Fractional Dynamic Modeling & Control: History, Definitions, Applications

2. DOCTORIALES 2017 Université de Bejaia du 1 er au 6 Aout 2017 Fractional Dynamic Modeling and Control with Applications Maamar Bettayeb Professor of Electrical and Computer Engineering and Vice Chancellor for Research and Graduate Studies University of Sharjah, UAE August 2nd, 2017

3. Outline • History of Fractional Calculus • Definitions of Fractional Derivatives • Laplace Transform for Fractional Order Derivatives • Fractional Order Differential Equations (FODE) • Models of Dynamic Systems • Approximation of Fractional Order Systems • Fractional Identification • IMC-PID-fractional-order-filter controllers design • Applications • Future Research

4. History • The term fractional calculus is not new. • It is a generalization of the ordinary differentiation and integration to non-integer (arbitrary) order. • The subject is as old as the calculus of differentiation and goes back to times when Leibniz, Gauss, and Newton invented this kind of calculation.

5. History (contd.) • In a letter to L`Hospital in 1695 Leibniz raised the following question: "Can the meaning of derivatives with integer order be generalized to derivatives with non-integer orders?" • The story goes that L`Hospital was somewhat curious about that question and replied by another question to Leibniz. "What if the order will be 1/2?"

6. History (contd.) • Leibniz in a letter dated September 30, 1695 replied: "It will lead to a paradox, from which one day useful consequences will be drawn." • The question raised by Leibniz for a fractional derivative was an ongoing topic in the last 300 years. • Several mathematicians contributed to this subject over the years.

8. Operator Dt  • A generalization of differential and integral operators :

10. Definitions of Fractional Derivative • Riemann-Liouville Definition • Grunwald-Letnikov Definition • Caputo Definition

12. Laplace Transform of Dt  • From Caputo definition

13. Fractional Derivative of Sine Function

14. LTI Systems Classification LTI Systems Non Integer Order Integer Order Non Commensurate Order Commensurate Order Rational Order Non Rational Order

15. Fractional Order Generalization to Complex Dynamics • Time Domain : From standard exponential evolution to generalized hyperbolic behaviour. • Frequency Domain : From entire order slopes to general slopes in frequency responses. • From short memory to long memory

23. Fractional Order Differential Equations

24. Analytical Time Domain Solution of FODE

26. Solving FODE : Laplace Method

27. Solving FODE : Numerical Solution

28. Solving FODE : Numerical Solution (contd.)

29. Solving FODE : Numerical Solution (contd.)

30. Models of Dynamic Systems

31. Transfer Function Representation

32. State Space Representation

33. State Space Representation (contd.)

34. Step/Impulse Responses

35. General Stability Condition

36. ? analytical equation Analytical method (Mittag Leffler function) Fractional system Continuous integer model continuousApproximation Discrete Approximation Several methods Discrete time integer model ! ! ! The integer model parameters are not explicitly given

37. continuousApproximations Using derivative function Using integral function ? ?

38. Grunwald-Letnikov definition For n real Laplace transform Fractional order derivative définition

39. Fractional order systems modeling Fractional order system U(t) Y(t) How to evaluate the output Y(t)? Solutions: • Analytic method (Impulse response expression) • Methods based on discrete models (Continuous to Discrete transformation: Euler, Tustin, Simpson…) • Methods based on continuous models

40. Methods based on continuous models These methods are based on the approximation of the fractional derivation/integration operator by an integer model.

41.  db  Bode diagram of the ideal integrator 1/sn Ideal Fractional Integrator 1/sn

42. Fractional Integration Operator In(s) The ideal integrator 1/sn is approximated by a fractional integration operator In(s) within a limited frequency bandwidth [ ωb, ωh]. This can be obtained using a recursive distribution of N poles and N zeros on [ ωb, ωh].

43. (1) fractional integration operator With: • α and η are the recursive parameters • ω’1 , ωN define the frequency range equivalent to [ ωb, ωh] • N is the cells number related to the approximation accuracy. • n is directly related to α and η by equation (1)

44. In(s) Bode Diagram In(s) acts with order n in the range [ωb, ωh ], and with order one outside it.

45. State Space Representation of In(s) In(s) block diagram Each variable x n+1 is related to the preceeding one xn by :

46. State Space Representation of In(s) The fractional integrator is approximated by an integer system of order N+1

47. Fractional system simulation The non-integer system Can be approximated by an integer order system of high dimension N+1 using the fractional integrator In(s)

48. Fractional system state space representation Overall system state space representation With:

49. Output residual: Model measurement Fractional system Identification Identification is performed using an output error method; Parameter vector θ =[a0 b0 n] for H(s) = b0/(a0+sn) Minimizing the quadratic criterion:

50. System + Criterion Model NLP ALGORITHM Principle of Ouput Error Algorithm Noise y(t) J y*(t) + u(t) -