Marta ZAGÓROWSKA, Jerzy BARANOWSKI, Waldemar BAUER, Tomasz DZIWIŃSKI, Paweł PIĄTEK AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY KRAKÓW, POLAND
Laypunov function method: non-integer order case Key words: Mittag-Leffler stability, Lyapunov direct method, supercapacitor
Eα (z) =
∞ X k=0
1
Introduction
Non-integer order systems (often called fractional systems) are a rapidly developing field in technical and mathematical sciences. Most focus is oriented on their properties (see for example [1, 3] and applications (see for example [2,7]). The goal of this paper is to highlight one of the interesting results from the first group. Lyapunov direct method provides a way to analyse the stability of dynamical systems without solving their differential equations. It is especially advantageous when the solution is difficult or even impossible to find with classical methods. Therefore, it is interesting to investigate extension of the method for non-integer order systems. Such extension relies heavily on a notion of Mittag-Leffler stability, which is presented along with the theorem, after which this paper is titled. Application of the theorem will be illustrated with an example of RC circuit with a supercapacitor.
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Mittag-Leffler stability can be considered a special case of Lyapunov stability. Before introducing formal definition, some basic concepts of non-integer calculus must be formulated. In this paper, systems described with the Caputo fractional derivative (CFD) are considered. CFD is given by the following formula 1 = Γ(p − n)
Zt
f (n) (τ )dτ (t − τ )p−n+1
(1)
t0
where
Z∞ Γ (x) =
sx−1 e−s ds
(3)
where α > 0 and z ∈ C. The Mittag-Leffler stability can be now formulated as follows [5]. Definition 1 (Mittag-Leffler stability). The solution of system C α t0 Dt x(t)
= f (t, x),
x(t0 ) = x0 ∈ Rn
is Mittag-Leffler stable if β kxk ≤ m x0 Eα − λ(t − t0 )α
(4)
(5)
where t0 is the initial time, α ∈ (0, 1), λ ≥ 0, b > 0, m(0) = 0, m(t) > 0 and m(x) is locally Lipschitz on x. It can be easily observed that Mittag-Leffler stability implies asymptotic stability. Usually the stability of equilibria is investigated. Definition 2 (Equilibrium of non-integer order system). The solution of equation (4) such that x(t) ≡ x0 = const is called the equilibrium. It can be observed, that x0 is an equilibrium of equation (4) iff f (t, x0 ) = 0, ∀t > t0 .
Mittag-Leffler stability
C p t0 Dt f (t)
zk Γ(kα + 1)
(2)
0
is the Gamma function and n − 1 < p < n. Also needed is the one parameter Mittag-Leffler function given by
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Lyapunov direct method
The direct Laypunov method can be extended to verify the Mittag-Leffler stability. The following theorem [6] presents this extension. Theorem 1 (Lyapunov direct method). Let x = 0 be an equilibrium of (4), let D ⊂ Rn be the domain containing the origin. Let V (t, z) : [0, ∞) × D → R be a continuously differentiable of order β and locally Lipshitz with respect to z function such that: α1 kzka ≤ V (t, z) ≤ α2 kzkab C β ≤ − α3 kzkab 0 Dt V (t, x(t))
(6) (7)
x(t)=z
where x(t) is given by (4), t ≥ 0,z ∈ D, β ∈ (0, 1), α1 , α2 , α3 , a, b are some positive constants. Then x = 0 is Mittag Leffler stable.
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Example
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Conclusions
[2] W. Bauer, J. Baranowski, and W. Mitkowski. Non-integer order PIα Dµ control ICU-MM. In Theorem 1 can now be illustrated with the following W. Mitkowski, J. Kacprzyk, and J. Baranowski, edexample. The RC system shown in figure 1 consists of itors, Advances in the Theory and Applications of a resistor and a supercapacitor [9] and can be described Non-integer Order Systems: 5th Conference on Nonwith the following non-integer order equation integer Order Calculus and Its Applications, Cracow, 1 Poland, pages 171–182. Springer, 2013. C α x(t), x(0) = x0 (8) 0 Dt x(t) = − [3] T. Kaczorek. Selected Problems of Fractional SysRC tems Theory. Lecture Notes in Control and InformaIt can be easily observed, that the solution is given tion Sciences. Springer, 2011. by (see for example [3, 7]) [4] H. Khalil. Nonlinear Systems. Prentice Hall PTR, 2002. 1 α t x0 (9) x(t) = Eα − [5] Y. Li, Y. Chen, and I. Podlubny. Mittag–leffler staRC bility of fractional order nonlinear dynamic systems. It will be shown that V (t, z) = |z| is the Lyapunov Automatica, 45(8):1965 – 1969, 2009. [6] Y. Li, Y. Chen, and I. Podlubny. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized mittag–leffler stability. Computers & Mathematics with Applications, 59(5):1810 – 1821, 2010. Fractional Differentiation and Its Applications. [7] P. Piątek and J. Baranowski. Investigation of FixedPoint Computation Influence on Numerical Solutions of Fractional Differential Equations. Acta Mechanica et Automatica, 5(2):101–107, 2011. Fig. 1: RC circuit [8] I. Podlubny. Fractional Differential Equations: An function for the system. It satisfies the inequality (6) Introduction to Fractional Derivatives, Fractional with α1 = 1, α2 = 1, a = 1, b = 1. If β = α then Differential Equations, to Methods of Their SoluC α C α 0 Dt V (t, x(t)) = 0 Dt |x(t)| = tion and Some of Their Applications. Mathematics in Science and Engineering. Elsevier Science, 1998. 1 α α t x0 | = =C D |E − α 0 t [9] P. Skruch and W. Mitkowski. Fractional-order RC models of the ultracapacitors. In W. Mitkowski, 1 α α J. Kacprzyk, and J. Baranowski, editors, Advances =C t |x0 | = 0 Dt Eα − RC in the Theory and Applications of Non-integer Or der Systems: 5th Conference on Non-integer Order 1 1 1 α Eα − t |x0 | = − |x(t)| = − Calculus and Its Applications, Cracow, Poland, pages RC RC RC 281–293. Springer, 2013. It fulfils inequality (7) with α3 = 1/RC.
Application of Lyapunov functions for analysis of stability of non-integer order systems is very interesting tool. It should be however noted, that definition of equilibrium substantially differs between integer and non-integer order systems. Especially stability of systems with controls requires special consideration and will be a topic of further research.
ACKNOWLEDGMENT Work realised in the scope of project titled ”Design and application of noniteger order subsystems in control systems”. Project was financed by National Science Centre on the base of decision no. DEC2013/09/D/ST7/03960.
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REFERENCES
[1] P. Bania and J. Baranowski. Laguerre polynomial approximation of fractional order linear systems. In W. Mitkowski, J. Kacprzyk, and J. Baranowski, editors, Advances in the Theory and Applications of Non-integer Order Systems: 5th Conference on Noninteger Order Calculus and Its Applications, Cracow, Poland, pages 171–182. Springer, 2013.
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STRESZCZENIE
W artykule przeanalizowano bezpośrednią metodę Lapunowa dla układów niecałkowitego rzędu. W tym celu zaprezentowano definicję stabilności Mittaga Lefflera oraz uogólnienie metody Lapunowa na takie układy dla szczególnego przypadku weryfikacji stabilności Mittaga Lefflera. Ostatnim elementem było zweryfikowanie zaprezentowanej teorii do przykładu systemu RC. mgr inż. Marta Zagórowska dr inż. Jerzy Baranowski mgr inż. Waldemar Bauer mgr inż. Tomasz Dziwiński dr inż. Paweł Piątek AGH Akademia Górniczo-Hutnicza Wydział Elektrotechniki, Automatyki, Informatyki i Inżynierii Biomedycznej Katedra Automatyki i Inżynierii Biomedycznej Al. Mickiewicza 30/B1 30-059 Kraków E-mail:
[email protected] [email protected] [email protected] [email protected] [email protected]
