Stability Issues of the Fractional Order Dynamics and Controls (I) WCICA 2010 Pre-Conference Workshop

1. Stability Issues of the Fractional Order Dynamics and Controls (I) WCICA 2010 Pre-Conference Workshop Yan LiSchool of Control Science and Engineering, Shandong University,Email: liyan_cse@sdu.edu.cn Web: http://liyanatsdu.blog.com/

2. Outline Part I • A brief history of fractional calculus • Very simple preliminaries of fractional calculus • What’s fractional calculus Part II • Why fractional calculus • Applications of fractional calculus

3. The concept of Fractional Calculus (calculus of integrals and derivatives of any arbitrary real or complex order) was raised in 1695 by Marquis de L’Hopital to Gottfried Wilhelm Leibniz: • On September 30th 1695, Leibniz replied to L’ Hopital

4. Who mentioned it in between 1695 and 1819? Euler in 1730, Lagrange in 1772, Laplace in 1812. • Until 1819, for y=x, S. F. Lacroix showed that • The question raised in 1695 was only partly answered 124 years later! • Don’t be scared, we’ll show you how easy it is.

5. Based on numerous efforts from many mathematicians, physicists, chemists, biologist, economists, engineers …… today, fractional calculus is an ongoing topic which draw lots of attentions from almost every branch of modern science. Where to begin? Power-law function Factorial & Gamma function Laplace transform & Convolution

6. Power-Law Function A power law is any polynomial relationship that exhibits the property of scale invariance. For example, where a and k are constants. For a constant c, which implies the scale invariance (scale free).

7. In complex domain (multi-valued) • In real domain (singularity at the origin) • In logarithm coordinates, log(f(x)) is shown in the following

8. The plots of log(f(cx)) in logarithm coordinates.

9. Cauchy’s integral and power-law function

10. Factorial & Gamma functions Q: Can we ``find a smooth curve that connects the points (n,n!)’’?

11. The Gamma function helps

12. Replace n by 0.5 and q>0, respectively

13. Laplace transform and Convolution Without loss of generality, let a=0, the above equation can be rewritten as where * denotes the convolution under meaning of the Laplace transform

14. Applying the Laplace transform to yields where we used Note here, H(x) denotes the unit step function which implies that we only need

15. Summary of the fractional-order integral

16. Fractional-Order Derivatives It’s time to define the fractional-order derivative . Let , it can be defined that In other words, 0.5=1-0.5. Moreover, let f(x)=x and a=0, the above equation leads to

17. Definition: Riemann-Liouville Fractional Derivative • It is obvious that • Let , we have • Moreover,

18. Let f(x) be a power-law function • Let f(x) be a constant C, i.e. • Moreover,

19. Other Properties of RL derivative For arbitrary p,q>0, it can be proved that

21. The Laplace transform of RL derivative • For example,

22. Other approaches of fractional-order derivatives • Caputo fractional-order derivative Recall the 0.5th order RL derivative (0.5=1-0.5) The 0.5th order Caputo derivative (0.5=-0.5+1)

23. Definition: Caputo fractional-order derivative • Note, using the integral by parts and let where function f(x) has n+ 1 continuous bounded derivatives.

24. The Laplace transform of Caputo derivative • For example,

25. Sequential Fractional Order Derivatives It follows from the above discussions that, for p,q>0, therefore, it’s meaningful to define sequential fractional order derivatives as where denotes either the RL or Caputo derivative.

26. Left and Right Fractional-Order Derivatives Figure: The left and right derivatives as operations on the “past” and the “future” of f(t). ------ Igor Podlubny, 1999.

27. Left and Right RL Fractional-Order Derivatives

28. Left and Right Caputo Fractional derivatives

29. Other approaches mainly include: • Erdelyi-Kober Type Fractional Integrals and Fractional Derivatives • Hadamard Type Fractional Integrals and Fractional Derivatives • Grunwald-Letnikov Fractional Derivatives • Riesz Fractional Integro-Differentiation • Fractal Fractional Derivative • Complex Fractional Derivatives

30. Two Remarks • Fractional-order or non-integer-order? As a phrase, fractional order means the order is a fractional number, irrational number is not included. Obviously, the definitions we discussed are “Non-Integer Order” ones, all the real or complex numbers can be included. In the previous references, the fractional calculus means the calculus of non-integer orders.

31. Why so Many Approaches(Definitions)? Is it included in the calculus(integer order calculus)? If not, is called the fractional-order calculus(non-integer order calculus). All the definitions of fractional calculus are somehow related. Fractional calculus is closely related to the Integral-differential equations and special functions.

32. Stability Issues of the Fractional Order Dynamics and Controls (II) WCICA 2010 Pre-Conference Workshop Yan LiSchool of Control Science and Engineering, Shandong University,Email: liyan_cse@sdu.edu.cn Web: http://liyanatsdu.blog.com/

33. After introducing the definitions of fractional calculus, another question to be answered could be Who cares about the fractional calculus? A simple investigation from ISI can partly answer it.

35. The development of fractional calulus

38. Not yet finished Two new fractional calculus books to be published this year:

39. How do we use fractional calculus correctly? Step 1: Physics phenomena Step 2: Basic laws Step 3: Constitutive equation

40. Fractional Constitutive Equations (viscoelastic model) The constitutive equation is

41. Remark: • Viscoelastic material models are the first successful applications of fractional calculus. • The fractional-order constitutive equation comes from the existence of the fractional-order dashpot. In other words, compared with integer-order models, fractional-order ones are not simple replacement of 1 by 0.5. • Any real fractional-order element?

43. Other real fractional-order systems include but not limited to: (I) For the diffusion of heat through a semi-infinite solid, the heat flow is equal to the half-derivative of the temperature. (II) The dynamical processes in fractals lead to the fractional order PDE, where the fractional orders depend on the fractal dimension. (III) The fractional order physics and so on.

45. Fractional-Order Systems

46. Stabilities of FO-LTI systems

49. The solutions of the RL and Caputo systems – the applications of the Mittag-Leffler function For the RL FO-LTI system applying Laplace transform to it yields

50. It follows from the invertibility of that Using inverse Laplace transform to the above equation yields is the Mittag-Leffler function in two parameters.