1. Stability Problems in Constrained Pendulum Systems
and
Time-Delayed Systems
Presented by
Prashanth Ramachandran
Major Advisors
Dr. Yitshak M. Ram and Dr. Marcio de Queiroz

2. Overview
Determining the boundary of stability of a mechanical system as a
function of a parameter
 ,xx f  T
21 xxx 
 t1x
 t2x

Position Vector
Scalar Parameter
Velocity Vector
System Stability
Constrained Double Pendulum Feedback Control System with Time Delay
pivotebetween thDistancec  TimeDelay,c

3. Stability - System Poles
 s
 s
planes 
RegionStable RegionUnstable
Linearizing
The system is stable when α 0 in s α i β
Response components of a system from pole locations
 s
 s
planes pole
zero
 ExcitationHarmonic
 s
 s
planes pole
zero
planes pole
zero
 ExcitationHarmonic  ExcitationHarmonic
Pole placement
  xAx   ,xx f
0xxx  State Vector
    0, xxf  xxA  System Matrix

4. Instability of a Constrained Pendulum System
 Stability - Constrained Double Pendulum
 Solution Strategy - Linear Perturbation Analysis
 An Interesting and Counter-Intuitive Phenomena
 Extension - Higher Dimensions
 Experiment

5. Solution Strategy
Linear Perturbation Analysis
Static Equilibrium Configuration
Non-Linear Differential Equations of Motion
Linearized Equations for Small Perturbations
About Static Equilibrium
D
Y
N
A
M
I
C
S
does not exist
1
(a)
(b)
Eigenvalue Analysis
 exists 

6. Model Definition
1
2 3
g
l
d
AO BO
l
l
1
2
3DATUM
g
Link 2
Link 1 Link 3
d
AO BO
Static Configuration Dynamic Configuration
Degrees of Freedom = 1
Constraints
  0sinsinsin 3211
 dl    0coscoscos 3212   l

7. Equations of Motion
Kinetic Energy
  1221
2
2
2
1
2
cos22
2
1
  mlT
Potential Energy due to Weight
)coscos2coscos2( 2121   mglV
      
      








0cosˆsinˆ
0cosˆsin1ˆsincos
0cosˆsin2ˆsincos2
33
2221
2
12112
1112
2
21221



gg
ggl
ggl


From Euler-Lagrange
1
1
cos
mg
F 
1
3
cos
mg
F 
  32 tanmgF
mgmg
3tan mg 3tan mg

8. Equations of Static Equilibrium
Omitting time derivatives, Substituting k
 for ,k
  for ,ˆ  for ˆ
 







  0cossin2 11
  












 
1
13
2
1
1
tan
,1
,
,2
,21sin




 d
      
      








0cosˆsinˆ
0cosˆsin1ˆsincos
0cosˆsin2ˆsincos2
33
2221
2
12112
1112
2
21221



gg
ggl
ggl


  0cossin1 22
 
0cossin 33
 
d 321
sinsinsin 
0coscoscos 321
 
Physical Meaning of Lagrange Multipliers!

9. Linearization Procedure
To describe small oscillations about equilibrium position
,kkk
  3,2,1k  ,ˆ t
   t
 ˆ
where ,k
 ,
 
 are infinitesimal
The essential rule of linearization
  ,1cos     sin
Eliminating higher order
,12
 
       
       
   












0sincoscossin
0sincoscos1sincos
0sincoscos2sincos2
33333
222222112
111111221






gg
ggl
ggl


,0coscoscos 332211
 
0sinsinsin 332211
 

10. Generalized Eigenvalue Problem
0  KM  where  T

 321
ε





















00sinsinsin
00coscoscos
sincos00
sincos00
sincos00
321
321
3333
2222
1111





gK
 
 



















00000
00000
00000
0001cos
000cos2
21
12


lM
3,2,1i   ,sincos3 iiii
i  
Try,   tt sinvε 
  02
 vMK 
Non-Trivial Solutions of v

11. Substituting the results of Static Equilibrium for






















00sin1sin
00cos0cos
sincoscos/100
100tan0
sincos00cos/1
11
11
111
1
111





gK

















00000
00000
00000
0001sin
000sin2
1
1


lM
l
g
1
1
3
2
cos
sin21





 T
1
2
111
3
11
sincos2tansin211sin21  v
Eigenvalue
Eigenvector
Results
3
2
 ,

12. Results
d
g
l2

lg2

simple
pendulum
compound
pendulum
lg  30cos2

Slope of Link 1 01 A
B C
D
1d
2 3
1dWhenCase 1:
l
g
2

31  dRange of distance
0dWhenCase 2:
m m
l 
60 60
G
O
Compound Pendulum
 6cos2

l
g

Slope of Link 1
6
1

 
I
Mg
 2
mM 2 2
2mlI 
 6cos  l

13. The Paradox
d
g
l2

lg2

simple
pendulum
compound
pendulum
lg  30cos2

2
 is negative
No Oscillations
52
AOBO
5874.0CRd
   

525.0sin 311
1 CRDD

5874.021 32
 CRdd
Could it be that the Linear Perturbation Analysis
failed in properly characterizing the problem?

14. Paradox Resolved
G
C
mg2
1F 2F
Free body diagram Equilibrium Positions
AO BO
P
Q
PG
QG
C
At static equilibrium, the sum of moments of all
external forces about any point should vanish.

15. Paradox Resolved
P
Q
PCQC
6.0d
QG
PG
1 2 3  
Stable configuration Q  78.62 40.73  00.138 2306.1 3669.1
Unstable configuration P  13.53 90  86.126 3333.1 0000.1
Stable and Unstable Configurations for 6.0d

16. Extension to Higher Dimension
Model of n masses and n+1 links
 1 nlhLength of each link nMm Value of each mass

17. System Dynamics
  kk
n
k
knmghV  coscos1
1
 

Potential Energy
       
 


n
k
n
ij
ijji
n
i
jnmhknmhT
1 1
1
1
222
cos11
2
1
 Kinetic Energy
   iiii in  sincos1  1,...,2,1  ni
where























00sinsinsin
00coscoscos
sincos00
sincos00
sincos00
321
321
3333
2222
1111











gK 33 
 nn
K
Stiffness

18. System Dynamics contd.
Mass Matrix 






00
0M
M 11 33 
 nn
M nn
11M
where













nnnn
n
n
mmm
mmm
mmm
h




21
22221
11211
11M     ijij jinm   cos,max1ni ,...,2,1 nj ,...,2,1
Example
     
     
     
      
















1coscoscos
cos2cos2cos2
coscos23cos3
coscos2cos34
434241
343231
242321
141312
11




hM
4n

19. Comments
  

























30cos...coscos
20sin...sinsin
1,...,2,10cossin1
121
121
3
2
1
nk
nkhd
nkkn
f
f
f
n
n
kk
n




f
KK
The static equilibrium data , and used in
evaluating and are obtained by the set of n+3 equations
k 1,...,2,1  nk  
K M
The solutions from the above system of equations substituted into the
mass and stiffness matrices yield the eigenvalue problem which
determine the natural frequency of the system.

20. Validation of Results
Example













0000
0000
0000
0001
hM















0000
0011
015.00
0105.0
gK
2l 0d1nIf , and
h
g
2
Eigenvalue
The system vibrates like a simple pendulum of length h=1

21. Validation of Results
Example
4l 0d3nIf , and
 
h
g
33
2
12
1 
h
g22
2   
h
g
33
2
12
3 




















00000
00000
00111
00122
00123
hM



















01111
15.1000
105.000
1005.00
10005.1
gK

22. Experiment
cm16.5Δ  cm16.5Δ  cm16.5Δ  cm21Δ  cm21Δ  cm21Δ 
mm165 587.0541.0  CRdd mm210 587.0688.0  CRdd
Symmetric Equilibrium Configuration Un-symmetric Equilibrium Configuration
mM 2 GISuppose and the moment of inertia is
mghV 2Potential Energy
2
2
2
4
 







l
mTP
22
2
 






m
I
mT G
B
4
22
2 ml
mvTP Kinetic Energy Substituting  v
The constrained pendulum and the bar string system are statically equivalent

23. Conclusions
The natural frequency of vibration of a system of pendulums has been
developed
 The pendulum system is stable for finite perturbations when d > dcr
and the configuration is always symmetric
 But when the absolute distance between the pivots OA and OB is
increased beyond dcr, the equilibrium configuration with Link 2 being
horizontal is no longer stable
 The counterintuitive phenomena of asymmetric equilibrium is
demonstrated by an experiment
A lumped parameter model for higher dimensions were developed and
the equilibrium configurations were provided
Ramachandran .P, Krishna S.G., and Ram Y. M. “Instability of a constrained pendulum system”,
American Journal of Physics, Vol. 79, Issue 4, pp. 395-400, April 2011

24. Stability Boundaries of Mechanical Controlled Systems
- Determination of Critical Time Delay
 Stability - Control Perspective
 Problem Definition - Time Delay
 Critical Time Delay in SIMO Controlled System
 SIMO System - Numerical Algorithm
 Critical Time Delay in MIMO Controlled System
 MIMO System - Numerical Algorithm
 Bisection - A Practical Approach

25. Vibration Control
Passive Control
Control Force
ΔKxxΔCKxxCxM  
Active (State Feedback) Control
 tu  tu
  xgxf TT
tu  
Governing Differential Equation
       tuttt bKxxCxM  












1
1
0
b

26. Problem Definition
State Feedback Control
ModelSystem
uBAxx 
LawControl
F
feedbackstateFull
u x
Block diagram of state feedback control
- Time Delay Ackermann’s Formula
 APψeF n
1

 Tn
BABAABBψ 12 
 
     
0
2
2
1
1
1
 







n
n
n
n
n
n
sss
pspssP
 1000 ne
There is an inherent time delay between the measure of
state and the application of the control force.

27. Problem Definition contd.
Governing Dynamics with Time delay
Modified Differential Equation
where
        tttt BuKxxCxM 
       ttt TT
xGxFu 
Separation of Variables
  t
et 
vx 
Transcendental Eigenvalue Problem
     0vGFBKCMR   TT
e  2
,

28. Literature Review
M. J. Satche (1949)
Graphical Stability test based on the Nyquist method
E. W. Kamen (1980) and A. Thowsen (1982)
• Conditions for asymptotic stability of delay difference equations
• Cumbersome for larger model order and retardation
J. H. Su (1994)
• Stability criteria to characterize the bound for time delay
• Matrix inequality with an optimization variable
• No analytical proof was available

29. Literature Review contd.
N. Olgac and N. Jalili (1999)
• Multiple delayed resonators to suppress tonal oscillations
• Stability charts were used to determine system behavior
N. Olgac and R. Sipahi’s erroneous solution (2002)
• Substitution for the transcendental term to determine the root crossing
• Concluded that only a finite number of purely imaginary roots exist
A. Singh and Y. M. Ram (2008)
• Theory of state estimation
• Inaccessibility of complete states
• Induced time delay results in undesirable condition number
Thus an analytical solution representing a bound of the
time delay that ensures system stability is missing

30. Motivation
 
 
τc Stable Unstable
λ is purely imaginary
τ is real
Problem can be stated as finding λ and τ,
     0,det,,1   Rf
  0,, 2
2  f
  0,, 2
3  f
Transcendental eigenvalue problem
(5i)*(-5i) + (5i)2 = 0
(2)*(2) - (2)2 = 0

31. Motivation contd.
Newton’s Method




































333
222
111
fff
fff
fff
J
Problem of
e.g. Maple Solution
 
   ;,, lambdaconjugatelambdatauf
lambdaconjugate

Error , invalid derivative
    ;*2*5*3:,, lambdaconjugatelambdataulambdaconjugatelambdatauf 
3τ + 5λ + 2λ
   ;,, lambdaconjugatelambdatauf
tau

3
   ;,, lambdaconjugatelambdatauf
lambda

5 - 2λ + 4 abs (1,λ)
λ signum (λ)
 
   lambdaconjugatelambdatauf
lambdaconjugate
,,


Not differentiable w. r. t.
complex variables

32. SIMO System
 tu
 tx1
 tx2
 tu tu
 tx1
 tx2
 tu
   0vgfbKCM   TT
e  2
First Order Realization






































0
0
v
v
bfbg
00
M0
0I
CK
I0

 
TT
e
A  B 
 e H y 0( )
(or)
Non-Trivial Solution
  0det  
HBA 
 e
0y  if and only if
Transcendental Characteristic Equation
        tuttt bKxxCxM 
nnn 1
,,, 
 bKCM

33. Solution Strategy
T
VUH   0...0diag
Singular Value Decomposition
T
VUH   0...0diag
The Transcendental Characteristic Equation becomes,
   0det  
 eT
VBAU  orthogonal, VU
Define
   VBAUQ   T  
 
 


Pe 
1det
det
Q
Q
is the leading Principal Submatrix of 1Q  Q
   P ln

34. Solution Strategy contd.
For any complex variable s
  2,1,0,2arglnln  kksiss 
Since –λτ is purely imaginary,
    1 PP
       
    
   
   
12 




DD
NN
DD
DNDN (or)         0  DDNN
 N  D
 N
In general, the polynomials and
 D
are not simply expressible in terms of the coefficients
of and .
General Formula
  01
2
2
12
12
2
2 ... nnnnnN n
n
n
n  
    01
2
2
32
32
22
22
12
12 ... ddddddD n
n
n
n
n
n  




 
  01
2
2
12
12
2
2 ...ˆ nnnnnN n
n
n
n  
    01
2
2
32
32
22
22
12
12 ...ˆ ddddddD n
n
n
n
n
n  




 

35. Generalized Solution for SIMO System
Then when λ is imaginary we have
        DDNN  ˆ,ˆ
   
,
2arg
k
k
kr
irP




 ...1,0,1...r
Example 1
1 1
5.0
1 1
 tu
 tx1  tx2
 tu
1 1
5.0
1 1
 tu
 tx1  tx2
 tu







10
01
M 






00
05.0
C 








11
12
K 






1
1
b 







2
1
f 








3
1
g,
First Order Realization Singular Value Decomposition

36. Example 1 contd.
With first order realization
IB 















0011
05.012
1000
0100
A















2131
2131
0000
0000
H
And singular value decomposition











 

1001
1001
0200
0020
2
1
U  00030 diagΣ

















423401
16832414
0353126
0351114
210
1
V

































2126621276
03106146
031022142
216622472
21262422977
4223816144
424348142
212642137
840
1
Q

37. Critical Time Delay - SIMO System
  ,15.035.0ˆ 234
 N   115.55ˆ 23
 D
e.g. let λ = 2*i
N(λ) = 5 – 3*i N(λ) = 5 + 3*i D(λ) = 9 + 3*i D(λ) = 9 - 3*i
We get the polynomial
   
 
 
 1
23
234
det
det
115.55
15.035.0
Q
Q




 



D
N
P
Therefore
R(λ) = λ8 + 6.75 λ6 – 3.5 λ4 – 74 λ2 – 120 = 0
i3985.22,1
Purely Imaginary Roots



ri2
1503.0  ,...1,0,1...,r

38. SIMO System - Numerical Algorithm
Exactness with Moderate Dimensions
 
   





 n
k
k
n
k
k
T
N 2
1
2
1
det



AVU
 
 







 12
1
12
1
det
n
k
k
n
k
k
D



Ψ
For Purely Imaginary 
 
   





 n
k
k
n
k
k
T
N 2
1
2
1
det



AVU
 
 







 12
1
12
1
det
n
k
k
n
k
k
D



Ψ

39. Moderate Dimensions Contd.
Example 2
5
1
1 1
 tu  tu
5 55
5
1
1 1 1
5
1
1 1
 tu  tu
5 55
5
1
1 1 1
IM 5



















10010
00000
00000
10010
00001
C






















55
5105
5105
5105
510
K


















1
1
0
0
0
b



















1
0
1
0
1
f

















1
0
0
0
1
g
.
i4961.22,1 Purely Imaginary Roots
713.11 
i9677.34,3 
Corresp. Critical Time Delays 0120.2 

40. MIMO SYSTEM
Transcendental Eigenvalue Problem (T.E.P)
     0vGFBKCMvR   TT
e  2
,
 
 
 


Pe 
1det
det
Q
Q
Since  TT
GFBH   has 1m singular values
Closed form solution
not possible
We define
, i ρψv i ρψ,,,, 

41. Solution Strategy
Since λ is purely imaginary 0
  0zP ,
The condition is that the real and imaginary part vanish simultaneously.
  







12
21
,
PP
PP
P 
    TT
BFBGKMP  sincos2
1      TT
BGBFCP  sincos2 
T.E.P is given by
         0ρψGFBKCM  iiii TT
sincos2








ρ
ψ
z

42. Lemmata
Lemma 1
For any real τ the eigenvalue s in P( s, τ ) has double symmetry property,
i.e., s and -s are also eigenvalues of P.
Proof
P1 ( s, τ ) = P1 ( -s, τ ) and P2 ( s, τ ) = -P2 ( -s, τ )
[ sin (z) = - sin (-z), cos (z) = cos (-z) ]






 12
21
PP
PP





 
12
21
PP
PP






 I
I
0
0






 I
I
0
0
=
the matrices P ( s, τ ) and P ( -s, τ ) are similarly congruent and share common
eigenvalues.
For real τ, cos ( τs) and sin (τs) in P ( s, τ ) may be their Taylor’s Series
expansions which means the eigenvalues are closed under conjugation.

43. Lemmata contd.
Lemma 2
Each real eigenvalue β of P( β, τ ) associated with real τ, is a repeated
eigenvalue with multiplicity p > 1.
Proof
The proof follows from the double symmetry property of β established in Lemma1.
Let us define
    0det,  P   0, 






For a certain real τ, the eigenvalue β is a root of φ ( τ, β = 0 with multiplicity
p > 1, then β is also a root of χ ( τ, β)= 0.
Ram Y. M., “A method for finding repeated roots in transcendental eigenvalue problems”,
Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical
Engineering Science, Vol. 222, pp. 1665-1671, 2008
Turnbull H. W., “Theory of equations”, 1947, pp. 61 (Oliver and Boyd, Edinburgh)

44. Examples
Employing Newton’s Method


















J 




















22
J
Examples 3 & 4
For Example 1 we start with a tolerance of convergence ɛ = 1 e -14 for the norm of
 T
  2
we obtain after 33 iterations
3985.2 6290.10
as in Example 1.
i3529.07690.1  i7355.00080.3 
2,2   i
which has no physical consequence.
we obtain after 11 iterations









ieeiee
iiee
20607.225131.325789.925470.2
1812.97411.69131369.1132528.6
J








45633.147815.6
62788.4118190.1
ee
ee
J

45. MIMO System - Numerical Algorithm
Double Eigenvalue  

,P
d
d
 

,P
d
d
Determinant of a matrix
Let L (ξ) be a matrix of dimension n x n.
Let Lk (ξ) be the matrix with its k-th column replaced by its derivative w.r.t ξ.
Let Lkr (ξ) be the matrix L (ξ) with its k-th and r-th columns replaced by their derivatives w.r.t ξ.
Then Lkk (ξ) is L (ξ) with its k-th column replaced by its second derivative.


n
k
k
d
d
1
L
L

  

 
1
1 11
2
2
2
n
k
n
kr
kr
n
k
kk
d
d
LL
L


46. Examples
Example 5
Suppose






 2
3
43
2


L 25
64  L
From the definitions for the determinant of a matrix,






 2
2
1
43
23


L 








83
23
2L 





 211
40
26


L 








83
23 2
12L 






83
03
22


L


n
k
k
d
d
1
L
L

  

 
1
1 11
2
2
2
n
k
n
kr
kr
n
k
kk
d
d
LL
L

Thus






1220
83
2
43
23 4
3
2
2
21













 LL
L
d
d
180
83
0
83
23
2
40
26
2
3
32
2
2212112
2




























LLL
L
d
d

47. Examples contd.
Example 6
Using the system from Example 2


















11
01
00
00
00
B






















01
10
01
10
01
F





















11
01
10
21
02
G
With an Initial Guess of β = 2 , τ = 1 and tolerance of convergence ɛ = 1 e-12, after 79 iterations
9164.2 5172.46
which correspond to,
i9164.2 8809.0

48. Bisection - A Practical Approach
Rewriting the T.E.P
       0vHE   ,
  KCME   2
   TT
GFBH     
 
 e,
where
For any purely imaginary λ
By varying λ over a certain range on the imaginary axis



ln

    0Im  
Bisection Strategy
    1ImIm
1


k
m
k
k 

 

,  mkk ,2,1,Im 

49. Bisection contd.
Example 7
Considering the system from Example 2
Varying λ over the interval [ 0, 6i ] and obtain functions τ1 (λ) and τ2 (λ)
         5.44,45.3,5.33,35.2,5.15.0
k 1 2 3 4 5
Λk 1.1192i 2.9164i 3.2511i 3.6573i 4.3572i
Τk 0.8804 0.8809 0.4293 1.4647 0.6702

50. Conclusions
The boundary of stability where an actively controlled mechanical
system may lose or gain stability is considered
 For a SIMO controlled system, the problem may be reduced using
SVD to that of finding the roots of a certain polynomial
A numerical algorithm for systems with small to moderate dimension
was developed
 However, the technique could not be extended for a MIMO system
since the rank of H > 1.
 Two numerical methods, one involving Newton’s iterations and the
other involving Bisection for multiple functions were developed.
Ramachandran .P and Ram Y. M. “Stability Boundaries of Mechanical Controlled System with
Time Delay ”, Journal of Mechanical Systems and Signal Processing, Vol. 27, pp. 523, February
2012

51. Acknowledgements
• Dr. Yitshak M. Ram, Dr. Marcio de Queiroz
• Advisory committee- Dr. Pang, Dr. Khonsari, Dr. Cai, Dr. Giaime
• Department of Mechanical Engineering, LSU

52. Selected References
1. Tadjbakhsh I.G., and Wang Y.M., “Transient Vibrations of a Taut Inclined Cable with a Riding
Accelerating Mass”, Journal of Nonlinear Dynamics, vol. 6, pp. 143-161, 1994
2. Ram Y.M., “A constrained eigenvalue problem and nodal and modal control of vibrating
systems”, Proceedings of the Royal Society of London Series A – Mathematical Physical and
Engineering Sciences, vol. 466, pp. 831-851, 2010
3. Irvine H.M., Cable Structures, The MIT Press, Cambridge, Massachusetts, 1981
4. Inman D.J., Engineering Vibration, Third Edition, Prentice Hall, Upper-Saddle River, N.J., 2007
5. Ziegler. H, Principles of Structural Stability, (Blaisdell, London, 1968)
6. Irvine H. M., and Caughey T. K., “The linear theory of free vibrations of a suspended cables”,
Proceedings of the Royal Society of London – Series A., Vol. 341, pp. 299-315, 1974
7. Feynman R. P., Leighton R. B., and Sands M., The Feynman Lectures on Physics,
(Pearson/Addison-Wesley, San Francisco, CA, 2006)
8. Ram Y. M., “A method for finding repeated roots in transcendental eigenvalue problems”,
Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical
Engineering Science, Vol. 222, pp. 1665-1671, 2008
9. Chen J. S., Li H. C., Ro W. C., “Slip-through of a heavy elastica on point supports”,
International Journal of Solids and Structures, Vol. 47, pp. 261-268, 2010
10. Singh A., State Feedback Control with Time Delay, Dissertation, Louisiana State University,
2009

53. Back-up
Property of Double Symmetry
T.E.P
   0vGKFCM    ss
eess2
Theorem 1
The poles of (3) are closed under conjugation. Equivalently we may say that the poles
of T.E.P are symmetric about the real axis of the complex plane.
Proof
 is  ψμv i
   
   0ψGFFCM
μGFFKCM








sincossin2
cossincos22
eee
eee
  
    0ψGFFKCM
μGFFCM




ieee
eeei




cossincos
sincossin2
22
 is  ψμv i
     
   
   0ψμGKFCM  
ieeii ii 

2
     
   
   0ψμGKFCM  
ieeii ii 

2
   
   0ψGFFCM
μGFFKCM








sincossin2
cossincos22
eee
eee
  
    0ψGFFKCM
μGFFCM




ieee
eeei




cossincos
sincossin2
22

54. Back-up contd.
In the degenerate uncontrolled - undamped case
  0vKM 2
s
Theorem 2
The poles of T.E.P have double symmetry. They are symmetric about the real and imaginary
axes of the complex plane.
Proof
 s  s
 s
planes pole planes pole
Property of Double Symmetry for S.D.O.F