scalar constant μ 4 0 such that 2 3 T SymfAðrX þ r X Þ þ BFb Cg M ðrX þ rX ÞT NT1 þC T Fb NT2 6 7 6 7 o 0; ð37Þ   μI μI 4 5    SymfJg  μI

CðrX þ rX Þ ¼ ZC;

ð38Þ

then the uncertain fractional order linear system (1) is asymptotically stabilizable via static output feedback control uðtÞ ¼ FyðtÞ. Moreover, a asymptotically stabilizing static output feedback gain matrix is given by F ¼ Fb Z  1 :

ð39Þ

Proof. Since the output matrix C satisfies Assumption 1, C is of full-low rank and rankðCÞ ¼ m. Noting that the matrix ðrX þ rX Þ is real and nonsingular (see [33]), then the matrix equation CðrX þ rX Þ ¼ ZC implies   m ZrankðZÞ Z rankðZCÞ ¼ rank CðrX þ rX Þ   Z rankf CðrX þ rX Þ ðrX þr X Þ  1 g ¼ rankðCÞ ¼ m;

α

d xðtÞ α ¼ ðA þ BFC þ ΔA þ ΔBFCÞxðtÞ dt ¼ ðA þ BFC þM Δðζ ÞðN 1 þ N2 FCÞÞxðtÞ

ð40Þ

that is, Z is nonsingular. If the static output feedback gain matrix is chosen as F ¼ Fb Z  1 , we have Fb ¼ FZ. Substituting Fb ¼ FZ into the LMI in (37), we have 2 3 SymfAðrX þ r X Þ þ BFZCg M ðrX þ rX ÞT N T1 þ C T ðFZÞT N T2 6 7   μI μI 4 5 o 0:    SymfJg  μI ð41Þ

ð45Þ

is asymptotical stable. Therefore, the uncertain fractional order linear system (1) via static output feedback control uðtÞ ¼ FyðtÞ is asymptotically stabilizable. This completes the proof. □ Theorem 6. Let 1 r α o 2; θ ¼ π  απ =2, Θ satisfy (9) and the output matrix C satisfy Assumption 1. If there exist matrices X 11 ¼ X T11 4 0; X 11 A Rmm ; X 22 ¼ X T22 40; X 22 A Rðn  mÞðn  mÞ ; Fb A Rlm and a real scalar constant μ 4 0 such that 2 3 SymfΘ  ðAX þ BFb CÞg Θ  M I 2  ðXN T1 þC T Fb T N T2 Þ 6 7 6 7 o 0 ð46Þ μI   μI 4 5    SymfI 2  Jg  μI with " X¼V

and

ð44Þ

X 11

0

0

X 22

# VT;

ð47Þ

where V satisfies (11), then the uncertain fractional order linear system (1) via static output feedback control uðtÞ ¼ FyðtÞ is asymptotically stabilizable. Moreover, a asymptotically stabilizing static output feedback gain matrix is given by 1 1 1 S U : F ¼ Fb USX 11

ð48Þ

Proof. The uncertain fractional order linear system (1) via static output feedback control uðtÞ ¼ FyðtÞ is asymptotically stabilizable if the closed-loop system α

d xðtÞ α ¼ ðA þ BFC þ ΔA þ ΔBFCÞxðtÞ dt ¼ ðA þ BFC þM Δðζ ÞðN 1 þ N2 FCÞÞxðtÞ

ð49Þ

is asymptotical stable. It follows from Lemma 3 that the closedloop system (49) is asymptotical stable if there exists X ¼ X T 40 such that SymfΘ  ððA þ BFC þ M Δðζ ÞðN 1 þ N 2 FCÞÞXÞg o0 ¼ SymfΘ  ðAX þ BFCX þ M Δðζ ÞðN 1 X þN 2 FCXÞÞgo 0:

ð50Þ

From Lemma 5 and (47), there exists X~ such that CX ¼ X~ C, where X~ ¼ USX 11 S  1 U  1 . Then inequality (50) can be rewritten as    ð51Þ SymfΘ  AX þBF X~ C þ M Δðζ Þ N1 X þ N 2 F X~ C g o 0:

204

Y. Ma et al. / ISA Transactions 53 (2014) 199–209

Similar to the proof of Theorem 2, inequality (51) is equivalent to that there exist matrices X; X~ and a real scalar μ 4 0 such that 2 3  T T T T ~ ~ 6 SymfΘ  ðAX þ BF X CÞg Θ  M I 2  ðXN 1 þ C F X N2 Þ 7 6 7 6 7 o 0: μI   μI 4 5    SymfI 2  Jg  μI ð52Þ Set Fb ¼ F X~ , inequality (52) is equivalent to 2 3 SymfΘ  ðAX þ BFb CÞg Θ  M I 2  ðXN T1 þ C T Fb T N T2 Þ 6 7 6 7 o 0: μI   μI 4 5    SymfI 2  Jg  μI

 μI

6 6 0 4 μI

Σ 22 ¼ 6 6

□

μI

0  μI 0

μI

0

0

0  SymfJg  μI 0

μI 0  SymfJg  μI

3 7 7 7; 7 5

ð59Þ

then the uncertain fractional order linear system (1) is asymptotically stabilizable via observer-based output feedback control uðtÞ ¼ K b x ðtÞ. Moreover, a asymptotically stabilizing control gain matrix K and observer gain matrix L are given by b ðrX 1 þ rX 1 Þ  1 ; K ¼K

ð53Þ This completes the proof.

2

L¼b LZ  1 :

ð60Þ

Proof. The uncertain fractional order linear system (1) is asymptotically stabilizable via observer-based output feedback control uðtÞ ¼ K b x ðtÞ if the closed-loop system α

4.2.2. Observer-based output feedback stabilization Here, we assume that the perturbations ΔA and ΔB are measurable (see [47]) and construct the following observer: 8 α > x ðtÞ :y bðtÞ ¼ C b x ðtÞ;

d xc ðtÞ α ¼ Ac xc ðtÞ dt

ð61Þ

is asymptotically stable. It follows from Lemma 2 that the closedloop system (61) is asymptotically stable if there exists X ¼ X n ¼ diagfX 1 ; X 2 g A C2n2n 4 0 such that

bðtÞ A Rn denotes where α is the fractional commensurate order, x m bðtÞ A R is the output vector of the state vector of the observer, y the observer, L A Rnm is the observer gain matrix to be designed and the other notations are the same as those in (1). Based on the observer (54), we consider the following observer-based output feedback control: uðtÞ ¼ K b x ðtÞ;

ð55Þ

where K A R

ð62Þ

ln

is the control gain matrix to be designed. Define " # b x ðtÞ eðtÞ ¼ xðtÞ  b x ðtÞ; xc ¼ ; eðtÞ " # LC Aþ ΔA þ ðB þ ΔBÞK ; Ac ¼ 0 A þ ΔA LC

It follows from the proof of Theorem 5 that Z is nonsingular. b ðrX 1 þ Using CðrX 2 þ r X 2 Þ ¼ ZC from (58) and substituting K ¼ K rX 1 Þ  1 ; L ¼ b LZ  1 into the LMI in (62), we have

then the closed-loop system is given as α

d xc ðtÞ α ¼ Ac xc ðtÞ: dt

ð56Þ

Our objective is to design the observer gain matrix L and the control gain matrix K such that the closed-loop system (56) is asymptotically stable. Next, observer-based output feedback stabilization results are given. Theorem 7. Let 0 o α o 1; θ ¼ ð1  αÞπ =2, r ¼ eθi and the output

matrix C satisfy Assumption 1. If there exist matrices X 1 ¼ b A Rlm , X n1 4 0; X 1 A Cnn ; X 2 ¼ X n2 4 0; X 2 A Cnn ; K b L A Rnm ; Z A Rmm and a real scalar constant μ 4 0 such that " #

Σ 11 Σ 12 o 0;  Σ 22

ð57Þ

CðrX 2 þ rX 2 Þ ¼ ZC;

ð58Þ

where ("

Σ 11 ¼ Sym

b LC

0

AðrX 2 þ rX 2 Þ  b LC

2 M Σ 12 ¼ 4 0

#)

b AðrX 1 þ rX 1 Þ þBK T

;

0

b NT ðrX 1 þr X 1 ÞT N T1 þ K 2

0

M

0

ðrX 2 þr X 2 ÞT N T1

3 5;

ð63Þ where ("

Σ 11 ¼ Sym

b AðrX 1 þr X 1 Þ þ BK

b LC

0

AðrX 2 þr X 2 Þ  b LC

#) ;

~ ðζ Þ ¼ I  Δðζ Þ; J~ ¼ I  J; ~ ¼ I 2  M; Δ M 2 2 " # b N K 0 ðrX þ rX Þ þ N 1 1 1 2 ~ N¼ : 0 N 1 ðrX 2 þ rX 2 Þ Similar to the proof of Theorem 1, the matrix inequality (63) holds if there exist matrices X 1 ¼ X n1 4 0; X 1 A Cnn ; X 2 ¼ b A Rlm , b X n2 4 0; X 2 A Cnn ; K L A Rnm ; Z A Rmm and a real scalar constant μ 4 0 such that 2 3 T ~ Σ M N~ 6 11 7 6  7 o 0: ð64Þ  μI μI 4 5    SymfJ~ g  μI This ends the proof.

□

Y. Ma et al. / ISA Transactions 53 (2014) 199–209

Theorem 8. Let 1 r α o 2; θ ¼ π  απ =2, Θ satisfy (9) and the output matrix C satisfy Assumption 1. If there exist matrices X 1 ¼ X T1 4 0; X 1 A Rnn ; X 11 ¼ X T11 40; b A Rlm ; b X 11 A Rmm ; X 22 ¼ X T22 40; X 22 A Rðn  mÞðn  mÞ ; K L A Rnm , and a real scalar constant μ 4 0 such that " #

Σ 11 Σ 12 o 0;  Σ 22

where

(

ð65Þ

"

Σ 11 ¼ Sym Θ 

b AX 1 þ BK 0



h

;

LC AX 2  b

T T T Σ 12 ¼ Θ  ðI2  MÞ I2  X 1 NT1 þ Kb N 2 00X 2 N 1

"

Σ 22 ¼ " X2 ¼ V

 μI

μI

X 11 0

μI

i

;

#

 SymfI 2  ðI 2  JÞg  μI # 0 VT ; X 22

; ð66Þ

V satisfies (11), then the uncertain fractional order linear system (1) is asymptotically stabilizable via observer-based output feedback control uðtÞ ¼ K b x ðtÞ. Moreover, a asymptotically stabilizing control gain matrix K and observer gain matrix L are given by b X  1; K ¼K 1

where (

"

Σ 11 ¼ Sym Θ 

b AX 1 þBK

b LC

0

LC AX 2  b

#) ;

~ ðζ Þ ¼ I  ðI  Δðζ ÞÞ; J~ ¼ I  ðI  JÞ; ~ ¼ Θ  ðI 2  MÞ; Δ M 2 2 2 2 " # b N 1 X 1 þN 2 K 0 N~ ¼ I 2  : 0 N1 X 2 Similar to the proof of Theorem 2, the matrix inequality (69) holds if there exist matrices X 1 ¼ X T1 4 0; X 1 A Rnn ; X 11 ¼ X T11 4 0; b A Rlm ; b X 11 A Rmm ; X 22 ¼ X T22 4 0; X 22 A Rðn  mÞðn  mÞ ; K L A Rnm , and a real scalar constant μ 4 0 such that 2 3 T ~ Σ M N~ 6 11 7 6  7 o 0:  μI μI 4 5 ~    SymfJ g  μI

#)

b LC

205

1 1 1 L¼b LUSX 11 S U :

Proof. The uncertain fractional order linear system (1) is asymptotically stabilizable via observer-based output feedback control uðtÞ ¼ K b x ðtÞ if the closed-loop system α

d xc ðtÞ α ¼ Ac xc ðtÞ dt

ð67Þ

is asymptotically stable. It follows from Lemma 3 that the closed-loop system (67) is asymptotically stable if there exists X ¼ X T ¼ diagfX 1 ; X 2 g A R2n2n 4 0 such that

This ends the proof.

□

5. Numerical examples In this section, several numerical examples are given to illustrate the effectiveness of the proposed method. The predictor–corrector approach for the numerical solution of fractional differential equations [48] is used. 5.1. Example 1 for the 0 o α o1 case Consider the output feedback stabilization problem of the uncertain fractional order system (1) with the parameters: 3 2 2 3 2:5 10  5 0:5 1 0 6 7 6 7 0:2 0 5; A ¼ 4  15 7:5  5 5; M ¼ 4  0:4 0:1  0:1 0:6 10 10  5

ð68Þ

2

Note that (66) means that there exists a matrix b ¼ KX 1 ; CX 2 ¼ X~ C, where X~ ¼ USX 11 S  1 U  1 . Let K the inequality (68) is equivalent to ( Sym Θ 

"

b ðA þ ΔAÞX 1 þ ðB þ ΔBÞK (

3 Σ 11 þ Sym Θ 

0

"

~ ðζ ÞNg ~Δ ~ o 0; 3 Σ 11 þ SymfM

#)

b LC ðA þ ΔAÞX 2  b LC

b M Δðζ ÞN 1 X 1 þ M Δðζ ÞN 2 K 0

X~ satisfying b L ¼ LX~ , then

o0

0 MΔðζ ÞN 1 X 2

3 2 3 2 3 0:5 1:5 2 1 1 6 7 6 7 6 7 N1 ¼ 4 0 0:5 2:5 5; B ¼ 4 2 5; N 2 ¼ 4 0:5 5; 0 0 2:5 0:5 1

1 1 1 C¼ ; J ¼ I 3 ; α ¼ 0:8: 0 2 2

ð70Þ

Firstly, we consider static output feedback control and use Theorem 5 to obtain

#) o0

ð69Þ

μ ¼ 32409:03; Fb ¼ ½  1:84 0:49; F ¼ ½  3:32  0:77;

206

Y. Ma et al. / ISA Transactions 53 (2014) 199–209 30

20 15

20

10 10

Imaginary Part

Imaginary Part

5 0 unstable region

stable region 5

0 unstable region

stable region 10

10 20

15 20

6

4

2

0

2

4

30

6

6

4

2

0

Real Part Fig. 1. The location of eigenvalues of the uncertain closed-loop system via output feedback control in Example 1.

x1

1

4

6

8

10

Fig. 3. The location of eigenvalues of the uncertain closed-loop system via observer-based output feedback control in Example 1.

Secondly, we consider observer-based output feedback control and use Theorem 7 to obtain

0 1

2

Real Part

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

μ ¼ 22955:77;

ð71Þ

b ¼ ½  2:25  3:86  2:92; K

ð72Þ

t

x2

1 0 1 2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

x3

2

2

2:24

0:29

6 b L ¼ 4 1:96 1:82

3

0:37 7 5;

ð73Þ

0:079

1 0 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

K ¼ ½  2:30 1:80  1:72;

ð74Þ

t Fig. 2. The time response of a selected closed-loop system via output feedback control in Example 1.

2

0:20 0:07  0:21i 6 0:49 X ¼ 4 0:07 þ 0:21i 0:15  0:04i 0:06  0:10i

0:70  0:33 Z¼ :  0:63 0:79

3

2

4:51

0:15 þ 0:04i 7 0:06 þ 0:10i 5;

2

0:56

So, according to Theorem 5, we can conclude that the uncertain fractional order system (1) with uðtÞ ¼ FyðtÞ and the parameters in (70) is asymptotically stabilizable. The location of eigenvalues of the uncertain fractional order system (1) with uðtÞ ¼ FyðtÞ and the parameters in (70) is shown in Fig. 1, which shows that all eigenvalues locate the asymptotical stable domain (see Lemma 1). The time response of a selected system in such uncertain fractional order closed-loop systems with A þ MFðζ ÞðI 3 þ JFðζ ÞÞ  1 N 1 þ ðB þ MFðζ ÞðI 3 þJFðζ ÞÞ  1 N 2 ÞFC ¼ Aþ MI 3 ðI 3 þJI 3 Þ  1 N 1 þ ðB þ MI3 ðI 3 þ JI 3 Þ  1 N 2 ÞFC 2 3  0:69 8:84  5:04 6 7 ¼ 4  20:91 4:12  8:28 5 6:96 8:40  7:43

is shown in Fig. 2, which shows that all its states asymptotically converge to zero.

3 2:96 0:73 7 5;

6:08 6 L ¼ 4 3:60

0:32

6 X 1 ¼ 4 0:09 þ0:39i 0:28 0:06i 2

0:19 6 X 2 ¼ 4 0:07 þ0:19i 0:14 0:04i Z¼

ð75Þ

1:94

0:09  0:39i

0:28 þ0:06i

3

0:68 0:04  0:43i

7 0:04 þ0:43i 5; 0:65

0:07  0:19i

0:14 þ0:04i

0:36 0:02  0:21i

0:67

 0:21

 0:62

0:53

:

ð76Þ

3

7 0:02 þ0:21i 5;

ð77Þ

0:49

ð78Þ

So, according to Theorem 7, we can conclude that the uncertain ^ and the parameters fractional order system (1) with uðtÞ ¼ K xðtÞ in (70) is asymptotically stabilizable. The location of eigenvalues ^ of the uncertain fractional order system (1) with uðtÞ ¼ K xðtÞ and the parameters in (70) is shown in Fig. 3, which shows that all eigenvalues locate the asymptotical stable domain (see Lemma 1). The time response of a selected system in such uncertain fractional order closed-loop systems with

Y. Ma et al. / ISA Transactions 53 (2014) 199–209

207

x1

1 0 1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3

3.5

4

4.5

5

3

3.5

4

4.5

5

t

x2

1 0 1 2

0

0.5

1

1.5

2

2.5

t

x3

2 1 0 1

0

0.5

1

1.5

2

2.5

is shown in Fig. 4, which shows that all its states asymptotically converge to zero.

t Fig. 4. The time response of a selected closed-loop system via observer-based output feedback control in Example 1.

5.2. Example 2 for the 1 r α o2 case Consider the output feedback stabilization problem of the uncertain fractional order system (1) with the parameters: 2 3 0 4 6 6 6 0 4 2 07 6 7 A¼6 7; 4  2  2 6 2 5

30

20

10 2

Imaginary Part

10

0:2 6 0 6 M¼6 4 0

0

0 2

10

0:1 6 0 6 N1 ¼ 6 4 0

unstable region

stable region 20

0 30

15

10

5

0

2

5

Real Part Fig. 5. The location of eigenvalues of the uncertain closed-loop system via static output feedback control in Example 2.

0

0:2

0

0

0:2

0:2

0

0:2

0

0

0:3 0:1 0 0 3

3

0 7 7 7; 0:2 5

0:2 3 2 3 0:4 0 1 7 627 0:5 0 7 6 7 7; B ¼ 6 7; 415 0:5 0 5 0

1

C¼

0 5

2

3

10

 20

15

0

20



0:1

4

1

3 2

0.5

1

0

x2

x1

4

0:2 6 0:1 7 6 7 N2 ¼ 6 7; 4 0:1 5

1.5

0.5

0 1

1 1.5

10

2 0

2

4

6

8

3

10

0

2

4

t

6

8

10

6

8

10

t

1.5

2

1

1

0.5 0

x4

x3

0 0.5

1

1 2

1.5 2

0

2

4

6

t

8

10

3

0

2

4

t

Fig. 6. The time response of a selected closed-loop system via static output feedback control in Example 2.

;

J ¼ I 4 ; α ¼ 1:2:

ð79Þ

208

Y. Ma et al. / ISA Transactions 53 (2014) 199–209

2

Firstly, we consider static output feedback control and use Theorem 6 to obtain

 11:25

6  22:30 6 ¼6 4  13:37

μ ¼ 0:64;

9:89

Fb ¼ ½  0:20  0:01; F ¼ ½ 1:98 0:06;

0:37  0:06 ; X 11 ¼  0:06 0:12

X 22 ¼

0:39

 0:02

 0:02

0:44

:

So, according to Theorem 6, we can conclude that the uncertain fractional order system (1) with uðtÞ ¼ FyðtÞ and the parameters in (79) is asymptotically stabilizable. The location of eigenvalues of the uncertain fractional order system (1) with uðtÞ ¼ FyðtÞ and the parameters in (79) is shown in Fig. 5, which shows that all eigenvalues locate the asymptotical stable domain (see Lemma 1). The time response of a selected system in such uncertain fractional order closed-loop systems with A þ MFðζ ÞðI 4 þ JFðζ ÞÞ  1 N 1 þ ðB þ MFðζ ÞðI 4 þJFðζ ÞÞ  1 N 2 ÞFC ¼ Aþ MI 4 ðI 4 þJI 4 Þ  1 N 1 þ ðB þ MI4 ðI 4 þ JI 4 Þ  1 N 2 ÞFC

6

2

0:09 6 0:03 6 b L¼6 4 0:02 0:06

Imaginary Part

0:04

3:89

2 0

0:03

3  0:33 0:07 7 7 7;  0:04 5 0:02

 0:03

0:21  0:06

0:18 0:05

 0:05 ; 0:03

X 11 ¼

8 20

10

0

10

20

Real Part Fig. 7. The location of eigenvalues of the uncertain closed-loop system via observer-based output feedback control in Example 2.

2.5

0.2

2

0

1.5

X 22 ¼

0:20 0:04

0:04 : 0:09

1

x2

x1

0.2 0.4

0.5

0.6

0

0.8

0.5 0

2

4

6

8

1

10

0

2

4

t

6

8

10

6

8

10

t

0.5

0.5 0

0

0.5

x4

x3

0.5 1

1 1.5

1.5 2

0:22

So, according to Theorem 8, we can conclude that the uncertain ^ fractional order system (1) with uðtÞ ¼ K xðtÞ and the parameters in (79) is asymptotically stabilizable. The location of eigenvalues of the ^ uncertain fractional order system (1) with uðtÞ ¼ K xðtÞ and the parameters in (79) is shown in Fig. 7, which shows that all eigenvalues locate the asymptotical stable domain (see Lemma 1).

0.4

1

3  0:07  0:09 7 7 7;  0:06 5

 0:05  0:03

 0:09

4 6

 0:09 0:24

 0:07

30

 0:08

3

0:20 6  0:09 6 X1 ¼ 6 4  0:05

2

40

3:94

0:03 7 7 7; 0 5

2

50

9:97

 36:95 7 7 7  16:75 5

b ¼ ½  0:30  0:26  0:20  0:34; K

4

60

 9:75  11:99

μ ¼ 0:60;

3:14 6 2:78 6 L¼6 4 1:10

8

10

 9:98  5:05

2 0

2

4

6

t

3

K ¼ ½  17:16  15:11 11:50  16:26;

unstable region

stable region

0:11

is shown in Fig. 6, which shows that all its states asymptotically converge to zero. Secondly, we consider observer-based output feedback control and use Theorem 8 to obtain

2

10

 12:66

1:02

8

10

2.5

0

2

4

t

Fig. 8. The time response of a selected closed-loop system via observer-based output feedback control in Example 2.

Y. Ma et al. / ISA Transactions 53 (2014) 199–209

The time response of a selected system in such uncertain fractional

order closed-loop systems with is shown in Fig. 8, which shows that all its states asymptotically converge to zero.

6. Conclusions In this paper, we have investigated the robust stability and stabilization of fractional order linear systems with positive real uncertainty. Firstly, we have presented sufficient conditions for the asymptotical stability of such uncertain fractional order systems. Secondly, we have obtained sufficient conditions for the asymptotical stabilization of such uncertain fractional order systems via state feedback control, static output feedback control and observer-based output feedback control and have given the design methods for these controllers. All the results are expressed in terms of LMIs, therefore they are very convenient to be used in practice. Finally, simulation results have demonstrated the effectiveness of the proposed theoretical results.

Acknowledgments This work was partially supported by the National Natural Science Foundation of China under Grants 60974002, 61221003 and 61374030. References [1] Podlubny I. Fractional differential equations. Academic Press: San Dieage; 1999. [2] Oustaloup A, Levron F, Mathieu B, Nanot FM. Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans Circuits Syst I: Fundam Theory Appl 2000;47(1):25–39. [3] Sabatier J, Agrawal OP, Machado JAT. Advances in fractional calculus: theoretical developments and applications in physics and engineering. Dordrecht, The Netherlands: Springer; 2007. [4] Ortigueira MD. An introduction to the fractional continuous-time linear systems: the 21st century systems. IEEE Circuits Syst Mag 2008;8(3):19–26. [5] Arena P, Caponetto R, Fortuna L, Porto D. Nonlinear noninteger order circuits and systems—an introduction. Singapore: World Scientific; 2000. [6] Caponetto R, Dongola G, Fortuna L, Petras I. Fractional order systems: modeling and control applications. Singapore: World Scientific; 2010. [7] Petras I. Fractional-order nonlinear systems: modeling, analysis and simulation. Dordrecht, The Netherlands: Springer; 2011. [8] Bagley RL, Calico RA. Fractional-order state equations for the control of viscoelastic damped structures. J Guid Control Dyn 1991;14(2):304–11. [9] Ionescu CM, Keyser RD. Relations between fractional-order model parameters and lung pathology in chronic obstructive pulmonary disease. IEEE Trans Biomed Eng 2009;56(4):978–87. [10] Ionescu CM, Muntean I, Machado JAT, Keyser RD. Modeling of the lung impedance using a fractional-order ladder network with constant phase elements. IEEE Trans Biomed Circuits Syst 2011;5(1):83–9. [11] Ionescu CM. Phase constancy in a ladder model of neural dynamics. IEEE Trans Syst Man Cybern Part A: Syst Hum 2012;42(6):1543–51. [12] Weber E. Linear transient analysis. New York: Wiley; 1956. [13] Jenson VG, Jeffreys GV. Mathematical methods in chemical engineering. 2nd ed. London: Academic Press; 1977.

209

[14] Podlubny I. Fractional-order systems and PI λ Dμ controllers. IEEE Trans Autom Control 1999;44(1):208–14. [15] Nakagava M, Sorimachi K. Basic characteristics of a fractance device. IEICE Trans Fundam Electr Commun Comput Sci 1992;E75-A(12):1814–8. [16] Westerlund S, Ekstam L. Capacitor theory. IEEE Trans Dielectr Electr Insulation 1994;1(5):826–39. [17] Oustaloup A, Mathieu B, Lanusse P. The crone control of resonant plants: application to a flexible transmission. Eur J Control 1995;1(2):113–21. [18] Hamamci SE. An algorithm for stabilization of fractional-order time delay systems using fractional-order PID controllers. IEEE Trans Autom Control 2007;52(10):1964–9. [19] Vinagre BM, Feliu V. Optimal fractional controllers for rational order systems: a special case of the Wiener–Hopf spectral factorization method. IEEE Trans Autom Control 2007;52(12):2385–9. [20] Luo Y, Chen YQ. Fractional order [proportional derivative] controller for a class of fractional order systems. Automatica 2009;45(10):2446–50. [21] Caponetto R, Dongola G, Fortuna L, Gallo A. New results on the synthesis of FO-PID controllers. Commun Nonlinear Sci Numer Simul 2010;15(4): 997–1007. [22] Wang DJ, Gao XL. H 1 design with fractional-order PD controllers. Automatica 2012;48(5):974–7. [23] Delavari H, Baleanu D, Sadati J. Stability analysis of caputo fractional-order nonlinear systems revisited. Nonlinear Dyn 2012;67:2433–9. [24] Li Y, Chen YQ, Podlubny I. Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 2009;45(8):1965–9. [25] Matignon D. Stability results for fractional differential equations with applications to control processing. In: IMACS, IEEE-SMC. Lille, France, 1996. p. 963–68. [26] Tavazoei MS, Haeri M. A note on the stability of fractional order systems. Math Comput Simul 2009;79(5):1566–76. [27] Ahn HS, Chen YQ. Necessary and sufficient stability condition of fractionalorder interval linear systems. Automatica 2008;44(11):2985–8. [28] Ahn HS, Chen YQ, Podlubny I. Robust stability test of a class of linear timeinvariant interval fractional-order system using Lyapunov inequality. Appl Math Comput 2007;187(1):27–34. [29] Chen YQ, Ahn HS, Podlubny I. Robust stability check of fractional order linear time invariant systems with interval uncertainties. In: IEEE international conference on mechatronics and automation 2005, vol. 1, 2005. p. 210–15. [30] Tan N, Özguven OF, Özyetkin MM. Robust stability analysis of fractional order interval polynomials. ISA Trans 2009;48(2):166–72. [31] Lu JG, Chen GR. Robust stability and stabilization of fractional-order interval systems: an LMI approach. IEEE Trans Autom Control 2009;54(6):1294–9. [32] Lu JG, Chen YQ. Robust stability and stabilization of fractional-order interval systems with the fractional order: the 0o α o 1 case. IEEE Trans Autom Control 2010;55(1):152–8. [33] Farges C, Moze M, Sabatier J. Pseudo-state feedback stabilization of commensurate fractional order systems. Automatica 2010;46(10):1730–4. [34] Farges C, Sabatier J, Moze M. Fractional order polytopic systems: robust stability and stabilisation. Adv Differ Equ 2011;35(1–10). [35] Lan YH, Zhou Y. LMI-based robust control of fractional-order uncertain linear systems. Comput Math Appl 2011;62(3):1460–71. [36] Li C, Wang JC. Robust stability and stabilization of fractional order interval systems with coupling relationships: the 0 o α o 1 case. J Frankl Inst 2012;349(7): 2406–2419. [37] Liao Z, Peng C, Li W, Wang Y. Robust stability analysis for a class of fractional order systems with uncertain parameters. J Frankl Inst 2011;348(6):1101–13. [38] Hosseinnia SH, Ghaderi AR, Mahmoudian M, Momani S. Sliding mode synchronization of an uncertain fractional order chaotic system. Comput Math Appl 2010;59(5):1637–43. [39] Lin TC, Lee TY. Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control. IEEE Trans Fuzzy Syst 2011;19(4):623–35. [40] Wang Z, Huang X, Shen H. Control of an uncertain fractional order economic system via adaptive sliding mode. Neurocomputing 2012;83:83–8. [41] Garcia G, Daafouz J, Bernussou J. Output feedback disk pole assignment for systems with positive real uncertainty. IEEE Trans Autom Control 1996;41(9): 1385–1391. [42] Haddad WM, Bernstein DS. Robust stabilization with positive real uncertainty: beyond the small gain theorem. Syst Control Lett 1991;17(3):191–208. [43] Haddad WM, Bernstein DS. Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle, and Popov theorems and their application to robust stability. Part I: continuous-time theory. Int J Robust Nonlinear Control 1993;3:313–39. [44] Shim D. Quadratic stability in the circle theorem or positivity theorem. Int J Robust Nonlinear Control 1996;6:781–8. [45] Sabatier J, Moze M, Farges C. On stability of fractional order systems. In: Proceedings of 3rd IFAC workshop on fractional differentiation and its applications, Ankara, Turkey, 2008. p. 1–13. [46] Ho DWC, Lu GP. Robust stabilization for a class of discrete-time non-linear systems via output feedback: the unified LMI approach. Int J Control 2003;76:105–15. [47] Wang Z, Unbehauen H. Robust H2 =H 1 state estimation for systems with error variance constraints: the continuous-time case. IEEE Trans Autom Control 1999;44(5):1061–5. [48] Diethelm K, Ford NJ, Freed AD. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 2002;29:3–22.