Backstepping control of cart pole system

Nonlinear and Adaptive Control of Cart Pole System has been discussed in this presentation.

1. Backstepping Control
of
Cart Pole System
Presented by
Shubhobrata Rudra
Master in Control System Engineering
Roll No: M4CTL 10-03
Under the Supervision of
Dr. Ranjit Kumar Barai

2. Content
Objectives of the Research
Modeling of the Physical Systems
Difficulties of the Controller Design
Backstepping Control
Stabilization of Inverted Pendulum
Anti Swing Operation of Overhead Crane
Adaptive Backstepping Control & its application on Inverted Pendulum
Conclusion & Scope of Future Research
References

3. Objective of the Research
Maintainthestabilityofaninvertedpendulummountedona
movingcartwhichistravellingthrougharailoffinitelength.
Enhancetrackingcontrolofanoverheadcrane(cartpole
systeminitsstableequilibrium)withguaranteedanti-swing
operation

4. Modeling of Cart Pole System
FTFlx
dt
d
mM sin
2
2 x,x fT 
V

5. Contd.
State Model of Inverted Pendulum:




Most of the
Nonlinearities
except the
friction T are the
functions of the
pendulum angle
x
2
If the angle of the
pendulum is
quite small we
can replace those
nonlinear terms.
Hence we can
realize a Linear
Modelfor small
angle deviation!!!
Hence Based on
Angular position
of Pendulum in
space it is
possible to divide
the total
operating region
in two different
zone

6. Difficulties of the Controller Design
ThesystemModelisquitecomplicatedandnonlinear.
Itisalmostimpossibletoobtainatruemodeloftherealsystemandifitis
achievedbymeansofsometediousmodeling,themodelwillbetoo
complextodesignacontrolalgorithmforit.
Thesystemhasgottwooutput,namelythemotionofthecartandthe
angleofthependulum.Itisaquitecomplicateddesignchallengeto
reshapethecontrolinputinsuchamannerthatcancontrolbothoutput
ofthecartpolesystemsimultaneously.

7. BACKSTEPPING
CONTROL

8. CONTENT
What is Backstepping?
Why Backstepping?
Different Cases of Stabilization Achieved by Backstepping
Backstepping: A Recursive Control Design Algorithm
New Research Ideas

9. What is Backstepping?
Stabilization Problem of Dynamical System
Design objective is to constructa control input u which ensures the
regulation of the state variables x(t)and z(t),for all x(0) and z(0).
Equilibriumpoint:x=0,z=0
Design objective can be achieved by making the above mentioned
equilibrium a GAS.

10. Contd.
Block Diagram of the system:

11. Contd.
First step of the designis to construct a control input for the scalar
subsystem
z can be considered as a control input to the scalar subsystem
Construction of CLF for the scalar subsystem
Control Law:
But z is only a state variable, it is not the control input.

12. Contd.
Only one can conclude the desired value of z as
Definition of Error variable e:
z is termed as the Virtual Control
Desired Value of z, α
s(x) is termed as stabilizing function.
System Dynamics in ( x, e) Coordinate:

13. Modified Block Diagram
Contd.
Feedback Control Law
α
sBackstepping
Signal -α
s

14. So the signal α
s(x) serve the purpose of feedback control law inside the block
and “backstep” -α
s(x) through an integrator.
Contd.
Feedback loop
with +α
s(x)
Backstepping of Signal -α
s(x)
Through integrator

15. Construction of CLF for the overall 2
nd
order system:
Derivative of V
a
A simple choice of Control Input u is:
With this control input derivative of CLF becomes:
Contd.

16. Consider the scalar nonlinear system
Control Law( using Feedback Linearization):
Resultant System:
EduradoD. Sontag Proposed a formula to avoid the Cancellationof these
useful nonlinearities.
Why Backstepping?
is it essential to
cancel out the
term ?
Not at
all!!!!
This is an Useful
Nonlinearity, it
has an Stabilizing
effect on the
system.

17. Sontag's Formula:
Control Law (Sontag’s Formula):
Control Law (using Backstepping):
Contd.
For large values
of x, the
control law
becomes
u≈sinx
So this control
law avoids the
cancellation of
useful
nonlinearities!
For higher
values of x
But this
formula leads a
complicated
control input
for
intermediate
values of x
0 0
0
42
g
x
V
for
g
x
V
for
g
x
V
g
x
V
f
x
V
f
x
V
u

18. Simulation Results: Stabilization of the Nonlinear Scalar plant
Contd.
Variation of x with time
Feedback
Linearization
Sontag’s
Formula
Backstepping
Control Law
Feedback Linearization
***Sontag’s Formula
+++Backstepping Control law
Control Effort variation with time

19. Contd.
IEEE Explore 1990-2003 Backsteppingin title
Conference
Paper
Journal
Paper
Ola HarkegardInternal seminar on Backstepping January 27, 2005

20. Different Cases of Stabilization
Achieved by Backstepping
Integrator Backstepping
Nonlinear Systems Augmented by a Chain of Integrator
Stable Nonlinear System Cascaded with a Dynamic System
Input Subsystem is a Linear System
Input Subsystem is a Nonlinear System
Nonlinear System connected with a Dynamic Block
Dynamic block connected with the system is a linear one
Dynamic block connected with the system is a Nonlinear one

21. Integrator Backstepping
Theorem of Integrator Backstepping:
If the nonlinear system satisfies certain assumption with z ЄR as its
control then
The CLF
depicts the control input u
 renders the equilibrium point x=0, z=0 is GAS.

Nonlinear System
Integrator

22. Chain of Integrator
Chain of integrator:
CLF
Nonlinear
System∫ ∫∫
K th
integrator

23. Stabilization of an unstable system
Stabilizing Function:
Choice of Control law:
Integrator Backstepping Exampleuz
xzxx


2
Simulation Results
The equilibrium point x=0, z=0 is a GAS.

24. u
Stabilization of Cascaded System
Stable nonlinear system cascaded with a Linear system
CLF
The Control Law:
Ensures the Equilibrium (x=0, z=0) is a GAS.yxgxfx
∫Czy
BuAzz


u
A, B, C are
FPR

25. Stable nonlinear system cascaded with a Nonlinear system
CLF
Control Law
Ensures the Equilibrium (x=0,z=0) is GAS.yxgxfx Czy
BuAzz


u()()yzxgzxfx ,+,= ()()
()zCy
uzβzηz
=
+=
Feedback Passive
System with U(z)
as a +ve Definite
Storage Function
u=K(z)+r(z)v
is a Feedback
Transformation
Such that the
resulting system is
Passivewith
Storage Function
U(z)
Contd.

26. System Dynamics:
Feedback Law:
Storage function:
Derivative of Storage Function:
Stabilization with Passivity an Example42
1 zxexx
z
 uzz
3
 42
1 zxexx
z
 uzz
3

uvzu
2 vzzvzzzU
4635 dzzUtzUdvy
tt
0
6
0
0

27. Control law
Simulation Results:
The equilibrium point x=0, z=0 is a GAS.
Contd. 32
xzu

-10 -5 0 5 10
-10
-5
0
5
10
x1(t)
x2(t)
Phase-Plane Portrait

28. Block Backstepping
Nonlinear system cascaded with a Linear Dynamic Block
Using the feedback transformation
The State equation of the system becomes
Control Law
Ensures the equilibrium point x=0, z=0 is GAS.yxgxfx Czy
BuAzz


u


Eigen values of the are the
zeros of the transfer function A0


Zero
Dynamics
Stable/Unstable
Nonlinear system
Minimum Phase
Linear System with
relative degree one

29. Nonlinear system cascaded with a Nonlinear Dynamic Block
Control Law:
Ensures the equilibrium x=0, z=0 is GAS.
Contd. yxgxfx zCy
uzxzxz

,,
u
Nonlinear System with relative
degree one
And the zero dynamics
subsystems is globally defined and
it is Input to state stable

30. Backstepping: A Recursive Control Design Algorithm
Backstepping Control law is a Constructive Nonlinear Design Algorithm
It is a Recursive control design algorithm.
It is applicable for the class of Systems which can be represented by
means of a lower triangular form.
In order of increasing complexity these type of nonlinear system can be
classified as
Strict Feedback System
Semi –Strict Feedback System
Block Strict Feedback Systems

31. Strict Feedback Systems:
Control Input:
CLF
Contd.








Lower Triangular Form

32. Semi Strict Feedback Systems:
CLF:
Control Input:
Contd.








Lower Triangular Form

33. Block Strict Feedback forms:
Contd. mmm
mmmmm
mmm
mmmmmm
kkk
kkkkk
XCy
uXXXxGXXXxFX
XCy
yXXXxGXXXxFX
XCy
yXXXxGXXXxFX
XCy
yXXxGXXxFX
XCy
yXxGXxFX
yxgxfx
,,,,,,,,
,,,,,,,,

,,,,,,,,

,,,,
,,
2121
111
121112111
121221
222
32122122
111
211111
1












34. Assumptions:
Each K subsystem with state and ,and input satisfies:
BSF-1: Its relative degree is one uniformly in
BSF-2: Its zero dynamics subsystem is ISS w.r.to
Sub-System Dynamics in transformed Co ordinate:
Contd. n
k
X k
y 1k
y 11
,,,
k
XXx kk
yXXx ,,,,
11
 11111
111
,,,,,,,,,,
,,,,,,
kkkkkkk
kkkkkk
k
k
k
yxyxgxyxf
yXXxGXXxFX
X
C
y

 kkk
kkk
k
k
kkkkkk
k
i i
k
k
yXXx
XXxFXXx
X
yXXxGXXxFXXx
X
,,,,,
,,,,,,
,,,,,,,,,
11
11
1111
1
1



 kkkk
yyyx ,,,,,,,
1111


35. The change of Coordinate Results:
Contd. mmm
mmmmm
mmm
mmmmmm
kkk
kkkkk
XCy
uXXXxGXXXxFX
XCy
yXXXxGXXXxFX
XCy
yXXXxGXXXxFX
XCy
yXXxGXXxFX
XCy
yXxGXxFX
yxgxfx
,,,,,,,,
,,,,,,,,

,,,,,,,,

,,,,
,,
2121
111
121112111
121221
222
32122122
111
211111
1










 mmmmm
mmkmmmm
mmmkmmmm
λyλyλyxλ
λyxλ
uλyλyxGλyλyxFy
yλyλyxGλyλyxFy
yλyλyxGλyλyxFy
yλyxGλyxFy
yxgxfx
,,,,,,,

,,
,,,,,,,,,,
,,,,,,,,,,

,,,,,,,,
,,,,
1111
111
1111
11111111111
322112221122
21111111
1










Strict Feedback
Form
Zero Dynamics

36. In 1993, I. Kanellakopoulos and P. T. Krein introduced the use of Integral
action along with the Backstepping control algorithm, which considerably
improves the steady-state controller performance [2].
It is possible to represent a complex nonlinear system as a combination of
two nonlinear subsystem, while each subsystem is in Block Strict Feedback
form. And if the zero dynamics of input subsystem is Input to State Stable
(ISS). Then it is possible to stabilize the system using Backstepping algorithm.
Integral Action along with Block Backstepping algorithm may gives a better
transient as well as steady state performance of the controller for complex
nonlinear plant.
New Research Ideas

37. STABILIZATION OF
INVERTED
PENDULUM

38. Content
Control Objective
Two Zone Control Theory of Inverted Pendulum
Design of Control Algorithm for Stabilization zone
Design of Control Algorithm for Swinging Zone
Schematic Diagram of Controller
Results of Real Time Experiment
Comparative Study and Conclusion

39. Control Objective
Design a control system
that keeps the pendulum
balanced and tracks the
cart to a commanded
position!!!
Maintain the Stability of
the Inverted Pendulum
when it is suffering with
external disturbances.

40. Two Zone Control Theory
Most of the nonlinearities (present in the state model of Inverted Pendulum)
are the function of pendulum angle in space.
Stabilization
Zone
Swinging
Zone
Unstable
Equilibrium
Point

41. Features of Two Zone Control Theory
Two independent controller can be used for two different zones.
One can use a linearize model of Inverted Pendulum in Stabilization zone
Linear model of the pendulum facilitates the design of more complex
control algorithm, which enhance the steady state performance of the
inverted pendulum.
While a less complicated algorithm can be used for the swinging zone
operation.
Designer can modify the algorithm independently for each zone and get a
optimal combination of controller for swinging and stabilization zone.

42. Linearize model of Inverted Pendulum
Choice of Control Variable::
Design of Control Algorithm for Stabilization Zone
The state model
of the system
not allows the
designer to
implement
backstepping
algorithm on it
It is possible to
represent the
system as a
combination of
two dynamic
block each of
them in block
strict feedback
system

43. Contd.
Choice of Stabilizing Function:
Choice of second error variable:
Derivative of z
1 and z
2
Integral action is introduced to
enhance the controller performance
in steady state operation

44. Choice of CLF:
Control Input:
Where
Derivative of CLF:
Contd.


Integral action reduces the steady
state error of the system.

45. List of the controller parameters
Where d
1=c
1+c
2& d
2=c
1c
2. k
1=1, c
1=c
2=50, c=0.001
Contd.

46. State model of the Inverted Pendulum:
Choice of Control variable:
Design of Control Algorithm for Swinging Zone

47. Choice of Stabilizing function:
Choice of second error variable:
Derivative of z
3 and z
4
Contd.

48. Choice of CLF:
Control Input:
Derivative of CLF:
Contd.

49. List of Controller’s Parameters
Contd.

50. List of Controller’s Parameters
k
2=0.1, d
3=c
3+c
4and d
4=c
3c
4+1, where c
3=c
4=20
Contd.

51. Schematic Diagram of Controller
Reference
Input
Linear
Backstepping
Controller
Nonlinear
Backstepping
Controller
Controller for Stabilization Zone
Controller for Swinging Zone
Inverted
Pendulum
Switch
ing
Mecha
nism
Control
Input
Switching
Law

52. Results of Real Time Experiment
Angle of the Inverted Pendulum
Pendulum reach its
stable position
within 4 seconds

53. Angular Velocity of the Inverted Pendulum
Contd.

54. Cart Movement with time
Contd.
The cart is able to
track the reference
trajectory within 15
seconds

55. Cart Velocity
Contd.

56. Voltage applied on the actuator motor
Contd.
Moderate Variation
of voltage reduces
the stress on
actuator motor

57. Angle of the Inverted Pendulum when it is suffering with external impact
Contd.
Pendulum regain its inverted position
within 3 seconds

58. Angular Velocity of the Pendulum
Contd.

59. Cart Position of the pendulum (suffering with an external impact)
Contd.
Cart Regain its
Desired trajectory
within 12 seconds

60. Cart Position of the pendulum (suffering with an external impact)
Contd.

61. Voltage applied on the actuator motor
Contd.

62. Comparative Study and Conclusions
Comparative study on the Pendulum angular position in space

63. Comparison of Cart tracking Performance
Contd.

64. Conclusion
Backstepping controller along with Integral action enhance the performance
of the steady state operation of the controller.
Nonlinear Backstepping controller ensure the enhance swing operation of
the Inverted Pendulum.
TheBacksteppingcontrolalgorithmhasanabilityofquicklyachievingthe
controlobjectivesandanexcellentstabilizingabilityforinvertedpendulum
systemsufferingwithanexternalimpact.
Theuseofintegral-actioninbacksteppingallowsustodealwithan
approximate(lessinformativeandlesscomplex)modeloftheoriginal
system;asaresultitreducesthecomputationtaskofthedesigner,but
offeringacontrollerwhichisabletoprovidesuccessfulcontroloperationin
spiteofthepresenceofmodelingerror

65. ANTISWING OPERATION
OF OVERHEAD CRANE

66. Control Objective
Two Zone Control Theory of Over Head Crane
Design of Control Algorithm for Stabile Tracking zone
Design of Control Algorithm for Anti-Swinging Zone
Schematic Diagram of Controller
Results of Real Time Experiment
Comparative Study and Conclusion
Content

67. Control Objective
Proper tracking of The
Cart Motion along a
reference/desired
trajectory.
Proper Antiswing
operation of pay load
during travel

68. Most of the nonlinearities (present in the state model of Overhead Crane)
are the function of payload angle in space.
Two Zone Control Theory
Stable Tracking
Zone
Anti Swing
Zone

69. Linearize model of Overhead Crane
Choice of Control Variable:
Design of Control Algorithm for Stable Tracking Zone
The Primary
objective of
design is to
control the
motion of the
cart along with
a reference
trajectory

70. Contd.
Choice of Stabilizing Function:
Choice of second error variable:
Derivative of z
1 and z
2
Integral action is introduced to
enhance the controller performance
in steady state operation

71. Contd.
Choice of CLF:
Control Input:
Where
Derivative of CLF:


Integral action reduces the steady
state error of the system.

72. List of Controller Parameters
Where d
1=c
1+c
2& d
2=c
1c
2. k
1=1, c
1=c
2=50, c=0.001
Contd.

73. In case of Anti swing operation the primary concern of the controller is to
reduce the oscillation of the pay load, & brings it back inside the stable region.
In case of Inverted Pendulum the controller tries to launch the pendulum
inside its stabilization zone.
So in case of Anti-swing operation the same controller which has been used
for Swinging operation can be utilized!!!!!!!
Design of Control Algorithm for Anti-Swinging Zone

74. Contd.
Same Control Algorithm is
being used to serve the
opposite purpose!!!
Swinging
Zone
Anti Swing
Zone
Inverted Pendulum Overhead Crane

75. Schematic Diagram of Controller
Reference
Input
Linear
Backstepping
Controller
Nonlinear
Backstepping
Controller
Controller for Stable Tracking Zone
Controller for Anti Swing Zone
Inverted
Pendulum
Switch
ing
Mecha
nism
Control
Input
Switching
Law
Overhead
Crane

76. Motion of the Cart
Results of Real Time Experiment
Steady state Tracking error reduces with time

77. Cart Velocity
Contd.

78. Payload Angular Position
Contd.
3.15

79. Payload Angular Velocity
Contd.

80. Cart Motion of the pendulum when suffering with an external impact
Contd.
The cart is able to
track the reference
trajectory within 15
seconds

81. Cart Velocity when suffering with an external impact
Contd.

82. Angle of the Payload when suffering with an external impact
Contd.
The angle of the
payload reduces
within 15 seconds

83. Angular Velocity of the Payload when suffering with an external impact
Contd.

84. Conclusion
Backstepping controller along with Integral action enhance the performance
of the steady state operation of the controller.
Nonlinear Backstepping controller ensures the proper anti-swing operation
of overhead crane. Here one can reuse the nonlinear controller which has
been used for swinging purpose of inverted pendulum.
Though the total control scheme is little bit complex than that of classical
PID controller. But in case of classical PID control it is not able to address
the problem of anti-swing operation properly.

85. Adaptive Backstepping
Control
and its Application on
Inverted Pendulum

86. Content
Adaptation as Dynamic Feedback
Adaptive Integrator Backstepping
Stabilization of an Inverted Pendulum
Robust Adaptive Backstepping
Simulation Results
Conclusion

87. Adaptation as Dynamic Feedback
Stabilization problem of a nonlinear system:
Static Control Law:
Augmented Lyapunov function:uxx
Θis an unknown
constant parameterxcxu
1
Θis an unknown
parameter so it is
impossible to use
this expression of
control
law, containing
unknown parameter
One Can use a
certainty
equivalence form
where θis replaced
by an estimate of θ, ˆ
Dynamic Control Law
γis adaptation
gain
Is the
parameter errorˆ
~

88. Derivative of Augmented Lyapunov function:
Update law:
Which ensures the negative definiteness of .
System dynamics:
Contd. 


~1~

~~1
2
1
xxxc
xxV
a xx
~
ˆ aV xx
xxcx


~
~
1

89. Block diagram of the Closed loop Adaptive system
Contd.

90. Adaptive Backstepping
Stabilization of 2
nd
order nonlinear system:
Stabilizing Function:
CLF:
Control law:uxx
xxx
22
1121

 11111
xxcx
s 2
2
2
1
2
1
2
1
xxxxV
sc xx
x
xxcu
s
s 212
1
122
θis an
unknown
parameter. So
θshould be
replaced by its
estimated
value. xx
x
xxcu
s
s 212
1
122
ˆ 2
2
2
1
2
1
2
1
zzxV
c

91. Error Dynamics:
Construction of Augmented Lyapunov Function:
Derivative of Augmented Lyapunov function:
Update Law :
Contd. ~

0

1
1
22
1
2
1
2
1
xz
z
c
c
z
z
dt
d 22
2
2
1
~
2
1
2
1
2
1~
, zzzV
a 
 ˆ
1~~
,,
22
2
22
2
1121 zzczczzV
a 22
ˆ
z


92. Block diagram of the closed loop Adaptive System:
Contd.

93. Adaptive Backstepping Control of Inverted Pendulum
Dynamics of the Cart Pole system:
Dynamics of the Pendulum Angle:
Where
State Space Representation:)(sincos tumlmlxcxmM
2 θxmlθmglθ)ml(I cossin 

2
Model is being
obtained
Lagrangian
Dynamics`tusincostansec
2
321
 gmM )(
2 ml
3 ml
mlI
mM
2
1 21
zz zk=uzzg -
21

(6.3.5.a)&
13111
zzzg cossec 1
2
2312 zzzzk sin-tan

94. Contd.
Reformed Equation of Control Input :
Definition of 1
st
error variable:
Stabilizing Function:
Choice of 2
nd
error variable:
Control Lyapunov Function:hzzgu
21
)( ()
()zg
zk
h= refe-
1 refrefecz

11 22 -zze
ref 2
2
2
12
2
1
2
1
eeV

95. Derivative of Lyapunov Function:
Control Input:
Augmented Lyapunov Function:h
g
u
eeceeeceeeeeV
ref

21112211122112 c
Contd. hecceczgu
ref
ˆ
-


2211
2
11
1 2
2
2
1
2
2
2
1
2
1
2
1
2
1
2
1
hg
g
eeV
a

96. Derivative of the Lyapunov function:
Parameter Update Law:
Contd. )-(}
ˆ
-)
ˆ
)()-(({-
dt
dh
eh
dt
gd
heccece
g
g
ececV
a
2
2
1
ref2211
2
12
2
22
2
11
1

1
1-  )
ˆ
)()-((
ˆ
heccece
dt
gd
ref2211
2
1211  22
e
dt
hd
ˆ

97. Robust Adaptive Backstepping
Difficulties for the designer of Adaptive Control
Mathematical Models are not free from Unmodeled Dynamics
Parameter Drift may occur in the time of real world
implementation
Noises are unavoidable in real time application.
Bounded disturbances may cause a high rate of adaptation
which leads to an unstable/undesirable system performance.

98. Contd.
Robust Adaptive
Control!!!!!
Different type of switching
techniques can be used to
prevent the abnormal
variation of the rate of
adaptation
A continuous Switching function is use to implement the Robustification
measures :
where 0g0
00
0
0
0
2g if
2g if
g if 0
g
gg
g
gg
g
ggs
ˆ
ˆ
ˆ
ˆ
ˆ ghecceceg
gsref
ˆˆˆ
12211
2
121
1
 heh
sh
ˆˆ
222
 0h0
00
0
0
0
2h if
2h if
h if 0
h
hh
h
hh
h
hhs
ˆ
ˆ
ˆ
ˆ
ˆ

99. Simulation Results
Angular variation of Pendulum

100. Disturbances Signal:
Contd.

101. Estimation of the Parameter g
Contd.

102. Parameter Estimation of h with time
Contd.

103. Conclusion & Scope of Future Research
This research presents an idea of using integral action along
with the backstepping control algorithms and achieves quite
satisfactory results in real time experiment.
One can employ Adaptive Block Backstepping algorithm to
obtain a more generalize controller for the cart pole system
A Robust Adaptive Block Backstepping control algorithm can
be designed to address the problem of motion control of a
cart pole system on inclined rail.

104. Questions
Polygonia interrogationis known as Question Mark

105. M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive
Control Design, New York; Wiley Interscience, 1995.
I. Kanellakopoulos and P. T. Krein, “Integral-action nonlinear control of
induction motors,” Proceedings of the 12
th
IFAC World Congress, pp. 251-
254, Sydney, Australia, July 1993.
H. K. Khalil, Nonlinear Systems, Prentice Hall, 1996.
J.J.E Slotine and W. LI, Applied Nonlinear Control, Prentice Hall, 1991
JhouJ. and Wen. C, Adaptive Backstepping Control of Uncertain
Systems, Springer-Verlag, Berlin Heidelberg, 2008.
A Isidori, Nonlinear control Systems, Second Edition, Berlin: Springer
Verlag, 1989.
References

106. K. J. Astrőmand K. Futura, “Swinging up a pendulum by energy control,”
Preprints 13
th
IFAC World Congress, pp: 37-42, 1996.
Furuta, K.: “Control of pendulum: From super mechano-system to human
adaptive mechatronics,” Proceedings of 42th IEEE Conference on Decision
and Control, pp. 1498–1507 (2003)
Angeli, D.: “Almost global stabilization of the inverted pendulum via
continuous state feedback,” Automatica, vol: 37 issue 7, pp 1103–1108
2001.
Aström, K.J., Furuta, K.: “Swing up a pendulum by energy control,”
Automatica, Vol: 36, issue 2, pp 287–295, 2000
Chung, C.C., Hauser, J.: “Nonlinear control of a swinging pendulum”.
AutomaticaVol: 36, pp 287–295 (2000)
References

107. Gordillo, F., Aracil, J.: “A new controller for the inverted pendulum on a
cart,”. Int. J. Robust Nonlinear Control Vol: 18, pp 1607–1621, 2008
S. J. Huang and C. L. Huang, “Control of an inverted pendulum using grey
prediction model,” IEEE Transaction on Industry Applications, Vol: 36 Issue:
2, pp 452-458, 2000
R. oltafiSaber, “Fixed point controllers and stabilization of the cart pole
system and the rotating pendulum,” Proceedings of the 38
th
IEEE
Conference on Decision and Control, Vol: 2, pp 1174-1181, 1999.
Q. Wei, et al, “Nonlinear controller for an inverted pendulum having
restricted travel,” Automatica,vol. 31, no 6, pp 841-850, 1995
Ebrahim. A and Murphy, G.V, “Adaptive Backstepping Controller Design of
an inverted Pendulum,” IEEE Proceedings of the Thirty-Seventh Symposium
on System Theory, pp. 172-174, 2005.
References

108. Lee, H.-H., 1998, “Modeling and Control of a Three-Dimensional Overhead
Crane,” ASME J. Dyn. Syst., Meas., Control, 120, pp. 471–476.
Kiss, B., Levine, J., and Mullhaupt, P., 2000, “A Simple Output Feedback PD
Controller for Nonlinear Cranes,” Proc. of the 39th IEEE Conf. on Decision
and Control, Sydney, Australia, pp. 5097–5101
Yang, Y., Zergeroglu, E., Dixon, W., and Dawson, D., 2001, “Nonlinear
Coupling Control Laws for an Overhead Crane System,” Proc. of the 2001
IEEE Conf. on Control Applications, Mexico City, Mexico, pp. 639–644.
Joaquin Collado, Rogelio Lozano, Isabelle Fantoni, “Control of convey-
crane based on passivity,” Proceedings of the American Control Conference
Chicago, Illinois, pp 1260-1264 June 2000
References

109. Thank you

113. Taken from Feedback Manual of Inverted Pendulum

114. Taken from Feedback Manual of Inverted Pendulum

115. Feedback Positive Real
•The triple (A,B,C) is feedback positive real (FPR)if there
exist a linear feedback transformation u = Kz + v such that
the following two conditions hold good
•A + BK is Hurwitz
•And there are matrices P > 0, Q ≥ 0 which satisfy
A sufficient condition for FPR is that there exists a gain row
vector K such that A + BK is Hurwitz, in other words the
transfer function is appositive real one , and the pair
(A + BK, C) is observable.

116. Passivity
The system
(i)
Is said to be feedback passive (FP) if there exists a feedback transformation.
(ii)
suchthattheresultingsystem,y=C(z)ispassivewithastoragefunctionU(z)
whichispositivedefiniteandradicallyunbounded:
Thesystemof(i)issaidtobefeedbackstrictlypassive(FSP)ifthefeedback
transformationofequation(ii)rendersitstrictlypassive:RuRzzCyuzzz
n
, 0,0C , , vzrzKu 0
0
zUtzUdvy
t

117. Thesystemof(3.5.35)issaidtobefeedbackstrictlypassive(FSP)ifthe
feedbacktransformationofequation(3.5.36)rendersitstrictlypassive:tt
dzzUtzUdvy
00
0