shubhobrata rudra presentation on backstepping control1

7/31/2019 Shubhobrata Rudra Presentation on Backstepping Control1

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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



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



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



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



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.



BACKSTEPPING

CONTROL



CONTENT

What is Backstepping?

Why Backstepping?

Different Cases of Stabilization Achieved by Backstepping

Backstepping: A Recursive Control Design Algorithm

New Research Ideas



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.



Contd.

Block Diagram of the system:



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.



Modified Block Diagram

Contd.

Feedback Control Law

αs Backstepping

Signal -αs



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



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.



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.



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



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



Contd.

IEEE Explore 1990-2003 Backstepping in title

Conference

Paper

Journal

Paper

Ola Harkegard Internal seminar on Backstepping January 27, 2005

ff f



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



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



Chain of Integrator

Chain of integrator:

CLF

Nonlinear

System∫ ∫ ∫ K th

integrator



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.



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



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.



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



Bl k B k t i



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



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



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



Strict Feedback Systems:

Control Input:

CLF

Contd.

Lower Triangular Form

C d



Semi Strict Feedback Systems:

CLF:

Control Input:

Contd.

Lower Triangular Form

C td



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


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



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



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


X C y


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



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



STABILIZATION OF

INVERTEDPENDULUM



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



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.



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



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



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



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



Choice of CLF:

Control Input:

Where

Derivative of CLF:

Contd.

Integral action reduces the steady

state error of the system.

Contd



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



State model of the Inverted Pendulum:

Choice of Control variable:

Design of Control Algorithm for Swinging Zone

Contd



Choice of Stabilizing function:



Contd.

Contd



Choice of CLF:

Control Input:

Derivative of CLF:

Contd.

Contd



List of Controller’s Parameters

Contd.

Contd



List of Controller’s Parameters

k2=0.1, d3=c3+c4 and d4=c3c4+1, where c3=c4=20

Contd.





Reference

Input

Linear

Backstepping

Controller

Nonlinear

Backstepping

Controller

Controller for Stabilization Zone

Controller for Swinging Zone

Inverted

Pendulum

Switching

Mecha

nism

ControlInput

Switching

Law




Angle of the Inverted Pendulum

Pendulum reach its

stable positionwithin 4 seconds

Contd



Angular Velocity of the Inverted Pendulum

Contd.

Contd



Cart Movement with time

Contd.

The cart is able to

track the reference

trajectory within 15seconds

Contd



Cart Velocity

Contd.

Contd. M d t V i ti



Voltage applied on the actuator motor

Contd. Moderate Variation

of voltage reduces

the stress on

actuator motor

Contd.



Angle of the Inverted Pendulum when it is suffering with external impact

Contd.

Pendulum regain its inverted position

within 3 seconds

Contd.



Angular Velocity of the Pendulum

Contd.

Contd.



Cart Position of the pendulum (suffering with an external impact)

Contd.

Cart Regain its

Desired trajectory

within 12 seconds

Contd.



Cart Position of the pendulum (suffering with an external impact)

Contd.

Contd.



Voltage applied on the actuator motor

C ti St d d C l i



Comparative Study and Conclusions

Comparative study on the Pendulum angular position in space

Contd.



Comparison of Cart tracking Performance

Conclusion



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



ANTISWING OPERATION

OF OVERHEAD CRANE

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



Comparative Study and Conclusion

Content

Control Objective



Control Objective

Proper tracking of The

Cart Motion along a

reference/desired

trajectory.Proper Antiswing

operation of pay load

during travel




Most of the nonlinearities (present in the state model of Overhead Crane)

are the function of payload angle in space.


Stable Tracking

Zone

Anti Swing

Zone

Design of Control Algorithm for Stable Tracking Zone



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.



Choice of Stabilizing Function:



Integral action is introduced toenhance the controller performance

in steady state operation

Contd.



Choice of CLF:

Control Input:

Where

Derivative of CLF:

Integral action reduces the steady

state error of the system.

Contd.



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



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.



Same Control Algorithm is

being used to serve the

opposite purpose!!!Swinging

Zone

Anti Swing

Zone

Inverted Pendulum Overhead Crane




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




Motion of the Cart

Steady state Tracking error reduces with time

Contd.



Cart Velocity

Contd.



Payload Angular Position

3.15

Contd.



Payload Angular Velocity

Contd.



Cart Motion of the pendulum when suffering with an external impact

The cart is able to

track the reference

trajectory within 15

seconds

Contd.



Cart Velocity when suffering with an external impact

Contd.



Angle of the Payload when suffering with an external impact

The angle of the

payload reduces

within 15 seconds

Contd.



Angular Velocity of the Payload when suffering with an external impact

Conclusion



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.



Adaptive BacksteppingControl

and its Application onInverted Pendulum

Content



Content

Adaptation as Dynamic Feedback

Adaptive Integrator Backstepping

Stabilization of an Inverted Pendulum

Robust Adaptive Backstepping

Simulation Results

Conclusion

Adaptation as Dynamic Feedback



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.



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.



Block diagram of the Closed loop Adaptive system

Adaptive Backstepping



Stabilization of 2nd order nonlinear system:


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.



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.



Block diagram of the closed loop Adaptive System:

Adaptive Backstepping Control of Inverted Pendulum

(6.3.5.a)



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.



Reformed Equation of Control Input :

Definition of 1st error variable:


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.



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.



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



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

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



Angular variation of Pendulum

Contd.



Disturbances Signal:

Contd.



Estimation of the Parameter g

Contd.



Parameter Estimation of h with time

Conclusion & Scope of Future Research



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



Questions

Polygonia interrogationis known as Question Mark

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.

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”.

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

References



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



Thank you



Taken from Feedback Manual of Inverted Pendulum



Taken from Feedback Manual of Inverted Pendulum

Feedback Positive Real



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



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



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

shubhobrata rudra presentation on backstepping control1

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