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Robust stability analysis for a class of fractional order systems with uncertain parameters
Journal of the Franklin Institute 348 (2011) 1101–1113 www.elsevier.com/locate/jfranklin
Robust stability analysis for a class of fractional order systems with uncertain parameters Zeng Liaon, Cheng Peng, Wang Li, Yong Wang Department of Automation, University of Science and Technology of China, Hefei 230027, China Received 10 March 2010; received in revised form 16 April 2011; accepted 21 April 2011 Available online 28 April 2011
Abstract The research of robust stability for fractional order linear time-invariant (FO-LTI) interval systems with uncertain parameters has become a hot issue. In this paper, it is the first time to consider robust stability of uncertain parameters FO-LTI interval systems, which have deterministic linear coupling relationship between fractional order and other model parameters. Linear matrix inequalities (LMI) methods are used, and a criterion for checking asymptotical stability of this class of systems is presented. One numerical illustrative example is given to verify the correctness of the conclusions. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Many researches have shown the existence of fractional order objects, such as viscoelastic materials, chaos, fractals and so on, which can be described more accurately using fractional order calculus than traditional integer order calculus [1–5]. Therefore, fractional order calculus has become a hot research issue in recent years [6–10]. In the system control fields, Podlubny firstly applied fractional differential equations and Laplace transform to system control problem, described control system using fractional order transfer function, carried out the unit step response expression and proposed the PIlDm controller [11,12]. Matignon systematically studied the properties of fractional order systems, including stability, controllability and observability [13–15]. Oustaloup studied the methods of design and implementation for CRONE controller and gained good n
Corresponding author. E-mail address: [email protected] (Z. Liao).
0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.04.012
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control effects in practice [16–19]. Vinagre et al. studied the descretization realization problems and extended the Tustin method to fractional order systems [20]. Fractional order systems have been attracting more and more experts and scholars in-depth exploration and study. Many people have studied the stability of fractional order systems and proposed several useful conclusions [21–26]. For fractional order linear time-invariant (FO-LTI) interval systems, the stability issues were discussed in Refs. [27–31]. In Ref. [27], a method based on the Lyapunov inequality was proposed. In Ref. [28], a stability criterion was proposed to check the stability of FO-LTI interval systems with small perturbation interval. In Ref. [29], a better stability criterion without perturbation interval size restricted was proposed. However, the calculation of these methods is complex because of eigenvalues calculation of uncertain FO-LTI interval systems. In order to reduce the complexity, the linear matrix inequalities (LMI) methods were used. The necessary and sufficient conditions based on LMI methods with fractional order 1rao2 and 0oao1 were proposed in Refs. [30,31] respectively. However, the papers mentioned above pretermit such a fact that the fractional order and other model parameters may have some coupling relationship in real systems. In system modeling or identification, the model parameters of fractional order systems depend on the value of fractional order chosen in advance and vary with fractional order. Therefore, there is some coupling relationship between fractional order and other model parameters of fractional order systems. If fractional order is perturbed in a deterministic interval, other model parameters of system will correlatively be perturbed at the same time. In this paper, it is the first time to discuss the robust stability issue of FO-LTI interval systems with some deterministic linear coupling relationship between fractional order and other model parameters. LMI methods are used under the condition that fractional order is 1rao2, and a criterion for determining asymptotic stability of system is presented. A numerical illustrative example is given to verify the correctness of conclusions. The rest of the paper is organized as follows: In Section 2, fractional order calculus and system model are presented. In Section 3, the stability criterion is derived. A numerical illustrative example is shown in Section 4. Finally, conclusions are drawn in Section 5.
2. Model description There are several definitions of fractional order calculus. Caputo’s definition of the fractional order differentiation plays an important role in fractional order systems, because the Laplace transform of Caputo’s definition allows utilization of initial values of classical integer order derivatives with clear physical interpretations [7]. In this paper, Caputo’s definition is applied and it can be written as Z t ðnÞ da 1 f ðtÞdt Da f ðtÞ ¼ a f ðtÞ ¼ , ðn1oaonÞ ð1Þ GðnaÞ 0 ðttÞaþ1n dt where G(x) is Gamma function. Eq. (1) denotes fractional order integration in case of ao0, and fractional order differentiation in case of a40. Because the stability domain of fractional order systems with fractional order 0oao1 is nonconvex set, which cannot be
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described using LMI methods, this paper focuses on the case that fractional order is 1rao2. If FO-LTI interval system does not have any coupling relationship between fractional order and other model parameters, it can be described by state space equation of the form da xðtÞ ¼ AxðtÞ dta
ð2Þ
where A 2 ½A,A ¼ f½aij : a ij raij raij ,1ri,jrng. If there is some coupling relationship in FO-LTI interval system, the perturbation of model parameters A can be considered as a function of variable a. Therefore, the FO-LTI interval system with some coupling relationship between A and a can be described by state space equation of the form da xðtÞ ¼ AxðtÞ dta
ð3Þ
where a 2 ½a1 ,a2 ,a1 ,a2 2 R,A ¼ AðaÞ 2 ½AðaÞ,AðaÞ ¼ f½aij ðaÞ9a ij ðaÞraij ðaÞraij ðaÞ,1ri,jrng What we discuss in the paper will focus on the linear coupling relationship between the perturbation scope of A(a) and fractional order a. Then the system model can be described by state space equation of the form d a0 þDa xðtÞ ¼ ½A0 þ kða0 þ DaÞDAxðtÞ dta0 þDa
ð4Þ
where DA=AM[dij]n n=[gij]n n[dij]n n,gijZ0, gijAR,1rdijr1, a0 and A0 are nominal values. kaDA shows that the perturbation scope of model parameters A varies with fractional order a linearly. k is the linear coefficient and AM is the maximal perturbation scope matrix. Dealing with perturbation item of system (4) using the method proposed in Ref. [30], we obtain the following equivalent equation: d a0 þDa xðtÞ ¼ ½A0 þ kða0 þ DaÞDA FA EA xðtÞ dxa0 þDa
ð5Þ
where a0=(a1þa2)/2, Da=(a2a1)/2 2
2
FA ¼ diagðd11 ,:::,d1n ,:::,dn1 ,:::,dnn Þ 2 Rn n pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi DA ¼ ½ g11 en1 ,. . ., g1n en1 ,:::, gn1 enn ,. . ., gnn enn nn2 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi EA ¼ ½ g11 en1 ,. . ., g1n enn ,. . ., gn1 en1 ,. . ., gnn enn Tn2 n epk 2 Rp denotes the column vectors with the kth element being 1 and all the others being 0. Eq. (5) has a unique solution [7], therefore it is meaningful to discuss the above problem, which fractional order a and other model parameters A have simple linear coupling relationship in this paper. If k=1/a and fractional order a does not be perturbed, Eq. (5) will degenerate into Eq. (2) and become a general FO-LTI interval system. If there is no parameter perturbation in the system, that is AM=0, Eq. (5) will degenerate further into a general FO-LTI system of which there have been several mature conclusions to check the stability.
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3. Stability analysis Before the stability analysis of FO-LTI interval system (5), several useful lemmas are introduced: Lemma 1. [25] Let AARn n be a deterministic real matrix. Then system (2) is asymptotically stable if and only if 9argðspecðAÞÞ94ap=2 where spec(A) is the spectrum of A. Lemma 2. [26] FO-LTI system with fractional order 1rao2 is asymptotically stable if and only if there exists a real symmetric positive definite matrix P40, PARn n, such that " X ðA,yÞ ¼
ðAP þ PAT Þsin y
ðAPPAT Þcosy
ðPAT APÞcos y
ðAP þ PAT Þsin y
# o0
where y=pap/2 Lemma 3. [30] For any matrix X and Y with appropriate dimensions, we have X T Y þ Y T X reX T X þ e1 Y T Y ,
for 8e40
Lemma 4. Let M(t), N(t), P(t) be symmetric matrix with appropriate dimensions, then "
# MðtÞ PðtÞ o0,8t40 PT ðtÞ NðtÞ
if and only if ( NðtÞo0 MðtÞPðtÞN 1 ðtÞPT ðtÞo0
,
8t40
Lemma 1. implies that the stability domain of fractional order systems depends on the fractional order a. The stability domain is located between boundaries, which are equal to 7pa/2. That is to say, the fractional order system is stable if and only if all eigenvalues of matrix A locate in the stability domain, as shown in Fig. 1. However, this method needs to compute all eigenvalues of system, and it is hard to yield all eigenvalues of FO-LTI interval system with uncertain parameters. If fractional order satisfied 1rao2, LMI methods can be utilized to determine system stability without computing eigenvalues. System (5) has a linear coupling relationship between fractional order a and model parameters A, therefore the eigenvalues distribution depends on fractional order a. If fractional order a changes, the stability domain and eigenvalues distribution will change at the same time, and then system stability may make some unpredictable changes. In the following paper, we will discuss this issue under these
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3
2
Imaginary axis
1 stability domain
0
θ=π-πα/2
unstability domain
-1
-2
-3 -3
-2
-1
0
1
2
3
Real axis Fig. 1. Stability domain of fractional order system with order a.
conditions. First, we prove Theorem 1 to determine whether system (5) is asymptotically stable. Theorem 1. FO-LTI interval system (5) with fractional order 1rao2 is asymptotically stable if there exist a symmetric positive definite matrix P40,PARn n and a real scalar constant e40, such that X ðA0 ,y0 Þo0 2
O1
ð6Þ PEAT 0
OT2
6O O 1 6 2 O¼6 6 EA P 0 4 0 EA P
eI 0
3 0 PEAT 7 7 7o0 7 0 5 eI
ð7Þ
where " X ðA0 ,y0 Þ ¼
ðA0 P þ PAT0 Þsin y0
ðA0 PPAT0 Þcosy0
ðPAT0 A0 PÞcosy0
ðA0 P þ PAT0 Þsin y0
O1 ¼ ðA0 P þ PAT0 Þ
#
sin y0 cosðy0 þ yM Þsin yM þ ek2 a22 DA DTA , cos y0
O2 ¼ ðPAT0 A0 PÞcosðy0 þ yM Þ y0 ¼ pða2 þ a1 Þp=4,
yM ¼ ða2 a1 Þp=4
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Proof. Suppose there exists a P40, e40, PARn n, such that 2 3 PEAT 0 O1 OT2 6 O O 0 PEAT 7 2 1 6 7 6 7o0 O¼6 E P 0 eI 0 7 4 A 5 0 EA P 0 eI Using Lemma 4, we obtain " #T " " # EA P 0 EA P O1 OT2 1 þe 0 EA P 0 O2 O1 Define
0 EA P
# o0
kða0 þ DaÞDA FA EA P cosy sin y " # " # F 0 P 0 E A A HermðX Þ9X þ X T ,F 9 ,N9 0 FA 0 EA P " # DA sin y DA cos y DðyÞ9 DA cosy DA sin y
XS ðyÞ9Herm
sin y
cos y
Substitute A=A0þkaDAFAEA into X(A,y), and then yield " # ðA0 P þ PA0 T Þsin y ðA0 PPA0 T Þcos y X ðA,yÞ ¼ ðPA0 T A0 PÞcos y ðA0 P þ PA0 T Þsin y sin y cosy þ Herm kaDA FA EA P cos y sin y sin y cosy ¼ X ðA0 ,yÞ þ Herm kaDA FA EA P cos y sin y ¼ X ðA0 ,yÞ þ XS ðyÞ For X(A0,y) of Eq. (8), substitute y=y0þDy and yield " # ðA0 P þ PAT0 Þsin y0 ðA0 PPAT0 Þcos y0 X ðA0 ,yÞ ¼ cosDy ðPAT0 A0 PÞcos y0 ðA0 P þ PAT0 Þsin y0 " # ðA0 P þ PAT0 Þcos y0 ðA0 PPAT0 Þsin y0 þ sin Dy ðPAT0 A0 PÞsin y0 ðA0 P þ PAT0 Þcosy0 ¼ X ðA0 ,y0 Þcos Dy þ X ðA0 ,p=2 þ y0 Þsin Dy We have X ðA0 ,y0 Þsin y0 þ X ðA0 ,p=2 þ y0 Þcos y0 ¼ X ðA0 ,p=2Þ
ð8Þ
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Thus X ðA0 ,yÞ ¼ X ðA0 ,y0 Þ
cosðy0 þ DyÞ sin Dy þ X ðA0 ,p=2Þ cosy0 cos y0
Because of Eq. (6) and the nature of negative definite matrix, ðA0 P þ PAT0 Þsiny0 is negative definite matrix. We have 1ra0o2, 1ra0þDao2, and then yield y0 ¼ pa0 p=2 2 ð0,p=2 Dy 2 ½yM ,yM ¼ ½pða1 a2 Þ=4,pða2 a1 Þ=4 2 ½p=4,p=4 y0 þ Dy 2 ½y0 yM ,y0 þ yM ¼ ½ppa2 =2,ppa1 =2 2 ð0,p=2 Thus, we can obtain sin y0 40,cos y0 40, sin yM rsin Dyrsin yM , cosðy0 yM ÞZ cosðy0 þ DyÞZcosðy0 þ yM Þ40and then A0 P þ PAT0 o0,which implies " # 0 A0 P þ PAT0 o0 X ðA0 ,p=2Þ ¼ 0 A0 P þ PAT0 Thus X(A0,y0) and X(A0,p/2) are negative definite matrix, because of the nature of negative definite matrix, we obtain cosðy0 þ DyÞ sin Dy þ X ðA0 ,p=2Þ cosy0 cos y0 cosðy0 þ yM Þ sin yM rX ðA0 ,y0 Þ X ðA0 ,p=2Þ cosy0 cos y0
X ðA0 ,yÞ ¼ X ðA0 ,y0 Þ
For XS(y) of Eq. (8), we have gij40, 1ri,jrn, then 2 Pn 0 j¼1 g1j " # 6 Pn T 0 0 DA DA 6 j¼1 g2j ¼6 DðyÞDðyÞT ¼ 6 ::: ::: 0 DA DTA 4 0 0
:::
0
:::
0
::: :::
::: Pn
3
j¼1 gnj
7 7 740 7 5
Then using Lemma 3 and the nature of positive definite matrix, we obtain sin y cosy XS ðyÞ ¼ Herm kða0 þ DaÞDA FA EA P cos y sin y ( " #" #) #" DA sin y DA cosy FA 0 0 EA P ¼ Herm kða0 þ DaÞ DA cos y DA sin y 0 FA 0 EA P ¼ ½kða0 þ DaÞDðyÞFNT þ kða0 þ DaÞDðyÞFN rek2 ða0 þ DaÞ2 DðyÞDðyÞT þ e1 N T N rek2 a22 DðyÞDðyÞT þ e1 N T N " #T " " # DA DTA 0 EA P 0 EA P 2 2 1 ¼ ek a2 þe T 0 DA DA 0 EA P 0
0 EA P
#
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Therefore, substitute X(A0,y) and XS(y) into Eq. (8), we obtain X ðA,yÞ ¼ X ðA0 ,yÞ þ XS ðyÞ cosðy0 þ yM Þ sin yM X ðA0 ,p=2Þ rX ðA0 ,y0 Þ cosy0 cosy0 " #T " " # T D D 0 E P 0 EA P A A A þek2 a22 þ e1 0 DA DTA 0 EA P 0 " #T " # " # EA P 0 EA P 0 O1 OT2 ¼ o0 þ e1 0 EA P 0 EA P O2 O1 Using Lemma 2, system (5) is asymptotically stable.
0
#
EA P
&
Theorem 1. enables to determine the asymptotical stability of FO-LTI interval systems with linear coupling relationship between fractional order and other model parameters. If k=1/a, Eq. (5) shows that the coupling relationship does not exist between fractional order and other model parameters. In addition, if there are no perturbations in fractional order, that is yM=0 or a1=a2, system (5) will degenerate into a commonly FO-LTI interval system, and the Theorem 1 will be the same as the one in Ref. [30] .When the system parameters are deterministic values, system (5) will degenerate further into a FO-LTI system, and the Theorem 1 will be the same as the criterion in Ref. [26]. Therefore, Theorem 1 proposed in this paper have a broader applicability in determining asymptotical stability of fractional order systems. 4. Numerical examples Consider the robust stability of the following FO-LTI interval system with linear coupling relationship between fractional order a and model parameters A: da xðtÞ ¼ AðaÞxðtÞ dta
ð9Þ
where a=a0þDaA[a1,a2]C[1,2), A(a)=A0þkaDA, DA=AM[dij]n n, 1rdijr1, and 2 3 2 3 1:35 0:35 1:25 0:25 0:25 0:05 6 7 6 7 2:45 1:1 5, AM ¼ 4 0:1 0:05 0:2 5 A0 ¼ 4 0 0:3 1:25 1:75 0:1 0:05 0:25
Case 1. If k=1/a and a has a deterministic value, system (9) will degenerate into a FO-LTI interval system without any coupling relationship in parameters. In this case, Theorem 1 is equivalent to the method mentioned in Ref. [30]. And we can determine whether system (9) is asymptotically stable using Theorem 1 or the method in Ref. [30]. For example, if a=1.5, we can obtain a feasible solution as follows: 2
3
0:8851
0:1545
0:0524
6 P ¼ 4 0:1545 0:0524
0:3275 0:0634
7 0:0634 540 e ¼ 0:7959 0:3476
Then system (9) is asymptotically stable.
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3
Imaginary axis
2
1
0
-1
-2
-3 -4
-3.5
-3
-2.5
-2
-1.5 -1 Real axis
-0.5
0
0.5
1
Fig. 2. Eigenvalues perturbation area of FO-LTI system without coupling relationship. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In Fig. 2, blue and green lines are, respectively, upper and lower boundaries of fractional order a=1.4 and a=1.5, green area is the eigenvalues perturbation area of A, which is computed using the methods in Ref. [29]. Because there is no coupling relationship between fractional order a and model parameters A, eigenvalues perturbation area only depends on A. No matter how fractional order a changes, the eigenvalues perturbation area does not occur any changes. Case 2. If ka1/a and a is perturbed in a deterministic interval [a1,a2], Eq. (9) shows that there is a linear coupling relationship between a and A. The LMI methods that have been presented so far in all literatures cannot easily determine the stability of system in this case. However, we can determine whether system (9) is asymptotically stable using Theorem 1 in the paper.
If k=1/1.5, aA[1.4,1.5], we can obtain a feasible solution as follows: 2 3 12:9016 2:8031 1:0655 6 7 P ¼ 4 2:8031 4:2923 0:9026 540 e ¼ 7:5092 1:0655 0:9026 4:9782 Thus, system (9) is asymptotically stable. In Fig. 3, green and blue areas are, respectively, eigenvalues perturbation areas with fractional order a=1.4 and a=1.5. Fig. 3 shows that eigenvalues perturbation area is extended when fractional order a is increased from 1.4 to 1.5. However, this area does not exceed the boundaries determined by fractional order a. In other words, we can always ensure that all eigenvalues of system are located in the stability domain no matter how
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3 order perturbation area
Imaginary axis
2
1 eigenvalue perturbation area
0
-1
-2
order perturbation area
-3 -4
-3.5
-3
-2.5
-2
-1.5 -1 Real axis
-0.5
0
0.5
1
Fig. 3. Eigenvalues perturbation area of FO-LTI system with linear coupling relationship and aA[1.4,1.5]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3
Imaginary axis
2
unstable eigenvalue
1
0
-1
-2
-3 -4
-3.5
-3
-2.5
-2
-1.5 -1 Real axis
-0.5
0
0.5
1
Fig. 4. Eigenvalues perturbation area of FO-LTI system with linear coupling relationship and aA[1.4,1.7]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
fractional order is perturbed in the deterministic interval. Therefore, this FO-LTI interval system with linear coupling relationship is always asymptotically stable in this case. If k=1/1.5, aA[1.4,1.7], the feasible solution does not exist, and system (9) is not asymptotically stable.
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In Fig. 4, green and blue areas are, respectively, eigenvalues perturbation areas with fractional order a=1.4 and a=1.7. Fig. 4 shows that eigenvalues perturbation area exceeds the boundaries determined by fractional order when fractional order a is increased from 1.4 to 1.7. We cannot ensure that all eigenvalues of system are located in the stability domain when fractional order is perturbed in the deterministic interval. Therefore, this FO-LTI interval system with linear coupling relationship is unstable in this case. Remark 1. If we ignore the coupling relationship between fractional order and other model parameters, system stability only depends on model parameters A and the maximum value of fractional order a no matter how fractional order a is perturbed in a deterministic interval. Remark 2. If we consider the coupling relationship between fractional order and other model parameters, fractional order perturbation will have great effects on the robust stability of FO-LTI interval system. In case 2, system stability not only depends on perturbation of model parameters A, but also the value of fractional order a in a deterministic interval. We must consider the upper and lower perturbation boundaries of fractional order a comprehensively. The perturbation areas of all eigenvalues will vary if fractional order a is perturbed in some interval. Coefficient ka denotes the change rate of perturbation area of all eigenvalues. Thus, when A and a are perturbed at the same time, perturbation area of all eigenvalues may exceed the stability boundary determined by fractional order a and the system will become unstable. In case 2, when the value of fractional order a is increased, the system becomes unstable from stable.
5. Conclusion This paper has firstly discussed the robust stability of FO-LTI interval system with certain linear coupling relationship between fractional order and other model parameters, and presented the criterion derived from LMI methods to determine whether this system is asymptotically stable. At present, there are many questions on robust stability of fractional order systems unsolved. The universal conclusion has not been presented yet. Therefore, fractional order systems still need more and more people to explore and study further. Acknowledgment This work was supported by the National Natural Science Foundation of China (no. 61004017, 60974103). References [1] P.J. Torvik, R.L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, Journal of Applied Mechanics 51 (2) (1984) 294–298. [2] J.J. de Esp ´ındola, C.A. Bavastri, E.M.O. Lopes, On the passive control of vibrations with viscoelastic dynamic absorbers of ordinary and pendulum types, Journal of the Franklin Institute 347 (1) (2010) 102–115. [3] X. Wu, J. Li, G. Chen, Chaos in the fractional order unified system and its synchronization, Journal of the Franklin Institute 345 (4) (2008) 392–401.
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[29] H.S. Ahn, Y.Q. Chen, Necessary and sufficient stability condition of fractional-order interval linear systems, Automatica 44 (11) (2008) 2985–2988. [30] J.G. Lu, G.R. Chen, Robust stability and stabilization of fractional-order interval systems: an LMI approach, IEEE Transactions on Automatic Control 54 (6) (2009) 1294–1299. [31] J.G. Lu, Y.Q. Chen, Robust stability and stabilization of fractional-order interval systems with the fractional order a: the 0oao1 case, IEEE Transactions on Automatic Control 55 (1) (2010) 152–158.
Robust stability analysis for a class of fractional order systems with uncertain parameters Zeng Liaon, Cheng Peng, Wang Li, Yong Wang Department of Automation, University of Science and Technology of China, Hefei 230027, China Received 10 March 2010; received in revised form 16 April 2011; accepted 21 April 2011 Available online 28 April 2011
Abstract The research of robust stability for fractional order linear time-invariant (FO-LTI) interval systems with uncertain parameters has become a hot issue. In this paper, it is the first time to consider robust stability of uncertain parameters FO-LTI interval systems, which have deterministic linear coupling relationship between fractional order and other model parameters. Linear matrix inequalities (LMI) methods are used, and a criterion for checking asymptotical stability of this class of systems is presented. One numerical illustrative example is given to verify the correctness of the conclusions. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Many researches have shown the existence of fractional order objects, such as viscoelastic materials, chaos, fractals and so on, which can be described more accurately using fractional order calculus than traditional integer order calculus [1–5]. Therefore, fractional order calculus has become a hot research issue in recent years [6–10]. In the system control fields, Podlubny firstly applied fractional differential equations and Laplace transform to system control problem, described control system using fractional order transfer function, carried out the unit step response expression and proposed the PIlDm controller [11,12]. Matignon systematically studied the properties of fractional order systems, including stability, controllability and observability [13–15]. Oustaloup studied the methods of design and implementation for CRONE controller and gained good n
Corresponding author. E-mail address: [email protected] (Z. Liao).
0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.04.012
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control effects in practice [16–19]. Vinagre et al. studied the descretization realization problems and extended the Tustin method to fractional order systems [20]. Fractional order systems have been attracting more and more experts and scholars in-depth exploration and study. Many people have studied the stability of fractional order systems and proposed several useful conclusions [21–26]. For fractional order linear time-invariant (FO-LTI) interval systems, the stability issues were discussed in Refs. [27–31]. In Ref. [27], a method based on the Lyapunov inequality was proposed. In Ref. [28], a stability criterion was proposed to check the stability of FO-LTI interval systems with small perturbation interval. In Ref. [29], a better stability criterion without perturbation interval size restricted was proposed. However, the calculation of these methods is complex because of eigenvalues calculation of uncertain FO-LTI interval systems. In order to reduce the complexity, the linear matrix inequalities (LMI) methods were used. The necessary and sufficient conditions based on LMI methods with fractional order 1rao2 and 0oao1 were proposed in Refs. [30,31] respectively. However, the papers mentioned above pretermit such a fact that the fractional order and other model parameters may have some coupling relationship in real systems. In system modeling or identification, the model parameters of fractional order systems depend on the value of fractional order chosen in advance and vary with fractional order. Therefore, there is some coupling relationship between fractional order and other model parameters of fractional order systems. If fractional order is perturbed in a deterministic interval, other model parameters of system will correlatively be perturbed at the same time. In this paper, it is the first time to discuss the robust stability issue of FO-LTI interval systems with some deterministic linear coupling relationship between fractional order and other model parameters. LMI methods are used under the condition that fractional order is 1rao2, and a criterion for determining asymptotic stability of system is presented. A numerical illustrative example is given to verify the correctness of conclusions. The rest of the paper is organized as follows: In Section 2, fractional order calculus and system model are presented. In Section 3, the stability criterion is derived. A numerical illustrative example is shown in Section 4. Finally, conclusions are drawn in Section 5.
2. Model description There are several definitions of fractional order calculus. Caputo’s definition of the fractional order differentiation plays an important role in fractional order systems, because the Laplace transform of Caputo’s definition allows utilization of initial values of classical integer order derivatives with clear physical interpretations [7]. In this paper, Caputo’s definition is applied and it can be written as Z t ðnÞ da 1 f ðtÞdt Da f ðtÞ ¼ a f ðtÞ ¼ , ðn1oaonÞ ð1Þ GðnaÞ 0 ðttÞaþ1n dt where G(x) is Gamma function. Eq. (1) denotes fractional order integration in case of ao0, and fractional order differentiation in case of a40. Because the stability domain of fractional order systems with fractional order 0oao1 is nonconvex set, which cannot be
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described using LMI methods, this paper focuses on the case that fractional order is 1rao2. If FO-LTI interval system does not have any coupling relationship between fractional order and other model parameters, it can be described by state space equation of the form da xðtÞ ¼ AxðtÞ dta
ð2Þ
where A 2 ½A,A ¼ f½aij : a ij raij raij ,1ri,jrng. If there is some coupling relationship in FO-LTI interval system, the perturbation of model parameters A can be considered as a function of variable a. Therefore, the FO-LTI interval system with some coupling relationship between A and a can be described by state space equation of the form da xðtÞ ¼ AxðtÞ dta
ð3Þ
where a 2 ½a1 ,a2 ,a1 ,a2 2 R,A ¼ AðaÞ 2 ½AðaÞ,AðaÞ ¼ f½aij ðaÞ9a ij ðaÞraij ðaÞraij ðaÞ,1ri,jrng What we discuss in the paper will focus on the linear coupling relationship between the perturbation scope of A(a) and fractional order a. Then the system model can be described by state space equation of the form d a0 þDa xðtÞ ¼ ½A0 þ kða0 þ DaÞDAxðtÞ dta0 þDa
ð4Þ
where DA=AM[dij]n n=[gij]n n[dij]n n,gijZ0, gijAR,1rdijr1, a0 and A0 are nominal values. kaDA shows that the perturbation scope of model parameters A varies with fractional order a linearly. k is the linear coefficient and AM is the maximal perturbation scope matrix. Dealing with perturbation item of system (4) using the method proposed in Ref. [30], we obtain the following equivalent equation: d a0 þDa xðtÞ ¼ ½A0 þ kða0 þ DaÞDA FA EA xðtÞ dxa0 þDa
ð5Þ
where a0=(a1þa2)/2, Da=(a2a1)/2 2
2
FA ¼ diagðd11 ,:::,d1n ,:::,dn1 ,:::,dnn Þ 2 Rn n pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi DA ¼ ½ g11 en1 ,. . ., g1n en1 ,:::, gn1 enn ,. . ., gnn enn nn2 pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi EA ¼ ½ g11 en1 ,. . ., g1n enn ,. . ., gn1 en1 ,. . ., gnn enn Tn2 n epk 2 Rp denotes the column vectors with the kth element being 1 and all the others being 0. Eq. (5) has a unique solution [7], therefore it is meaningful to discuss the above problem, which fractional order a and other model parameters A have simple linear coupling relationship in this paper. If k=1/a and fractional order a does not be perturbed, Eq. (5) will degenerate into Eq. (2) and become a general FO-LTI interval system. If there is no parameter perturbation in the system, that is AM=0, Eq. (5) will degenerate further into a general FO-LTI system of which there have been several mature conclusions to check the stability.
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3. Stability analysis Before the stability analysis of FO-LTI interval system (5), several useful lemmas are introduced: Lemma 1. [25] Let AARn n be a deterministic real matrix. Then system (2) is asymptotically stable if and only if 9argðspecðAÞÞ94ap=2 where spec(A) is the spectrum of A. Lemma 2. [26] FO-LTI system with fractional order 1rao2 is asymptotically stable if and only if there exists a real symmetric positive definite matrix P40, PARn n, such that " X ðA,yÞ ¼
ðAP þ PAT Þsin y
ðAPPAT Þcosy
ðPAT APÞcos y
ðAP þ PAT Þsin y
# o0
where y=pap/2 Lemma 3. [30] For any matrix X and Y with appropriate dimensions, we have X T Y þ Y T X reX T X þ e1 Y T Y ,
for 8e40
Lemma 4. Let M(t), N(t), P(t) be symmetric matrix with appropriate dimensions, then "
# MðtÞ PðtÞ o0,8t40 PT ðtÞ NðtÞ
if and only if ( NðtÞo0 MðtÞPðtÞN 1 ðtÞPT ðtÞo0
,
8t40
Lemma 1. implies that the stability domain of fractional order systems depends on the fractional order a. The stability domain is located between boundaries, which are equal to 7pa/2. That is to say, the fractional order system is stable if and only if all eigenvalues of matrix A locate in the stability domain, as shown in Fig. 1. However, this method needs to compute all eigenvalues of system, and it is hard to yield all eigenvalues of FO-LTI interval system with uncertain parameters. If fractional order satisfied 1rao2, LMI methods can be utilized to determine system stability without computing eigenvalues. System (5) has a linear coupling relationship between fractional order a and model parameters A, therefore the eigenvalues distribution depends on fractional order a. If fractional order a changes, the stability domain and eigenvalues distribution will change at the same time, and then system stability may make some unpredictable changes. In the following paper, we will discuss this issue under these
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3
2
Imaginary axis
1 stability domain
0
θ=π-πα/2
unstability domain
-1
-2
-3 -3
-2
-1
0
1
2
3
Real axis Fig. 1. Stability domain of fractional order system with order a.
conditions. First, we prove Theorem 1 to determine whether system (5) is asymptotically stable. Theorem 1. FO-LTI interval system (5) with fractional order 1rao2 is asymptotically stable if there exist a symmetric positive definite matrix P40,PARn n and a real scalar constant e40, such that X ðA0 ,y0 Þo0 2
O1
ð6Þ PEAT 0
OT2
6O O 1 6 2 O¼6 6 EA P 0 4 0 EA P
eI 0
3 0 PEAT 7 7 7o0 7 0 5 eI
ð7Þ
where " X ðA0 ,y0 Þ ¼
ðA0 P þ PAT0 Þsin y0
ðA0 PPAT0 Þcosy0
ðPAT0 A0 PÞcosy0
ðA0 P þ PAT0 Þsin y0
O1 ¼ ðA0 P þ PAT0 Þ
#
sin y0 cosðy0 þ yM Þsin yM þ ek2 a22 DA DTA , cos y0
O2 ¼ ðPAT0 A0 PÞcosðy0 þ yM Þ y0 ¼ pða2 þ a1 Þp=4,
yM ¼ ða2 a1 Þp=4
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Proof. Suppose there exists a P40, e40, PARn n, such that 2 3 PEAT 0 O1 OT2 6 O O 0 PEAT 7 2 1 6 7 6 7o0 O¼6 E P 0 eI 0 7 4 A 5 0 EA P 0 eI Using Lemma 4, we obtain " #T " " # EA P 0 EA P O1 OT2 1 þe 0 EA P 0 O2 O1 Define
0 EA P
# o0
kða0 þ DaÞDA FA EA P cosy sin y " # " # F 0 P 0 E A A HermðX Þ9X þ X T ,F 9 ,N9 0 FA 0 EA P " # DA sin y DA cos y DðyÞ9 DA cosy DA sin y
XS ðyÞ9Herm
sin y
cos y
Substitute A=A0þkaDAFAEA into X(A,y), and then yield " # ðA0 P þ PA0 T Þsin y ðA0 PPA0 T Þcos y X ðA,yÞ ¼ ðPA0 T A0 PÞcos y ðA0 P þ PA0 T Þsin y sin y cosy þ Herm kaDA FA EA P cos y sin y sin y cosy ¼ X ðA0 ,yÞ þ Herm kaDA FA EA P cos y sin y ¼ X ðA0 ,yÞ þ XS ðyÞ For X(A0,y) of Eq. (8), substitute y=y0þDy and yield " # ðA0 P þ PAT0 Þsin y0 ðA0 PPAT0 Þcos y0 X ðA0 ,yÞ ¼ cosDy ðPAT0 A0 PÞcos y0 ðA0 P þ PAT0 Þsin y0 " # ðA0 P þ PAT0 Þcos y0 ðA0 PPAT0 Þsin y0 þ sin Dy ðPAT0 A0 PÞsin y0 ðA0 P þ PAT0 Þcosy0 ¼ X ðA0 ,y0 Þcos Dy þ X ðA0 ,p=2 þ y0 Þsin Dy We have X ðA0 ,y0 Þsin y0 þ X ðA0 ,p=2 þ y0 Þcos y0 ¼ X ðA0 ,p=2Þ
ð8Þ
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Thus X ðA0 ,yÞ ¼ X ðA0 ,y0 Þ
cosðy0 þ DyÞ sin Dy þ X ðA0 ,p=2Þ cosy0 cos y0
Because of Eq. (6) and the nature of negative definite matrix, ðA0 P þ PAT0 Þsiny0 is negative definite matrix. We have 1ra0o2, 1ra0þDao2, and then yield y0 ¼ pa0 p=2 2 ð0,p=2 Dy 2 ½yM ,yM ¼ ½pða1 a2 Þ=4,pða2 a1 Þ=4 2 ½p=4,p=4 y0 þ Dy 2 ½y0 yM ,y0 þ yM ¼ ½ppa2 =2,ppa1 =2 2 ð0,p=2 Thus, we can obtain sin y0 40,cos y0 40, sin yM rsin Dyrsin yM , cosðy0 yM ÞZ cosðy0 þ DyÞZcosðy0 þ yM Þ40and then A0 P þ PAT0 o0,which implies " # 0 A0 P þ PAT0 o0 X ðA0 ,p=2Þ ¼ 0 A0 P þ PAT0 Thus X(A0,y0) and X(A0,p/2) are negative definite matrix, because of the nature of negative definite matrix, we obtain cosðy0 þ DyÞ sin Dy þ X ðA0 ,p=2Þ cosy0 cos y0 cosðy0 þ yM Þ sin yM rX ðA0 ,y0 Þ X ðA0 ,p=2Þ cosy0 cos y0
X ðA0 ,yÞ ¼ X ðA0 ,y0 Þ
For XS(y) of Eq. (8), we have gij40, 1ri,jrn, then 2 Pn 0 j¼1 g1j " # 6 Pn T 0 0 DA DA 6 j¼1 g2j ¼6 DðyÞDðyÞT ¼ 6 ::: ::: 0 DA DTA 4 0 0
:::
0
:::
0
::: :::
::: Pn
3
j¼1 gnj
7 7 740 7 5
Then using Lemma 3 and the nature of positive definite matrix, we obtain sin y cosy XS ðyÞ ¼ Herm kða0 þ DaÞDA FA EA P cos y sin y ( " #" #) #" DA sin y DA cosy FA 0 0 EA P ¼ Herm kða0 þ DaÞ DA cos y DA sin y 0 FA 0 EA P ¼ ½kða0 þ DaÞDðyÞFNT þ kða0 þ DaÞDðyÞFN rek2 ða0 þ DaÞ2 DðyÞDðyÞT þ e1 N T N rek2 a22 DðyÞDðyÞT þ e1 N T N " #T " " # DA DTA 0 EA P 0 EA P 2 2 1 ¼ ek a2 þe T 0 DA DA 0 EA P 0
0 EA P
#
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Therefore, substitute X(A0,y) and XS(y) into Eq. (8), we obtain X ðA,yÞ ¼ X ðA0 ,yÞ þ XS ðyÞ cosðy0 þ yM Þ sin yM X ðA0 ,p=2Þ rX ðA0 ,y0 Þ cosy0 cosy0 " #T " " # T D D 0 E P 0 EA P A A A þek2 a22 þ e1 0 DA DTA 0 EA P 0 " #T " # " # EA P 0 EA P 0 O1 OT2 ¼ o0 þ e1 0 EA P 0 EA P O2 O1 Using Lemma 2, system (5) is asymptotically stable.
0
#
EA P
&
Theorem 1. enables to determine the asymptotical stability of FO-LTI interval systems with linear coupling relationship between fractional order and other model parameters. If k=1/a, Eq. (5) shows that the coupling relationship does not exist between fractional order and other model parameters. In addition, if there are no perturbations in fractional order, that is yM=0 or a1=a2, system (5) will degenerate into a commonly FO-LTI interval system, and the Theorem 1 will be the same as the one in Ref. [30] .When the system parameters are deterministic values, system (5) will degenerate further into a FO-LTI system, and the Theorem 1 will be the same as the criterion in Ref. [26]. Therefore, Theorem 1 proposed in this paper have a broader applicability in determining asymptotical stability of fractional order systems. 4. Numerical examples Consider the robust stability of the following FO-LTI interval system with linear coupling relationship between fractional order a and model parameters A: da xðtÞ ¼ AðaÞxðtÞ dta
ð9Þ
where a=a0þDaA[a1,a2]C[1,2), A(a)=A0þkaDA, DA=AM[dij]n n, 1rdijr1, and 2 3 2 3 1:35 0:35 1:25 0:25 0:25 0:05 6 7 6 7 2:45 1:1 5, AM ¼ 4 0:1 0:05 0:2 5 A0 ¼ 4 0 0:3 1:25 1:75 0:1 0:05 0:25
Case 1. If k=1/a and a has a deterministic value, system (9) will degenerate into a FO-LTI interval system without any coupling relationship in parameters. In this case, Theorem 1 is equivalent to the method mentioned in Ref. [30]. And we can determine whether system (9) is asymptotically stable using Theorem 1 or the method in Ref. [30]. For example, if a=1.5, we can obtain a feasible solution as follows: 2
3
0:8851
0:1545
0:0524
6 P ¼ 4 0:1545 0:0524
0:3275 0:0634
7 0:0634 540 e ¼ 0:7959 0:3476
Then system (9) is asymptotically stable.
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3
Imaginary axis
2
1
0
-1
-2
-3 -4
-3.5
-3
-2.5
-2
-1.5 -1 Real axis
-0.5
0
0.5
1
Fig. 2. Eigenvalues perturbation area of FO-LTI system without coupling relationship. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In Fig. 2, blue and green lines are, respectively, upper and lower boundaries of fractional order a=1.4 and a=1.5, green area is the eigenvalues perturbation area of A, which is computed using the methods in Ref. [29]. Because there is no coupling relationship between fractional order a and model parameters A, eigenvalues perturbation area only depends on A. No matter how fractional order a changes, the eigenvalues perturbation area does not occur any changes. Case 2. If ka1/a and a is perturbed in a deterministic interval [a1,a2], Eq. (9) shows that there is a linear coupling relationship between a and A. The LMI methods that have been presented so far in all literatures cannot easily determine the stability of system in this case. However, we can determine whether system (9) is asymptotically stable using Theorem 1 in the paper.
If k=1/1.5, aA[1.4,1.5], we can obtain a feasible solution as follows: 2 3 12:9016 2:8031 1:0655 6 7 P ¼ 4 2:8031 4:2923 0:9026 540 e ¼ 7:5092 1:0655 0:9026 4:9782 Thus, system (9) is asymptotically stable. In Fig. 3, green and blue areas are, respectively, eigenvalues perturbation areas with fractional order a=1.4 and a=1.5. Fig. 3 shows that eigenvalues perturbation area is extended when fractional order a is increased from 1.4 to 1.5. However, this area does not exceed the boundaries determined by fractional order a. In other words, we can always ensure that all eigenvalues of system are located in the stability domain no matter how
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3 order perturbation area
Imaginary axis
2
1 eigenvalue perturbation area
0
-1
-2
order perturbation area
-3 -4
-3.5
-3
-2.5
-2
-1.5 -1 Real axis
-0.5
0
0.5
1
Fig. 3. Eigenvalues perturbation area of FO-LTI system with linear coupling relationship and aA[1.4,1.5]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3
Imaginary axis
2
unstable eigenvalue
1
0
-1
-2
-3 -4
-3.5
-3
-2.5
-2
-1.5 -1 Real axis
-0.5
0
0.5
1
Fig. 4. Eigenvalues perturbation area of FO-LTI system with linear coupling relationship and aA[1.4,1.7]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
fractional order is perturbed in the deterministic interval. Therefore, this FO-LTI interval system with linear coupling relationship is always asymptotically stable in this case. If k=1/1.5, aA[1.4,1.7], the feasible solution does not exist, and system (9) is not asymptotically stable.
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In Fig. 4, green and blue areas are, respectively, eigenvalues perturbation areas with fractional order a=1.4 and a=1.7. Fig. 4 shows that eigenvalues perturbation area exceeds the boundaries determined by fractional order when fractional order a is increased from 1.4 to 1.7. We cannot ensure that all eigenvalues of system are located in the stability domain when fractional order is perturbed in the deterministic interval. Therefore, this FO-LTI interval system with linear coupling relationship is unstable in this case. Remark 1. If we ignore the coupling relationship between fractional order and other model parameters, system stability only depends on model parameters A and the maximum value of fractional order a no matter how fractional order a is perturbed in a deterministic interval. Remark 2. If we consider the coupling relationship between fractional order and other model parameters, fractional order perturbation will have great effects on the robust stability of FO-LTI interval system. In case 2, system stability not only depends on perturbation of model parameters A, but also the value of fractional order a in a deterministic interval. We must consider the upper and lower perturbation boundaries of fractional order a comprehensively. The perturbation areas of all eigenvalues will vary if fractional order a is perturbed in some interval. Coefficient ka denotes the change rate of perturbation area of all eigenvalues. Thus, when A and a are perturbed at the same time, perturbation area of all eigenvalues may exceed the stability boundary determined by fractional order a and the system will become unstable. In case 2, when the value of fractional order a is increased, the system becomes unstable from stable.
5. Conclusion This paper has firstly discussed the robust stability of FO-LTI interval system with certain linear coupling relationship between fractional order and other model parameters, and presented the criterion derived from LMI methods to determine whether this system is asymptotically stable. At present, there are many questions on robust stability of fractional order systems unsolved. The universal conclusion has not been presented yet. Therefore, fractional order systems still need more and more people to explore and study further. Acknowledgment This work was supported by the National Natural Science Foundation of China (no. 61004017, 60974103). References [1] P.J. Torvik, R.L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, Journal of Applied Mechanics 51 (2) (1984) 294–298. [2] J.J. de Esp ´ındola, C.A. Bavastri, E.M.O. Lopes, On the passive control of vibrations with viscoelastic dynamic absorbers of ordinary and pendulum types, Journal of the Franklin Institute 347 (1) (2010) 102–115. [3] X. Wu, J. Li, G. Chen, Chaos in the fractional order unified system and its synchronization, Journal of the Franklin Institute 345 (4) (2008) 392–401.
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