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1. ALARM LAB STABILITY of LINEAR TIME INVARIANT TIME DELAYED SYSTEMS (LTI-TDS) CLUSTER TREATMENT OF CHARACTERISTIC ROOTS (CTCR) Prof. Nejat Olgac University of Connecticut (860) 486 2382

2. ALARM LAB Overview : 1) Cluster Treatment of Characteristic Roots (CTCR) paradigm. Overview of the progress. A unique paradigm “Cluster Treatment of Characteristic Roots” (“Direct Method” as it was called first) was introduced in Santa-Fe IFAC 2001 – plenary address. We report an overview of the paradigm and the progress since. Retarded LTI-TDS case is reviewed. 2) Practical Applications from vibration control to target tracking. MDOF dynamics are considered with time delayed control. The analysis of dynamics for varying time delays using the Direct Method and corresponding simulations are presented.

3. ALARM LAB Overview and Progress CLUSTER TREATMENT OF CHARACTERISTIC ROOTS (CTCR) (earlier named “Direct Method”)

4. ALARM LAB Problem statement Stability analysis of the Retarded LTI systems where x(n1), A, B (nn) constant, +

5. ALARM LAB Characteristic Equation: transcendental retarded system with commensurate time delays ak(s) polynomials of degree (n-k) in s and real coefficients

6. ALARM LAB Proposition 1 (IEEE-TAC, May 2002; SIAM Cont-Opt 2006) For a given LTI-TDS, there can only be a finite number (< n2 ) of imaginary roots {c} (distinct or repeated). Assume that these roots are somehow known, as:

7. ALARM LAB Clustering feature # 1

8. ALARM LAB Is invariant of . Clustering feature #2. Proposition 2. (IEEE-TAC, May 2002; Syst. Cont. Letters 2006)Invariance of root tendency For a given time delay system, crossing of the characteristic roots over the imaginary axis at any one of the ck’s is always in the same direction independent of delay.

9. ALARM LAB Root clustering features #1 and #2

10. ALARM LAB D-Subdivision Method Using the two propositions

11. ALARM LAB Explicit function for the number of unstable roots, NU • U(, k1) = A step function •  is the ceiling function • NU(0) is from Routh array. • k1, smallest  corresponding to ck , k=1..m, • k = k, - k,-1, k=1..m • RT(k) , k=1..m NU=0 >>> Stability

12. ALARM LAB exact mapping for Finding all the crossings exhaustively? Rekasius (80), Cook et al. (86), Walton et al. (87), Chen et al (95), Louisell (01)

13. ALARM LAB Re-constructed CE=CE(s,T) 2n-degree polynomial without transcendentality

14. ALARM LAB i) Stability analysis for  = 0 ii) Stability analysis for  > 0 Additional condition  R21(T) b0 > 0 Necessary condition  R1 (T) = 0 Routh-Hurwitz array For s =  i

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16. ALARM LAB i) Stability for  = 0+ Routh-Hurwitz ii) Stability for  > 0 D-subdivision method (continuity argument) Summary: Direct Method for Retarded LTI-TDS NU ( ) Non-sequentially evaluated. An interesting feature to determine the control gains in real time (synthesis).

17. ALARM LAB Stable for  = 0  NU(0)=0 An example study n=3; i) for  = 0

18. ALARM LAB • Apply Routh-Hurwitz array on CE(s,T) • Extract R1(T), R21(T) and b0 • Find Tc   from R1(T) = 0 • Check positivity condition R21(Tc)b0 > 0 • If positivity holds, ii) for   0 Rekasius transformation;

19. ALARM LAB Proposition 1; R1(T) = 4004343.44 T9 - 541842.39 T8 - 1060480.49 T7 -78697.71 T6 - 15015.61 T5 + 1216.09 T4 + 401.12 T3 -10.25 T2 + 0.11 T -0.11 = 0 Numer(R21) = 11261902.54 T8 - 2692164.60 T7 - 2626804 T6 +19682.38T5 -76010.04 T4 + 7184.05 T3 - 644.70 T2 + 4.80 T - 2.76 Denom(R21) = 12535.51 T6 - 4843.52 T5 - 5284.07 T4 - 760.01 T3 - 168.68 T2 - 6.84 T - 0.4 b0 = 23.2

20. ALARM LAB Exact mapping for

21. ALARM LAB Proposition 2;

22. ALARM LAB Stability outlook Pocket 1 Pocket 2

23. ALARM LAB 2 c3 c1 50 c4 0 40 304010 30 Stable c1 = 15.503 rad / s 20 c3 = 3.034 c5 = 2.11 10 0 c2 = 0.84 c4 = 2.912 0 2 4 6 8  [sec] Explicit function NU():

24. ALARM LAB Time trace of x2 state as  varies

25. ALARM LAB Interesting feature Root locus plot (partial):

26. ALARM LAB PRACTICAL APPLICATIONS of CLUSTER TREATMENT OF CHARCATERISTIC ROOTS (CTCR)

27. ALARM LAB ACTIVE VIBRATION SUPPRESSION WITH TIME DELAYED FEEDBACK (ASME Journal of Vibration and Acoustics 2003)

28. ALARM LAB x12 u2 k12 k22 m12 m22 k11 k21 c2 m21 m11 u1 k10 k20 c1

29. ALARM LAB Characteristic equation MIMO Dynamics:

30. ALARM LAB Mapping scheme

31. ALARM LAB Stability Pocket Stability table using NU ()

32. ALARM LAB [rad / s] Control with delay ( = 250 ms) Control with no delay No control Frequency Response  |x12|[dB]

33. ALARM LAB TARGET TRACKING WITH DELAYED CONTROL

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36. ALARM LAB STABILITYTABLE

37. ALARM LAB MATLAB SIMULATION ANSIM ANIMATION

38. ALARM LAB SIMULATIONRESULTS

39. ALARM LAB • CONCLUSION • Cluster treatment of the characteristic roots / as a numerically simple, exact, efficient and exhaustive method for LTI-TDS. • Many practical applications are under study.

40. ALARM LAB Acknowledgement Former and present graduate students Brian Holm-Hansen, M.S Hakan Elmali, Ph.D. Martin Hosek, Ph.D. Nader Jalili, Ph.D. Mark Renzulli, M.S. Chang Huang, M.S. Rifat Sipahi, Ph.D. Ali Fuat Ergenc, Ph.D. Hassan Fazelinia, Ph.D. Emre Cavdaroglu. M.S.

41. ALARM LAB Funding NSF NAVSEA (ONR) ELECTRIC BOAT ARO PRATT AND WHITNEY SEW Eurodrive FOUNDATION (German) SIKORSKY AIRCRAFT CONNECTICUT INNOVATIONS Inc. GENERAL ELECTRIC