Proceeding of the IEEE International Conference on Automation and Logistics

Zhengzhou, China, August 2012

Stabilization of Singular Fractional-Order Systems·

A Linear Matrix Inequality Approach

Xiaona Songl,2 and Leipo Liu 1 I. Electronic and Information Engineering College

Henan University of Science and Technology

471003 Luoyang, China

2. Luoyang Optoelectro Technology Development Center

471009 Luoyang, China

xiaona_97@ 163.com liuleipo123@yahoo.com.cn

Abstract-In this study, the problems of stability and stabi­lization for singular fractional-order (SFO) systems have been studied. For the stability problem, conditions are given such that the SFO system is regular and stable; while for the stabilization problem, we design a state feedback control law which guarantees the resulting closed-loop system is stable. In terms of linear matrix inequality, an explicit expression for the desired state feedback control is given. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.

Index Terms-Singular systems, fractional-order systems, linear matrix inequality

I. INTRODUCTION

Singular systems, which are also referred to as descriptor

systems, implicit systems, generalized state-space systems

and so on [7], [9], have been widely studied by many authors

in the past years. This is due not only to the theoretical

interest but also to the extensive application of such system in

large-scale systems, economic systems, power systems, and

other areas [7], [9]. With respect to the problem of stability

and stabilization for the singular systems, much attention

has been focused on the problem of robust stability and

robust stabilization for continuous-time singular systems [5],

[20], [21] and discrete-time singular systems [19], [22]. The

problem of robust stability for interval descriptor systems has

also been presented in [12]. But it should be pointed out that

all the existing results are for normal state space system.

On the other hand, fractional-order (FO) systems have

attracted increasing interest [I], [2], [16], [18]. This is mainly

due to the fact that many real-world physical systems are

better characterized by FO differential equations [15]. The

analysis of stability and stabilization for FO linear time

invariant (FO-LTI) system have been widely investigated, and

there have been many results [8], [II], [13], [17], [23]. For

interval FO-LTI systems, the stability and the controllability

problems have been addressed for the first time in [14] and

[6], respectively. Recently, the stability analysis of FO-LTI

systems with order 1 � a < 2 is converted into the domain

978-1-4673-0364-4112/$31.00 ©2012 IEEE 19

Zhen Wang3

3. College of Information Science and Engineering

Shandong University of Science and Technology

266590 Qingdao, China

wangzhen. sd@gmail.com

of ordinary systems which is well established and well

understood in [18]; while in [16], a necessary and sufficient

LMI condition for stability analysis of FO-LTI system with

order 0 < a < 1 is given. But for singular fractional-order

(SFO) system, the problems of stability and the stabilization

are rarely attacked.

The main contribution of this paper include the analysis of

the stability and stabilization condition for SFO LTI systems.

The Caputo definition for fractional derivative is adopted. The

purpose of the stability problem is to give condition such that

the SFO system is regular and stable; while the aim of the

stabilization is to design a state feedback control law such

that the resulting closed-loop system is regular and stable. In

terms of linear matrix inequality eLMI), sufficient conditions

for the solvability of the stability and stabilization problem

for SFO LTI system of 0 < a < 2 are proposed. When

the LMI is feasible, an explicit expression of a desired state

feedback controller is also given.

Notations: Throughout this paper, for real symmetric

matrices X and Y, the notation X � Y (respectively,

X > Y) means that the matrix X -Y is positive semidefinite

(respectively, positive definite). The notation NIT represents

the transpose of the matrix lVI. Inxn denotes the nxn identity

matrix. In symmetric block matrices, "*" is used as an ellipsis

for terms induced by symmetry. Matrices, if not explicitly

stated, are assumed to have appropriate dimensions. Sym(X) denotes the expression X + XT.

II. PRELIMINARIES AND PROBLEM FORMULATION

In this paper, we adopt the following Caputo definition for

fractional derivative, which allows utilization of initial values

of classical integer-order derivatives with known physical

interpretations [4], [15]

D"f(t) = d"f(t)

=

1 it f(n)(T)dT (I)

dta r(n-a) 0 (t-T)a+l-n'

where n is an integer satisfying n - 1 < a � n.

Considering the following SFO LTI system:

EDCtx(t) = Ax(t) + Bu(t) , 0 < a < 2, (2)

where a is the time fractional derivative order. x(t) ERn is

the state, u(t) E Rrn is the control input. The matrix E E Rnxn is singular, we shall assume that rank E = r < n. A and B are known real constant matrices with appropriate

dimensions.

Without loss of generality, we suppose E, A and B have

the following form

E= [ ci The nominal unforced SFO system of (2) can be written as

EDCtx(t) = Ax(t) . (4)

Definition I : I) The SFO system (4) is said to be regular if det(sCt E-A)

is not identically zero.

2) The SFO system (4) is said to be impulse free if (4) is

regular and deg(det(sCtE - A)) = rankE. Definition II :

The SFO system (2) is said to be stabilizable if there

exists a linear state feedback control law u(t) = Kx(t), K E RTnxn such that the closed-loop system is regular,

impulse free and stable in the sense of Definition I. In this

case,

u(t) = Kx(t), (5)

is said to be a state feedback control law for system (2).

Generally speaking, there are two kinds of stabilization

problems for singular continuous-time systems. One is to

determine state feedback controllers such that the c1osed­

loop system is regular, impulse-free and stable. The other is

to design state feedback controllers to make the closed-loop

system regular and stable [22]. In this paper, we deal with

the second stabilization problem for the SFO system (2).

III. MAIN RESULTS

In this section, we give a solution to the stability analysis

and the stabilization problems formulated in the previous

part, by using a strict LMI approach. We first give the

following results which will be used in the proof of our main

results.

Lemma 1: [i8J The FO-LTI system Dqx = Ax (1 � q < 2) is asymptotically stable if and only if the LTI system,

� [ A sin q7r x-

2 - -Acos q;

is asymptotically stable.

Acos q; ] _

Asin q; x, (6)

Lemma 2: [3J Integer order system x(t) = Ax(t) is

asymptotically exponentially stable if and only if there exists

20

a positive definite matrix PES, where S denotes the set of

symmetric matrices, such that:

ATp + PA<O. (7)

Theorem 1: The system (4) with order 1 � a < 2 is

regular and stable if and only if the following conditions are

satisfied ..

I) A4 is invertible.

2) There exits a symmetric matrix P > 0, the following

LMI satisfied

sym{8 Q9 (Ap)} < 0, (8)

where

8= [ sin Ct27r _ cos (�7r (9)

Proof' For the system as follows

EDCtx(t) = Ax(t) . (10)

Let f(s) = 1 sCtE - A I, then we can prove that f(s) is an

analytical function with respect to s on the whole complex

plane. When s is large enough, 1 sCt E - A IY!O 0, it follows

from f (s) is an analytical function that (sCt E - A) -1 exists

almost everywhere on comlex plane [10].

Therefore, det( sO: E-A) is not identically zero, and system

(10) is regular.

Now, we will present the stable condition.

Let

x(t) = [ ����� ] , then, the SFO system (10) can be decomposed as

where

From (12), we can derive

A1X1(t) + A2X2(t) , A3X1(t) + A4X2(t) ,

This, together with (1l), we have

(I l)

(12)

(13)

(14)

It is easy to see that the stability condition of the SFO system

(10) is equivalent to that of the system (14). In view of this,

next we shall find the stability condition of the system (14).

U sing the Lemma 1, the SFO LTI system Dn Xl (t) =

(AI - A2A4l A3)Xl(t) = AX1(t) (1 ::; a < 2) is asymptot­

ically stable if and only if the LTI system,

Xl t - - 2 � [ A sin mr

( ) - -Acos (�7f

is asymptotically stable.

A cos 0i27f ] _

A- . n7f Xl(t) , sm2 (15)

So, the stability condition of (10) is finally equivalent to

that of the system (15).

According to Lemma 2, the system (15) is asymptotically

exponentially stable if and only if there exists a positive

definite matrix PES, where S denotes the set of symmetric

matrices, such that:

[ (Ap + pAT) sin (�7f (P AT - Ap) cos (�7f - - 2 < 0

(Ap - pAT) cos n7f ] (AP + PAT) sin (�7f ,

(16)

which is equivalent to there exits P > 0 such that

sym{8 Q9 (Ap)) < 0, (17)

which is the stable condition. This completes the proof.

Now, we are in a position to present a solution to the state

feedback control problem.

Theorem 2: The system (2) with order 1 ::; a < 2 can

be stabilized by the state feedback controller (5), if there

exist the matrices G, L2 and symmetric matrix P > 0, the

following conditions are satisfied

I) A4 + B2K2 is invertible.

2) the following LMI satisfied

where

Proof Let

AlP + B1G - 2L2, L1A3P = L1B2G,

B2G, (A2 + B1K2)(A4 + B2K2) -1.

from (IS), we can derive

where

On the other hand, for the system

EDOi x(t) = Ax(t) + Bu(t) ,

(IS)

(19)

(20)

(21 )

(22)

(23)

(24)

(26)

21

We design the state feedback controller as in Definition II,

and the controller K has the following form:

(27)

Then, using the expression in (3), and (27), the SFO system

(26) can be decomposed as

(AI + B1Kl)Xl(t) +(A2 + B1K2)X2(t) (2S)

o (A3 + B2Kdxl(t) +(A4 + B2K2)X2(t)

From (29), we can derive

This, together with (28), one can obtain

where

Akl Al + B1Kl, Ak2 = A2 + B1K2,

Ak3 A3 + B2Kl, Ak4 = A4 + B2K2,

(29)

(30)

Using the same method in the stability part, according

to the Lemma I, the stability condition of system (31) is

equivalent to that the system as follows:

£1 (t) =

k sm 2 [ A . C>7f -Ak cos (�7f

Ak cos C>27f ] A (t) A . C>7f Xl . ksm2 (32)

Therefore, the stability condition of (26) is finally equivalent

to that of the system (32).

According to Lemma 2, the system (32) is asymptotically

exponentially stable if and only if there exists a positive

definite matrix PES, where S denotes the set of symmetric

matrices, such that:

[ (AkP + PAn sin ';7f (P A[ - AkP) cos (�7f

(AkP - PAn cos ';7f ] (AkP + PAn sin (�7f

which is equivalent to there exits P > ° such that

< 0,

(33)

(34)

which is the same to (24). This completes the proof.

Theorem 3: [l6) Fractional system DVx(t) = Ax(t) of

order ° < v < 1 is rc> asymptotically stable if and only if

there exist positive definite matrices Xl = X{ E cnxn and

X2 = X� E cnxn such that

(35)

where

r = e.i(l-v)-!f. (36)

From Theorem 3, we can obtain the following theorem

easily.

Theorem 4: FO system (4) of order 0 < v < 1 is

roc asymptotically stable if and only if there exist positive

definite matrices Xl = X; E cnxn and X2 = X� E cnxn such that

1) A4 is invertible.

2) The following LMI satisfied

(37)

where

Proof' Using the similar method in the Proof of the

Theorem I and the Theorem 3, we can obtain the condition

directly. This completes the proof.

Theorem 5: SFO system (2) of order 0 < a < 1 can

be stabilized by (5), if there exist the matrices G, K2 and

symmetric matrix X > 0, such that

I) B2 and A4 + B2K2 is invertible

2) The following LMI satisfied

where

G = LA3X, L = (A2+B1K2)(A4 +B2K2)-1, A3 = B2Kl. (40)

Proof' For system

EDOCx(t) = Ax(t) + Bu(t), 0 < a < 1, (41)

using the similar method in the Proof of the Theorem 2, we

can obtain that the system above is asymptotically stable if

and only if the system

D"'Xl(t) = [Akl-Ak2Ak4l Adxl(t) = AkXl(t), 0 < a < 1,

(42)

is asymptotically stable.

According to Theorem 4, and let Xl = X2 = X, we can

obtain that the system (42) is asymptotically stable if and

only if the following matrix inequality satisfied:

1 1f(1-a) [AklX - Ak2Ak4 Ak3X] sin

2 < O. (43)

From (40), we can obtain the following LMI

22

1f(1-a) 1 sin 2

Sym(A1X + B1B:; A3X - 2G) < 0, (44)

which is the stabilization condition. This completes the proof.

IV. NUMERICAL EXAMPLE

Consider the stabilization problem for the SFO system (2)

of order 0 < a < 2 with the following parameters:

a 1.2, Al=

A3 3 � ] , 0

Bl -1 0

0 -1

[ � -1 ] 5 ' A2 = [

A4 = [ -5

-1 �6 ] , ] , [ -2 0

B2 = 0 -1

-5 0 ] , 0 -2

] . Obviously, when u(t) = 0, the system (2) is unstable

because the eigenvalues of A are {-4.9515, -2.2321 + 2.3433i, -2.2321- 2.3433i, 4.4158}, which are outside the

stable area. The purpose is to design a state feedback control

law such that the closed-loop system is stable. Now, using

Matlab LMI Control Toolbox to solve the LMI (18), the

asymptotically stabilizing state-feedback gain matrix is ob­

tained as

o -3

0.6873 -2.2901

-0.0023 ] -11.1605 .

While, for the SFO system (2) with the following parameters:

0.8, Al = [ -3 �8

] , A2 = [ -1 -2 ] a 0 -1 -2 '

A3 [ -3 � ] , [ -1 � ] , 0 A4 = -1

Bl [ -1 �1

] , [ -3 0 ] . 0 B2 = 0 -1

when u(t) 0, the system (2) is unstable because the

eigenvalues of A are { 5.1629, 0.7183,-4.4833,-7.3979}. Then, using Matlab LMI Control Toolbox to solve the LMI

(39), we can obtain the state-feedback gain matrix. With the

state feedback controller, the closed-loop system is stable.

The state response of the SFO system of order a = 1.2 are

given in Fig. I, while Fig. 2 shows the state response of the

SFO system of order a = 0.8. From these simulation results, it can be seen the designed

state feedback controller ensures the asymptotic stability of

the SFO system.

V. CONCLUSION

The problems of stability and stabilization for SFO system

have been studied. In terms of LMI, sufficient conditions

for the stability and stabilization of the SFO system have

been established. The proposed state feedback control law

guarantees that the closed-loop system is stable.

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23

-0.5

¥ -1

-1.5 1

-2 _,

-2.5

- 3 oL-------------��------------�,O--------------�,5 Time(s)

Fig.I. State response Xi (t), i= 1,2,3,4. (a= 1.2)

-

---',

1.5

-0.5 oL-------------�--------------�, o--------------�,5 Time(s)

Fig.2. State response Xi(t), i=I,2,3,4. (a=O.8)