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Signal Processing 86 (2006) 2611–2618 www.elsevier.com/locate/sigpro
Robust stability check of fractional order linear time invariant systems with interval uncertainties YangQuan Chena,, Hyo-Sung Ahna, Igor Podlubnyb a
Department of Electrical and Computer Engineering, Center for Self-Organizing and Intelligent System, Utah State University, Logan, UT-84322-4160, USA b Department of Informatization and Control of Processes, Technical University of Kosice, B. Nemcovej 3, 04200 Kosice, Slovak Republic Received 12 April 2005; received in revised form 29 October 2005; accepted 6 December 2005 Available online 9 March 2006
Abstract For uncertain fractional-order linear time invariant (FO-LTI) systems with interval coefficients described in state space form, the robust stability check problem is solved for the first time in this paper. Both the checking procedure and the Matlab code are presented with two illustrative examples. The conservatism is shown to be small. r 2006 Elsevier B.V. All rights reserved. Keywords: Fractional order linear dynamic systems; Robust stability; Interval linear time invariant systems; Interval matrix
1. Introduction In smart mechatronic systems, it was advocated in [1] that, fractional order calculus will play an important role. Fractional order dynamic systems and controls, which are based on fractional order calculus [2–5], have been gaining increasing attention in research communities [6–10]. Pioneering works in applying fractional calculus in dynamic systems and controls include [11–14] while some recent developments can be found in [15–17]. A comparative introduction of four fractional order controllers can be found in [18]. Stability and controllability concepts are fundamental to any dynamic control systems including Corresponding author. Tel.: +1 435 7970148(O); fax: +1 435 7973054. E-mail addresses: [email protected], [email protected] (Y.Q. Chen). URL: http://mechatronics.ece.usu.edu/foc/.
fractional order control systems [19,20]. In [21–26], stability results of fractional order control systems were presented while in [27], the first discussion about the controllability of fractional order control systems can be found. For interval fractional-order linear time invariant (FO-LTI) systems, the first result on stability was discussed in [28] and further in [29] with even interval uncertainties (in the fractional order!). Recently, the controllability issue for interval FO-LTI systems has been addressed for the first time in [30]. In this paper, we focus on the stability test method for interval FO-LTI systems. Note that, in [28,29], the available results were largely based on experimentally verified Kharitonov-like procedure for SISO (single-input singleoutput) FO-LTI systems in transfer function form. For uncertain FO-LTI systems with interval coefficients described in state space form, however, the robust stability check problem is still open to our best knowledge. For the first time, this paper will present both the robust stability checking
0165-1684/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2006.02.011
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procedure and the Matlab code with two illustrative examples.
similar discussions in [28,29] and the references therein, the stability condition for da xðtÞ=dta ¼ AxðtÞ is clearly
2. FO-LTI interval system and the robust stability test problem
min j argðli ðAÞÞj4ap=2;
We consider the following FO-LTI system with interval uncertainties in the parameters considered
Then, our robust stability test task for FO-LTI interval systems amounts to examining if
da xðtÞ ¼ AxðtÞ þ BuðtÞ, (1) dta where a is a non-integer number; A 2 AI ¼ ½A; A and B 2 BI ¼ ½B; B. We adopt the following Caputo definition for fractional derivative of order a of any function f ðtÞ, because the Laplace transform of the Caputo derivative allows utilization of initial values of classical integer-order derivatives with known physical interpretations [31,5] Z t ðnÞ da f ðtÞ 1 f ðtÞ dt a D f ðtÞ ¼ ¼ , (2) a dt Gða nÞ 0 ðt tÞaþ1n
min j argðli ðAÞÞj4ap=2;
where n is an integer satisfying n 1oapn. Note that in certain sense, Caputo derivative is considered as more practical over other fractional derivatives such as Riemann–Liouville because initial conditions such as f ð0Þ; f 0 ð0Þ; f 00 ð0Þ can be represented by Caputo derivative. See [32] for a detailed justification. On the other hand, since stability has nothing to do with initial conditions, the results of this paper apply straightforwardly when other fractional derivatives are used. The robust controllability test problem has been solved in [30]. Since in this paper we are concerned with the robust stability test problem, we can simply consider the uncontrolled part of the FO-LTI interval system da xðtÞ=dta ¼ AxðtÞ. With no interval uncertainty and for a ¼ 1, it is well-known that the stability condition of an LTI system x_ ¼ Ax, x 2 RN1 is
fi ¼ j arctanðzi =si Þj;
Reðli ðAÞÞo0;
i ¼ 1; 2; . . . ; N,
(3)
where li ðAÞ is the ith eigenvalue of A. When 0oao1, internal and external stability conditions are given in [22], and recently LMI conditions for both 0oao1 and 1ao2 were suggested by [33]. In fact, condition (3) is equivalent to the following: min j argðli ðAÞÞj4p=2; i
i ¼ 1; 2; . . . ; N.
Both of the above conditions tell the same ‘‘lefthalf-plane (LHP)’’ rule for all the eigenvalues of A. Let us return to the fractional order case and consider no interval uncertainty. Following the
i
i
i ¼ 1; 2; . . . ; N.
i ¼ 1; 2; . . . ; N; 8A 2 AI .
3. Interval eigenvalue problem As shown in the preceding section, for the robust stability check of the uncertain fractional system, it is required to calculate the arguments of phase of eigenvalues. In case when there is no model uncertainty, it is easy to find the argument of phase of each eigenvalue. That is, by simply calculating i ¼ 1; . . . ; N,
(4)
where N is number of eigenvalues of A, si ¼ Refli g and zi ¼ Imfli g of eigenvalue li , and finding the minimum fi such as f ¼ infff1 ; . . . ; fN g,
(5)
if f 4ap=2, then the fractional system is considered robust stable. However, with interval model uncertainty, it is not easy to find (4) because li is not a fixed point in complex plane, instead it is a cluster of infinite points. For illustration, let us see Fig. 1. The left figure shows eigenvalues without model uncertainty; as shown in this figure, the argument of phase can be easily found. But the right figure shows the eigenvalues of model uncertain system; as shown in this figure, it is difficult to find the minimum phase angle, because there is infinite number of eigenvalues. However, fortunately, there is an existing result for interval eigenvalue boundary of interval matrix system. Although not directly applicable for our robust stability test, in this paper, we managed to make use of this existing method for finding strong boundaries of eigenvalues of interval uncertain system. According to [34], the lower and upper boundaries of cluster of eigenvalues can be found although there is a conservatism. However, based on our experience, we found that the conservatism is small. In what follows, this interval eigenvalue bound estimation method is briefly explained.
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2.5 Without interval model uncertainty
2
eigenvalue
1.5
With interval model uncertainty cluster of eigenvalues
1.5
1
1
0.5
0.5
φ
0
Real axis
-0.5
φ
0
Real axis
Imaginary axis
-0.5
-1 -1.5 -2 -2.5 -2.5 -2 -1.5 -1 -0.5
Imaginary axis
2
2613
-1 -1.5 -2 0
0.5
1
1.5
2
2.5
-2.5 -2.5 -2 -1.5 -1 -0.5
0
0.5
1
1.5
2
2.5
Fig. 1. Left: no interval uncertainty. Right: with interval uncertainty.
3.1. Boundaries of eigen-clusters For bounding eigenvalues of model uncertain fractional system, the following interval matrix is defined:
eigenvectors are denoted as vi and ui , respectively, and they are normalized such as: vi uj ¼ dij ; i; j ¼ 1; . . . ; N. Then, from [34], separately considering the real part and imaginary part of eigenvectors such as
AI ¼ ½Ac DA; Ac þ DA ¼ ½A; A,
vre i ¼ Reðvi Þ;
vim i ¼ Imðvi Þ,
ure i ¼ Reðui Þ;
uim i ¼ Imðui Þ,
where Ac is a center matrix (nominal plant without uncertainty), DA is a radius matrix corresponding to interval uncertainty, and A is lower boundary matrix, and A is upper boundary matrix. Then, interval eigenvalues corresponding to interval matrix are defined as LI ¼ fl 2 C : Ax ¼ lx; xa0; A 2 AI g. I
Clearly, L is a set composed of infinite number of elements. In fact, it is impossible to find exact boundaries of LI in complex plane, so in this paper, an existing result, which finds the upper and lower boundaries of real part and imaginary part of eigenvalue set separately, is utilized. The following is a summary of [34,35]. Throughout the paper, it is assumed that in given fractional interval system such as Da x ¼ AI x;
A I 2 AI
AI is N N real square matrix. Now, assuming A and A are given, let us define center matrix and radius matrix in the following ways: AþA AA ; DA ¼ . 2 2 Furthermore, let us denote the left and right eigenvectors corresponding to center matrix Ac by L and R where row vectors of L are left eigenvectors and column vectors of R are right eigenvectors corresponding to eigenvalues li . Left and right Ac ¼
we have two main results summarized in the following lemmas: Lemma 1. Define a sign calculation operator evaluated at Ac such as re im im T Pi :¼sgn½ðure i vi ui vi Þ , re im im are eigenvectors correwhere ure i , vi , ui , and vi sponding to ith eigenvalue of Ac . If Pi is constant for all AI , AI 2 AI , then the lower and upper boundaries of the real part of ith interval eigenvalue are calculated as c re i lre i ¼ Oi ðA DA P Þ,
where Ore i ðÞ is an operator for selecting the ith real eigenvalue of ðÞ and C ¼ A B are ckj ¼ akj bkj , and as re c i lre i ¼ Oi ðA þ DA P Þ.
Lemma 2. Defining a sign calculation operator evaluated at Ac such as im im re T Qi :¼sgn ðure i v i þ ui v i Þ , if Qi is constant for all AI , AI 2 AI , then the lower and upper boundaries of the imaginary part of ith interval eigenvalue are calculated as c i im lim i ¼ Oi ðA DA Q Þ,
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where Oim i ðÞ is an operator for selecting the ith imaginary eigenvalue, and as
4.1. Example 1
c i im lim i ¼ Oi ðA þ DA Q Þ.
So, using Lemmas 1 and 2, it is easy to calculate the lower and upper boundaries of interval eigenvalue separately in real part and imaginary part. From above lemma, if Pi and Qi , i ¼ 1; . . . ; N are calculated, then, interval ranges of eigenvalues are finally calculated as re im im lIi 2 LIi :¼f½lre i ; li þ j½li ; li g,
where j represents imaginary part. Note that Deif’s algorithm given above also has been used in [35] for the performance comparison of the perturbation based method.
3.2. Stability check using minimum argument of phase As shown in right figure of Fig. 1, there is infinite number of arguments of phases of an interval fractional system. From (5), since the stability condition is given as f 4ap=2, if we find sufficient condition for this, the stability can be checked. For calculating f , the following procedure can be used. P1. Calculate Pi and Qi for i ¼ 1; . . . ; N. re im im P2. Calculate lre for all i , li , li , and li i 2 f1; 2; . . . ; Ng. P3. Find arguments of phase of four points such as
P4. P5. P6. P7.
im f1i ¼ ffðlre i ; li Þ;
im f2i ¼ ffðlre i ; li Þ,
f3i
f4i
¼
im ffðlre i ; li Þ;
4. Two illustrative examples
¼
im ffðlre i ; li Þ
in the complex plane. Find fi ¼ inffjf1i j; jf2i j; jf3i j; jf4i jg. Repeat procedures P3 and P4 for i ¼ 1; . . . ; N. Find f ¼ infffi ; i ¼ 1; . . . ; Ng. If f 4ap=2, then the fractional interval system is robust stable. Otherwise, the stability of system cannot be guaranteed.
Appendix shows the Matlab code of above algorithm. For confirming instability of FO-LTI interval system, the following procedure can be considered: Defining fzi ¼ supfjf1i j; jf2i j; jf3i j; jf4i jg, if there exists i such that fzi pap=2 and f pap=2, then the system is robust unstable.
Let us check robust stability of the following fractional interval system: Da x ¼ AI x,
(6)
where A 2 A ¼ A; A with 0 1 1:1 0:55 2:2 B C 0:45 0:55 A, A ¼ @ 0:9 I
I
0:55
2:25
1:08
0:9
0:45
1:8
0
B A ¼ @ 1:1 0:45
0:55 2:75
1
C 0:45 A 1:32
and a ¼ 0:45. Eigenvalues of center matrix Ac ¼ A þA=2 are calculated as 1:7486, 1:2243þ j1:5597, and 1:2243 j1:5597. In P1, Pi and Qi are calculated as 0 1 0 1 0 0 0 1 1 1 B C B C P1 ¼ @ 1 1 1 A; Q1 ¼ @ 0 0 0 A, 0 0 0 1 1 1 0
1
1
1
1
B P2 ¼ @ 1 1 1 1
C 1 A; 1
0
1 1 C 1 A;
1 1 B 3 P ¼@ 1 1 1 1
1
0
1
B Q2 ¼ @ 1 1 0
1 B 3 Q ¼ @ 1 1
1 1 1
1
1
C 1 A, 1
1 1 1 C 1 1 A. 1 1
1
Observe that in Q , all elements are zero. This is due to the fact that there is no interval uncertainty in imaginary part of the first eigenvalue of center matrix (see Fig. 2). From P2, the interval ranges of eigenvalues are found as lI1 2 LI1 :¼f½2:0228; 1:4826 þ j½0; 0g, lI2 2 LI2 :¼f½1:0557; 1:3986 þ j½1:3209; 1:7859g, lI3 2 LI3 :¼f½1:0557; 1:3986 þ j½1:7859; 1:3209g. These interval boundaries are drawn in Fig. 2 with circle marks. Now, we find that ap=2 ¼ 0:7069 and from procedures P3–P6, f ¼ 0:7568. So, since f ¼ 0:75684ap=2 ¼ 0:7069, we conclude that the fractional interval system is robust stable. Fig. 2 also shows random test results within interval
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With Interval model uncertainty
2.5 cluster of eigenvalues
2
2
1.5
1.5 1
α π/2 = 0.7069
boundary of interval eigenvalues
0.5
0.5
φ∗ =0.7568
0
0
Real axis
-0.5
φ∗ =0.6769
α π /2 = 0.7069
Real axis
-0.5 Imaginary axis
-1 -2 -1.5 -2.5 -2.5
-2
-1.5
-1
-0.5
-1
Imaginary axis
1
2615
-1.5 -2
random tests
0
0.5
1
1.5
2
2.5
Fig. 2. Boundaries of interval eigenvalue for Example 1.
-2.5 -2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 3. Boundaries of interval eigenvalue for Example 2.
boundary boxes. For the validation of the algorithm, 1000 random plants are selected and at each random plant, the eigenvalues are plotted in Fig. 2. This random test result shows that the calculated interval boundaries are reliable and not so conservative. 4.2. Example 2 Let us check robust stability of the following fractional interval system: I
a
D x ¼ A x, I
(7) I
where A 2 A ¼ ½A; A with 0 1 1:1 0:55 2:2 B C 0:45 0:55 A, A ¼ @ 0:9 0:55 0
2:25
0:9 0:45 B 0:55 A ¼ @ 1:1 0:45 2:75
1:35 1
1:8 C 0:45 A 1:65
and a ¼ 0:45. Eigenvalues of center matrix Ac ¼ A þA=2 are calculated as 1:6932, 1:3466þ j1:4990, and 1:3466 j1:4990. From Matlab code of Appendix, f ¼ 0:6769. So, since f oap=2 ¼ 0:7069, the fractional interval system is considered unstable. Fig. 3 shows random test and boundaries of interval eigenvalues. But, as shown in this figure, from random test, the system could be still stable. So, there may exist conservatism in our method, however as shown in Figs. 2 and 3, the conservatism could be very small. This conservatism is due to the
fact that the boundaries of interval eigenvalues are calculated in real part and imaginary part separately. If we can find the more tight interval ranges of eigenvalue cluster of interval matrix such as interval radius, the result could be less conservative. But, currently, we are not able to find this kind of tight boundaries.
5. An extension and a discussion on further research efforts Here, we present a simple extension of our algorithm to the following FO-LTI system with interval uncertainties both in the parameters and in the order: da xðtÞ ¼ AxðtÞ, dta
(8)
where A 2 AI :¼½A; A and a 2 aI :¼½a; a. From our previous discussions, it is clear and straightforward that the robust stability test task for FO-LTI interval systems amounts to examining if min j argðli ðAÞÞj4ap=2; i
i ¼ 1; 2; . . . ; N; 8A 2 AI .
This can also be done easily by the code in the Appendix. Further research efforts include the stabilizability issue for da xðtÞ=dta ¼ AxðtÞ þ BuðtÞ. In particular, we are interested in the following general setting with fractional order state feedback.
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EðDa xÞ ¼ Ax þ Bu, b
y ¼ CðD xÞ,
ð9Þ ð10Þ
where E 2 EI :¼½E c DE; E c þ DE ¼ ½E; E; A 2 AI :¼½Ac DA; Ac þ DA ¼ ½A; A and B 2 BI :¼ ½Bc DB; Bc þ DB ¼ ½B; B. a and b are real numbers and furthermore, a 2 aI :¼½a; a. The following interesting research questions could be asked:
What is the robust stability condition of (9) with u ¼ 0? If the system is unstable, how to design a control input u such that the close-loop system is stable with full states measured? If the system is unstable, how to design the control input u such that the close-loop system is stable if only the output y are measured.
We believe, based on the result of this paper, the above questions can be eventually answered. 6. Conclusions In this paper, we have presented, for the first time, a robust stability test procedure and Matlab code for uncertain fractional-order linear time invariant (FO-LTI) systems with interval coefficients
described in state space form. Two illustrative examples are included to show that the conservatism is practically small.
Acknowledgements This project has been funded in part by the National Academy of Sciences under the Collaboration in Basic Science and Engineering Program/ Twinning Program supported by Contract no. INT0002341 from the National Science Foundation. The contents of this publication do not necessarily reflect the views or policies of the National Academy of Sciences or the National Science Foundation, nor does mention of trade names, commercial products or organizations imply endorsement by the National Academy of Sciences or the National Science Foundation. The authors are grateful to the comments from anonymous reviewers which improved the presentation of this paper. The conference version of this paper was presented at The 2005 IEEE International Conference on Mechatronics and Automation held at the Sheraton Fallsview Hotel & Conference Centre, Niagara Falls, Ontario, Canada from July 29 to August 1, 2005.
Appendix
clear all % Input-1: size of square A matrix n ¼ 3; % Input-2: Nominal center matrix: Ac Ac ¼ ½1; 0:5; 2:0; 1:0; 0:5; 0:5;... 0:5; 2:5; 1:5 ; % center matrix % Input-3: Interval radius matrix: Ar Ar ¼ abs(0.1*Ac) ; % radius matrix % Input-4: Fractional nalpha alpha ¼ 0.45; % Main function [R, D] ¼ eig(Ac) ; L ¼ inv(R) ; for i ¼ 1:1:n; Pi ¼ ( sign(real(R(:,i))*real(L(i,:))... - imag(R(:,i))*imag(L(i,:))) )’ ; Qi ¼ ( sign(real(R(:,i))*imag(L(i,:))... + imag(R(:,i))*real(L(i,:))) )’; Lambda_real_min ¼ real(eig(Ac - Ar.*Pi)); Lambda_real_max ¼ real(eig(Ac + Ar.*Pi));
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lambda_i_real_min(i) ¼ Lambda_real_min(i); lambda_i_real_max(i) ¼ Lambda_real_max(i); Lambda_imag_min ¼ imag(eig(Ac - Ar.*Qi)); Lambda_imag_max ¼ imag(eig(Ac + Ar.*Qi)); lambda_i_imag_min(i) ¼ Lambda_imag_min(i); lambda_i_imag_max(i) ¼ Lambda_imag_max(i); end % Argument of phase for i ¼ 1:1:n eig_i_j(1,i) ¼ lambda_i_real_min(i)... + j*lambda_i_imag_min(i); eig_i_j(2,i) ¼ lambda_i_real_min(i)... + j*lambda_i_imag_max(i); eig_i_j(3,i) ¼ lambda_i_real_max(i)... + j*lambda_i_imag_max(i); eig_i_j(4,i) ¼ lambda_i_real_max(i)... + j*lambda_i_imag_min(i); end jj ¼ 0; for k ¼ 1:1:n; for i ¼ 1:1:4; jj ¼ jj+1; abs_phase(jj) ¼ abs(angle(eig_i_j(i,k))); end end % Output-1: minimum phase % min_arg_phase ¼ min(abs_phase); % Output-2: system stability % Fractional_alpha ¼ alpha*pi/2; if min_arg_phase4Fractional_alpha display(’System is stable’); else display(’The robust stability is not guaranteed’); end
References [1] Y.Q. Chen, D. Xue, H. Dou, Fractional calculus and biomimetic control, in: Proceedings of the First IEEE International Conference on Robotics and Biomimetics (RoBio04), Shengyang, China, August 2004, IEEE, pages robio2004–347. [2] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [3] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives and Some of Their Applications, Nauka i technika, Minsk, 1987. [4] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[5] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999. [6] B.J. Lurie, Three-parameter tunable tilt-integral-derivative (TID) controller, US Patent US5371670, 1994. [7] I. Podlubny, Fractional-order systems and PIl Dm -controllers, IEEE Trans. Automat. Control 44 (1) (1999) 208–214. [8] A. Oustaloup, B. Mathieu, P. Lanusse, The CRONE control of resonant plants: application to a flexible transmission, European J. Control 1 (2) (1995). [9] A. Oustaloup, X. Moreau, M. Nouillant, The CRONE suspension, Control Eng. Practice 4 (8) (1996) 1101–1108. [10] H.F. Raynaud, A. Zergaı¨ noh, State-space representation for fractional order controllers, Automatica 36 (2000) 1017–1021.
ARTICLE IN PRESS 2618
Y.Q. Chen et al. / Signal Processing 86 (2006) 2611–2618
[11] S. Manabe, The non-integer integral and its application to control systems, Japanese Institute of Electrical Engineers J. 80 (860) (1960) 589–597. [12] S. Manabe, The non-integer integral and its application to control systems, Electrotechnical J. Japan 6 (3–4) (1961) 83–87. [13] A. Oustaloup, Fractional order sinusoidal oscillators: optimization and their use in highly linear FM modulators, IEEE Trans. Circuits and Systems 28 (10) (1981) 1007–1009. [14] M. Axtell, E.M. Bise, Fractional calculus applications in control systems, in: Proceedings of the IEEE 1990 National Aerospace and Electronics Conference, New York, USA, 1990, pp. 563–566. [15] B.M. Vinagre, Y.Q. Chen, Lecture notes on fractional calculus applications in automatic control and robotics, in: B.M. Vinagre, Y.Q. Chen (Eds.), The 41st IEEE CDC2002 Tutorial Workshop # 2, pp. 1–310. http://mechatronics. ece.usu.edu/foc/cdc02_tw2_ln.pdf, Las Vegas, Nevada, USA, 2002. [16] J.A. Tenreiro Machado, Fractional calculus and applications, Nonlinear Dynamics 29 (March 2002) 1–385 (special issue). [17] M. Duarte Ortigueira, J.A. Tenreiro Machado, Fractional signal processing and applications, Signal Processing 83 (11) (November 2003) 2285–2480 (special issue). [18] D. Xue, Y.Q. Chen, A comparative introduction of four fractional order controllers, in: Proceedings of The 4th IEEE World Congress on Intelligent Control and Automation (WCICA02), Shanghai, China, June 2002, IEEE, New York, pp. 3228–3235, [19] W.J. Rugh, Linear Systems Theory, Prentice-Hall, Englewood Cliffs, NJ, 1993. [20] M. Vidyasagar, A characterization of eAt and a constructive proof of the controllability criterion, IEEE Trans. Automat. Control 60 (2) (1971) 370–371. [21] S.B. Skaar, A.N. Michel, R.K. Miller, Stability of viscoelastic control systems, IEEE Trans. Automat. Control 33 (4) (April 1988) 348–357. [22] D. Matignon, Stability result on fractional differential equations with applications to control processing, in: IMACS-SMC Proceedings, Lille, France, July 1996, pp. 963–968. [23] D. Matignon, Stability properties for generalized fractional differential systems, in: D. Matignon, G. Montseny (Eds.), Proceedings of the Colloquium FDS’98: Fractional Differential Systems: Models, Methods and Applications, vol. 5, Paris, December 1998, pp. 145–158 [24] D. Matignon, Stability properties for generalized fractional differential systems, in: ESAIM: Proceedings, vol. 5, pp. 145–158. URL: http://www.emath.fr/Maths/Proc/Vol.5/, December 1998.
[25] D. Matignon, Generalized fractional differential and difference equations: stability properties and modelling issues, in: Proceedings of the Mathematical Theory of Networks and Systems symposium (MTNS’98), Padova, Italy, July 1998, pp. 503–506. [26] C. Bonnet, J.R. Partington, Coprime factorizations and stability of fractional differential systems, Systems Control Lett. (2000) 41. [27] D. Matignon, B. d’Andre´a Novel, Some results on controllability and observability of finite-dimensional fractional differential systems, Proceedings of the Computational Engineering in Systems and Application Multiconference, Lille, France, vol. 2, IMACS, IEEE-SMC, July 1996, pp. 952–956. [28] I. Petra´sˇ , Y.Q. Chen, B.M. Vinagre, Robust stability test for interval fractional order linear systems, in: V.D. Blondel, A. Megretski (Eds.), Unsolved Problems in the Mathematics of Systems and Control, vol. 208–210, Princeton University Press, Princeton, NJ, 2004 (Chapter 6.5). [29] I. Petra´sˇ , Y.Q. Chen, B.M. Vinagre, I. Podlubny, Stability of linear time invariant systems with interval fractional orders and interval coefficients, in: Proceedings of the International Conference on Computation Cybernetics (ICCC04), Viena Technical University, Viena, Austria, 8/30-9/1 2005, pp. 1–4, [30] Y.Q. Chen, H. Ahn, D. Xue, Robust controllability of interval fractional order linear time invariant systems, in: Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, paper# DETC2005-84744, Long Beach, CA, September 24–28, 2005, pp. 1–9. [31] M. Caputo, Linear models of dissipation whose q is almost frequency independent-II, Geophys. J. R. Astronom. Soc. 13 (1967) 529–539. [32] B.N. Narahari Achar, C. Lorenzo, T. Hartley, Initialization issues of the Caputo fractional derivative, in: Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, paper# DETC2005-84348, Long Beach, CA, USA, September 24–28, 2005. [33] M. Moze, J. Sabatier, LMI tools for stability analysis of fractional systems, in: Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, paper# DETC2005-85182, Long Beach, CA, USA, September 24–28, 2005. [34] A.S. Deif, The interval eigenvalue problem, Z. Angew. Math. Mech. 71 (1) (1991) 61–64. [35] Z. Qiu, P.C. Mu¨ller, A. Frommer, An appliximation method for the standard interval eigenvalue problem of real nonsymmetric interval matrices, Comm. Numer. Methods Eng. 17 (2001) 239–251.
Signal Processing 86 (2006) 2611–2618 www.elsevier.com/locate/sigpro
Robust stability check of fractional order linear time invariant systems with interval uncertainties YangQuan Chena,, Hyo-Sung Ahna, Igor Podlubnyb a
Department of Electrical and Computer Engineering, Center for Self-Organizing and Intelligent System, Utah State University, Logan, UT-84322-4160, USA b Department of Informatization and Control of Processes, Technical University of Kosice, B. Nemcovej 3, 04200 Kosice, Slovak Republic Received 12 April 2005; received in revised form 29 October 2005; accepted 6 December 2005 Available online 9 March 2006
Abstract For uncertain fractional-order linear time invariant (FO-LTI) systems with interval coefficients described in state space form, the robust stability check problem is solved for the first time in this paper. Both the checking procedure and the Matlab code are presented with two illustrative examples. The conservatism is shown to be small. r 2006 Elsevier B.V. All rights reserved. Keywords: Fractional order linear dynamic systems; Robust stability; Interval linear time invariant systems; Interval matrix
1. Introduction In smart mechatronic systems, it was advocated in [1] that, fractional order calculus will play an important role. Fractional order dynamic systems and controls, which are based on fractional order calculus [2–5], have been gaining increasing attention in research communities [6–10]. Pioneering works in applying fractional calculus in dynamic systems and controls include [11–14] while some recent developments can be found in [15–17]. A comparative introduction of four fractional order controllers can be found in [18]. Stability and controllability concepts are fundamental to any dynamic control systems including Corresponding author. Tel.: +1 435 7970148(O); fax: +1 435 7973054. E-mail addresses: [email protected], [email protected] (Y.Q. Chen). URL: http://mechatronics.ece.usu.edu/foc/.
fractional order control systems [19,20]. In [21–26], stability results of fractional order control systems were presented while in [27], the first discussion about the controllability of fractional order control systems can be found. For interval fractional-order linear time invariant (FO-LTI) systems, the first result on stability was discussed in [28] and further in [29] with even interval uncertainties (in the fractional order!). Recently, the controllability issue for interval FO-LTI systems has been addressed for the first time in [30]. In this paper, we focus on the stability test method for interval FO-LTI systems. Note that, in [28,29], the available results were largely based on experimentally verified Kharitonov-like procedure for SISO (single-input singleoutput) FO-LTI systems in transfer function form. For uncertain FO-LTI systems with interval coefficients described in state space form, however, the robust stability check problem is still open to our best knowledge. For the first time, this paper will present both the robust stability checking
0165-1684/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2006.02.011
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procedure and the Matlab code with two illustrative examples.
similar discussions in [28,29] and the references therein, the stability condition for da xðtÞ=dta ¼ AxðtÞ is clearly
2. FO-LTI interval system and the robust stability test problem
min j argðli ðAÞÞj4ap=2;
We consider the following FO-LTI system with interval uncertainties in the parameters considered
Then, our robust stability test task for FO-LTI interval systems amounts to examining if
da xðtÞ ¼ AxðtÞ þ BuðtÞ, (1) dta where a is a non-integer number; A 2 AI ¼ ½A; A and B 2 BI ¼ ½B; B. We adopt the following Caputo definition for fractional derivative of order a of any function f ðtÞ, because the Laplace transform of the Caputo derivative allows utilization of initial values of classical integer-order derivatives with known physical interpretations [31,5] Z t ðnÞ da f ðtÞ 1 f ðtÞ dt a D f ðtÞ ¼ ¼ , (2) a dt Gða nÞ 0 ðt tÞaþ1n
min j argðli ðAÞÞj4ap=2;
where n is an integer satisfying n 1oapn. Note that in certain sense, Caputo derivative is considered as more practical over other fractional derivatives such as Riemann–Liouville because initial conditions such as f ð0Þ; f 0 ð0Þ; f 00 ð0Þ can be represented by Caputo derivative. See [32] for a detailed justification. On the other hand, since stability has nothing to do with initial conditions, the results of this paper apply straightforwardly when other fractional derivatives are used. The robust controllability test problem has been solved in [30]. Since in this paper we are concerned with the robust stability test problem, we can simply consider the uncontrolled part of the FO-LTI interval system da xðtÞ=dta ¼ AxðtÞ. With no interval uncertainty and for a ¼ 1, it is well-known that the stability condition of an LTI system x_ ¼ Ax, x 2 RN1 is
fi ¼ j arctanðzi =si Þj;
Reðli ðAÞÞo0;
i ¼ 1; 2; . . . ; N,
(3)
where li ðAÞ is the ith eigenvalue of A. When 0oao1, internal and external stability conditions are given in [22], and recently LMI conditions for both 0oao1 and 1ao2 were suggested by [33]. In fact, condition (3) is equivalent to the following: min j argðli ðAÞÞj4p=2; i
i ¼ 1; 2; . . . ; N.
Both of the above conditions tell the same ‘‘lefthalf-plane (LHP)’’ rule for all the eigenvalues of A. Let us return to the fractional order case and consider no interval uncertainty. Following the
i
i
i ¼ 1; 2; . . . ; N.
i ¼ 1; 2; . . . ; N; 8A 2 AI .
3. Interval eigenvalue problem As shown in the preceding section, for the robust stability check of the uncertain fractional system, it is required to calculate the arguments of phase of eigenvalues. In case when there is no model uncertainty, it is easy to find the argument of phase of each eigenvalue. That is, by simply calculating i ¼ 1; . . . ; N,
(4)
where N is number of eigenvalues of A, si ¼ Refli g and zi ¼ Imfli g of eigenvalue li , and finding the minimum fi such as f ¼ infff1 ; . . . ; fN g,
(5)
if f 4ap=2, then the fractional system is considered robust stable. However, with interval model uncertainty, it is not easy to find (4) because li is not a fixed point in complex plane, instead it is a cluster of infinite points. For illustration, let us see Fig. 1. The left figure shows eigenvalues without model uncertainty; as shown in this figure, the argument of phase can be easily found. But the right figure shows the eigenvalues of model uncertain system; as shown in this figure, it is difficult to find the minimum phase angle, because there is infinite number of eigenvalues. However, fortunately, there is an existing result for interval eigenvalue boundary of interval matrix system. Although not directly applicable for our robust stability test, in this paper, we managed to make use of this existing method for finding strong boundaries of eigenvalues of interval uncertain system. According to [34], the lower and upper boundaries of cluster of eigenvalues can be found although there is a conservatism. However, based on our experience, we found that the conservatism is small. In what follows, this interval eigenvalue bound estimation method is briefly explained.
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2.5 Without interval model uncertainty
2
eigenvalue
1.5
With interval model uncertainty cluster of eigenvalues
1.5
1
1
0.5
0.5
φ
0
Real axis
-0.5
φ
0
Real axis
Imaginary axis
-0.5
-1 -1.5 -2 -2.5 -2.5 -2 -1.5 -1 -0.5
Imaginary axis
2
2613
-1 -1.5 -2 0
0.5
1
1.5
2
2.5
-2.5 -2.5 -2 -1.5 -1 -0.5
0
0.5
1
1.5
2
2.5
Fig. 1. Left: no interval uncertainty. Right: with interval uncertainty.
3.1. Boundaries of eigen-clusters For bounding eigenvalues of model uncertain fractional system, the following interval matrix is defined:
eigenvectors are denoted as vi and ui , respectively, and they are normalized such as: vi uj ¼ dij ; i; j ¼ 1; . . . ; N. Then, from [34], separately considering the real part and imaginary part of eigenvectors such as
AI ¼ ½Ac DA; Ac þ DA ¼ ½A; A,
vre i ¼ Reðvi Þ;
vim i ¼ Imðvi Þ,
ure i ¼ Reðui Þ;
uim i ¼ Imðui Þ,
where Ac is a center matrix (nominal plant without uncertainty), DA is a radius matrix corresponding to interval uncertainty, and A is lower boundary matrix, and A is upper boundary matrix. Then, interval eigenvalues corresponding to interval matrix are defined as LI ¼ fl 2 C : Ax ¼ lx; xa0; A 2 AI g. I
Clearly, L is a set composed of infinite number of elements. In fact, it is impossible to find exact boundaries of LI in complex plane, so in this paper, an existing result, which finds the upper and lower boundaries of real part and imaginary part of eigenvalue set separately, is utilized. The following is a summary of [34,35]. Throughout the paper, it is assumed that in given fractional interval system such as Da x ¼ AI x;
A I 2 AI
AI is N N real square matrix. Now, assuming A and A are given, let us define center matrix and radius matrix in the following ways: AþA AA ; DA ¼ . 2 2 Furthermore, let us denote the left and right eigenvectors corresponding to center matrix Ac by L and R where row vectors of L are left eigenvectors and column vectors of R are right eigenvectors corresponding to eigenvalues li . Left and right Ac ¼
we have two main results summarized in the following lemmas: Lemma 1. Define a sign calculation operator evaluated at Ac such as re im im T Pi :¼sgn½ðure i vi ui vi Þ , re im im are eigenvectors correwhere ure i , vi , ui , and vi sponding to ith eigenvalue of Ac . If Pi is constant for all AI , AI 2 AI , then the lower and upper boundaries of the real part of ith interval eigenvalue are calculated as c re i lre i ¼ Oi ðA DA P Þ,
where Ore i ðÞ is an operator for selecting the ith real eigenvalue of ðÞ and C ¼ A B are ckj ¼ akj bkj , and as re c i lre i ¼ Oi ðA þ DA P Þ.
Lemma 2. Defining a sign calculation operator evaluated at Ac such as im im re T Qi :¼sgn ðure i v i þ ui v i Þ , if Qi is constant for all AI , AI 2 AI , then the lower and upper boundaries of the imaginary part of ith interval eigenvalue are calculated as c i im lim i ¼ Oi ðA DA Q Þ,
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where Oim i ðÞ is an operator for selecting the ith imaginary eigenvalue, and as
4.1. Example 1
c i im lim i ¼ Oi ðA þ DA Q Þ.
So, using Lemmas 1 and 2, it is easy to calculate the lower and upper boundaries of interval eigenvalue separately in real part and imaginary part. From above lemma, if Pi and Qi , i ¼ 1; . . . ; N are calculated, then, interval ranges of eigenvalues are finally calculated as re im im lIi 2 LIi :¼f½lre i ; li þ j½li ; li g,
where j represents imaginary part. Note that Deif’s algorithm given above also has been used in [35] for the performance comparison of the perturbation based method.
3.2. Stability check using minimum argument of phase As shown in right figure of Fig. 1, there is infinite number of arguments of phases of an interval fractional system. From (5), since the stability condition is given as f 4ap=2, if we find sufficient condition for this, the stability can be checked. For calculating f , the following procedure can be used. P1. Calculate Pi and Qi for i ¼ 1; . . . ; N. re im im P2. Calculate lre for all i , li , li , and li i 2 f1; 2; . . . ; Ng. P3. Find arguments of phase of four points such as
P4. P5. P6. P7.
im f1i ¼ ffðlre i ; li Þ;
im f2i ¼ ffðlre i ; li Þ,
f3i
f4i
¼
im ffðlre i ; li Þ;
4. Two illustrative examples
¼
im ffðlre i ; li Þ
in the complex plane. Find fi ¼ inffjf1i j; jf2i j; jf3i j; jf4i jg. Repeat procedures P3 and P4 for i ¼ 1; . . . ; N. Find f ¼ infffi ; i ¼ 1; . . . ; Ng. If f 4ap=2, then the fractional interval system is robust stable. Otherwise, the stability of system cannot be guaranteed.
Appendix shows the Matlab code of above algorithm. For confirming instability of FO-LTI interval system, the following procedure can be considered: Defining fzi ¼ supfjf1i j; jf2i j; jf3i j; jf4i jg, if there exists i such that fzi pap=2 and f pap=2, then the system is robust unstable.
Let us check robust stability of the following fractional interval system: Da x ¼ AI x,
(6)
where A 2 A ¼ A; A with 0 1 1:1 0:55 2:2 B C 0:45 0:55 A, A ¼ @ 0:9 I
I
0:55
2:25
1:08
0:9
0:45
1:8
0
B A ¼ @ 1:1 0:45
0:55 2:75
1
C 0:45 A 1:32
and a ¼ 0:45. Eigenvalues of center matrix Ac ¼ A þA=2 are calculated as 1:7486, 1:2243þ j1:5597, and 1:2243 j1:5597. In P1, Pi and Qi are calculated as 0 1 0 1 0 0 0 1 1 1 B C B C P1 ¼ @ 1 1 1 A; Q1 ¼ @ 0 0 0 A, 0 0 0 1 1 1 0
1
1
1
1
B P2 ¼ @ 1 1 1 1
C 1 A; 1
0
1 1 C 1 A;
1 1 B 3 P ¼@ 1 1 1 1
1
0
1
B Q2 ¼ @ 1 1 0
1 B 3 Q ¼ @ 1 1
1 1 1
1
1
C 1 A, 1
1 1 1 C 1 1 A. 1 1
1
Observe that in Q , all elements are zero. This is due to the fact that there is no interval uncertainty in imaginary part of the first eigenvalue of center matrix (see Fig. 2). From P2, the interval ranges of eigenvalues are found as lI1 2 LI1 :¼f½2:0228; 1:4826 þ j½0; 0g, lI2 2 LI2 :¼f½1:0557; 1:3986 þ j½1:3209; 1:7859g, lI3 2 LI3 :¼f½1:0557; 1:3986 þ j½1:7859; 1:3209g. These interval boundaries are drawn in Fig. 2 with circle marks. Now, we find that ap=2 ¼ 0:7069 and from procedures P3–P6, f ¼ 0:7568. So, since f ¼ 0:75684ap=2 ¼ 0:7069, we conclude that the fractional interval system is robust stable. Fig. 2 also shows random test results within interval
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With Interval model uncertainty
2.5 cluster of eigenvalues
2
2
1.5
1.5 1
α π/2 = 0.7069
boundary of interval eigenvalues
0.5
0.5
φ∗ =0.7568
0
0
Real axis
-0.5
φ∗ =0.6769
α π /2 = 0.7069
Real axis
-0.5 Imaginary axis
-1 -2 -1.5 -2.5 -2.5
-2
-1.5
-1
-0.5
-1
Imaginary axis
1
2615
-1.5 -2
random tests
0
0.5
1
1.5
2
2.5
Fig. 2. Boundaries of interval eigenvalue for Example 1.
-2.5 -2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 3. Boundaries of interval eigenvalue for Example 2.
boundary boxes. For the validation of the algorithm, 1000 random plants are selected and at each random plant, the eigenvalues are plotted in Fig. 2. This random test result shows that the calculated interval boundaries are reliable and not so conservative. 4.2. Example 2 Let us check robust stability of the following fractional interval system: I
a
D x ¼ A x, I
(7) I
where A 2 A ¼ ½A; A with 0 1 1:1 0:55 2:2 B C 0:45 0:55 A, A ¼ @ 0:9 0:55 0
2:25
0:9 0:45 B 0:55 A ¼ @ 1:1 0:45 2:75
1:35 1
1:8 C 0:45 A 1:65
and a ¼ 0:45. Eigenvalues of center matrix Ac ¼ A þA=2 are calculated as 1:6932, 1:3466þ j1:4990, and 1:3466 j1:4990. From Matlab code of Appendix, f ¼ 0:6769. So, since f oap=2 ¼ 0:7069, the fractional interval system is considered unstable. Fig. 3 shows random test and boundaries of interval eigenvalues. But, as shown in this figure, from random test, the system could be still stable. So, there may exist conservatism in our method, however as shown in Figs. 2 and 3, the conservatism could be very small. This conservatism is due to the
fact that the boundaries of interval eigenvalues are calculated in real part and imaginary part separately. If we can find the more tight interval ranges of eigenvalue cluster of interval matrix such as interval radius, the result could be less conservative. But, currently, we are not able to find this kind of tight boundaries.
5. An extension and a discussion on further research efforts Here, we present a simple extension of our algorithm to the following FO-LTI system with interval uncertainties both in the parameters and in the order: da xðtÞ ¼ AxðtÞ, dta
(8)
where A 2 AI :¼½A; A and a 2 aI :¼½a; a. From our previous discussions, it is clear and straightforward that the robust stability test task for FO-LTI interval systems amounts to examining if min j argðli ðAÞÞj4ap=2; i
i ¼ 1; 2; . . . ; N; 8A 2 AI .
This can also be done easily by the code in the Appendix. Further research efforts include the stabilizability issue for da xðtÞ=dta ¼ AxðtÞ þ BuðtÞ. In particular, we are interested in the following general setting with fractional order state feedback.
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EðDa xÞ ¼ Ax þ Bu, b
y ¼ CðD xÞ,
ð9Þ ð10Þ
where E 2 EI :¼½E c DE; E c þ DE ¼ ½E; E; A 2 AI :¼½Ac DA; Ac þ DA ¼ ½A; A and B 2 BI :¼ ½Bc DB; Bc þ DB ¼ ½B; B. a and b are real numbers and furthermore, a 2 aI :¼½a; a. The following interesting research questions could be asked:
What is the robust stability condition of (9) with u ¼ 0? If the system is unstable, how to design a control input u such that the close-loop system is stable with full states measured? If the system is unstable, how to design the control input u such that the close-loop system is stable if only the output y are measured.
We believe, based on the result of this paper, the above questions can be eventually answered. 6. Conclusions In this paper, we have presented, for the first time, a robust stability test procedure and Matlab code for uncertain fractional-order linear time invariant (FO-LTI) systems with interval coefficients
described in state space form. Two illustrative examples are included to show that the conservatism is practically small.
Acknowledgements This project has been funded in part by the National Academy of Sciences under the Collaboration in Basic Science and Engineering Program/ Twinning Program supported by Contract no. INT0002341 from the National Science Foundation. The contents of this publication do not necessarily reflect the views or policies of the National Academy of Sciences or the National Science Foundation, nor does mention of trade names, commercial products or organizations imply endorsement by the National Academy of Sciences or the National Science Foundation. The authors are grateful to the comments from anonymous reviewers which improved the presentation of this paper. The conference version of this paper was presented at The 2005 IEEE International Conference on Mechatronics and Automation held at the Sheraton Fallsview Hotel & Conference Centre, Niagara Falls, Ontario, Canada from July 29 to August 1, 2005.
Appendix
clear all % Input-1: size of square A matrix n ¼ 3; % Input-2: Nominal center matrix: Ac Ac ¼ ½1; 0:5; 2:0; 1:0; 0:5; 0:5;... 0:5; 2:5; 1:5 ; % center matrix % Input-3: Interval radius matrix: Ar Ar ¼ abs(0.1*Ac) ; % radius matrix % Input-4: Fractional nalpha alpha ¼ 0.45; % Main function [R, D] ¼ eig(Ac) ; L ¼ inv(R) ; for i ¼ 1:1:n; Pi ¼ ( sign(real(R(:,i))*real(L(i,:))... - imag(R(:,i))*imag(L(i,:))) )’ ; Qi ¼ ( sign(real(R(:,i))*imag(L(i,:))... + imag(R(:,i))*real(L(i,:))) )’; Lambda_real_min ¼ real(eig(Ac - Ar.*Pi)); Lambda_real_max ¼ real(eig(Ac + Ar.*Pi));
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lambda_i_real_min(i) ¼ Lambda_real_min(i); lambda_i_real_max(i) ¼ Lambda_real_max(i); Lambda_imag_min ¼ imag(eig(Ac - Ar.*Qi)); Lambda_imag_max ¼ imag(eig(Ac + Ar.*Qi)); lambda_i_imag_min(i) ¼ Lambda_imag_min(i); lambda_i_imag_max(i) ¼ Lambda_imag_max(i); end % Argument of phase for i ¼ 1:1:n eig_i_j(1,i) ¼ lambda_i_real_min(i)... + j*lambda_i_imag_min(i); eig_i_j(2,i) ¼ lambda_i_real_min(i)... + j*lambda_i_imag_max(i); eig_i_j(3,i) ¼ lambda_i_real_max(i)... + j*lambda_i_imag_max(i); eig_i_j(4,i) ¼ lambda_i_real_max(i)... + j*lambda_i_imag_min(i); end jj ¼ 0; for k ¼ 1:1:n; for i ¼ 1:1:4; jj ¼ jj+1; abs_phase(jj) ¼ abs(angle(eig_i_j(i,k))); end end % Output-1: minimum phase % min_arg_phase ¼ min(abs_phase); % Output-2: system stability % Fractional_alpha ¼ alpha*pi/2; if min_arg_phase4Fractional_alpha display(’System is stable’); else display(’The robust stability is not guaranteed’); end
References [1] Y.Q. Chen, D. Xue, H. Dou, Fractional calculus and biomimetic control, in: Proceedings of the First IEEE International Conference on Robotics and Biomimetics (RoBio04), Shengyang, China, August 2004, IEEE, pages robio2004–347. [2] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [3] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives and Some of Their Applications, Nauka i technika, Minsk, 1987. [4] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[5] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 1999. [6] B.J. Lurie, Three-parameter tunable tilt-integral-derivative (TID) controller, US Patent US5371670, 1994. [7] I. Podlubny, Fractional-order systems and PIl Dm -controllers, IEEE Trans. Automat. Control 44 (1) (1999) 208–214. [8] A. Oustaloup, B. Mathieu, P. Lanusse, The CRONE control of resonant plants: application to a flexible transmission, European J. Control 1 (2) (1995). [9] A. Oustaloup, X. Moreau, M. Nouillant, The CRONE suspension, Control Eng. Practice 4 (8) (1996) 1101–1108. [10] H.F. Raynaud, A. Zergaı¨ noh, State-space representation for fractional order controllers, Automatica 36 (2000) 1017–1021.
ARTICLE IN PRESS 2618
Y.Q. Chen et al. / Signal Processing 86 (2006) 2611–2618
[11] S. Manabe, The non-integer integral and its application to control systems, Japanese Institute of Electrical Engineers J. 80 (860) (1960) 589–597. [12] S. Manabe, The non-integer integral and its application to control systems, Electrotechnical J. Japan 6 (3–4) (1961) 83–87. [13] A. Oustaloup, Fractional order sinusoidal oscillators: optimization and their use in highly linear FM modulators, IEEE Trans. Circuits and Systems 28 (10) (1981) 1007–1009. [14] M. Axtell, E.M. Bise, Fractional calculus applications in control systems, in: Proceedings of the IEEE 1990 National Aerospace and Electronics Conference, New York, USA, 1990, pp. 563–566. [15] B.M. Vinagre, Y.Q. Chen, Lecture notes on fractional calculus applications in automatic control and robotics, in: B.M. Vinagre, Y.Q. Chen (Eds.), The 41st IEEE CDC2002 Tutorial Workshop # 2, pp. 1–310. http://mechatronics. ece.usu.edu/foc/cdc02_tw2_ln.pdf, Las Vegas, Nevada, USA, 2002. [16] J.A. Tenreiro Machado, Fractional calculus and applications, Nonlinear Dynamics 29 (March 2002) 1–385 (special issue). [17] M. Duarte Ortigueira, J.A. Tenreiro Machado, Fractional signal processing and applications, Signal Processing 83 (11) (November 2003) 2285–2480 (special issue). [18] D. Xue, Y.Q. Chen, A comparative introduction of four fractional order controllers, in: Proceedings of The 4th IEEE World Congress on Intelligent Control and Automation (WCICA02), Shanghai, China, June 2002, IEEE, New York, pp. 3228–3235, [19] W.J. Rugh, Linear Systems Theory, Prentice-Hall, Englewood Cliffs, NJ, 1993. [20] M. Vidyasagar, A characterization of eAt and a constructive proof of the controllability criterion, IEEE Trans. Automat. Control 60 (2) (1971) 370–371. [21] S.B. Skaar, A.N. Michel, R.K. Miller, Stability of viscoelastic control systems, IEEE Trans. Automat. Control 33 (4) (April 1988) 348–357. [22] D. Matignon, Stability result on fractional differential equations with applications to control processing, in: IMACS-SMC Proceedings, Lille, France, July 1996, pp. 963–968. [23] D. Matignon, Stability properties for generalized fractional differential systems, in: D. Matignon, G. Montseny (Eds.), Proceedings of the Colloquium FDS’98: Fractional Differential Systems: Models, Methods and Applications, vol. 5, Paris, December 1998, pp. 145–158 [24] D. Matignon, Stability properties for generalized fractional differential systems, in: ESAIM: Proceedings, vol. 5, pp. 145–158. URL: http://www.emath.fr/Maths/Proc/Vol.5/, December 1998.
[25] D. Matignon, Generalized fractional differential and difference equations: stability properties and modelling issues, in: Proceedings of the Mathematical Theory of Networks and Systems symposium (MTNS’98), Padova, Italy, July 1998, pp. 503–506. [26] C. Bonnet, J.R. Partington, Coprime factorizations and stability of fractional differential systems, Systems Control Lett. (2000) 41. [27] D. Matignon, B. d’Andre´a Novel, Some results on controllability and observability of finite-dimensional fractional differential systems, Proceedings of the Computational Engineering in Systems and Application Multiconference, Lille, France, vol. 2, IMACS, IEEE-SMC, July 1996, pp. 952–956. [28] I. Petra´sˇ , Y.Q. Chen, B.M. Vinagre, Robust stability test for interval fractional order linear systems, in: V.D. Blondel, A. Megretski (Eds.), Unsolved Problems in the Mathematics of Systems and Control, vol. 208–210, Princeton University Press, Princeton, NJ, 2004 (Chapter 6.5). [29] I. Petra´sˇ , Y.Q. Chen, B.M. Vinagre, I. Podlubny, Stability of linear time invariant systems with interval fractional orders and interval coefficients, in: Proceedings of the International Conference on Computation Cybernetics (ICCC04), Viena Technical University, Viena, Austria, 8/30-9/1 2005, pp. 1–4, [30] Y.Q. Chen, H. Ahn, D. Xue, Robust controllability of interval fractional order linear time invariant systems, in: Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, paper# DETC2005-84744, Long Beach, CA, September 24–28, 2005, pp. 1–9. [31] M. Caputo, Linear models of dissipation whose q is almost frequency independent-II, Geophys. J. R. Astronom. Soc. 13 (1967) 529–539. [32] B.N. Narahari Achar, C. Lorenzo, T. Hartley, Initialization issues of the Caputo fractional derivative, in: Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, paper# DETC2005-84348, Long Beach, CA, USA, September 24–28, 2005. [33] M. Moze, J. Sabatier, LMI tools for stability analysis of fractional systems, in: Proceedings of ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, paper# DETC2005-85182, Long Beach, CA, USA, September 24–28, 2005. [34] A.S. Deif, The interval eigenvalue problem, Z. Angew. Math. Mech. 71 (1) (1991) 61–64. [35] Z. Qiu, P.C. Mu¨ller, A. Frommer, An appliximation method for the standard interval eigenvalue problem of real nonsymmetric interval matrices, Comm. Numer. Methods Eng. 17 (2001) 239–251.
