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Stabilization of Singular Fractional-Order Systems: A Linear Matrix Inequality Approach

Slide- 1 /46. Stabilization of Singular Fractional-Order Systems: A Linear Matrix Inequality Approach. Xiaona Song 3-4-09. Slide- 2 /48. Outline. Ⅰ 、 Previous Research Ⅱ 、 Recent Research A 、 Introduction B 、 Main results C 、 Example

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Stabilization of Singular Fractional-Order Systems: A Linear Matrix Inequality Approach

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  1. Slide-1/46 Stabilization of Singular Fractional-Order Systems:A Linear Matrix Inequality Approach Xiaona Song 3-4-09

  2. Slide-2/48 Outline • Ⅰ、Previous Research • Ⅱ、Recent Research A、Introduction B、Main results C、Example D、Conclusions • Ⅲ、Further Research

  3. Slide-3/46 • Model: Takagi-Sugeno (T- S) fuzzy model • Method: Linear matrix inequality (LMI) • Tools: Matlab LMI toolbox, Simulink

  4. Slide-4/46 Ⅰ、Previous Research • 1、Stability condition • 2、H_infinity/Guaranteed cost output feedback controller design • 3、Examples • 4、Conclusions

  5. Slide-5/46 1、Stability condition • Continuous T-S Model with state and distributed delays:

  6. Slide-6/46 • Discrete T-S Model with state and input delays:

  7. Slide-7/46 Construct Lyapunov Functions as Follows:

  8. Slide-8/46

  9. Slide-9/46 2、Output feedback controller design

  10. Slide-10/46 • Using some special mathematical methods, in terms of linear matrix inequality (LMI), we can obtain the parameters of the output feedback controllers.

  11. Slide-11/46 • 1) For continuous time fuzzy systems, a fuzzy dynamic output feedback controller which ensures the robust asymptotic stability of the closed-loop system and guarantees an H_infinity norm bound constraint on disturbance attenuation for all admissible uncertainties.

  12. Slide-12/46 • 2) For discrete time fuzzy systems, a piecewise output feedback controller which ensures the robust asymptotic stability and minimizes the guaranteed cost of the closed-loop uncertain system.

  13. Slide-13/46 3、Examples

  14. Slide-14/46 Continuous-time state response

  15. Slide-15/46 Discrete-time state response

  16. Slide-16/46 4、Conclusions • 1) Robust H-infinity control for uncertain T-S fuzzy systems with distributed delays via output feedback controllers • 2)Robust guaranteed cost output feedback control for uncertain discrete fuzzy systems with state and input delays

  17. Slide-17/46 Ⅱ、Recent Research • The stability and stabilization for singular fractional-order systems were realized. The model is : Where a is the fractional commensurate order. x(t) ∈ Rn is the state, u(t) ∈ Rm is the control input. The matrix E ∈ Rn×nmay be singular, we shall assume that rank E=r ＜ n. A and B are known real constant matrices with appropriate dimensions.

  18. Slide-18/46 A、Introduction • 1) singular system • 2) fractional-order system • 3) singular fractional-order system stability condition

  19. Slide-19/46 1）Singular system • Singular systems present a much wider class of systems than normal systems. The state space equation is as follows:

  20. Slide-20/46 • 1、The robust stability and the control for singular systems have been addressed in many papers. • 2、For singular fractional-order systems, the problem of stability and stabilization are seldom studied.

  21. Slide-21/46 2）Fractional-order system • On the other hand, fractional-order control systems have attracted increasing interest. This is mainly due to the fact that many real-world physical systems are better characterized by fractional-order equations.

  22. Slide-22/46 • 1、The problem of stability and the stabilization for fractional-order normal systems. Stability region of FO-LTI system with order 0<q≤1

  23. Slide-23/46 Stability region of FO-LTI system with order 1 ≤ q <2 2、Few paper has done the research about the stability and the stabilization of singular fractional-order systems.

  24. Slide-24/46 3）Singular fractional-order system stability condition The Caputo definition for fractional derivative : where n is an integer satisfying n-1 < a ≤ n.

  25. Slide-25/46 • The singular fractional-order LTI system is: 0 < a < 2, and the matrix E∈Rn×n has rank d (d<n)

  26. Slide-26/46 B、Main results • Generally speaking, there are two kinds of stabilization problems for singular continuous-time systems. 1、regular, impulse-free and stable 2、regular and stable The second stabilization problems for singular fractional-order systems was considered.

  27. Slide-27/46 • Regular The singular fractional-order system is said to be regular if det (saE-A) is not identically zero.

  28. Slide-28/46 Proof: Set f (s) =︱saE -A︱ (A is nonsingular), Then, we can prove that f (s) is an analytical function with respect to s on the whole complex plane, which implies the isolation of the zeros. And, when s=0, f (0)=(-1)n︱A︱≠0. Therefore, (saE -A) -1 exists almost everywhere on s∈ C.

  29. Slide-29/46 • Stable (1) where Then theSFO system (1) can be decomposed as: (2) (3)

  30. Slide-30/46 (4) The SFO-LTI system (4) is asymptotically stable if and only if the LTI system, is asymptotically stable.

  31. Slide-31/46 • When 0 <a ≤ 1, The SFO-LTI system (4) is asymptotically stable if and only if the LTI system, is asymptotically stable.

  32. Slide-32/46

  33. Slide-33/46 • Theorem 1:The system (1) is asymptotically stable if there exists a matrix V and symmetric matrix X > 0,the following LMI satisfied where

  34. Slide-34/46 (5) • Stabilization where (6) (7)

  35. Slide-35/46 (7a) Using the Lemma 1, The SFO-LTI system (7a) is asymptotically stable if and only if the LTI system, (8) is asymptotically stable.

  36. Slide-36/46 Lemma 2: The system (8) is asymptotically stable if there exist a matrix V and symmetric matrix X > 0, the following LMI satisfied: (9)

  37. Slide-37/46 where

  38. Slide-38/46 Let (10) (11) (12) Then the matrix inequality can be changed to the following LMI ：

  39. Slide-39/46 • Theorem 2:The singular fractional -order system (5) is asymptotically stabilized by a state feedback controller u(t) = kx(t) if there exist matrices G, L1,L2G, L2L and symmetric matrix X > 0,the following LMI satisfied

  40. Slide-40/46 C、Example Given:

  41. Slide-41/46 Using the Matlab toolbox, we can get the parameter of the state feedback controller:

  42. Slide-42/46 Fig. 3 State response

  43. Slide-43/46 D、Conclusions • 1、 In terms of LMIs, sufficient conditions for the stability and stabilization of the singular fractional-order system have been established. • 2、The proposed state feedback control law guarantees that the closed-loop system is stable.

  44. Slide-44/46 Ⅲ、Further research • 1、For the singular fractional order system: • output feedback controller design • H_infinity and guaranteed cost control.

  45. Slide-45/46 2、Singular fractional-order interval state space is as follows: are lower / upper boundaries of the uncertain A and B in elementwise sense. Stability and stabilization of the above system.

  46. Slide-46/46 3、Singular fractional-order interval state space is as follows: are lower/upper boundaries of the uncertain E in elementwise sense. Stability and stabilization of the interval system.

  47. Slide-47/46 4、Singular fractional-order fuzzy state space is as follows: Stability and stabilization of the above singular FO fuzzy system.

  48. Slide-48/46 Thanks Questions?

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