STABILITY of

LINEAR TIME INVARIANT TIME DELAYED SYSTEMS (LTI-TDS)

CLUSTER TREATMENT OF CHARACTERISTIC ROOTS (CTCR)

Prof. Nejat OlgacUniversity of Connecticut

(860) 486 2382

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.

Overview and Progress

CLUSTER TREATMENT OF CHARACTERISTIC ROOTS (CTCR)

(earlier named “Direct Method”)

Stability analysis of the Retarded LTI systems

where x(n1), A, B (nn) constant, +

Problem statement

)()()( ttt BxAxx

Characteristic Equation:

• transcendental• retarded system with commensurate time delays

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

00)(

)(...)()(

)det(),(

0

0)1(

1

n

k

skk

snn

snn

s

esa

saesaesa

essCE BAI

Proposition 1 (IEEE-TAC, May 2002; SIAM Cont-Opt 2006)

For a given LTI-TDS, there can only be afinite number (< n2 ) of imaginary roots {c}

(distinct or repeated). Assume that these roots are somehow known, as:

is ck mk ...1

c1

c2

::

cm

m1

m2

::

m

21

22

::

2

11

12

::

1 {

k}

...1,...11,, ,2

mk

ckkkk

Clustering feature # 1

Proposition 2. (IEEE-TAC, May 2002; Syst. Cont. Letters 2006)

Invariance of root tendencyFor 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.

...1,...1

])(Re[sgn

mk

isk

k

ckk

ck

sTR

Is invariant of . Clustering feature #2.

Root clustering features #1 and #2

c1

c2

::

cm

m1

m2

::

m

21

22

::

2

11

12

::

1 {

k}

RT1

RT2

::

RTm

D-Subdivision Method

Using the two propositions

m1

m2

m3

m4

:

m

21

22

23

:

2

11

12

::

1

31

32

33

34

::::

3

. . .

k =

k =1...m =1...

cm2

51

21

31

52

11

22

32

53

33

34

::

=

Sequence

Explicit function for the number of unstable roots, NU

m

kk

k

k kRTUNUNU1

11 )(),()()0()(

• 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

0,2

0,1

00

1

1

1

ckk

ckk

k

for

NU=0 >>> Stability

Finding all the crossings exhaustively?

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

Chen et al (95), Louisell (01)

TTs

Tse s ,

1

1

...2,1,0])([tan2 1

T

,isexact mapping for

Re-constructed CE=CE(s,T)

2n-degree polynomial without transcendentality

n

k

kk Ts

Tssa

0

0)1

1()(

n

k

kknk TsTssa

0

0)1()1()(

n

k

kk sTbTsCE

2

0

0)(),(

Routh-Hurwitz array

s2n s2n-1

s2n-2

s2n-3

: : : sn

sn-1

sn-2

: :

s2

s1

s0

b2n b2n-1

R2[2n-2, 1] R2 [2n-3, 1]

: : :

R2 [n, 1] R2 [n-1, 1] R2 [n-2, 1]

: :

R21(T) R1(T)

R0 [0, 1] = b0

b2n-2 b2n-3

R2 [2n-2, 2] R2 [2n-3, 2]

: : :

R2 [n, 2] R2 [n-1, 2] R2 [n-2, 2]

: :

b0

b0

bn (n even) bn-1 (n even)

R2 [2n-2, n/2] R2 [2n-3, n/2]

: : :

R2 [n, n/2]

. . .

. . .

. . .

. . . : : :

. . .

. . .

. . . : :

. . .

. . .

. . .

. . . : : :

. . .

ii) Stability analysis for > 0

Necessary condition R1 (T) = 0

Additional condition R21(T) b0 > 0

For s = i

)(21

0

TR

b

i) Stability analysis for = 0

sn

sn-1

sn-2

::s2

s1

s0

an

an-1

R1[n-2, 1]::

R1 [2, 1]R1 [1, 1]R1 [0, 1]

an-2

an-3

R1 [n-2, 2]::

R1 [2, 2]R1 [1, 2]

a0 (n even). . .. . .. . .::

. . .

Tc1

Tc2

::

Tcm

c1

c2

::

cmm1

m1

m2

::

m

21

22

::

2

11

12

::

1 {

k}

Summary: Direct Method for Retarded LTI-TDS

i) Stability for = 0+ Routh-Hurwitz

ii) Stability for > 0 D-subdivision method

(continuity argument)

NU ( ) Non-sequentially evaluated.

An interesting feature to determine the control gains in real time (synthesis).

An example study

)(

602

512

3.701.79.5

)(

412

213

15.131

)(

33

txtxtx

n=3;

i) for = 0

9.2

22.)(

210

321

3.716.209.6

)(3

2,1

i

rootsChartxtx

Stable for = 0 NU(0)=0

ii) for 0

0)()()()(),( 012

23

3 saesaesaesasCE sss

0)(),(62

0

n

j

jj sTbTsCE

Rekasius transformation;

• 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, )(21

0

cc TR

b

Proposition 1; {Tc} =

-0.4269

-0.1332

0.0829

0.0953

0.6233

15.5030

0.8407

2.1100

3.0347

2.9123

{c} =

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

...2,1,0])([tan2 1

TExact mapping is for

0.87253.84896.8254

:

m

7.208214.681

::

2

0.22190.62721.0325

:

1 {k}

{Tc} =

-0.4269

-0.1332

0.0829

0.0953

0.6233

15.5030

0.8407

2.1100

3.0347

2.9123

{c} =

Proposition 2;

mkisk

k

ck

sTR ...1,])(Re[sgn

1

RT1

RT2

RT3

RT4

RT5

+1-1+1-1+1

=

[sec]

RT Stable / UnstableNU()

[rad/sec]

T

0 - - - -

S, NU=0

.1624 1 3.0347 .0829

U, NU=2

.1859 -1 2.9123 .0953

S, NU=0

.2219 1 15.5032 -.4269

U, NU=2

.6272 1 15.5032 -.4269

U, NU=4

.8725 1 2.1109 .6233

U, NU=6

1.0325 1 15.5032 -.4269

U, NU=8

1.4378 1 15.5032 -.4269

Stability outlook

Pocket 1

Pocket 2

0 2 4 6 8

0

10

20

30

40

50

[sec]

Num

ber

of u

nsta

ble

root

s

c4 = 2.912

c5 = 2.11c3 = 3.034

c1 = 15.503 rad / sStable

2

30 40 10

0

c4

c3 c1

c2 = 0.84

Explicit function NU():

Time trace of x2 state as varies

15.5030

0.8407

2.1100

3.0347

2.9123

{c} = [rad/sec]

Root locus plot (partial):

Interesting feature

PRACTICAL APPLICATIONSof

CLUSTER TREATMENT OF CHARCATERISTIC ROOTS (CTCR)

ACTIVE VIBRATION SUPPRESSION WITHTIME DELAYED FEEDBACK

(ASME Journal of Vibration and Acoustics 2003)

mNkkkkkk

msNcc

kgmmmm

/2,2,4,2,4,2

/9.1,2.2

15.0,2.0,15.0,2.0

222120121110

21

22211211

m11m21

c1

m12 m22

k12

k10 k20

k22

c2

u2

u1

k11 k21

x12

)sin(0 tff

)(

888879.2227.6561.1507.5748.1321.4144.1202.71

00000000

62.1457.3570.1780.6768.2913.6979.1699.54

00000000

62.1457.3570.1780.6768.2913.6979.1699.54

00000000

79.2227.6561.1507.5748.1321.4144.1202.71

00000000

112001000110

10000000

033.1367.124067.12000

00100000

0067.12067.1267.26033.13

00001000

110000101130

00000010

txxx

) (txBxAxMIMO Dynamics:

0)()()(),( 2210 ss esaesasasCE

Characteristic equation

Mapping scheme

0.30711.1444

:

4

0.54411.9766

:

3

2.00524.1137

:

1

2.00284.0665

:

2

503.7

386.4

044.3

98.2

}{

1

1

1

1

4

3

2

1

RT

RT

RT

RT

Stability table using NU ()

Stability Pocket

[sec]RT Number of

Unstable Roots

[rad/sec]

0 0

0.3071 +1 7.5032

2

0.5441 +1 4.3864

4

2.0028 -1 3.0446

8

2.0052 -1 2.98

6

Frequency Response

[rad / s]

Control with delay

( = 250 ms)

Control with no delay

No control

|x12| [dB]

TARGET TRACKING

WITH DELAYED CONTROL

ERROR DYNAMICS

)t(zzz BA

tar

tar

tar

tar

yy

yy

xxxx

z

/mc/mk

/mc/mk

yy

xx

00

1012

00

1110

A

/mk2/mk1

mk2mk1

yy

xx

00

0000

00//

0000

B

SYSTEM PARAMETERS

m=1, kx=30.5, cx=2.8, k1x=-5.5, k2x=3 ky=40, cy=2, k1y=-0.4, k2y=-2.4

TARGET DYNAMICS

1) Helical )cos(51020)( tttx )5sin(107100)( ttty

2) Circular )4sin(100200)( ttx )4cos(100200)( tty

STABILITY TABLE

Time Delay (sec) Stability Chart0

Stable0.2036

Unstable0.463

Stable0.9323

Unstable1.3368

Stable1.6609

Unstable2.2107

Stable2.3896

Unstable

MATLAB SIMULATION

ANSIM ANIMATION

SIMULATION RESULTS

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.

AcknowledgementFormer and present graduate students

Brian Holm-Hansen, M.SHakan 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.

Funding

NSFNAVSEA (ONR)

ELECTRIC BOATARO

PRATT AND WHITNEYSEW Eurodrive FOUNDATION (German)

SIKORSKY AIRCRAFTCONNECTICUT INNOVATIONS Inc.

GENERAL ELECTRIC