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
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”)
Problem statement
Stability analysis of the Retarded LTI systems
x (t )  Ax (t )  Bx(t   )
where x(n1), A, B  (nn) constant, +
Characteristic Equation:
CE ( s,  )  det( sI  A  Be s )
 an ( s) e  ns  an 1 ( s) e ( n 1)s  ...  a0 ( s)
n
  ak ( s) e  ks  0   0
k 0
• 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)
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:
s   ck i
k  1...m
c1
c2
:
:
cm
21
22
:
:
2
m1
m2
:
:
m
 k   k ,   k , 1 
2
 ck
11
12
:
:
1
, k  1... m ,   1... 
Clustering feature # 1
{k}
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.
R Tk
  ck
  k
s
 sgn [ Re (

s  ck i
  k
k  1... m ,   1... 
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
k =
k =1...m
 =1...
2
21
 cm
m3
m4
:
m
...
31
32
33
34
:
:
:
:
3
22
11
=
23
:
2
12
:
:
1
51
21
31
52
11
22
32
53
33
34
:
:
 Sequence
Explicit function for the number of unstable roots, NU
m
NU ( )  NU (0)   (
k 1
   k1
) U ( , k1 )  RT (k )
 k
• U(, k1) = A step function
0

1
2

0     k1
for
   k1 ,  ck  0
   k1 ,  ck  0
•  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
Finding all the crossings exhaustively?
Rekasius (80), Cook et al. (86), Walton et al. (87),
Chen et al (95), Louisell (01)
e s 
1  Ts
1  Ts
   , T  
exact mapping for
s   i ,   
2
  0,1,2...


[tan 1 (T )   ]
Re-constructed CE=CE(s,T)
1  Ts k
ak ( s ) (
) 0

1  Ts
k 0
n
n
nk
k
a
(
s
)
(
1

Ts
)
(
1

Ts
)
0
 k
k 0
2n
CE ( s, T )   bk (T ) s k  0
k 0
2n-degree polynomial without transcendentality
Routh-Hurwitz array
ii)
i) Stability analysis for  =>00
sn
2n
ssn-1
n-2
ss2n-1
s2n-2
:
2n-3
s:
s:2
s:1
s:0
sn
sn-1
sn-2
:
:
s2
s1
s0
an
. . . a0 (n even)
an-2
...
b0
b2n
bn (n even)
b2n-2 . . .. . .
an-1
an-3
b2n-11] R1 [n-2,
b2n-32] . . .. . . bn-1 (n even) . . .
R1[n-2,
R2[2n-2,
1] R2 [2n-2,
2] : . . . R2 [2n-2, n/2] . . .
:
:
R2 [2n-3,
1] R2 [2n-3,
2] : . . . R2 [2n-3, n/2] . . .
:
:
:
:
R1 [2,: 1]
R1 [2,:2] . . . :
:
:
:
R1 [1,: 1]
R1 [1,:2]
:
:
:
:
R1 [0,: 1]
...
R2 [n, 1]
R2 [n, n/2]
R2 [n, 2] . . .
R2 [n-1, 1] R2 [n-1, 2] . . .
R2 [n-2, 1] R2 [n-2, 2] . . .
For s =  i
:
:
:
:
:
:
Additional condition  R21(T) b0 > 0
R21(T)
b0
R1(T)
Necessary condition  R1 (T) = 0
R0 [0, 1] = b0

b0
R21 (T )

Tc1
Tc2
:
:
Tcm

m1
c1
c2
:
:
cm
m1
m2
:
:
m
21
22
:
:
2
11
12
:
:
1
{k}
Summary: Direct Method for Retarded LTI-TDS
i) Stability for  = 0+
ii) Stability for  > 0
Routh-Hurwitz
D-subdivision method
(continuity argument)
NU ( )
Non-sequentially evaluated.
An interesting feature to determine the
control gains in real time (synthesis).
An example study
n=3;
  1 13.5  1
 5.9 7.1  70.3




x (t )   3  1  2 x(t )   2
1
5  x(t   )
 2  1  4
 2
0
6 

33


i) for  = 0
 6.9 20.6  71.3
  2  2i


x (t )    1
2
3  x(t )  Char. roots  1, 2
3  2.9
 0

1
2 

Stable for  = 0  NU(0)=0
ii) for   0
CE ( s, )  a3 ( s ) e 3 s  a2 ( s ) e 2 s  a1 ( s ) e  s  a0 ( s )  0
Rekasius transformation;
CE( s, T ) 
2 n 6
 b (T ) s
j 0
j
j
0
• 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, c

b0
R21 (Tc )
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
Proposition 1;
{Tc} =
-0.4269
15.5030
-0.1332
0.8407
0.0829
 {c} =
2.1100
0.0953
3.0347
0.6233
2.9123
Exact mapping
{Tc} =

2

[tan 1 (T )   ]
-0.4269
15.5030
-0.1332
0.8407
0.0829
 {c} =
2.1100
0.0953
3.0347
0.6233
2.9123
0.8725
3.8489
6.8254
:
m
  0,1,2... for s   i
7.2082
14.681
:
:
2
0.2219
0.6272
1.0325
:
1
{k}
Proposition 2;
s
R Tk  sgn [ Re (


RT1
RT2
RT3
RT4
RT5
s  ck i
  k 1
=
+1
-1
+1
-1
+1
) ] , k  1... m
Stability outlook

[sec]
0
RT Stable / Unstable

NU()
[rad/sec]
-
-
-
T
-
Pocket 1
S, NU=0
.1624
1
3.0347
.0829
2.9123
.0953
U, NU=2
.1859
-1
Pocket 2
S, NU=0
.2219
1
15.5032
-.4269
15.5032
-.4269
2.1109
.6233
15.5032
-.4269
15.5032
-.4269
U, NU=2
.6272
1
U, NU=4
.8725
1
U, NU=6
1.0325
1
U, NU=8
1.4378
1
Explicit function NU():
Number of unstable roots
2
50
40
c3
c1
c4
0
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
 [sec]
8
Time trace of x2 state as  varies
Root locus plot (partial):
15.5030
0.8407
{c} =
2.1100
[rad/sec]
3.0347
2.9123
Interesting feature
PRACTICAL APPLICATIONS
of
CLUSTER TREATMENT OF
CHARCATERISTIC ROOTS (CTCR)
ACTIVE VIBRATION SUPPRESSION WITH
TIME DELAYED FEEDBACK
(ASME Journal of Vibration and Acoustics 2003)
x12
u2
k12
m12
k11
k22
m22
k21
c2
m11
m21
u1
k10
f  f 0 sin(  t )
k20
c1
m 11  0.2 , m 12  0.15 , m 21  0.2 , m 22  0.15 kg
c1  2.2 , c2 1.9 N s / m
k 10  2 , k 11  4 , k 12  2 , k 20  4 , k 21  2 , k 22  2 N / m
MIMO Dynamics: x  A x  B x (t   )
x 
1
0
0
0
0
0
0 
 0



30

11
10
0
0
0
0
11


 0
0
0
1
0
0
0
0 


0
12.67
0
0 
13.33 0  26.67  12.67
 0
0
0
0
0
1
0
0 

 0
0
0
12.67  40  12.67 13.33 0 


0
0
0
0
0
0
1 
 0
 0
11
0
0
10
0
 20  1188

x
0
0
0
0
0
0
0 
 0



71
.
02

12
.
44
41
.
21
13
.
48
57
.
07
15
.
61

65
.
27

22
.79 

 0
0
0
0
0
0
0
0 


29.68
67.80
17.70  35.57  14.62 
  54.99  16.79 69.13
 0
0
0
0
0
0
0
0 

 54.99
16.79  69.13  29.68  67.80  17.70 35.57
14.62 


0
0
0
0
0
0
0 
 0
 71.02
12.44  41.21  13.48  57.07  15.61 65.27
22.79 88

Characteristic equation
CE (s, )  a0 (s)  a1 (s) e
 s
 a2 (s) e
2 s
0
x(t  )
Mapping scheme
 2.98 




3.044 


{ }  

4.386






7
.
503


2.0052
4.1137
2.0028
4.0665
0.3071
1.1444
:
4
0.5441
1.9766
:
3
:
2
:
1
 RT1  1




 RT2  1




 RT  1
 3





RT


1
 4

Stability table using NU ()

RT
[sec]
Unstable Roots
0
0.3071
Number of

[rad/sec]
Stability Pocket
0
+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]
|x12| [dB]
Control with Control with
delay
no delay
( = 250 ms)
No control
TARGET TRACKING
WITH DELAYED CONTROL
ERROR DYNAMICS
z  A z  B z( t  )
1 
1
1
 0

 k /m  c /m
0
0
x
x

A
 2
1 
0
1


 k y /m  c y /m 
0
 0
 x  x tar 
 x  x tar 

z
y

y

tar 
 y  y 
tar 

0 
0
0
 0

k1 / m k2 / m
0
0
x
x

B
 0
0 
0
0


/m
k2
/m
k1
0
0
y
y


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
2) Circular
x(t )  20  10t  5 cos(t )
x(t )  200  100 sin( 4t )
y (t )  100  7t  10 sin( 5t )
y (t )  200  100 cos( 4t )
STABILITY TABLE
Time Delay (sec)
0
0.2036
0.463
0.9323
1.3368
1.6609
2.2107
2.3896
Stability Chart
Stable
Unstable
Stable
Unstable
Stable
Unstable
Stable
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.
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.
Funding
NSF
NAVSEA (ONR)
ELECTRIC BOAT
ARO
PRATT AND WHITNEY
SEW Eurodrive FOUNDATION (German)
SIKORSKY AIRCRAFT
CONNECTICUT INNOVATIONS Inc.
GENERAL ELECTRIC