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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. ALARM LAB. Overview :

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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

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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.

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Overview and Progress

CLUSTER TREATMENT OF CHARACTERISTIC ROOTS (CTCR)

(earlier named “Direct Method”)

problem statement

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Problem statement

Stability analysis of the Retarded LTI systems

where x(n1), A, B (nn) constant, +

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Characteristic Equation:

transcendental

retarded system with commensurate time delays

ak(s) polynomials of degree (n-k) in s and real coefficients

proposition 1 ieee tac may 2002 siam cont opt 2006

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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:

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Clustering feature # 1

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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.

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Root clustering features #1 and #2

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D-Subdivision Method

Using the two propositions

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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

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exact mapping for

Finding all the crossings exhaustively?

Rekasius (80), Cook et al. (86), Walton et al. (87),

Chen et al (95), Louisell (01)

re constructed ce ce s t 2n degree polynomial without transcendentality

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Re-constructed CE=CE(s,T)

2n-degree polynomial without transcendentality

routh hurwitz array

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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

summary direct method for retarded lti tds

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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).

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Stable for  = 0  NU(0)=0

An example study

n=3;

i) for  = 0

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  • 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;

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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

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Exact mapping

for

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Proposition 2;

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Stability outlook

Pocket 1

Pocket 2

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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():

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Time trace of x2 state as  varies

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Interesting feature

Root locus plot (partial):

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PRACTICAL APPLICATIONS

of

CLUSTER TREATMENT OF CHARCATERISTIC ROOTS (CTCR)

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ACTIVE VIBRATION SUPPRESSION WITH

TIME DELAYED FEEDBACK

(ASME Journal of Vibration and Acoustics 2003)

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x12

u2

k12

k22

m12

m22

k11

k21

c2

m21

m11

u1

k10

k20

c1

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Characteristic equation

MIMO Dynamics:

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Mapping scheme

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Stability Pocket

Stability table using NU ()

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[rad / s]

Control with delay

( = 250 ms)

Control with no delay

No control

Frequency Response

|x12|[dB]

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TARGET TRACKING

WITH DELAYED CONTROL

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STABILITYTABLE

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MATLAB SIMULATION

ANSIM ANIMATION

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SIMULATIONRESULTS

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  • 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.
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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.

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Funding

NSF

NAVSEA (ONR)

ELECTRIC BOAT

ARO

PRATT AND WHITNEY

SEW Eurodrive FOUNDATION (German)

SIKORSKY AIRCRAFT

CONNECTICUT INNOVATIONS Inc.

GENERAL ELECTRIC