1. Stability of Fractional Order Systems Advisor : Dr. N. Pariz Consultant : Dr. A. Karimpour Presenter : H. Malek Msc. Student of Control Engineering Ferdowsi University of Mashhad

2. Topics:  Fundamentals of Fractional Calculus  Fractional Operators  Solution of Fractional Order Equations  Concepts of Fractional Order Operators  Stability of Linear Fractional Order Systems  Stability of Nonlinear Fractional Order Systems Stability of Fractional Order Systems

3. Definition of Gamma Function: Fundamentals of F.C. Definition of Factorial Function:

4. Fundamentals of F.C. Definition of Mittag-Lefler Function: G. M. Mittag-Lefler

5. Topics: Fundamentals of Fractional Calculus Fundamentals of Fractional Calculus  Fractional Operators  Solution of Fractional Order Equations  Concepts of Fractional Order Operators  Stability of Linear Fractional Order Systems  Stability of Nonlinear Fractional Order Systems Stability of Fractional Order Systems

6. Def. of Fractional Operators  Riemann-Liouville Definition: IfThen Insomuch Then

7. Def. of Fractional Operators  Grunwald-Letnikov Definition: IfThen Insomuch Then

8.  Caputo Definition: Def. of Fractional Operators Like Riemann-Liouville Definition except : Insomuch Then  Miller-Ross Definition:

9. Def. of Fractional Operators  An Example of Fractional Derivative IfThen If Then

10. Topics: Fundamentals of Fractional Calculus Fundamentals of Fractional Calculus Fractional Operators Fractional Operators  Solution of Fractional Order Equations  Concepts of Fractional Order Operators  Stability of Linear Fractional Order Systems  Stability of Nonlinear Fractional Order Systems Stability of Fractional Order Systems

11. Laplace Transformation of Fractional Derivatives Fractional Order Equations 1- According to Reimann Def. 2- According to the Caputo Def. 3-Grunwald Def. Laplace Transformation of Mittag-Lefler Function

12. Fractional Order Equations Solutions of Linear Fractional Order Equations : Example : Solutions of Nonlinear Fractional Order Equations : Adomian Method Diethelm Method

13. Topics: Fundamentals of Fractional Calculus Fundamentals of Fractional Calculus Fractional Operators Fractional Operators Solution of Fractional Order Equations Solution of Fractional Order Equations  Concepts of Fractional Order Operators  Stability of Linear Fractional Order Systems  Stability of Nonlinear Fractional Order Systems Stability of Fractional Order Systems

14. Concepts of F.O. Operators Geometrical Concept of Fractional Integral :

15. Concepts of F.O. Operators Physical Concept of Fractional Integral : 123 4 56 7 8 123 4 56 7 The Fractional order Integral of velocity of a vehicle that its real time and its local time are not the same is the actual distance that it move. Cosmic Time Homogenous Time The Fractional order Derivative of the local velocity of a vehicle that its real time and its local time are not the same is the actual velocity that it has. Distance N= =Distance O Relation between Real time and Local time

16. Application of F.O. Operators Finding a curve such that the time it takes for P to go towards the origin is independent to the start point. Abel Problem: Water Passage Problem: Finding a proper shape for water passage of reservoir such that the velocity of water flow be a function of height of passage.

17. Topics: Fundamentals of Fractional Calculus Fundamentals of Fractional Calculus Fractional Operators Fractional Operators Solution of Fractional Order Equations Solution of Fractional Order Equations Concepts of Fractional Order Operators Concepts of Fractional Order Operators  Stability of Linear Fractional Order Systems  Stability of Nonlinear Fractional Order Systems Stability of Fractional Order Systems

18. Stability of L.F.O. Systems General Form of Linear Fractional Order Equations : Conmesurate Rational Stability Analysis of Linear Fractional Order Equations : 1-Direct Method :

19. Stability of L.F.O. Systems Stability Analysis of Linear Fractional Order Equations : For the system of conmesurate order systems: In the special case and for integer order systems: 2-Eign Value Method :

20. 3-Argument Principle for Studying Stability Stability of L.F.O. Systems Stability Analysis of Linear Fractional Order Equations :

21. Topics: Fundamentals of Fractional Calculus Fundamentals of Fractional Calculus Fractional Operators Fractional Operators Solution of Fractional Order Equations Solution of Fractional Order Equations Concepts of Fractional Order Operators Concepts of Fractional Order Operators Stability of Linear Fractional Order Systems Stability of Linear Fractional Order Systems  Stability of Nonlinear Fractional Order System Stability of Fractional Order Systems

22. Stability of NL.F.O. Systems Stability Analysis with linearization method: Linearization Which IfThen the considered system is asymptotically stable. “ The Fractional order systems, are stable at least as same as their equivalent system in integer order. ”

23. Stability of NL.F.O. Systems Example: Equilibrium Points:

24. Stability of NL.F.O. Systems Stability Analysis with direct lyapunov method: Lyapunov stability is the primary method of testing the stability of nonlinear systems, or linear systems with uncertainty or reliability problems. Lyapunov stability is the primary method of testing the stability of nonlinear systems, or linear systems with uncertainty or reliability problems. It is more general than other tests for stability. It does not depend on testing the roots of Eigen values or of testing poles. It is more general than other tests for stability. It does not depend on testing the roots of Eigen values or of testing poles. It involves finding a “Lyapunov function” for a system. If such a function exists, then the system is stable. A related result shows that if a similar function exists, it is possible to show that a system is unstable. It involves finding a “Lyapunov function” for a system. If such a function exists, then the system is stable. A related result shows that if a similar function exists, it is possible to show that a system is unstable.

25. Stability of NL.F.O. Systems Stability Analysis with direct lyapunov method: The most important part of this approach is finding a Lyapunov function that it should be satisfy some conditions: The most important part of this approach is finding a Lyapunov function that it should be satisfy some conditions: 1- 2- Comment: If in the Then the E.P. is asymptotic stable. If in the Then the E.P. is stable.

26. Stability of NL.F.O. Systems Stability Analysis with direct lyapunov method: Nonlinear Fractional Order Systems = Lyapunov Theorem can’t be applied, because of its order = Second kind of Convolution Volterra Integral Equation

27. Stability of NL.F.O. Systems Stability Analysis with direct lyapunov method: Lyapunov function candidate:

28. Stability of NL.F.O. Systems Stability Analysis with direct lyapunov method: Example:

29. Stability of NL.F.O. Systems Stability Analysis with direct lyapunov method: Theorem: Ifandand when Then the E.P. of is asymptotic stable. Example: 1- 2-

30. Suggestion In the following of applying the lyapunov theorem on the nonlinear fractional order systems, some subject can be suggested : 1.Finding the region(s) of attraction in the nonlinear fractional systems. 2.Proving the instability theorem, global stability theorem, … and other theorems that related to the nonlinear integer order systems. 3.Finding the controller based on the lyapunov function. 4.Applying this approach to the linear fractional order systems.

31. Thank you! Special Thanks to Dr. Pariz Special Thanks to Dr. Karimpour Special Thanks to Prof. Vahidian Thanks to Dr. Chen Thanks to Prof. Podlubny Thanks to Prof. Diethelm

32. To be Continued…!