Download - 3 modelling of physical systems

ME2142/ME2142E Feedback Control Systems1

Modelling of Physical Systems

The Transfer Function

Modelling of Physical Systems

The Transfer Function

ME2142/ME2142E Feedback Control SystemsME2142/ME2142E Feedback Control Systems


Differential EquationsDifferential Equations

Differential equation is linear if coefficients are constants or functions only of time t.

Linear time-invariant system: if coefficients are constants.

Linear time-varying system: if coefficients are functions of time.

Differential equation is linear if coefficients are constants or functions only of time t.

Linear time-invariant system: if coefficients are constants.

Linear time-varying system: if coefficients are functions of time.

PlantU YPlantU Y

In the plant shown, the input u affects the response of the output y.In general, the dynamics of this response can be described by a differential equation of the form

In the plant shown, the input u affects the response of the output y.In general, the dynamics of this response can be described by a differential equation of the form

ubdtdub

dtudb

dtudbya

dtdya

dtyda

dtyda

m

m

m

m

n

n

n

n 01

1

101

1

1


Newton’s Law

f is applied force, nm is mass in Kgx is displacement in m.

Newton’s Law

f is applied force, nm is mass in Kgx is displacement in m.

mf

x

Mechanical Systems – Translational SystemsMechanical Systems – Translational Systems

Mechanical Systems – Fundamental LawMechanical Systems – Fundamental Law

Modelling of Physical Dynamic SystemsModelling of Physical Dynamic Systems

xmmaf or

0 xmf

xm

D’Alembert’s Principle


T is applied torque, n-mJ is moment of inertia in Kg-m2

is displacement in radiansis the angular speed in rad/s

T is applied torque, n-mJ is moment of inertia in Kg-m2

is displacement in radiansis the angular speed in rad/s

JT


Mechanical Systems – Torsional SystemsMechanical Systems – Torsional Systems

JJT 0 JT

J

or


Rotational:

T are external torques applied on the torsional spring, n-m

G is torsional spring constant, n-m/rad

Rotational:

T are external torques applied on the torsional spring, n-m

G is torsional spring constant, n-m/rad

1 2

Translational:

f is tensile force in spring, nK is spring constant, n/m

Translational:

f is tensile force in spring, nK is spring constant, n/m

f

x1x2

f

K


Mechanical Systems - springsMechanical Systems - springs

)( 21 xxKf Important: Note directions and signs

)( 21 GT


Translational:

f is tensile force in dashpot, nb is coefficient of damping, n-s/m

Translational:

f is tensile force in dashpot, nb is coefficient of damping, n-s/m

f

x1x2

f

.

b

.

f

x1x2

f

.

b

.


Mechanical Systems – dampers or dashpotsMechanical Systems – dampers or dashpots

)( 21 xxbf

Rotational:

T is torque in torsional damper, n-mb is coefficient of torsional damping,

n-m-s/rad

Rotational:

T is torque in torsional damper, n-mb is coefficient of torsional damping,

n-m-s/rad

21

)( 21 bT


1 2

f

x1x2

f

K


Using superposition for linear systemsUsing superposition for linear systems

Due to x1: 1Kxf

2Kxf Due to x2:

)( 21 xxKf Due to both x1 and x2 :

2GT Due to :2

Due to : 1GT 1

)( 21 GTDue to both and :1 2


Translational damperTranslational damper

f

x1x2

f

.

b

.

f

x1x2

f

.

b

.


Rotational damper: Rotational damper:

21

Using superposition for linear systemsUsing superposition for linear systems

Due to : 1xbf 1x

Due to :2x 2xbf

)( 21 xxbf Due to both and :1x 2x

2bT Due to :2

Due to : 1bT 1

)( 21 bTDue to both and :1 2



Example Example

Since m = 0, givesmaf 0 ds ff

Since and

Thus

Or

ybfd )( yxKf s

0)( ybyxK

KxKyyb

xy

b KA

Derive the differential equation relating the output displacement y to the input displacement x.

Derive the differential equation relating the output displacement y to the input displacement x.

Free-body diagram at point A,A fsfd

Note: Direction of fs and fd shown assumes they are tensile.

Note: Direction of fs and fd shown assumes they are tensile.


The transfer function of a linear time invariant system is defined as the ratio of the Laplace transform of the output (response) to the Laplace transform of the input (actuating signal), under the assumption that all initial conditions are zero.

The transfer function of a linear time invariant system is defined as the ratio of the Laplace transform of the output (response) to the Laplace transform of the input (actuating signal), under the assumption that all initial conditions are zero.

The Transfer FunctionThe Transfer Function

Previous ExampleAssuming zero conditions and taking Laplace transforms of both sides we have

Transfer Function

This is a first-order system.

Previous ExampleAssuming zero conditions and taking Laplace transforms of both sides we have

Transfer Function

This is a first-order system.

KxKyyb

)()()( sKXsKYsbsY

KbsK

sXsYsG

)()()(



Example Example

Free-Body diagram

givesmaf ods xmff

m

fs fd

xo

m

K b

xi

xo

)()()()()(2 sKXsbsXsKXsbsXsXms iiooo

ooioi xmxxbxxK )()(

iiooo KxxbKxxbxm

Thus

Or

And

KbsmsKbs

sXsX

sGi

o

2)()(

)(Transfer Function . This is a second-order system.

For the spring-mass-damper system shown on the right, derive the transfer function between the output xo and the input xi.

For the spring-mass-damper system shown on the right, derive the transfer function between the output xo and the input xi.

Note: fs and fd assumed to be tensile.


Capacitance

Or

Complex impedance

Capacitance

Or

Complex impedance

eqC

Ceq

dtdeC

dtdqi

)(sECI

)/(1 sCX c

cIXsC

IE 1

e i C


Electrical ElementsElectrical Elements Resistance

Units of R: ohms ( )

Resistance

Units of R: ohms ( )

iRe

Rei

e i R

Inductance

Units of L: Henrys (H)

Or

Inductance

Units of L: Henrys (H)

Or

dtdiLe

t

teL

i0

d1

)(sLIIXE L

e i L

IRE



Electrical Circuits- Kirchhoff’s LawsElectrical Circuits- Kirchhoff’s Laws

Current Law:

The sum of currents entering a node is equal to that leaving it.

Current Law:

The sum of currents entering a node is equal to that leaving it.

0 i

Voltage Law:

The sum algebraic sum of voltage drops around a closed loop is zero.

Voltage Law:

The sum algebraic sum of voltage drops around a closed loop is zero.

0 e



Electrical Circuits- ExamplesElectrical Circuits- Examples

RC circuit: Derive the transfer function for the circuit shown,

and

giving

This is a first-order transfer function.

RC circuit: Derive the transfer function for the circuit shown,

and

giving

This is a first-order transfer function.

ci IXIRE

co IXE

)/(1)/(1

sCRsC

XRX

EE

c

c

i

o

11

RCs

eii C

R

eo



Electrical Circuits- ExamplesElectrical Circuits- Examples

RLC circuit:

and

giving

This is a second-order transfer function.

RLC circuit:

and

giving

This is a second-order transfer function.

cLi IXIXIRE

co IXE

)/(1)/(1

sCsLRsC

XXRX

EE

cL

c

i

o

11

2

RCsLCs

eii C

R

eo

L



Operational Amplifier – Properties of an ideal Op AmpOperational Amplifier – Properties of an ideal Op Amp

Gain A is normally very large so that compared withother values, is assumed small, equal to zero.Gain A is normally very large so that compared withother values, is assumed small, equal to zero.

)( 12 vvAvo )( 12 vv

The input impedance of the Op Amp is usually very high (assumed infinity) so that the currents i1 and i2 are very small, assumed zero.

The input impedance of the Op Amp is usually very high (assumed infinity) so that the currents i1 and i2 are very small, assumed zero.

Two basic equation governing the operation of the Op Amp

and

Two basic equation governing the operation of the Op Amp

and 0,0 21 ii2112 or0)( vvvv



Operational Amplifier – ExampleOperational Amplifier – Example

For the Op Amp, assume i1=0 and vs=v+=0.For the Op Amp, assume i1=0 and vs=v+=0.

-

+

vi i1=0voZi

Zf

ii

if

S

Then orThen or0 fi ii 0f

o

i

i

ZV

ZV

ThereforeTherefore

i

f

i

o

ZZ

sVsV

)()(

ii

fo V

RZ

V



Operational Amplifier – ExampleOperational Amplifier – Example

-

+

vi i1=0voZi

Zf

ii

if

S

ii

fo V

ZZ

V

For the following sCRZ ff

1

sKK

CsRRR

RZ

Vi

pii

f

i

f

i

o 1V


Permanent Magnet DC Motor Driving a LoadPermanent Magnet DC Motor Driving a Load

For the dc motor, the back emf is proportional to speed and is given by where is the voltage constant. The torque produced is proportional to armature current and is given by where is the torque constant.

For the dc motor, the back emf is proportional to speed and is given by where is the voltage constant. The torque produced is proportional to armature current and is given by where is the torque constant.

eK

eKiKT t tK

Relevant equations:Relevant equations: eaa KdtdiLiRe

iKT t bdtdJT

e i

Ra La

eK J

bT

Note: By considering power in = power out, can show that Ke=KtNote: By considering power in = power out, can show that Ke=Kt


End

Download - 3 modelling of physical systems

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