This document summarizes research on the stability of constrained pendulum systems and time-delayed systems. It presents an overview and discusses:
1) Using linear perturbation analysis to determine the stability boundary of a mechanical system as a parameter varies.
2) Modeling a constrained double pendulum feedback control system with time delay and analyzing its stability based on system poles.
3) Research findings that show a constrained pendulum system can become unstable, even when the distance between pivots is reduced, which is counterintuitive.
Robust Fuzzy Output Feedback Controller for Affine Nonlinear Systems via T–S ...Mostafa Shokrian Zeini
This presentation concerns the design of a robust H_∞ fuzzy output feedback controller for a class of affine nonlinear systems with disturbance via Takagi-Sugeno (T–S) fuzzy bilinear model. The parallel distributed compensation (PDC) technique is utilized to design a fuzzy controller. The stability conditions of the overall closed loop T-S fuzzy bilinear model are formulated in terms of Lyapunov function via linear matrix inequality (LMI). The control law is robustified by H_∞ sense to attenuate external disturbance. Moreover, the desired controller gains can be obtained by solving a set of LMI.
Energy-Based Control of Under-Actuated Mechanical Systems - Remotely Driven A...Mostafa Shokrian Zeini
This presentation concerns the energy-based swing-up control for a remotely driven acrobot (RDA) which is a 2-link planar robot with the first link being underactuated and the second link being remotely driven by an actuator mounted at a fixed base through a belt.
Robust Fuzzy Output Feedback Controller for Affine Nonlinear Systems via T–S ...Mostafa Shokrian Zeini
This presentation concerns the design of a robust H_∞ fuzzy output feedback controller for a class of affine nonlinear systems with disturbance via Takagi-Sugeno (T–S) fuzzy bilinear model. The parallel distributed compensation (PDC) technique is utilized to design a fuzzy controller. The stability conditions of the overall closed loop T-S fuzzy bilinear model are formulated in terms of Lyapunov function via linear matrix inequality (LMI). The control law is robustified by H_∞ sense to attenuate external disturbance. Moreover, the desired controller gains can be obtained by solving a set of LMI.
Energy-Based Control of Under-Actuated Mechanical Systems - Remotely Driven A...Mostafa Shokrian Zeini
This presentation concerns the energy-based swing-up control for a remotely driven acrobot (RDA) which is a 2-link planar robot with the first link being underactuated and the second link being remotely driven by an actuator mounted at a fixed base through a belt.
Linear quadratic regulator and pole placement for stabilizing a cart inverted...journalBEEI
The system of a cart inverted pendulum has many problems such as nonlinearity, complexity, unstable, and underactuated system. It makes this system be a benchmark for testing many control algorithm. This paper presents a comparison between 2 conventional control methods consist of a linear quadratic regulator (LQR) and pole placement. The comparison indicated by the most optimal steps and results in the system performance that obtained from each method for stabilizing a cart inverted pendulum system. A mathematical model of DC motor and mechanical transmission are included in a mathematical model to minimize the realtime implementation problem. From the simulation, the obtained system performance shows that each method has its advantages, and the desired pendulum angle and cart position reached.
Controller design of inverted pendulum using pole placement and lqreSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Design of an adaptive state feedback controller for a magnetic levitation sy...IJECEIAES
This paper presents designing an adaptive state feedback controller (ASFC) for a magnetic levitation system (MLS), which is an unstable system and has high nonlinearity and represents a challenging control problem. First, a nonadaptive state feedback controller (SFC) is designed by linearization about a selected equilibrium point and designing a SFC by pole-placement method to achieve maximum overshoot of 1.5% and settling time of 1s (5% criterion). When the operating point changes, the designed controller can no longer achieve the design specifications, since it is designed based on a linearization about a different operating point. This gives rise to utilizing the adaptive control scheme to parameterize the state feedback controller in terms of the operating point. The results of the simulation show that the operating point has significant effect on the performance of nonadaptive SFC, and this performance may degrade as the operating point deviates from the equilibrium point, while the ASFC achieves the required design specification for any operating point and outperforms the state feedback controller from this point of view.
Equation of motion of a variable mass system2Solo Hermelin
This is the second of three presentations (self content) for derivation of equations of motions of a variable mass system containing moving solids (rotors, pistons,..) and elastic parts. It uses the Reynolds' Transport Theorem. It is recommended to see the first presentation before this one. Each presentation uses a different method of derivation.
The presentation is at undergraduate (physics, engineering) level.
Please sent comments for improvements to solo.hermelin@gmail.com. Thanks!
For more presentations on different subjects please visit my website at http://www.solohermelin.com
MODELLING AND SIMULATION OF INVERTED PENDULUM USING INTERNAL MODEL CONTROLJournal For Research
The internal model control (IMC) philosophy relies on the internal model principle, which states that control can be achieved only if the control system encapsulates, either implicitly or explicitly, some representation of the process to be controlled. In particular, if the control scheme is developed based on an exact model of the process, then perfect control is theoretically possible. Transfer function of Inverted Pendulum is selected as the base of design, which examines IMC controller. Matlab/simulink is used to simulate the procedures and validate the performance. The results shows robustness of the IMC and got graded responses when compared with PID. Furthermore, a comparison between the PID and IMC was shows that IMC gives better response specifications.
Attitude Control of Satellite Test Setup Using Reaction WheelsA. Bilal Özcan
A reaction wheel is A type of flywheel used primarily by spacecraft for attitude control without using fuel for rockets or other reaction devices.It bases on the principle of angular momentum transfer. That is Newton’s third law of action-reaction.
1st paper: https://www.researchgate.net/publication/338119144_ATTITUDE_CONTROL_OF_SATELLITE_TEST_SETUP_USING_REACTION_WHEELS
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
It gives how states are representing in various canonical forms and how it it is different from transfer function approach. and finally test the system controllability and observability by kalman's test
Linear quadratic regulator and pole placement for stabilizing a cart inverted...journalBEEI
The system of a cart inverted pendulum has many problems such as nonlinearity, complexity, unstable, and underactuated system. It makes this system be a benchmark for testing many control algorithm. This paper presents a comparison between 2 conventional control methods consist of a linear quadratic regulator (LQR) and pole placement. The comparison indicated by the most optimal steps and results in the system performance that obtained from each method for stabilizing a cart inverted pendulum system. A mathematical model of DC motor and mechanical transmission are included in a mathematical model to minimize the realtime implementation problem. From the simulation, the obtained system performance shows that each method has its advantages, and the desired pendulum angle and cart position reached.
Controller design of inverted pendulum using pole placement and lqreSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Design of an adaptive state feedback controller for a magnetic levitation sy...IJECEIAES
This paper presents designing an adaptive state feedback controller (ASFC) for a magnetic levitation system (MLS), which is an unstable system and has high nonlinearity and represents a challenging control problem. First, a nonadaptive state feedback controller (SFC) is designed by linearization about a selected equilibrium point and designing a SFC by pole-placement method to achieve maximum overshoot of 1.5% and settling time of 1s (5% criterion). When the operating point changes, the designed controller can no longer achieve the design specifications, since it is designed based on a linearization about a different operating point. This gives rise to utilizing the adaptive control scheme to parameterize the state feedback controller in terms of the operating point. The results of the simulation show that the operating point has significant effect on the performance of nonadaptive SFC, and this performance may degrade as the operating point deviates from the equilibrium point, while the ASFC achieves the required design specification for any operating point and outperforms the state feedback controller from this point of view.
Equation of motion of a variable mass system2Solo Hermelin
This is the second of three presentations (self content) for derivation of equations of motions of a variable mass system containing moving solids (rotors, pistons,..) and elastic parts. It uses the Reynolds' Transport Theorem. It is recommended to see the first presentation before this one. Each presentation uses a different method of derivation.
The presentation is at undergraduate (physics, engineering) level.
Please sent comments for improvements to solo.hermelin@gmail.com. Thanks!
For more presentations on different subjects please visit my website at http://www.solohermelin.com
MODELLING AND SIMULATION OF INVERTED PENDULUM USING INTERNAL MODEL CONTROLJournal For Research
The internal model control (IMC) philosophy relies on the internal model principle, which states that control can be achieved only if the control system encapsulates, either implicitly or explicitly, some representation of the process to be controlled. In particular, if the control scheme is developed based on an exact model of the process, then perfect control is theoretically possible. Transfer function of Inverted Pendulum is selected as the base of design, which examines IMC controller. Matlab/simulink is used to simulate the procedures and validate the performance. The results shows robustness of the IMC and got graded responses when compared with PID. Furthermore, a comparison between the PID and IMC was shows that IMC gives better response specifications.
Attitude Control of Satellite Test Setup Using Reaction WheelsA. Bilal Özcan
A reaction wheel is A type of flywheel used primarily by spacecraft for attitude control without using fuel for rockets or other reaction devices.It bases on the principle of angular momentum transfer. That is Newton’s third law of action-reaction.
1st paper: https://www.researchgate.net/publication/338119144_ATTITUDE_CONTROL_OF_SATELLITE_TEST_SETUP_USING_REACTION_WHEELS
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
It gives how states are representing in various canonical forms and how it it is different from transfer function approach. and finally test the system controllability and observability by kalman's test
Evaluation of Vibrational Behavior for A System With TwoDegree-of-Freedom Und...IJERA Editor
Analysis of the vibrational behavior of a system is extremely important, both for the evaluation of operating conditions, as performance and safety reason. The studies on vibration concentrate their efforts on understanding the natural phenomena and the development of mathematical theories to describe the vibration of physical systems. The purpose of this study is to evaluate an undamped system with two-degrees-of-freedom and demonstrate by comparing the results obtained in the experimental, numerical and analytical modeling the characteristics that describe a structure in terms of its natural characteristics. The experiment was conducted in PUC-MG where the data were acquired to determine the natural frequency of the system. We also developed an experimental test bed for vibrations studies for graduate and undergraduate students. In analytical modeling were represented all the important aspects of the system. In order, to obtain the mathematical equations is used MATLAB to solve the equations that describe the characteristics of system behavior. For the simulation and numerical solution of the system, we use a computational tool ABAQUS. The comparison between the results obtained in the experiment and the numerical was considered satisfactory using the exact solutions. This study demonstrates that calculation of the adopted conditions on a system with two-degrees-of-freedom can be applied to complex systems with many degrees of freedom and proved to be an excellent learning tool for determining the modal analysis of a system. One of the goals is to use the developed platform to be used as a didactical experiment system for vibration and modal analysis classes at PUC Minas. The idea is to give the students an opportunity to test, play, calculate and confirm the results in vibration and modal analysis in a low-cost platform
This study deals with the active control of the dynamic response of a string with fixed ends and mass
loaded by a point mass. It has been controlled actively by means of a feed forward control method. A point mass of a
string is considered as a vibrating receiver which be forced to vibrate by a vibrating source being positioned on the
string. By analyzing the motion of a string, the equation of motion for a string was derived by using a method of
variation of parameters. To define the optimal conditions of a controller, the cost function, which denotes the dynamic
response at the point mass of a string was evaluated numerically. The possibility of reduction of a dynamic response
was found to depend on the location of a control force, the magnitude of a point mass and a forcing frequency
Big Bang- Big Crunch Optimization in Second Order Sliding Mode ControlIJMTST Journal
In this article, Second order sliding mode with Big Bang- Big Crunch optimization technique is employed
for nonlinear uncertain system.The sliding surface describes the transient behavior of a system in sliding
mode. Frequently, PD- type sliding surface is chosen as a hyperplane in the system state space.An integral
term incorporated in the sliding surface expression that resulted in a type of PID sliding surface as hyperbolic
function for alleviating chattering effect. The sliding mode control law is derived using direct Lyapunov
stability approach and asymptotic stability is proved theoretically. Here, novel tuning scheme is introduced for
estimation of PID sliding surface coefficients, due to which it reduces the reaching time as well as disturbance
effect.The simulation results are presented to make a quantitative comparison with the traditional sliding
mode control. It is demonstrated that the proposed control law improves the tracking performance of system
dynamic model in case of external disturbances and parametric uncertainties.
The paper deals with the problem of control of continuous-time linear systems by the dynamic
output controllers of order equal to the plant model order. The design procedure is based on a
solution of the set of linear matrix inequalities and ensures the closed-loop stability using
Lyapunov approach. Numerical examples are given to illustrate the design procedure and
relevance of the methods as well as to validate the performances of the proposed approach
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
Computational Motor Control: Optimal Control for Stochastic Systems (JAIST su...hirokazutanaka
This is lecure 5 note for JAIST summer school on computational motor control (Hirokazu Tanaka & Hiroyuki Kambara). Lecture video: https://www.youtube.com/watch?v=XS7MDRMPQfU
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 thDistancec 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
3DATUM
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
13. The Paradox
d
g
l2
lg2
simple
pendulum
compound
pendulum
lg 30cos2
2
is negative
No Oscillations
52
AOBO
5874.0CRd
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
6.0d
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 6.0d
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
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 2Potential 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
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...0diag
Singular Value Decomposition
T
VUH 0...0diag
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
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
40. MIMO SYSTEM
Transcendental Eigenvalue Problem (T.E.P)
0vGFBKCMvR TT
e 2
,
Pe
1det
det
Q
Q
Since TT
GFBH has 1m 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