pendul invers rotativ

7/28/2019 Pendul Invers Rotativ

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Analele Universit ii Constantin Brncui din Trgu Jiu, Seria Inginerie, Nr. 3/2010

Annals of the Constantin Brncui University of Trgu Jiu, Engineering Series, Issue 3/2010

230

PROIECTAREA PRIN METODALQRI METODA DIRECT

LYAPUNOV A SISTEMULUI DECONTROL AL PROCESULUIPENDUL INVERS ROTATIV

Marian Popescu,UniversityConstantin Brncui, Tg-Jiu,

ROMANIA

Adrian Runceanu,UniversityConstantin Brncui, Tg-Jiu,

ROMANIAConstantin Cercel,UniversityConstantin Brncui, Tg-Jiu,

ROMANIA

Rezumat: n cadrul acestei lucrri se vaprezenta implementarea unui sistem de control aprocesului real pendul invers rotativ. Acestsistem prezint restricii puternice de timp real itotodat precizia comenzii calculate trebuie s fiect mai mare, datorit faptului c sistemul are nplan vertical_Up un singur punct de echilibruinstabil. Algoritmii implementai sunt bazai pe ostructur de reglare cu reacie dup variabilele destare i proiectarea s-a realizat prin metodaLQR(Linear Quadratic Regulator) i prin metodadirect Lyapunov, realizndu-se astfel i ocomparaie ntre cele dou metode.

Cuvinte cheie: Lyapunov, LQR, pendulinvers rotativ

1. DESCRIEREA PROCESULUIPENDUL INVERS ROTATIV

Pendulul invers rotativ este utilizat

foarte mult n domeniul sistemelor de control

pentru a ilustra idea tehnologiei controlului

automat. Fiind un sistem neliniar i instabil,

practic are 2 puncte de echilibru n plan

vertical, unul stabil vertical-Down i altul

instabil vertical-Up, este foarte util n testarea

noilor dezvoltri n domeniul sistemelorneliniare. Ca i construcie, este format dintr-

un bra acionat de un motor de c.c. ce se

rotete n plan orizontal i are ataat la capt

LQR DESIGN METHOD ANDLYAPUNOV DIRECT METHODPROCESS CONTROL SYSTEM

"ROTARY INVERTED

PENDULUM"

Marian Popescu,UniversityConstantin Brncui, Tg-Jiu,

ROMANIA

Adrian Runceanu,UniversityConstantin Brncui, Tg-Jiu,

ROMANIAConstantin Cercel,UniversityConstantin Brncui, Tg-Jiu,

ROMANIA

Abstract: This paper will present theimplementation of a real process control systeminverted pendulum rotating. This system haspowerful real-time restrictions and also calculatedthe accuracy of the order must be as large becausethe system is planning vertical_Up one unstableequilibrium point. Implemented algorithms arebased on a reaction after adjusting structure statevariables and design was done by the method

LQR (Linear Quadratic Regulator) and Lyapunovdirect method, thus achieving a comparisonbetween the two methods.

Keywords: Lyapunov method, LQRmethod, rotational inverted pendulum

1. INVERTED PENDULUM ROTARYPROCESS DESCRIPTION

Rotary inverted pendulum is used very much

in control systems to illustrate the idea of

automatic control technology. Being an

unstable nonlinear system and basically has

two vertical equilibrium points, one stable

and one unstable vertical-Down-Up-down,

is very useful in testing new developments

in nonlinear systems. As construction

consists of an arm driven by a DC motor

that rotates horizontally and is attached to

the end of the pendulum itself that rotates in

a vertical plane perpendicular to the actuator

arm, as in Fig.1.


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231

pendulul propriu-zis care se rotete ntr-un

plan vertical perpendicular pe braul de

acionare, ca nFig.1.

Datorit faptului c pendulul invers

rotativ are 2 grade de libertate rotaionale i

numai un element de execuie, acest sistemintr n categoria sistemelor subacionate.

Ideea meninerii pendulului n pozitiavertical-Up const n proiectarea unui sistem

de control care s balanseze, printr-o coreciecontinu a comenzii la motorul de acionare,

pendulul n jurul punctului de echilibru

instabil din planul vertical. Motorul care

realizeaz acionarea este conectat mecanic la

braul pendulului printr-un mecanism de roidinate care realizeaz o multiplicare a

turaiei i totodat conecteaz ipoteniometrul de msur a poziiei braului.

2.MODELAREA PENDULULUI NPOZIIE VERTICAL-UP

Pornind de la Fig.1. a pendulului poziionat

n plan vertical-Up, vom avea proieciamicrii pendulului n planul XOY ca n

Fig.2. Modelarea micrii pendulului se varealiza folosind ecuaiile Euler-Lagrange i

vom considera pentru simplificare ca avnd

centrul de greutate chiar n vrful pendulului.

Because the rotary inverted

pendulum has two rotational degrees of

freedom and only one element of

performance, this system falls into the

category of under operated systems. Theidea of maintaining the pendulum in

vertical-up position is to design a control

system to balance, through a continuous

adjustment to the engine control actuators,

the pendulum around the equilibrium

unstable plane. Performing drive motor is

connected mechanically to the arm by a gear

mechanism that achieves a multiplication of

speed and also connect and position

measuring potentiometer arm.

2.MATHEMATICAL MODELING FORVERTICAL_UP PENDULUM

Based on Fig.1. the pendulum positioned

vertically-Up, we have movement in the

plane projection xOy as Fig.2. Modeling of

pendulum motion will be made using the

Euler-Lagrange equations and we consider

for simplicity as being the center of gravity

even at the top of the pendulum.

X

Fig.1. Vertical_Up

pendulum.

Y

Z

O

L

l

PWM

P


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Lagrangeanul sistemului este: Lagrangean system is:

coscossin2

1)(

2

1)(

2

1

coscossin2

1)(

2

1

2

1

2

1

2222222

22222222

glmLlmlmlmJLmJ

glmLlmlmlmJLmJL

ppppppb

ppppppb

++++=

=++++=

&&&&&

&&&&&&

(1)

Pentru coordonata generalizat avem

ecuaia Lagrange:For generalized coordinates we have

Lagrange equation:

=

LL

dt

d

&(2)

unde -este cuplul total care acioneaz

asupra axei de rotaie n direcia creterii lui

. Acesta reprezint cuplul exercitat demotor- m care trebuie s nving cuplul de

frecare.

where - is the total torque acting on the

axis of rotation towards higher . This

engine is the torque m - the torque requiredto overcome friction.

&bm C= (3)

unde bC -este coeficientul de frecare vscoas

n jurul axei de rotaie pentru unghiul .

Folosind relaiile (1),(2), (3) prima ecuaie de

micare a braului pendulului devine:

where bC - is the viscous friction coefficient

around of rotation axis for angle.

Using (1),(2), (3) first motion equation of

pendulum arm is:

&&&&&&&&&& bmpppppb CLlmlmLlmlmLmJ =++++ sincossin2cossin)(22222 (4)

Similar, pentru a 2-a coordonat generalizat

-unghiul braului avem ecuaia Lagrange:Idem, for second generalized coordinate -

angle of arm has Lagrange equation:

=

LL

dt

d

&(5)

unde -este cuplul total care acioneaz n

jurul axei de rotaie a braului n direcia

creterii unghiului . Considerm

Where, - is the total torque for acting

around of rotation axis of arm in direction of

rising the angle . Consider &

pC= ,

X

Y

O

L

lsin

1

1

Fig.2. Projection in xOy plan

Pendulum motion in the horizontalplaneLsin

Lcos


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&

pC= , unde pC este coeficientul de

frecare vscoas a pendulului n jurul axei de

rotaie cu unghiul .

Astfel, folosind relaiile (1),(5), obinem a 2-a

ecuaie de micare:

where pC is the viscous friction coefficient

around of rotation axis for angle.

So, using (1), (5) we obtain the second

motion equation:

=++ sinsincossinsincos)(222 glmLlmlmLlmLlmlmJ ppppppp

&&&&&&&&&

&&&&&& pppppp CglmlmlmJLlm =++ sincossin)(cos222 (6)

Astfel, din relaiile (4) i (6) se obin ecuaiile

de micare pentru sistemul pendul inversrotativ n poziia vertical-Up:

So, from (4) and (6) we obtain the motion

equation for rotation invers pendulum in

up-vertical position

(7)sincossin)(cos

sincossin2cossin)(

222

22222

=++

=++++

&&&&&&

&&&&&&&&&&

pppppp

bmpppppb

CglmlmlmJLlm

CLlmlmLlmlmLmJ

3. LINIARIZAREA SISTEMULUI NJURUL UNUI PUNCT STAIONAR DEFUNCIONARE,CAZUL PENDUL-UP

Pornind de la relaiile (7) ce reprezint

ecuaiile pentru bra i pentru pendul npoziia vertical-Up, vom liniariza aceste

ecuaii n jurul unui punct staionar de

funcionare 00 , , caracterizat de

urmtoarele relaii:

3. SYSTEM LINEARIZATIONAROUND A STATIONARY POINTOPERATION, THE CASE PENDUL_UP

Starting from the relations(7) what are the

equations for the pendulum arm and the

vertical_Up position, these equations will be

linearized around a steady operating

point 00 , ,characterized by the following

relations:

)()( 0 tt += ; )()( 0 tt += (8)

Poziia 0 -este unghiul unde trebuie inut

braul n plan orizontal i implicit pendulul s

fie n echilibru vertical_Up. Astfel ref =0

este chiar mrimea de referin pentru sistem.

The position 0 -is the angle where the arm

should be held horizontally and thus be in

balance pendulum vertical_Up. Thus

ref =0 is the reference for the system.

=

=

=

=

=+=

+=

)()(

)()(

)()(

)()(

)()()(

)()(

0

0

tt

tt

tt

tt

ttt

tt

&&&&

&&&&

&&

&&

(9)

Ecuaiile de stare(7), liniarizate n jurulpunctului staionar devin:

The state equation(7), linearized around the

stationary point is:

u

eab

de

eab

db

x

x

x

x

eab

ce

eab

hb

eab

fb

eab

cb

eab

ha

eab

fa

x

x

x

x

+

=

0

0

0100

0

0001

0

2

2

4

3

2

1

222

222

4

3

2

1

&

&

&

&

(10)


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234

unde: where:

glmhCflmJeR

Kd

R

KKCcLlmbLmJa pppp

a

t

a

btbppb ==+==+==+= ;;;;;);(

22

4. PROIECTAREA STRUCTURIISISTEMULUI DE CONTROL PRINMETODA LQR

Implementarea structurii de reglare cu reaciedup variabilele de stare este prezentat n

Fig.3. curspunsurile n timp real prezentate

nFig.4.

4. STRUCTURE OF CONTROLSYSTEM DESIGN WITH METHODLQR

Implementation of the control structure with

state feedback is presented in Fig.3. with

real-time responses shown inFig.4.

A,B,C,D

u(t)=-kx(t) )(t

Fig.3.Structura sistemului de reglare cu reacie dup variabilele de stare

K1

K2

K3

K4

x1

)(t&

x2

)(tx3

)(t&

)(t

x4

(

0= ref

0 =ref

State Feed-back

-


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5. PROIECTAREA STRUCTURIISISTEMULUI DE REGLARE PRINMETODA DIRECT LYAPUNOV

Ideea de baz n liniarizarea Intrare-Ieire

este gsirea unei relaii directe ntre mrimea

de comand i ieirea sistemului. O metodde a gsi aceast relaie este s derivm

succesiv mrimea de ieire pn aparemrimea de comand direct n ecuaie.

Procednd n aceast manier vom avea:

5. STRUCTURE OF CONTROLSYSTEM DESIGN WITH DIRECTLYAPUNOV METHOD

The basic ideea in state feedback

linearization is finding a direct relationship

between control signal and output signal of

the system. One way to find this relatioship

is successively derive the output signal until

control signal appear direct in the equation.

Thus we have:

&& =y ; )R

KLlcosm()det(

1a

tp ub

My +== &&&&

Se poate observa c intrarea u apare direct

n ieire la a 2-a derivare a ieirii i astfelvom liniariza sistemul (7) Intrare-Ieire prin

reacie, impunnd o comand virtual v de

forma:

It may be noted that the input signal u

appers directly in the output signal for

second derivative of otput signal and we

achieved fedback input-output liniarization

of the system (7) and impose a form of

virtual command:

)R

KLlcosm(

)det(

1

a

tp ub

Myv +=== &&&& (11)

=4x -poziie bra

&=3x -vitez bra

=2x -poziie pendul

&

=1x -vitez pendul

dreaptatreaptref = -referin poziie bra

u-comanda calculat

Fig.4. Rspunsul sistemului real obinut pentru variaia treapt a referinei ref


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236

Comanda real aplicat la motorul deacionare va fi de forma:

The real command applied to the motor

drive will be as:

cos

)det(

LlmR

K

bvMu

pa

t

= (12)

Structura sistemului este prezentat nFig.5. The system structure is presented inFig.5.

Comanda obinut pentru sistemul pendul

invers rotativ prin proiectare cu metoda a

doua Lyapunov este:

The command obtained for the system

rotary inversed pendulum is:

pendul

model

matematic

neliniar

k0

k1

k3

k3

k3

C1

C2

C3

comanda

neliniar

&=2x

=3x

&=4x

&=4x

=3x

0

- -

v1 v=v1+v2u

Fig.5.Structura sistemului proiectat prin metoda a doua Lyapunov

&&

k4

k4

k4

D1

D2

D3

&=2x

=3x

&=4x

k4 D1=1x

v=v1+v2+v3

-

4443342241144333232134130 xDkxDkxDkxDkxCkxCkxCkxkxkv =


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237

CONCLUZII

Cerinele de timp real ale acestui

proces sunt foarte puternice i acest lucru anecesitat realizarea unei distribuii mai mari a

prelucrrii. Implementarea a fost realizat cao structur de reglare cu reacie dup

variabilele de stare, iar proiectarea

algoritmului de reglare a fost realizat prin

metoda LQR i metoda a doua Lyapunov.

Prin realizarea unei comparaii ale celor 2rspunsuri, metoda direct Lyapunov, fiind o

metod bazat pe o funcie energetic,realizeaz stabilizarea pendulului n poziievertical-Up cu o robustee mai mare n jurul

poziiei de echilibru, pe cnd metoda LQR,dei stabilizeaz mai bine pendulul, are un

domeniul mic de stabilitate n jurul punct

staionar de funcionare.

BIBLIOGRAFIE[1] Astrm K.A., Murray R.M.,

FEEDBACK SYSTEMS-AN

CONCLUSIONS

Real-time requirements of this

process are very strong and this has

necessitated the achievement of a greater

distribution of processing. The

implementation was done as feedback input-

output linearization and control algorithm

designed was done by LQR method and the

second Lyapunov method. By a comparison

of the two responses, Lyapunov direct

method, is a method besed on an energy

basis, the stabilization pendulumvertical_Up with a more robust around the

equilibrium position, while the LQR

method, although better stabilizes pendulum

has a small field of stability around the

stationary point of operation

REFERENCES[1] Astrm K.A., Murray R.M.,

FEEDBACK SYSTEMS-AN

INTRODUCTION FOR SCIENTISTS

Fig.6.Rspunsul sistemului real pendul-invers


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238

INTRODUCTION FOR SCIENTISTS AND

ENGINEERS, 2008, Princeton University

Press, ISBN-13: 978-0-691-13576-2, ISBN-

10: 0-691-13576-2

[2] C.J.A. van Kats, NONLINEARCONTROL OF A FURUTA ROTARYINVERTED PENDULUM, DCT reportnr.2004.69, 2004, TU/e Bachelor Final

Project Report

[3] Ionete C., Manole E., Surlea D., xPCMULTITASKING CONTROL FOR TWO

QUANSER EXPERIMENTS, 9th

International Carpathian Control Conference-

ICCC2008, Sinaia, Romnia, 25-28 Mai

2008, pag.263-266, ISBN 978-973-746-897-0

[4] Popescu D., Rsvan V., Danciu D.,SOME ASPECTS RGARDING THE PIO

THEORY, 9th International Carpathian

Control Conference-ICCC2008, Sinaia,

Romnia, 25-28 Mai 2008, pag.525-530,

ISBN 978-973-746-897-0

[5] Rsvan V., Popescu D., Danciu D.,

NETWORKS AND

SYNCHRONIZATION, A XIII-a ediie

International symposium on systems theory

SINTES 13, 18-20 octombrie 2007, Craiova,

pag.164-168, ISBN:978-973-742-839-4

AND ENGINEERS, 2008, Princeton

University Press, ISBN-13: 978-0-691-

13576-2, ISBN-10: 0-691-13576-2

[2] C.J.A. van Kats, NONLINEAR

CONTROL OF A FURUTA ROTARYINVERTED PENDULUM, DCT reportnr.2004.69, 2004, TU/e Bachelor Final

Project Report

[3] Ionete C., Manole E., Surlea D., xPCMULTITASKING CONTROL FOR TWO

QUANSER EXPERIMENTS, 9th

International Carpathian Control

Conference- ICCC2008, Sinaia, Romnia,

25-28 Mai 2008, pag.263-266, ISBN 978-

973-746-897-0

[4] Popescu D., Rsvan V., Danciu D.,SOME ASPECTS RGARDING THE PIO

THEORY, 9th International Carpathian

Control Conference-ICCC2008, Sinaia,

Romnia, 25-28 Mai 2008, pag.525-530,

ISBN 978-973-746-897-0

[5] Rsvan V., Popescu D., Danciu D.,

NETWORKS AND

SYNCHRONIZATION, A XIII-a ediie

International symposium on systems theory

SINTES 13, 18-20 octombrie 2007, Craiova,

pag.164-168, ISBN:978-973-742-839-4

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