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
 Maintain the stability of an inverted pendulum mounted on a
moving cart which is travelling through a rail of finite length.
 Enhance tracking control of an overhead crane (cart pole
system in its stable equilibrium) with guaranteed anti-swing
operation

4. Modeling of Cart Pole System
F
T f x ,x V

d2
M m 2 x l sin F T
dt

5. Contd.
 State Model of Inverted Pendulum: If the angle of on
Hence Based the
Angular position
pendulum is
ofMost of thein
Pendulum
quite small we
Nonlinearities
can space it those
replace is
nonlinear the
except divide
possible toterms.
friction T are the
Hence total
the we can
functions region
realize a of the
operatingLinear
pendulum small
Model for angle
in two different
x2
zone
angle deviation!!!

6. Difficulties of the Controller Design
 The system Model is quite complicated and nonlinear.
 It is almost impossible to obtain a true model of the real system and if it is
achieved by means of some tedious modeling, the model will be too
complex to design a control algorithm for it.
 The system has got two output, namely the motion of the cart and the
angle of the pendulum. It is a quite complicated design challenge to
reshape the control input in such a manner that can control both output
of the cart pole system simultaneously.

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 construct a control input u which ensures the
regulation of the state variables x(t) and z(t), for all x(0) and z(0).
Equilibrium point: 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 design is 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, αs(x) is termed as stabilizing function.
 System Dynamics in ( x, e) Coordinate:

13. Contd.
 Modified Block Diagram
Feedback Control Law
Backstepping αs
Signal -αs

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

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

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

17. Contd.
 Sontag's Formula:
2 4
V V V
f f g
x x x V
for g 0
u V x
g
x
V
0 for g 0
x
So this control
But this
 Control Law (Sontag’s Formula): For large values
formula leads the
law avoids a
of x, the
complicated of
cancellation
control law
useful
control input
becomes
nonlinearities!
for
u≈sinx
For higher
intermediate
 Control Law (using Backstepping): values ofof x
values x

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

19. Contd.
 IEEE Explore 1990-2003 Backstepping in title
Conference
Paper
Journal
Paper
Ola Harkegard Internal 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:
Nonlinear System
Integrator
If the nonlinear system satisfies certain assumption with z Є R as its
control then
 The CLF
depicts the control input u
 renders the equilibrium point x=0, z=0 is GAS.

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

23. Integrator Backstepping Example
 Stabilization ofResults
Simulation an unstable system
x x2
 xz

z u
 Stabilizing Function:
 Choice of Control law:
The equilibrium point x=0, z=0 is a GAS.

24. Stabilization of Cascaded System
 Stable nonlinear system cascaded with a Linear system
u 
z Az Bu
u
y Cz ∫ 
x f x gxy
 CLF
A, B, C are
 The Control Law:
FPR
 Ensures the Equilibrium (x=0, z=0) is a GAS.

25. Contd.
 Stable nonlinear system cascaded with a Nonlinear system
u 
zz= η(Az+ βBuu
z) (z)
x = ff (x , z )g x (y, z ) y
x x +g x
y y C(z)
= Cz
Feedback Passive
u=K(z)+r(z)v
 CLF SystemFeedback
is a with U(z)
as a +ve Definite
Transformation
Storage Function
Such that the
 Control Law resulting system is
Passive with
Storage Function
U(z)
 Ensures the Equilibrium (x=0,z=0) is GAS.

26. Stabilization with Passivity an Example
 System Dynamics:

x x 1 ez x2 z 4
u

z z 3u z 3u

z 
x x 1 ez x2 z 4
 Feedback Law:
u z2 v
t t
y v d U zt U z0 z6 d
 Storage function: 0 0
 Derivative of Storage Function:
U z z5 z 3v z6 z 4v

27. Contd.
 Control law
u z2 x3
 Simulation Results:
Phase-Plane Portrait
10
5
x2(t)
0
-5
-10
-10 -5 0 5 10
x1(t)
The equilibrium point x=0, z=0 is a GAS.

28. Block Backstepping
 Nonlinear system cascaded with a Linear Dynamic Block
u 
z Az Bu

x f x gxy
y Cz
 Using the feedback transformation
Stable/Unstable
 The State equation of the system becomes Nonlinear system
 Control Law
Minimum Phase
Linear Systemof the A0 are the
Eigen values with
relative degree one function
zeros of the transfer
Zero
Dynamics
 Ensures the equilibrium point x=0, z=0 is GAS.

29. Contd.
 Nonlinear system cascaded with a Nonlinear Dynamic Block
u 
z x, z x, z u

x f x gxy
y C z
 Control Law:
Nonlinear System with relative
degree one
And the zero dynamics
subsystems is globally defined and
it is Input to state stable
 Ensures the equilibrium x=0, z=0 is GAS.

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. Contd.
 Strict Feedback Systems:
Lower Triangular Form
 Control Input:
 CLF

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

33. Contd.
 Block Strict Feedback forms:

x f x g x y1

X 1 F1 x, X 1 G1 x, X 1 y2
y1 C1 X 1

X2 F2 x, X 1 , X 2 G2 x, X 1 , X 2 y3
y2 C2 X 2


Xk Fk x, X 1 , X 2 ,, X k G2 x, X 1 , X 2 ,, X k yk 1
yk Ck X k


X m 1 Fm 1 x, X 1 , X 2 ,, X m 1 Gm 1 x, X 1 , X 2 ,, X m 1 ym
ym 1 Cm 1 X m 1

Xm Fm x, X 1 , X 2 ,, X m Gm x, X 1 , X 2 ,, X m u
ym Cm X m

34. Contd.
 Assumptions:
n
Each K subsystem with state X k and yk ,and input yk 1 satisfies:
 BSF-1: Its relative degree is one uniformly in x, X1 ,, X k 1
 BSF-2: Its zero dynamics subsystem is ISS w.r.to x, X1 ,, X k 1 , yk
 Sub-System Dynamics in transformed Co ordinate:
Ck

yk X k Fk x, X 1 ,, X k Gk x, X 1 ,, X k yk 1
Xk
f k x, y1 , 1 ,, xk , k g k x, y1 , 1 ,, xk , k yk 1
k 1
 k
x, X 1 ,, X k Fk x, X 1 ,, X k Gk x, X 1 ,, X k yk
k 1
i 1 Xi
k
x, X 1 ,, X k Fk x, X 1 ,, X k
Xk
x, X 1 ,, X k 1 , yk , k
x, y1 , 1 ,, yk 1 , k 1 , yk , k

35. Contd.
 The change of Coordinate Results:
Strict Feedback

x x x x
f
 f g x y1y1
g x Form

X 1 F1, x, X,1 λ G1 x, X 1 x2 y

y1 F x y
1 1 G y,
1 1 1 , λ1 y2
y1 C1 X 1

y2 F x, y , λ , y2 , λ2 G2 x, y1 , λ1 , y2 , λ2 y3
 2 1 1
X 2 F2 x, X 1 , X 2 G2 x, X 1 , X 2 y3
 y2 C2 X 2

ym F x, y1 , λ1 , , ym 1 , λm Gk x, y1 , λ1 ,  , ym 1 , λm ym
1 m 1 1 1 1

Xk Fk x, X 1 , X 2 ,, X k G2 x, X 1 , X 2 ,, X k yk
Fm x, y1 , λ1 , , ym , λm Gk x, y1 , λ1 , , ym , λm u
1

ym
yk Ck X k

 
X m 1 x,Fy 1, x, X 1 , X 2 ,, X m 1 Gm 1 x, X 1 , X 2 ,, X m 1 ym
λ1 m λ
1 1 Zero Dynamics
ym 1 Cm 1 X m 1


Xm Fm x, X 1 , X 2 ,, X m Gm x, X 1 , X 2 ,, X m u

λm ym Cm, X1 , λ1 ,  , ym
x ym , λm 1 , ym , λm
1

36. New Research Ideas
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.

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

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
Unstable Zone
Equilibrium
Point
Swinging
Zone

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

43. Contd.
 Choice of Stabilizing Function:
 Choice of second error variable:
 Derivative of z1 and z2
Integral action is introduced to
enhance the controller performance
in steady state operation

44. Contd.
 Choice of CLF:
 Control Input:
Where Integral action reduces the steady
state error of the system.
 Derivative of CLF:

45. Contd.
 List of the controller parameters
Where d1=c1+c2 & d2=c1c2. k1=1, c1=c2=50, c=0.001

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

47. Contd.
 Choice of Stabilizing function:
 Choice of second error variable:
 Derivative of z3 and z4

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

49. Contd.
 List of Controller’s Parameters

50. Contd.
 List of Controller’s Parameters
k2=0.1, d3=c3+c4 and d4=c3c4+1, where c3=c4=20

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

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

53. Contd.
 Angular Velocity of the Inverted Pendulum

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

55. Contd.
 Cart Velocity

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

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

58. Contd.
 Angular Velocity of the Pendulum

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

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

61. Contd.
 Voltage applied on the actuator motor

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

63. Contd.
 Comparison of Cart tracking Performance

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.
 The Backstepping control algorithm has an ability of quickly achieving the
control objectives and an excellent stabilizing ability for inverted pendulum
system suffering with an external impact.
 The use of integral-action in backstepping allows us to deal with an
approximate (less informative and less complex) model of the original
system; as a result it reduces the computation task of the designer, but
offering a controller which is able to provide successful control operation in
spite of the presence of modeling error

65. ANTISWING OPERATION
OF OVERHEAD CRANE

66. Content
 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

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

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

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

70. Contd.
 Choice of Stabilizing Function:
 Choice of second error variable:
 Derivative of z1 and z2
Integral action is introduced to
enhance the controller performance
in steady state operation

71. Contd.
 Choice of CLF:
 Control Input:
Where Integral action reduces the steady
state error of the system.
 Derivative of CLF:

72. Contd.
 List of Controller Parameters
 Where d1=c1+c2 & d2=c1c2. k1=1, c1=c2=50, c=0.001

73. Design of Control Algorithm for Anti-Swinging Zone
 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!!!!!!!

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

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

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

77. Contd.
 Cart Velocity

78. Contd.
 Payload Angular Position
3.15

79. Contd.
 Payload Angular Velocity

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

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

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

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

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:

x x u
 Static Control Law:
Dynamic Control Law
u x c1 x Θ is an unknown
γ is adaptation
constant parameter
gain
Θ ~ an unknown
is
ˆ
Can use a Is the
Oneparameter so it is
 Augmented Lyapunov function: parameter error
impossible to use
certainty
equivalence form of
this expression
control
where θ is replaced
law, containing
by an estimate of θ,
ˆ
unknown parameter

88. Contd.
 Derivative of Augmented Lyapunov function:
 1 ~~

Va 
xx
2 ~ 1~

c1 x x x
 Update law:

ˆ 
~
x x
 Which ensures the negative definiteness of 
Va.
 System dynamics:
~

x c1 x x

~
x x

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

90. Adaptive Backstepping
 Stabilization of 2nd order nonlinear system:

x1 x2 1 x1

x2 2 x u
θ is an
 Stabilizing Function: unknown
x1 c1 x1 x1 parameter. So
s 1
θ should be
replaced by its
 CLF: estimated
1 2 11 2 1 2 2 value.
Vc Vc
x x x1 z x2 sz2
x
2 2 21 2
 Control law:
u c
u x2
c2 x2 s x1
x 1
s s
x 2 x2 1 2 x ˆ x
2 s x1 1 2
x1

91. Contd.
 Error Dynamics:
d z1 c1 1 z1 0 ~
dt z2 1 c2 z2 2 x
 Construction of Augmented Lyapunov Function:
~ 1 2 1 2 1 ~2
Va z , z1 z2
2 2 2
 Derivative of Augmented Lyapunov function:
 Update Law :

a z1 , z2 , ~ ˆ c1 z1 2 zc2 z2
2 2 ~ 
1ˆ
V 2 z2 2

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

93. Adaptive Backstepping Control of Inverted Pendulum
(6.3.5.a)
 Dynamics of the Cart Pole system:
M m  cx
x  ml cos  ml sin 2 u (t )
(I 
ml 2 )θ mgl sin θ mlcos θ
x
 Dynamics of the Pendulum Angle:
 cos  2 sin
1 sec 2 tan 3 ut
Model is being
Where obtained
 State Space Representation: I ml 2 Lagrangian
M1 m Dynamics`
ml

z1 z2
(M m) g

2
g z1 z2=u - k z
3 ml
g z1 1 sec z1 3 cos z1 & k z 2 tan z1 - 2
3 z 2 sin z1

94. Contd.
 Reformed Equation of Control Input :
u 
g ( z1 ) z2 h
 Definition of 1st error variable:
k (z)
e1 - h=
ref g(z)
 Stabilizing Function:
zref c1e1 
ref
 Choice of 2nd error variable:
e2 zref - z2
 Control Lyapunov Function:
1 2 1 2
V2 e1 e2
2 2

95. Contd.
 Derivative of Lyapunov Function:
 ref u
 
V2 e1e1 e2e2 e1 c1e1 e2 e2 c1 c1e1 e2 h
g
 Control Input:
 ˆ
u g z1 1 - c12 e1 c1 c2 e2 ref h
 Augmented Lyapunov Function:
1 2 1 2 1 1 2
Va e1 e2 g2 h
2 2 2 1g 2 2

96. Contd.
 Derivative of the Lyapunov function:
 2 2 g 2  ˆ
ˆ 1 dg } h (e - 1 dh )
Va -c1e1 - c2e2 {e2 ((1 - c1 )e1 (c1 c2 )e2 ref h) - 2
g 1 dt 2 dt
 Parameter Update Law:
ˆ
dg 2  ˆ
1e2 ((1 - c1 )e1 (c1 c2 )e2 ref h)
dt
ˆ
dh
e
2 2
dt

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.
 A continuous Switching function is use to implement the Robustification
Different type of switching
measures : techniques can be used to
prevent the abnormal

ˆ
g 2
c2Robust Adaptive
e2  ˆ ˆ
1e2 1 c1 e1 c1 h
variation of the rate of 1
ref gs g
 Control!!!!!
adaptation
ˆ
h ˆ
2 e2 2 shh
where
0 ˆ
if h h0
0 ˆ
if g g 0
ˆ
h h0
ˆ
g g0 if h 0 ˆ
h 2h0
gs g0 if g 0 ˆ
g 2 g0 hs h0
ˆ
h
ˆ
g
ˆ
if h 2h0
g0 ˆ
if g 2 g 0 h0

99. Simulation Results
 Angular variation of Pendulum

100. Contd.
 Disturbances Signal:

101. Contd.
 Estimation of the Parameter g

102. Contd.
 Parameter Estimation of h with time

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. References
 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 12th 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
 Jhou J. 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.

106. References
 K. J. Astrőm and K. Futura, “Swinging up a pendulum by energy control,”
Preprints 13th 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”.

107. References
 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. oltafi Saber, “Fixed point controllers and stabilization of the cart pole
system and the rotating pendulum,” Proceedings of the 38th 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

108. References
 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

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)

z z z u, y C z , C 0 0, z Rn , u R
Is said to be feedback passive (FP) if there exists a feedback transformation.
u K z r zv (ii)
such that the resulting system, y = C(z) is passive with a storage function U(z)
which is positive definite and radically unbounded:
t
y v d U zt U z0
0
The system of (i) is said to be feedback strictly passive (FSP) if the feedback
transformation of equation (ii) renders it strictly passive:

117. The system of (3.5.35) is said to be feedback strictly passive (FSP) if the
feedback transformation of equation (3.5.36) renders it strictly passive:
t t
y v d U zt U z0 z d
0 0