STABILITY and APPLICATIONS

1
STABILITY and APPLICATIONS of LINEAR TIME
INVARIANT TIME DELAYED SYSTEMS (LTI-TDS)
Prof. Nejat Olgac University of Connecticut (860)
486 2382
2
Overview 1) The Root Clustering paradigm and
the DIRECT METHOD. Overview of the progress. A
new paradigm The Cluster Treatment of
Characteristic Roots 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
Overview and Progress CLUSTER TREATMENT OF
CHARACTERISTIC ROOTS and DIRECT METHOD
4
Problem statement

• Stability analysis of the Retarded LTI systems
• where x(n?1), A, B ?? (n?n) constant, ???

5

• Characteristic Equation
• transcendental
• retarded system with commensurate time delays
• ak(s) polynomials of degree (n-k) in s and real
  coefficients

6
Proposition 1 (IEEE-TAC, May 2002)
For a given LTI-TDS, there are only a finite
number of imaginary roots ?c (distinct or
repeated) . Assume that these roots are
exclusively determined, as
7
Clustering feature 1
8
Proposition 2. (IEEE-TAC, May 2002) Invariance
of root tendency
For a given time delay system, crossing of the
characteristic roots over the imaginary axis at
any one of the ?cks is always in the same
direction independent of delay.
9
Root cluster features 1 and 2
10
D-Subdivision Method
Using the two propositions
11
DIRECT METHOD 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 ,
  k1..m,
• ??k ?k,? - ?k,?-1 , k1..m
• RT(k) , k1..m

NU0 gtgtgt Stability
12

• Finding the crossings?
• Rekasius (80), Cook et al. (86), Walton et al.
  (87),
• Chen et al (95), Louisell (01)

13

• Re-constructed CECE(s,T)
• 2n-degree polynomial without transcendentality

14

• Routh-Hurwitz array

For s ? i
15
(No Transcript)
16

• Summary Direct Method for Retarded LTI-TDS

i) Stability for ? 0 Routh-Hurwitz ii)
Stability for ? gt 0 D-subdivision method
(continuity argument)
NU (? ) Non-sequentially evaluated. An
interesting feature to determine the control
gains in real time (synthesis).
17
An example study
n3
i) for ? 0
18
ii) for ? ? 0
Rekasius transformation
19
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
(No Transcript)
21
Proposition 2
22
Stability outlook
Pocket 1
Pocket 2
23
Explicit function NU(?)
24
Time trace of x2 state as ? varies
25
Root locus plot (partial)
26
PRACTICAL APPLICATIONS of DIRECT METHOD
27
ACTIVE VIBRATION SUPPRESSION WITH TIME DELAYED
FEEDBACK (ASME Journal of Vibration and Acoustics
2003)
28
(No Transcript)
29
MIMO Dynamics
30
Mapping scheme
31
Stability table using NU (?)
32
Frequency Response
?
x12 dB
33
TARGET TRACKING WITH DELAYED CONTROL
34
(No Transcript)
35
(No Transcript)
36
STABILITY TABLE
37
MATLAB SIMULATION ANSIM ANIMATION
38
SIMULATION RESULTS
39

• CONCLUSION
• Cluster treatment of the characteristic roots /
  DIRECT METHOD as a numerically simple and
  practicable method for LTI-TDS.
• Many practical applications are under study.

40
Acknowledgement Former and present graduate
students Brian Holm-Hansen, Ph.D. 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.
41
Funding NSF NAVSEA (ONR) ELECTRIC BOAT PRATT AND
WHITNEY SEW Eurodrive FOUNDATION
(German) SIKORSKY AIRCRAFT CONNECTICUT
INNOVATIONS Inc. GENERAL ELECTRIC