自動控制原理課件：CH2-1-Time Domain Models

,CH2MathematicalModeling

Thischapteraimstodevelop：

MathematicalModels->SystemAnalysis->SystemDesign

1.Mathematicalmodels

Amathematicalmodelisthedescriptionof

relationshipsamongthevariablesincontrolsystems,

anditcanmanydifferentformssuchaslookuptables,

differential/differenceequationsetc.

i

?Staticmodels:instaticconditions(allordersofderivatives

=0),thefunctionamongthevariablesdoesnotchangewith

time,usuallyrepresentedbyalgebraicequations.

?Dynamicmodels:thefunctionchangeswithtime,usually

representedbydifferentialordifferenceequations.

2.ModelingMethods

-->

-今SystemIdentification

.ModelRepresentations

■T'ixfilDTTifiiffierential/Differenceequation,statespace

?nsferfunction,Blockdiagram,signalflow

■FrequSsnsaFrequencyresponsefunction

2

Inthischapter：

■TimeDomainModels(Differentialequation)

■FrequencyDomainModels(Transferfunction)

■JBlockDiagram

■JSignalFlow

TimeDomainModels

Inthissection:

■Differentialequationsforlinearsystems

■Obtaindifferentialequations

■Solutiontodifferentialequations

■Superpositionprincipleforlinearsystems

■Linearizationofnonlineardifferentialequations

■Motionmodes

4

1TimeDomainModels

Ex2-l.lMass-Spring-DamperSystem

Input：r(t)

Output:y

Newton'sLawofMotion:

r(l)

5

1TimeDomainModels

Ex2-1.2RLCPassiveNetwork

LROInputVoltage:監。)

o—■

OutputVoltage:Uo(t)

二〃。(，)

KirchhoffTheorem:

o--------------------

RLCNetwork0di(t)1r

dt"

r+Ri(t)=u(J)

=才J力’

getridof?),input-outputfunctionisobtainedas:

…d2U(t)ncduc⑴/、/、

LC-------n---4-RC--------4-u(/)=〃/⑺

dt2dto

6

E

Ex2-1.3MotorControl

7

Et^Rme_DomairrModefe

8

E

iHRme_DomairrModefs

9

JHRme_DomatnModels

First-ordersystem

10

E

2ObtainDifferentialEquations

Stepstosetupdifferentialequations:

1.Analyzeprinciples,drawblockdiagram

2.Setupdifferentialequationforeachcomponent

3.Getridofintermediatevariables

4.Writeintoinput-outputformat

ii

E

2ObtainDifferentialEquations

Ex2-1.4(Bookexample2-5)

12

E

2ObtainDifferentialEquations

AmplifierI:

AmplifierII:

PowerAmplifier:

Motor:

Gearbox:

MeteringMotor:

WriteinStandardForm:

13

E

SimilarityPrinciple

Similarsystem:differentsystemswithsameDE

Similarvariable:possesssimilarroleinDE

MechanicalElectricalHydroThermal

SystemSystemSystemSystem

0

VelocityvIQh

ResistancefRRR

SpringCoef.KCCC

MassmL

14

E

----3SuperpositiorrPrincipte

r（060）

r

i（。+々（，）Cx（/）+°2（t）---superposition

”（t）Ac^（，）——homogeneity

Linearsystem:systemwithsuperpositionandhomogeneity

E

4SolutiontoDifferentialEquation

SolvingLTIdifferentialequationsusingLaplace

approach:

TakeLaplacetransformforeachterminequation(payclose

attentiontoinitialconditions);

ObtainLaplacetransferfunctionsfi*omtheresultingalgebraic

equation;

TakeinverseLaplacetransformfbroutputtoobtainitstime

domainrepresentation.Thisisexactlythesolution.

E

4SolutiontoDifferentialEquation

Ex2-1.5(Bookexample2-6)

2

du(t)dun(t)/、/、

<dt2dt°八

luO(0)=0.1,z(0)=0.1,〃1.⑺=1(。

TakeLaplacetransformonbothside:

2

L[o-oo-i1oooi

sU(s)su(G)u(0)+[sU(s)-u(0)>U(s)

17

4SolutiontoDifferentialEquation

n/x_U八s)s〃j(0)+%(0)+〃°(0)

'X『+S+1+/+s+i

10.15+0.2

——|—

2

s(sS1)§2+s+1

++

1O.k+O.2

%(%)=￡ik(s)]=L+

S(S9+5+1)S9+5+1

=1+1.5"°5'sin(0.866,一1200)+0.2。一°”sin(0.866/+30°)

ZeroinitialresponseZeroinputresponse

UnityStepResponse

E

5LinearizationMethod

Tangentialorsmallperturbationmethod:

EquilibriumA(x0,y0)Taylorexpansion:

y\dx)

>=/(%)=/(%)+(x一%)

A

+1(d2f(x)\

-o2+

2!d￡7(xx)

/xo

機x)

ox/'(%)+(x一%)

xoXIdt4

y-y=\可3

(x-xo)A=A

°I\d；t)

yKx

19

E

-5LinearizationMethod

Ex2-1.6SinglePendulumModel

Torqueonmass:

T=MgLsin0

T.，dsin?

??T-TxMgL(9-0o)

odid0

e=O,T。=)