Post on 05-Apr-2018
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Backstepping Controlof
Cart Pole System
Presented by
Master in Control System Engineering
Roll No: M4CTL 10-03
Under the Supervision of
Dr. Ranjit Kumar Barai
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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
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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
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Contd.
State Model of Inverted Pendulum:
Most of the
Nonlinearities
except the
friction T are the
functions of the
pendulum angle
x2
If the angle of the
pendulum isquite small we
can replace those
nonlinear terms.
Hence we can
realize a LinearModel for small
angle deviation!!!
Hence Based on
Angular positionof Pendulum in
space it is
possible to divide
the total
operating region
in two different
zone
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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 theangle 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.
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BACKSTEPPING
CONTROL
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CONTENT
What is Backstepping?
Why Backstepping?
Different Cases of Stabilization Achieved by Backstepping
Backstepping: A Recursive Control Design Algorithm
New Research Ideas
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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.
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Contd.
Block Diagram of the system:
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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.
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Modified Block Diagram
Contd.
Feedback Control Law
αs Backstepping
Signal -αs
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So the signal αs(x) serve the purpose of feedback control law inside the block
and “backstep” -αs(x) through an integrator.
Contd.
Feedback loop
with + αs(x)Backstepping of Signal -αs(x)
Through integrator
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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:
Contd.
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Consider the scalar nonlinear system
Control Law( using Feedback Linearization):
Resultant System:
Edurado D. Sontag Proposed a formula to avoid the Cancellation of these
useful nonlinearities.
Why Backstepping?
is it essential to
cancel out the
term ?
Not atall!!!!This is an Useful
Nonlinearity, it
has an Stabilizing
effect on the
system.
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Sontag's Formula:
Control Law (Sontag’s Formula):
Control Law (using Backstepping):
Contd.
For large values
of x, the
control law
becomesu≈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 xV for
g x
V for
g x
V
g x
V f
x
V f
x
V
u
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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
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Contd.
IEEE Explore 1990-2003 Backstepping in title
Conference
Paper
Journal
Paper
Ola Harkegard Internal seminar on Backstepping January 27, 2005
ff f
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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
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Integrator Backstepping
Theorem of Integrator Backstepping:
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.
Nonlinear System
Integrator
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Chain of Integrator
Chain of integrator:
CLF
Nonlinear
System∫ ∫ ∫ K th
integrator
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Stabilization of an unstable system
Stabilizing Function:
Choice of Control law:
Integrator Backstepping Example
u z
xz x x
2
Simulation Results
The equilibrium point x=0, z=0 is a GAS.
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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.
y xg x f x ∫ Cz y
Bu Az z
u
A, B, C are
FPR
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Stable nonlinear system cascaded with a Nonlinear system
CLF
Control Law
Ensures the Equilibrium (x=0,z=0) is GAS.
y xg x f x Cz y
Bu Az z
u y z xg z x f x ,+,=( ) ( )
( ) zC y
u z β zη z
=
+=
Feedback PassiveSystem with U(z)
as a +ve Definite
Storage Function
u=K(z)+r(z)vis a Feedback
Transformation
Such that the
resulting system is
Passive withStorage Function
U(z)
Contd.
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System Dynamics:
Feedback Law:
Storage function:
Derivative of Storage Function:
Stabilization with Passivity an Example
421 z xe x x z
u z z 3421 z xe x x z u z z 3
u
v zu 2
v z z v z z z U 4 6 3 5
d z zU t zU d v y
t t
0
6
0
0
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Bl k B k t i
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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.
y xg x f x Cz y
Bu Az z
u
Eigen values of the are thezeros of the transfer function
A0
Zero
Dynamics
Stable/Unstable
Nonlinear system
Minimum Phase
Linear System withrelative degree one
d
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Nonlinear system cascaded with a Nonlinear Dynamic Block
Control Law:
Ensures the equilibrium x=0, z=0 is GAS.
Contd.
y xg x f x
zC y
u z x z x z
,, u
Nonlinear System with relative
degree one
And the zero dynamicssubsystems is globally defined and
it is Input to state stable
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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
C d
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Strict Feedback Systems:
Control Input:
CLF
Contd.
Lower Triangular Form
C d
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Semi Strict Feedback Systems:
CLF:
Control Input:
Contd.
Lower Triangular Form
C td
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Block Strict Feedback forms:
Contd.
mmm
mmmmm
mmm
mmmmmm
k k k
k k k k k
X C y
u X X X xG X X X xF X
X C y
y X X X xG X X X xF X
X C y
y X X X xG X X X xF X
X C y
y X X xG X X xF X
X C y
y X xG X xF X
y xg x f x
,,,,,,,,
,,,,,,,,
,,,,,,,,
,,,,
,,
2121
111
121112111
121221
222
32122122
111
211111
1
C td
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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.
nk X k y 1k y
11 ,,, k X X x
k k y X X x ,,,, 11
11111
111
,,,,,,,,,,
,,,,,,
k k k k k k k
k k k k k k
k
k k
y x y xg x y x f
y X X xG X X xF X X
C y
k k k
k k k
k
k
k k k k k k
k
i i
k k
y X X x
X X xF X X x X
y X X xG X X xF X X x X
,,,,,
,,,,,,
,,,,,,,,,
11
11
1111
1
1
k k k k y y y x ,,,,,,,
1111
C td
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The change of Coordinate Results:
Contd.
mmm
mmmmm
mmm
mmmmmm
k k k
k k k k k
X C y
u X X X xG X X X xF X
X C y
y X X X xG X X X xF X
X C y
y X X X xG X X X xF X
X C y
y X X xG X X xF X
X C y
y X xG X xF X
y xg x f x
,,,,,,,,
,,,,,,,,
,,,,,,,,
,,,,
,,
2121
111
121112111
121221
222
32122122
111
211111
1
mmmmm
mmk mmmm
mmmk mmmm
λ y λ y λ y x λ
λ y x λ
u λ y λ y xG λ y λ y xF y
y λ y λ y xG λ y λ y xF y
y λ y λ y xG λ y λ y xF y
y λ y xG λ y xF y
y xg x f x
,,,,,,,
,,
,,,,,,,,,,
,,,,,,,,,,
,,,,,,,,
,,,,
1111
111
1111
11111111111
322112221122
21111111
1
Strict Feedback
Form
Zero Dynamics
h d
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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
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STABILIZATION OF
INVERTEDPENDULUM
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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
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Control Objective
Design a control systemthat 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.
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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
F t f T Z C t l Th
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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.
D i f C l Al i h f S bili i Z
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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
Contd
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Contd.
Choice of Stabilizing Function:
Choice of second error variable:
Derivative of z1 and z2
Integral action is introduced toenhance the controller performance
in steady state operation
Contd
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Choice of CLF:
Control Input:
Where
Derivative of CLF:
Contd.
Integral action reduces the steady
state error of the system.
Contd
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List of the controller parameters
Where d 1=c1+c2 & d 2=c1c2. k1=1, c1=c2=50, c=0.001
Contd.
D i f C t l Al ith f S i i Z
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State model of the Inverted Pendulum:
Choice of Control variable:
Design of Control Algorithm for Swinging Zone
Contd
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Choice of Stabilizing function:
Choice of second error variable:
Derivative of z3 and z4
Contd.
Contd
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Choice of CLF:
Control Input:
Derivative of CLF:
Contd.
Contd
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List of Controller’s Parameters
Contd.
Contd
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List of Controller’s Parameters
k2=0.1, d3=c3+c4 and d4=c3c4+1, where c3=c4=20
Contd.
Schematic Diagram of Controller
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Schematic Diagram of Controller
Reference
Input
Linear
Backstepping
Controller
Nonlinear
Backstepping
Controller
Controller for Stabilization Zone
Controller for Swinging Zone
Inverted
Pendulum
Switching
Mecha
nism
ControlInput
Switching
Law
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Results of Real Time Experiment
Angle of the Inverted Pendulum
Pendulum reach its
stable positionwithin 4 seconds
Contd
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Angular Velocity of the Inverted Pendulum
Contd.
Contd
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Cart Movement with time
Contd.
The cart is able to
track the reference
trajectory within 15seconds
Contd
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Cart Velocity
Contd.
Contd. M d t V i ti
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Voltage applied on the actuator motor
Contd. Moderate Variation
of voltage reduces
the stress on
actuator motor
Contd.
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Angle of the Inverted Pendulum when it is suffering with external impact
Contd.
Pendulum regain its inverted position
within 3 seconds
Contd.
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Angular Velocity of the Pendulum
Contd.
Contd.
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Cart Position of the pendulum (suffering with an external impact)
Contd.
Cart Regain its
Desired trajectory
within 12 seconds
Contd.
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Cart Position of the pendulum (suffering with an external impact)
Contd.
Contd.
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Voltage applied on the actuator motor
C ti St d d C l i
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Comparative Study and Conclusions
Comparative study on the Pendulum angular position in space
Contd.
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Comparison of Cart tracking Performance
Conclusion
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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 anapproximate (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
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ANTISWING OPERATION
OF OVERHEAD CRANE
Content
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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
Control Objective
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Control Objective
Proper tracking of The
Cart Motion along a
reference/desired
trajectory.Proper Antiswing
operation of pay load
during travel
Two Zone Control Theory
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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
Design of Control Algorithm for Stable Tracking Zone
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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
Contd.
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Choice of Stabilizing Function:
Choice of second error variable:
Derivative of z1 and z2
Integral action is introduced toenhance the controller performance
in steady state operation
Contd.
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Choice of CLF:
Control Input:
Where
Derivative of CLF:
Integral action reduces the steady
state error of the system.
Contd.
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List of Controller Parameters
Where d 1=c1+c2 & d 2=c1c2. k1=1, c1=c2=50, c=0.001
Design of Control Algorithm for Anti-Swinging Zone
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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!!!!!!!
g g g g
Contd.
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Same Control Algorithm is
being used to serve the
opposite purpose!!!Swinging
Zone
Anti Swing
Zone
Inverted Pendulum Overhead Crane
Schematic Diagram of Controller
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g
Reference
Input
Linear
Backstepping
Controller
Nonlinear
Backstepping
Controller
Controller for Stable Tracking Zone
Controller for Anti Swing Zone
Inverted
Pendulum
Switch
ing
Mecha
nism
ControlInput
Switching
Law
OverheadCrane
Results of Real Time Experiment
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Motion of the Cart
Steady state Tracking error reduces with time
Contd.
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Cart Velocity
Contd.
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Payload Angular Position
3.15
Contd.
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Payload Angular Velocity
Contd.
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Cart Motion of the pendulum when suffering with an external impact
The cart is able to
track the reference
trajectory within 15
seconds
Contd.
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Cart Velocity when suffering with an external impact
Contd.
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Angle of the Payload when suffering with an external impact
The angle of the
payload reduces
within 15 seconds
Contd.
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Angular Velocity of the Payload when suffering with an external impact
Conclusion
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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.
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Adaptive BacksteppingControl
and its Application onInverted Pendulum
Content
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Content
Adaptation as Dynamic Feedback
Adaptive Integrator Backstepping
Stabilization of an Inverted Pendulum
Robust Adaptive Backstepping
Simulation Results
Conclusion
Adaptation as Dynamic Feedback
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p y
Stabilization problem of a nonlinear system:
Static Control Law:
Augmented Lyapunov function:
u x x
Θ is an unknown
constant parameter xc xu 1
Θ is an unknown
parameter so it is
impossible to use
this expression of control law,
containing unknown
parameter
One Can use a
certainty
equivalence formwhere θ is replaced
by an estimate of θ,
ˆ
Dynamic Control Law
γ is adaptation
gain
Is the
parameter error
ˆ~
Contd.
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Derivative of Augmented Lyapunov function:
Update law:
Which ensures the negative definiteness of .
System dynamics:
~1~
~~1
2
1 x x xc
x xV a
x x ~ˆ
aV
x x
x xc x
~
~1
Contd.
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Block diagram of the Closed loop Adaptive system
Adaptive Backstepping
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Stabilization of 2nd order nonlinear system:
Stabilizing Function:
CLF:
Control law:
u x x
x x x
22
1121
11111 x xc xs
2
2
2
12
1
2
1 x x x xV s c
x x
x
x x cus
s 212
1
122
θ is an
unknown
parameter. Soθ should be
replaced by its
estimated
value.
x x x
x xcu ss 212
1
122ˆ
2
2
2
12
1
2
1 z z xV c
Contd.
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Error Dynamics:
Construction of Augmented Lyapunov Function:
Derivative of Augmented Lyapunov function:
Update Law :
~ 0
11
22
1
2
1
2
1
x z z
cc
z z
dt d
22
2
2
1
~
2
1
2
1
2
1~,
z z zV a
ˆ
1~~,, 22
2
22
2
1121 z zc zc z zV a 22ˆ z
Contd.
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Block diagram of the closed loop Adaptive System:
Adaptive Backstepping Control of Inverted Pendulum
(6.3.5.a)
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Dynamics of the Cart Pole system:
Dynamics of the Pendulum Angle:
Where
State Space Representation:
)(sincos t umlml xc xm M 2
θ xmlθ mglθ )ml(I cossin 2
Model is being
obtained
Lagrangian
Dynamics`
t u sincostansec2
321
gm M )( 2
ml3
ml
ml I m M
2
1
21 z z zk =u z zg -21
&
13111z z zg cossec
1
2
2312z z z zk sin-tan
Contd.
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Reformed Equation of Control Input :
Definition of 1st error variable:
Stabilizing Function:
Choice of 2nd error variable:
Control Lyapunov Function:
h z zgu 21 )(
( )
( ) z g
z k h =
ref e -1
ref ref ec z 11
22 - z ze ref
2
2
2
122
1
2
1eeV
Contd.
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Derivative of Lyapunov Function:
Control Input:
Augmented Lyapunov Function:
h
gueeceeeceeeeeV ref
21112211122112 c
heccec zgu ref ˆ
-
2211211 1
2
2
2
1
2
2
2
12
1
2
1
2
1
2
1hg
geeV a
Contd.
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Derivative of the Lyapunov function:
Parameter Update Law:
)-(}ˆ
-)ˆ)()-(({-dt
dheh
dt
gd heccece
g
gececV a
2
2
1
ref 2211212
222
211
1
11-
)ˆ)()-((ˆ
heccecedt
gd
ref 2211
2
1211
22edt
hd
ˆ
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Contd.
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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
ggg
gg
g
ggs
ˆ
ˆ
ˆ
ˆ
ˆ
ghecceceg gsref ˆˆ
ˆ 12211
2
1211
heh shˆˆ
222
0h0
00
0
0
0
2hif
2h if
hif 0
h
hhh
hh
h
hhs
ˆ
ˆ
ˆ
ˆ
ˆ
Simulation Results
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Angular variation of Pendulum
Contd.
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Disturbances Signal:
Contd.
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Estimation of the Parameter g
Contd.
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Parameter Estimation of h with time
Conclusion & Scope of Future Research
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This research presents an idea of using integral action along
with the backstepping control algorithms and achieves quitesatisfactory 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.
Questions
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Questions
Polygonia interrogationis known as Question Mark
References
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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.
References
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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”.
References
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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
References
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Lee, H.-H., 1998, “Modeling and Control of a Three-Dimensional OverheadCrane,” ASME J. Dyn. Syst., Meas., Control , 120, pp. 471 –476.
Kiss, B., Levine, J., and Mullhaupt, P., 2000, “A Simple Output Feedback PDController for Nonlinear Cranes,” Proc. of the 39th IEEE Conf. on Decisionand Control , Sydney, Australia, pp. 5097 –5101
Yang, Y., Zergeroglu, E., Dixon, W., and Dawson, D., 2001, “NonlinearCoupling Control Laws for an Overhead Crane System,” Proc. of the 2001IEEE 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 ConferenceChicago, Illinois, pp 1260-1264 June 2000
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Thank you
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Taken from Feedback Manual of Inverted Pendulum
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Taken from Feedback Manual of Inverted Pendulum
Feedback Positive Real
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Feedback Positive Real
•The triple (A,B,C) is feedback positive real (FPR) if thereexist 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 thetransfer function is appositive real one , and the pair(A + BK, C) is observable.
Passivity
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The system
(i)
Is said to be feedback passive (FP) if there exists a feedback transformation.
(ii)
such that the resulting system, y = C(z) is passive with a storage function U(z)
which is positive definite and radically unbounded:
The system of (i) is said to be feedback strictly passive (FSP) if the feedback
transformation of equation (ii) renders it strictly passive:
Ru R z zC yu z z z
n
, 0,0C , ,
v zr zK u
0
0
zU t zU d v y
t
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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
d z zU t zU d v y00
0
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