Robustness of dynamic output feedback control for singular perturbation systems

1. Robustness of dynamic output feedback control for singular perturbation systems Speaker : 陳彥谷 E-mail : n1694472@ccmail.ncku.edu.tw

2. outline • Model description • Output feedback controller design • Computation for up bound • Conclusion

3. Model description Consider a linear time-invariant singular perturbation system : (1) where , and is a small positive scalar representing the parasitic element . The reduced order model is obtained by setting

4. Reduced order model : (2a) (2b) where A general dynamic output feedback compensator of order s , is used to obtained the desired performance of closed- loop system for the reduced order model : (3a) (3b)

5. Since (2b) has direct transmission term , it is required is nonsingular . The reduced order closed-loop system can be described : (4) where

6. It is assumed that the choose of will guarantee that Compared with actual closed-loop system from (1) and (3) (5) where

7. The reduced order dynamic output feedback control for the reduced order model must satisfy the following conditions : (6a) (6b) for any sufficiently small where Remark : (1) (2) (3) If (i) (ii) then (6a) is satisfied .

8. Controller design Consider a reduced order closed-loop system that be written : (7) where

9. Assumption 1 : is controllable and observable is controllable and observable Assumption 2 : and or Assumption 3 : All the desired eigenvalues are distinct and are not included in the open-loop system

10. For system , if (8) an almost arbitrary set of distinct closed-loop poles are assignable by static output feedback . Consider fast subsystem , the characteristic polynominal is where

11. A necessary and sufficient condition for being an eigenvalue of the closed-loop system is that vanishes at . Thus , This implies that for some nonnull vector , we have or (9)

12. where (10) Remark : (1) (2) (3)

13. Similarly , the additional (m-p) eigenvalues may be assigned In the following way : (11)

14. (12) Eq (10) is the solution of static output feedback under the constraint Eq (12) , in other work , the number of free parameters is . The difference between the number calculated from Eq (9) and the dimension of the constraint Eq (12)

15. For slow subsystem (13) (14)

16. The number of free parameters under constrains Eq (14) is (15) The simultaneous solution of static output feedback gain matrices forsystems andis Eq (12) subject to the constraints Eq (11) , Eq (13) and Eq (14) . The free parameters in Eq (12) under the constraints Eq (11) , Eq (16) , and Eq (14) must be greater than or equal to zero . (16)

17. Computation for upper bound Theorem 1 : Given the original singular perturbed system (1) and its related reduced order system (2) where exists , if Assumptions 1-3 are valid and degree condition is satisfied , then the dynamic controller (3) design by () can stabilize both (1) and (2) for all , where is the largest positive scalar such that (17)

18. Conclusion • If system (1) satisfy assumption 1-3 ,and exists , • dynamic output feedback controller can be designed on • the basis ofreduced order closed-loop model (4) . • (2) For all , the controller that is designed from • reduced order closed-loop model (4) can be guaranteed • actual closed-loop model stable .

19. (3) Controller design procedure (i) 檢查開迴路系統是否滿足Assumption 1-2 和 (8) (ii) 根據Eq (16)決定控制器階數 s ，並找出降階閉迴路系 統之狀態空間表示式 Eq(7)。 (iii) 決定系統閉迴路之極點與特徵向量(滿足Assumption 3)，由Eq (10) , Eq (13)與 Eq (15)求得 與 (iiii) 由求得之 與 ，帶回原閉迴路系統 (5) ，並檢 查使否滿足 (6a) 與 (6b) ，若不滿足，則控制器階數 加1，重複步驟(ii)到(iiii)

20. Thanks for your attention