Stability Problems in Constrained Pendulum Systemsand

Time-Delayed Systems

Presented byPrashanth Ramachandran

Major AdvisorsDr. Yitshak M. Ram and Dr. Marcio de Queiroz

Overview

Determining the boundary of stability of a mechanical system as a function of a parameter

,xx f T21 xxx

t1x

t2x

Position Vector

Scalar ParameterVelocity Vector

System Stability

Constrained Double Pendulum Feedback Control System with Time Delay

pivot ebetween th Distancec TimeDelay,c

Stability - System Poles

s

s

planes

Region Stable RegionUnstable

Linearizing

The system is stable when α 0ins α iβ

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

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

Solution Strategy

Linear Perturbation Analysis

Static Equilibrium Configuration

Non-Linear Differential Equations of Motion

Linearized Equations for Small PerturbationsAbout Static Equilibrium

DYNAMICS

does not exist

1

(a)

(b)

Eigenvalue Analysis

exists

Model Definition

1

2 3

gl

d

AO BO

l

l1

23DATUM

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

Equations of Motion

Kinetic Energy

122122

21

2 cos2221 mlT

Potential Energy due to Weight

)coscos2coscos2( 2121 mglV

0cosˆsinˆ

0cosˆsin1ˆsincos

0cosˆsin2ˆsincos2

33

22212

12112

1112221221

gg

ggl

ggl

From Euler-Lagrange

11 cos

mgF 1

3 cosmgF

32 tanmgF

mgmg

3tan mg 3tan mg

Equations of Static Equilibrium

Omitting time derivatives, Substituting k for ,k for ,̂ for ̂

0cossin2 11

1

13

2

11

tan,1

,,2

,21sin

d

0cosˆsinˆ

0cosˆsin1ˆsincos

0cosˆsin2ˆsincos2

33

22212

12112

1112221221

gg

ggl

ggl

0cossin1 22 0cossin 33

d 321 sinsinsin 0coscoscos 321

Physical Meaning of Lagrange Multipliers!

Linearization Procedure

To describe small oscillations about equilibrium position

,kkk 3,2,1k ,ˆ t t ˆ

where ,k , are infinitesimal

The essential rule of linearization ,1cos sin

Eliminating higher order

,12

0sincoscossin0sincoscos1sincos0sincoscos2sincos2

33333

222222112

111111221

gggglggl

,0coscoscos 332211 0sinsinsin 332211

Generalized Eigenvalue Problem

0 KM where T 321ε

00sinsinsin00coscoscos

sincos00sincos00sincos00

321

321

3333

2222

1111

gK

0000000000000000001cos000cos2

21

12

lM

3,2,1i ,sincos3 iiii i

Try, tt sinvε

02 vMK

Non-Trivial Solutions of v

Substituting the results of Static Equilibrium for

00sin1sin00cos0cos

sincoscos/100100tan0

sincos00cos/1

11

11

111

1

111

gK

0000000000000000001sin000sin2

1

1

lM

lg

1

13

2

cossin21

T

1

2

111

3

11 sincos2tansin211sin21 v

Eigenvalue

Eigenvector

Results

32 ,

Results

d

gl2

lg2

simple pendulum

compoundpendulum

lg 30cos2

Slope of Link 1 01 A

B C

D

1d

2 3

1dWhenCase 1:

lg

2

31 dRange of distance

0dWhenCase 2:

m m

l

60 60

G

O

Compound Pendulum 6cos2

lg

Slope of Link 161

IMg 2

mM 2 22mlI 6cos l

The Paradox

d

gl2

lg2

simple pendulum

compoundpendulum

lg 30cos2

2 is negative

No Oscillations

52 AOBO

5874.0CRd

525.0sin 3111 CRDD

5874.021 32 CRdd

Could it be that the Linear Perturbation Analysis failed in properly characterizing the problem?

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.

Paradox Resolved

PQ

PCQC

6.0d

QGPG

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 6.0d

Extension to Higher Dimension

Model of n masses and n+1 links

1 nlhLength of each link nMm Value of each mass

System Dynamics

kk

n

kknmghV coscos1

1

Potential Energy

n

k

n

ijijji

n

ijnmhknmhT

1 1

1

1

222 cos1121 Kinetic Energy

iiii in sincos1 1,...,2,1 ni

where

00sinsinsin00coscoscos

sincos00sincos00sincos00

321

321

3333

2222

1111

gK 33 nnK

Stiffness

System Dynamics contd.

Mass Matrix

000M

M 11 33 nnM nn11M

where

nnnn

n

n

mmm

mmmmmm

h

21

22221

11211

11M ijij jinm cos,max1ni ,...,2,1 nj ,...,2,1

Example

1coscoscoscos2cos2cos2coscos23cos3coscos2cos34

434241

343231

242321

141312

11

hM

4n

Comments

30cos...coscos20sin...sinsin

1,...,2,10cossin1

121

121

3

2

1

nknkhd

nkkn

f

ff

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.

Validation of Results

Example

0000000000000001

hM

00000011015.000105.0

gK

2l 0d1nIf , and

hg

2Eigenvalue

The system vibrates like a simple pendulum of length h=1

Validation of ResultsExample

4l 0d3nIf , and

hg33

212

1 hg22

2 hg33

212

3

0000000000001110012200123

hM

0111115.1000105.000

1005.0010005.1

gK

Experimentcm 16.5Δ cm 16.5Δ cm 16.5Δ

cm 21Δ cm 21Δ cm 21Δ

mm 165 587.0541.0 CRdd mm 210 587.0688.0 CRdd

Symmetric Equilibrium Configuration Un-symmetric Equilibrium Configuration

mM 2 GISuppose and the moment of inertia is

mghV 2Potential Energy

22

2

4

lmTP22

2

mImT G

B4

222 mlmvTP Kinetic Energy Substituting v

The constrained pendulum and the bar string system are statically equivalent

ConclusionsThe natural frequency of vibration of a system of pendulums has been

developed

The pendulum system is stable for finite perturbations when d > dcrand 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

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

Vibration Control

Passive Control

Control Force

ΔKxxΔCKxxCxM

Active (State Feedback) Control tu tu

xgxf TTtu

Governing Differential Equation

tuttt bKxxCxM

110

b

Problem Definition

State Feedback Control

Model System

uBAxx

Law Control

F

feedback state Full

u x

Block diagram of state feedback control

- Time Delay Ackermann’s Formula

APψeF n1

Tn BABAABBψ 12

0

22

11

1

n

nn

nn

n

ssspspssP

1000 ne

There is an inherent time delay between the measure of state and the application of the control force.

Problem Definition contd.

Governing Dynamics with Time delay

Modified Differential Equation

where

tttt BuKxxCxM

ttt TT xGxFu

Separation of Variables tet vx

Transcendental Eigenvalue Problem

0vGFBKCMR TTe 2,

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

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

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,, 22 f

0,, 23 f

Transcendental eigenvalue problem

(5i)*(-5i) + (5i)2 = 0

(2)*(2) - (2)2 = 0

Motivation contd.Newton’s Method

333

222

111

fff

fff

fff

J

Problem ofe.g. Maple Solution

; ,,

lambdaconjugatelambdatauflambdaconjugate

Error ,  invalid derivative

; *2*5*3: ,, lambdaconjugatelambdataulambdaconjugatelambdatauf

3τ + 5λ + 2λ

; ,, lambdaconjugatelambdatauftau

3 ; ,, lambdaconjugatelambdatauf

lambda

5 - 2λ + 4 abs (1,λ)λ signum (λ)

lambdaconjugatelambdatauflambdaconjugate

,,

Not differentiable w. r. t. complex variables

SIMO System tu

tx1 tx2

tu tu

tx1 tx2

tu

0vgfbKCM TTe 2

First Order Realization

00

vv

bfbg00

M00I

CKI0

TTe

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

Solution Strategy

TVUH 0...0diag

Singular Value Decomposition

TVUH 0...0diag

The Transcendental Characteristic Equation becomes,

0det eT VBAU orthogonal, VU

Define

VBAUQ T

Pe

1detdet

QQ

is the leading Principal Submatrix of 1Q Q

P ln

Solution Strategy contd.

For any complex variable s

2,1,0 ,2arglnln kksiss

Since –λτ is purely imaginary,

1 PP

12

DDNN

DDDNDN (or) 0 DDNN

N D N

In general, the polynomials and

Dare not simply expressible in terms of the coefficients

of and .

General Formula

012

212

122

2 ... nnnnnN nn

nn

012

232

3222

2212

12 ... ddddddD nn

nn

nn

012

212

122

2 ...ˆ nnnnnN nn

nn

012

232

3222

2212

12 ...ˆ ddddddD nn

nn

nn

Generalized Solution for SIMO System

Then when λ is imaginary we have

DDNN ˆ ,ˆ

,2argk

kkr

irP

...1,0,1...r

Example 1

1 1

5.01 1

tu

tx1 tx2

tu1 1

5.01 1

tu

tx1 tx2

tu

1001

M

0005.0

C

1112

K

11

b

2

1f

31

g,

First Order Realization Singular Value Decomposition

Example 1 contd.

With first order realization

IB

001105.01210000100

A

21312131

00000000

H

And singular value decomposition

1001100102000020

21U 00030 diagΣ

4234011683241403531260351114

2101V

212662127603106146031022142216622472

212624229774223816144424348142212642137

8401 Q

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

detdet

115.5515.035.0

QQ

DNP

ThereforeR(λ) = λ8 + 6.75 λ6 – 3.5 λ4 – 74 λ2 – 120 = 0

i 3985.22,1

Purely Imaginary Roots ri21503.0 ,...1,0,1...,r

SIMO System - Numerical Algorithm

Exactness with Moderate Dimensions

n

kk

n

kk

T

N 2

1

2

1det

AVU

12

1

12

1det

n

kk

n

kk

D

Ψ

For Purely Imaginary

n

kk

n

kk

T

N 2

1

2

1det

AVU

12

1

12

1det

n

kk

n

kk

D

Ψ

Moderate Dimensions Contd.

Example 2

5

11 1

tu tu5 555

1

1 1 15

11 1

tu tu5 555

1

1 1 1

IM 5

1001000000000001001000001

C

555105

51055105

510

K

11000

b

10101

f

10001

g

. i 4961.22,1 Purely Imaginary Roots

713.11

i 9677.34,3

Corresp. Critical Time Delays 0120.2

MIMO SYSTEM

Transcendental Eigenvalue Problem (T.E.P)

0vGFBKCMvR TTe 2,

Pe

1detdet

QQ

Since TT GFBH has 1m singular values

Closed form solution

not possible

We define

, i ρψv i ρψ,,,,

Solution Strategy

Since λ is purely imaginary 0

0zP ,

The condition is that the real and imaginary part vanish simultaneously.

12

21,PPPP

P

TT BFBGKMP sincos21 TT BGBFCP sin cos 2

T.E.P is given by

0ρψGFBKCM iiii TT sincos 2

ρψ

z

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

PPPP

12

21

PPPP

I

I0

0

I

I0

0=

the matrices P ( s, τ ) and P ( -s, τ ) are similarly congruent and share commoneigenvalues.

For real τ, cos ( τs) and sin (τs) in P ( s, τ ) may be their Taylor’s Seriesexpansions which means the eigenvalues are closed under conjugation.

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)

Examples

Employing Newton’s Method

J

22J

Examples 3 & 4For 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

ieeieeiiee

20607.225131.325789.925470.21812.97411.69131369.1132528.6

J

45633.147815.662788.4118190.1

eeee

J

MIMO System - Numerical Algorithm

Double Eigenvalue

,Pdd

,P

dd

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

kkd

d1L

L

1

1 112

2

2n

k

n

krkr

n

kkkd

dLL

L

Examples

Example 5

Suppose

2

3

432L 25 64 L

From the definitions for the determinant of a matrix,

2

2

1 4323L

8323

2L

211 40

26

L

8323 2

12L

8303

22 L

n

kkd

d1L

L

1

1 112

2

2n

k

n

krkr

n

kkkd

dLL

L

Thus

1220832

4323 4

3

2

2

21

LLL

dd

180830

83232

4026

2

332

2

2212112

2

LLL

Ldd

Examples contd.

Example 6Using the system from Example 2

1101000000

B

0110

0110

01

F

11011021

02

G

With an Initial Guess of β = 2 , τ = 1 and tolerance of convergence ɛ = 1 e-12, after 79 iterations

9164.2 5172.46which correspond to,

i9164.2 8809.0

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

1ImIm1

k

m

kk

, mkk ,2,1,Im

Bisection contd.

Example 7Considering 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

ConclusionsThe 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

Acknowledgements

• Dr. Yitshak M. Ram, Dr. Marcio de Queiroz

• Advisory committee- Dr. Pang, Dr. Khonsari, Dr. Cai, Dr. Giaime

• Department of Mechanical Engineering, LSU

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

Back-upProperty of Double Symmetry

T.E.P

0vGKFCM ss eess2

Theorem 1The poles of (3) are closed under conjugation. Equivalently we may say that the polesof T.E.P are symmetric about the real axis of the complex plane.

Proof is ψμv i

0ψGFFCM

μGFFKCM

sincos sin 2cossin cos 22

eeeeee

0ψGFFKCM

μGFFCM

ieeeeeei

cossin cos sincos sin 2

22

is ψμv i

0ψμGKFCM ieeii ii 2 0ψμGKFCM ieeii ii 2

0ψGFFCM

μGFFKCM

sincos sin 2cossin cos 22

eeeeee

0ψGFFKCM

μGFFCM

ieeeeeei

cossin cos sincos sin 2

22

Back-up contd.

In the degenerate uncontrolled - undamped case

0vKM 2s

Theorem 2The poles of T.E.P have double symmetry. They are symmetric about the real and imaginaryaxes of the complex plane.Proof

s s

s

planes pole planes pole

Property of Double Symmetry for S.D.O.F