1. Robust Control of Linear and Non-Linear Dynamical
System by using Extended State Observer
Presented by: Kaliprasad Mahapatro Advisor: Prof Milind E. Rane
Department of Electronics & Telecommunication Engg.
Vishwakarma Institute of Technology, Pune
KALIPRASAD A. MAHAPATRO Vishwakarma Institute of Technology, Pune July 26, 2014 1 / 48

2. Contents I
1 Robust Control
2 Extended State Observer
3 Plant Dynamics
4 Linear and Nonlinear Plants
5 Motion Control
6 Magnetic Levitation
7 Flexible Link
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3. Contents II
8 Inverted Pendulum
9 Applications
10 Conclusion
11 State of Art
12 Acknowledge
KALIPRASAD A. MAHAPATRO Vishwakarma Institute of Technology, Pune July 26, 2014 3 / 48

4. Robust Control Expectation and Problem
1 Expectation
High Precision Control
with Minimum Convergence Time
2 Problem
Uncertainty due to some unknown dynamics, parameter variations,
external disturbances are prime parameters present in engineering
systems.
Due to their non-linear characteristic it becomes very difficult to
control this unknown parameter.
KALIPRASAD A. MAHAPATRO Vishwakarma Institute of Technology, Pune July 26, 2014 4 / 48

5. Robust Control Solution
The key solution to achieve robust control is to first estimate the
uncertainty and then implement the appropriate control law
based on estimation.
Estimation of state along with uncertainties will serve better
solution for implementing control signal as dependency on
practical plant reduces
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6. Extended State Observer Innovation
Extended State Observer(ESO)
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7. Extended State Observer Innovation
1 Owing to the great advances in non-linear control theory, the
observer-based controller has become one of the most commonly
used schemes in industrial applications [WG03] [ARD08].
2 The extended state observer (ESO) has high efficiency in
accomplishing the non-linear dynamic estimation.
3 ESO serves the best estimation by extending the internal and
external disturbance to a rank new state and then apply a special
non-smooth nonlinear error feedback to achieve state
tracking [ZGG12].
KALIPRASAD A. MAHAPATRO Vishwakarma Institute of Technology, Pune July 26, 2014 7 / 48

8. Extended State Observer Innovation
1 Owing to the great advances in non-linear control theory, the
observer-based controller has become one of the most commonly
used schemes in industrial applications [WG03] [ARD08].
2 The extended state observer (ESO) has high efficiency in
accomplishing the non-linear dynamic estimation.
3 ESO serves the best estimation by extending the internal and
external disturbance to a rank new state and then apply a special
non-smooth nonlinear error feedback to achieve state
tracking [ZGG12].
KALIPRASAD A. MAHAPATRO Vishwakarma Institute of Technology, Pune July 26, 2014 7 / 48

9. Extended State Observer Innovation
1 Owing to the great advances in non-linear control theory, the
observer-based controller has become one of the most commonly
used schemes in industrial applications [WG03] [ARD08].
2 The extended state observer (ESO) has high efficiency in
accomplishing the non-linear dynamic estimation.
3 ESO serves the best estimation by extending the internal and
external disturbance to a rank new state and then apply a special
non-smooth nonlinear error feedback to achieve state
tracking [ZGG12].
KALIPRASAD A. MAHAPATRO Vishwakarma Institute of Technology, Pune July 26, 2014 7 / 48

10. Plant Dynamics
Linear and Nonlinear Plants
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11. Linear and Nonlinear Plants Miscellaneous Plants
Table: Dissertation Flow Structure
Plant Dynamics Application Description
Motion control setup Robotics vehicle Plant dynamics
(2nd order system)
Magnetic levitation Bullet train Design of ESO
(3rd order system) (n+1)th order
Flexible Joint system Robotics arm
(4th order system) Controller Design
Flexible Link system Space robots
(4th order system) Application
Inverted Pendulum Humanoid Robot Classical
(4th order system) launch pad of missile Control
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12. Motion Control
ECP-220
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13. Motion Control Mathematical Model
˙z1 = z2
˙z2 =
−c∗
d
J∗
d
z2 +
TD
J∗
d
α(x)
+
1
J∗
d
β(x)
u (1)
J∗
d = combined inertia c∗
d = shaft friction coefficient
TD = Torque disturbance u = control voltage
Figure: Plant dynamic model ECP220
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14. Motion Control Feedback Linearization control law
[SL91] feedback control law can be designed as υ = α(x)+β(x)u
u =
υ −α(x)
β(x)
(2)
υ = ¨υc +k1(υc −z1)+k2( ˙υc −z2) (3)
Figure: Feedback Linearization
u =
υ −a0
b0
where a0 = α(x)+d b0 = β(x)+d (4)
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15. Motion Control Extended State Observer (ESO)
Extended State Observer(ESO) for ECP-220
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16. Motion Control Innovation
Figure: Block diagram for FL+ESO
u =
(
υ
k1(υc − ˆz1)−k2 ˆz2 − ˆz3 −a0)
b0
(5)
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17. Motion Control Mathematical Interpretation of ESO



˙ˆz1 = ˆz2 +β1g1(e)
˙ˆz2 = ˆz3 +β2g2(e)+b0u
˙ˆz3 = β3g3(e)−→ Lumped disturbances
y = z1
e = y− ˆz1−→ error
(6)
βi is the observer gain
where gi is
gi(e,αi,δ) =
| e |αi ,| e |> δ
e
δ1−αi
,| e |≤ δ
(7)
for
Nonlinear ESO α = [1 0.5 0.25]
Linear ESO α = [1 1 1]
δ is a small number used to limit the gain
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18. Motion Control Result Analysis of NESO and LESO
TD = 10% and b0 = 38 but actually it is b0 = 23.2/Kg−m2
Figure: Tracking for Step input
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19. Magnetic Levitation
MagLev
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20. Magnetic Levitation Problem Statement
Problem?
Use a voltage control electromagnet to suspend a ferromagnetic ball in
air at an height of 3mm and maintain the desired trajectory in-spite of
uncertainty and disturbances by using Feedback Linearization and
Extended State Observer.
Figure: MagLev model
Quanser- Canada
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21. Magnetic Levitation Problem Statement


˙x1
˙x2
˙x3

 =



−(R
L )x1
x3
g− 1
m
x2
1
b0+b1y+b2y2+b3y3


+


1
L
0
0

u
Considering another space coordination zi as stated in [SL91]
˙z =



˙x2
˙x3
d
dt (g− 1
m
x2
1
b0+b1y+b2y2+b3y3 )


[z]T
=





˙x2
˙x3
L3
f h(x)
α(x)
+Lg(L2
f h(x))
β(x)
u





[z]T
Now as stated [SL91] considering a non linear feedback control law
υ = α(x)+β(x)u such that z = φ(x) and the new input υ satisfy a linear
time invariant relation
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22. Magnetic Levitation Problem Statement
Figure: Block Diagram
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23. Magnetic Levitation Problem Statement
Due to uncertainty and disturbances
uFL =
k1(υc −z1)−k2z2 −k3z3 −a0 −d
b0
(8)
uESO =
1
b0
(−a0 +k1(υc − ˆz1)−k2 ˆz2 −k3 ˆz3 − ˆz4) (9)
(a) FL (b) ESO+FL
Figure: Estimation of states with 80% uncertainty
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24. Flexible Link
Flexible Link
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25. Flexible Link Distinguish between Flexible Joint and Link
Flexible Joint Flexible Link
Bulky Light weight manipulators
Spring Tension No Spring
Less Complex Lightweight slender manipulators,
increasing the complexity
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26. Flexible Link Distinguish between Flexible Joint and Link




˙θ
˙α
¨θ
¨α



 =





0 0 1 0
0 0 0 1
0
Kstif f
Jeq
−A1 0
0 −
Kstif f+Jarm
JeqJarm
A1 0









θ
α
˙θ
˙α



+




0
0
B1
−B1



Vm (10)
where A1
ηmηgKt KmK2
g +BeqRm
JeqRm
and B1
ηmηgKt Kg
JeqRm
.
State space represented in (10) can be simplified in z domain as
˙z1 = z2
˙z2 = z3
˙z3 = z4
˙z4 = −
Kstif f A1
Jeq
z2 −
Kstif f (Jeq +Jarm)
JarmJeq
z3 −A1z4
α
+
Kstif f B1
Jarm
β
Vm
y = z1 (11)
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27. Flexible Link Distinguish between Flexible Joint and Link
Table: Numerical values of parameters of flexible link system
Symbols Meaning Value
Kstif f modeled stiffness 0.7883 N.m/rad
Jeq gear inertia 0.000931 kg.m2
Jarm link inertia 0.0026 kg.m2
Beq viscous damping coeff 0.0015
ηm motor efficiency 0.69
ηg gearbox efficiency 0.9
Kt torque constant 0.00767 N.m
Kg motor gear ratio 70
Km back-EMF constant 0.00767 V/rpm
Rm armature resistance 2.6 Ω
ωc natural frequency 20.1 rad/s
m mass of link 0.065 kg
l length of link 0.3 m
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28. Flexible Link Distinguish between Flexible Joint and Link
Figure: Basic Feedback Linearization block
u =
1
β
[
....
υ c +k1(υc −z1)+k2( ˙υc −z2)
+ k3( ¨υc −z3)+k4(
...
υ c −z4)−α] (12)
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29. Flexible Link Distinguish between Flexible Joint and Link
Figure: ESO and FL block for estimating states and uncertainty
u =
1
b0
[
....
υ c +k1(υc − ˆz1)+k2( ˙υc − ˆz2)
+ k3( ¨υc − ˆz3)+k4(
...
υ c − ˆz4)−a0 − ˆz5] (13)
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30. Flexible Link Distinguish between Flexible Joint and Link
Figure: States and its estimate in flexible link
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31. Inverted Pendulum
Inverted Pendulum
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32. Inverted Pendulum Problem Statement
To balance the rod at a 0◦ or at a 7◦
Figure: Inverted Pendulum Apparatus
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33. Inverted Pendulum Nonlinear & Linear Mathematical Model
d
dt




θ
˙θ
x
˙x



 =







z2
2m1z3z4z2−m2lcgsin(z1)−m1gz3cos(z1)−m1l0z3z2
2+F(t)l0
(m1l2
0 −J0)
z4
−J0
m1
F(t)−J0z3z2
2−gJ0sin(z1)−2m1z3l0z4z2+l0(m1l0+m2lc)gsin(z1)+m1gl0z3cos(z1)
m1l2
0 −J0







(14)
By inspection we get
˙z = Az+BF(t);
where z1 = θ, z2 = ˙θ = ˙z1, z3 = x, z4 = ˙x = ˙z3
z =




θ
˙θ
x
˙x



;A =




0 1 0 0
m2lcg
J∗ 0 m1g
J∗ 0
0 0 0 1
(J∗−m2l0lc)g
J∗ 0 −m1l0g
J∗ 0



;B =
1
J∗




0
−l0
0
Joe
m1



;
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34. Inverted Pendulum Linear-quadratic (LQ) state-feedback regulator
uval = kpf cmdpos −k1enc1pos −k3enc2pos
−k2(enc1pos − pastpos1)−k4(enc2pos − pastpos2) (15)
kpf = hardware gain, k1,2,3,4= lqr constant gains
(a) Simulation (b) Hardware
Figure: Step response of an inverted pendulum by lqr technique
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35. Applications
Practical Applications
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36. Applications Plants and its Application
Figure: Flexible Joint
Figure: Flexible Joint Application
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37. Applications Plants and its Application
Figure: Flexible Link
Figure: Flexible Link Application
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38. Applications Plants and its Application
Figure: Inverted Pendulum in Military
Figure: Inverted Pendulum in Humanoid
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39. Conclusion
Conclusion
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40. Conclusion
1 Model dependent trajectory tracking control for linear and
nonlinear plant is possible by using Extended State Observer
(ESO).
2 If we assumed calibration error in position sensor by adjusting
proper observer gains better estimation of position and other
relevant states of plant can be done.
3 Lumped Uncertainty and Disturbances are well estimated by
ESO in model dependent plant
4 All the aforementioned details are proved mathematically and via
simulation to be stable and also validated experimentally on
various plant. Which inherently proves the robustness.
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41. Conclusion
Publication
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42. Conclusion Published Journal
TITLE - A Novel Approach for Internet Congestion Control Using an
Extended State Observer.
JOURNAL - International Journal of Electronics and Communication
Engineering & Technology
TITLE - Comparative Analysis of Linear and Non-linear Extended
State Observer with Application to Motion Control
JOURNAL - IEEE Conference on Convergence of Technology
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43. Conclusion Communicated Journal
TITLE - Estimating and Compensating Wide Range of Uncertainties in
MagLev by using Extended State Observer.
JOURNAL - Journal of Systems and Control
TITLE - Extended State Observer based Control of Flexible Link
Manipulator in presence of Unknown Payload Dynamics
JOURNAL - Transaction on Institute of Measurement and Control
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44. State of Art
State of Art
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45. State of Art
State of Art
1 SAGE Journal of Systems and Control Engineering
2 IEEE Transactions on Industrial Electronics
3 IEEE Transactions on Control Systems Technology
4 IEEE Transactions on Magnetics
5 American Control Conference
6 IEEE/ASME Transaction on Mechatronics
7 Elsevier Automatica
8 International Symposium on Magnetic Bearings
9 ISA Transactions
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46. State of Art
References I
[ARD08] B. X. S. Alexander, Richard Rarick, and Lili Dong.
A novel application of an extended state observer for high
performance control of nasa’s hss flywheel and fault
detection.
American Control Conference, pages 5216 – 5221, June
2008.
[SL91] Jean Jacques E. Slotine and Weiping Li.
Applied Nonlinear Control.
Prentice Hall, New Jersey, U.S.A, 1st edition, 1991.
[WG03] Weiwen Wang and Zhiqiang Gao.
A comparison study of advanced state observer design
techniques.
In Proceeding of the American Control Conference, pages
4754 – 4759, Denver, Colorado, 2003.
KALIPRASAD A. MAHAPATRO Vishwakarma Institute of Technology, Pune July 26, 2014 44 / 48

47. State of Art
References II
[ZGG12] Qing Zheng, Linda Q. Gao, and Zhiqiang Gao.
On validation of extended state observer through analysis
and experimentation.
Journal of Dynamic Systems, Measurement, and Control,
ASME, 134:024505–1 – 024505–6, March 2012.
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48. Acknowledge
Profound Thanks
Prof. Milind E. Rane
Prof. S. R. Bandewar
Prof. A. M. Chopde
Staff of Department of E&TC Engg for their kind support.
KALIPRASAD A. MAHAPATRO Vishwakarma Institute of Technology, Pune July 26, 2014 46 / 48

49. Acknowledge
Formal Thanks
Matlab
Simulink
LATEX
Template- Beamer
Package- Cambridge-US
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50. Acknowledge
Thanks
THANK YOU
By- KALIPRASAD A. MAHAPATRO
Gr.No.12M042
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