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Stability Issues of Fractional-Order Dynamic Systems and Controls

Stability Issues of Fractional-Order Dynamic Systems and Controls. Yan Li and YangQuan Chen Center for Self-Organizing and Intelligent Systems (CSOIS) Department of Electrical and Computer Engineering, Utah State University 4120 Old Main Hill, Logan, UT 84322, USA Email: yqchen@ieee.org.

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Stability Issues of Fractional-Order Dynamic Systems and Controls

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  1. Stability Issues of Fractional-Order Dynamic Systems and Controls Yan Li and YangQuan Chen Center for Self-Organizing and Intelligent Systems (CSOIS) Department of Electrical and Computer Engineering, Utah State University 4120 Old Main Hill, Logan, UT 84322, USA Email: yqchen@ieee.org W8: Applied Fractional Calculus in Controls and Signal Processing Pre-conference full day workshop on CDC 2010 – Room 207, 12/14/2010

  2. Dr. Yan Li is now an Associate Professor at the College of Automation Science and Engineering, Shandong University, China. Slides were prepared by Dr. Yan Li based on his research done at Center for Self-Organizing and Intelligent Systems (CSOIS), Utah State University from 2007 to 2010. He can be reached at liyan.sdu@gmail.com

  3. Part I • What's fractional calculus • Fractional-order systems • Fractional-order element networks • Part II • Fractional-order iterative learning control • Fractional-order universal adaptive control • Fractional-order optimal control • Mittag-Leffler stability of nonlinear systems • Part III: Conclusions and Future Works

  4. Stability Issues of Fractional-Order Dynamic Systems and Controls (I) Yan Li and YangQuan Chen Center for Self-Organizing and Intelligent Systems (CSOIS) Department of Electrical and Computer Engineering, Utah State University 4120 Old Main Hill, Logan, UT 84322, USA Email: yqchen@ieee.org W8: Applied Fractional Calculus in Controls and Signal Processing Pre-conference full day workshop on CDC 2010 – Room 207, 12/14/2010

  5. Part I-1: What's fractional calculus 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

  6. Who mentioned it in between 1695 and 1819? Euler in 1730, Lagrange in 1772, Laplace in 1812, and so on. 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.

  7. Cauchy’s integral and Gamma function

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

  9. The Gamma function helps

  10. Replace n by 0.5 and q>0, respectively, we have This defines the fractional-order integral!

  11. Another explanation using the Laplace transform When a=0, the above equation can be rewritten as where * denotes the convolution under meaning of the Laplace transform

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

  13. Remarks • In Laplace domain, the first-order and pth-order integral are corresponding to and , respectively. • The heredity of fractional-order integral is shown in the kernel function – the power-law function. • The convolution is a special case of the Boltzmann superposition principle. Therefore, if the f(x) is the input of a system and the output is , then we can say that the system is a power-law one, where the order is -p.

  14. 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

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

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

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

  18. The Laplace transform of RL derivative For example,

  19. Remark It follows from the above definition that the fractional -order derivative is a non-local one. In other words, it depends on the whole history of the integral. An easy question: Why 0.5=1-0.5=-0.5+1? Answer: The commutative law of addition However Riemann-Liouville ?

  20. 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)

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

  22. 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.

  23. 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.

  24. Left and Right RL Fractional-Order Derivatives

  25. Left and Right Caputo Fractional derivatives

  26. Other approaches include but not limit to: 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

  27. 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.

  28. Why so Many Approaches (Definitions)? • Verify if it is included in the calculus (integer order calculus)? If not, it 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. The same case happens in the matrix theory. How many definitions of a matrix norm? Induced norm, Frobenius norm, Max norm, Schatten norm, Consistent norm, …

  29. Part I-2: Fractional-order systems In this presentation, we mainly focus on the fractional-order state-space and the fractional-order nonlinear systems, where the fractional-order operator can be either the Riemann-Liouville one or the Caputo one. We first discuss the fractional-order state-space system: where denotes the system order, u is the control effort, y is the system output, and A,B,C have proper dimensions.

  30. Note: The controllability and observability of the above system are the same with the integer-order cases.

  31. Stabilities of FO-LTI systems

  32. The stable domain of the FO-LTI system.

  33. 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

  34. It follows from the invertibility of that Using inverse Laplace transform to the above equation yields is the Mittag-Leffler function in two parameters. It should be noted that the above system is Lp stable.

  35. For the Caputo FO-LTI system applying Laplace transform to it yields is the Mittag-Leffler function. The above system is NOT Lp stable.

  36. The non-linear cases are shown as follows:

  37. Note: The constant x0 is an equilibrium point of the above fractional-order dynamic system, if and only if

  38. Part I-3: Fractional-order element networks How to use fractional calculus correctly? Step 1: Physics phenomena Step 2: Basic laws Step 3: Constitutive equation

  39. Fractional Constitutive Equations (Riemann-Liouville viscoelastic model) The constitutive equation is

  40. Distributed order or Variable order models

  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?

  42. 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.

  43. Stability Issues of Fractional-Order Dynamic Systems and Controls (II) Yan Li and YangQuan Chen Center for Self-Organizing and Intelligent Systems (CSOIS) Department of Electrical and Computer Engineering, Utah State University 4120 Old Main Hill, Logan, UT 84322, USA Email: yqchen@ieee.org W8: Applied Fractional Calculus in Controls and Signal Processing Pre-conference full day workshop on CDC 2010 – Room 207, 12/14/2010

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