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LaplaceLaplace TransformationTransformation446446 -- 55

Prof. Neil A.Prof. Neil A. DuffieDuffieUniversity of WisconsinUniversity of Wisconsin --MadisonMadison

Neil A. Neil A. DuffieDuffie , 1996, 1996All rights reserved.All rights reserved.

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LaplaceLaplace TransformationTransformation

L [f (t )] = f (t )e -st dt0

Transform:Transform:

Differential equationsDifferential equations

Algebraic equationsAlgebraic equations

Functions of time (step, impulse, sine, etc.)Functions of time (step, impulse, sine, etc.)

ss is a new, complex variableis a new, complex variable

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Mechanical System VariablesMechanical System Variables

(t)(t)

v(t)

e(t),i(t)

(t) ,(t)

T(t)

(t) = rotational position(t) = rotational position(t) = rotational velocity(t) = rotational velocity

T(t) = motor torqueT(t) = motor torque

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Mechanical System ParametersMechanical System Parameters

(t)(t)

v(t)

e(t),i(t)

(t) ,(t)

T(t)

J = motor inertiaJ = motor inertiaKK tt = torque constant= torque constant

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Electrical System VariablesElectrical System Variables

(t)(t)

v(t)

e(t),i(t)

(t) ,(t)

T(t)

i(t) = motor currenti(t) = motor currente(t) = amplifier output voltagee(t) = amplifier output voltage

v(t) = amplifier input voltagev(t) = amplifier input voltage

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Electrical System ParametersElectrical System Parameters

(t)(t)

v(t)

e(t),i(t)

(t) ,(t)

T(t)

R = motor resistanceR = motor resistance

L = motor inductanceL = motor inductance

KKee = back emf constant= back emf constantKKvv = tachometer gain= tachometer gain

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Motor Motor --Amplifier EquationsAmplifier Equations

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Transforms of EquationsTransforms of Equations

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Transformed ModelTransformed Model -- VelocityVelocity

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Transformed ModelTransformed Model -- PositionPosition

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Unit Step FunctionUnit Step Function

00

f(t)f(t)

u(t)u(t)

L [u (t )] = u (t )e -st dt0

11

tt

L [u (t )] = e-st

dt0

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Unit Step FunctionUnit Step Function

00

f(t)f(t)

u(t)u(t)11

tt

L [u (t )] = 1s

L [u (t )] = e-s

-s - e -s 0-s

L [u (t )] = e -st-s 0

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Unit Impulse FunctionUnit Impulse Function

00tt

(t)(t)

00tt

(t)(t)

tt

tt

11

00tt

(t)(t)

tt

tt11

tt

(t)(t)tt11

00 tt

"strength""strength"(Area) = 1(Area) = 1

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00tt

(t)(t)

tt

tt11

f(t)f(t)

Area = 1Area = 1

L [(t )] = (t )e -st dt0

L [(t )] = limt 01

t e- st

dt0

t

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00tt

(t)(t)

tt

tt11

f(t)f(t)

strength = 1strength = 1

L [(t )] = limt 0

e -s t - e -s 0- s

t = lim

t 01 - e -s t

stUse L'Hopital's ruleUse L'Hopital's rule

L [(t )] = limt 0

e - st

-s t0

t

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00tt

(t)(t)

tt

tt11

f(t)f(t)

Differentiate numerator andDifferentiate numerator anddenominator with respect todenominator with respect to t t

strength = 1strength = 1

L [(t )] = limt 0

se -s ts = lim t 0

e -s t

L [ (t )] = 1

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Exponential FunctionExponential Function

00

tt

f(t)f(t)

11

L [e -t ] = e -t e -st dt0

L [e-t

] = e-(

1

+ s)

tdt

0

f (t ) = e -t

l

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Exponential FunctionExponential Function

00 tt

f(t)f(t)

11

L [e -t ] = e -(1 + s )

-(1 + s ) - e -(

1 + s )0-(1 + s )

L [e -t ] = 11 + s = s + 1

L [e -t ] = e -(1 + s )t

-(1 + s ) 0

f (t ) = e -t

l k

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Nonlinear Tank SystemNonlinear Tank System

qq ii(t)(t)

qq 00 (t)(t)

h(t)h(t) Tank area = ATank area = A

(t)(t)

ValveValveTankTank OutletOutlet

flowflow

InletInlet

flowflow

q 0 (t) = k(t) h(t)Nonlinear behavior of valve flow:Nonlinear behavior of valve flow:

l k d lN li T k M d l

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Nonlinear Tank ModelNonlinear Tank Model

q s (t) = q i (t) q o (t)

q s (t) = A dh(t)dt

q o (t) = k(t) h(t)

Adh(t)

dt = q i (t) k(t) h(t)

Adh(t)

dt +k(t) h(t) = q i (t)

Li i d M d l f V l FlLi i d M d l f V l Fl

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Linearized Model of Valve FlowLinearized Model of Valve Flow

q o = k hq i = qo

h = q i

k

2

q o (t) q o +q o

.h((t) ) + q o

h .h(h(t) h )

q o (t) q o +k h ( (t) ) k

2 h (h(t) h )

Li i d T k M d lLi i d T k M d l

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Linearized Tank ModelLinearized Tank Model

q o (t) q o +k h ( (t) ) +k

2 h (h(t) h )

q o (t) k h +k h (t) k h

+k

2 h h(t) k

2 h h

q o (t) k h (t) +k

2 h h(t) 12 k h

Adh(t)

dt q i (t) k h (t) k

2 h h(t) +12 k h

T f d T k S t M d lT f d T k S t M d l

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Transformed Tank System ModelTransformed Tank System Model

Adh(t)

dt +k

2 h h(t) q i (t) k h (t) +12 k h

A dh(t)dt + k2 h h(t) q i (t) k h (t) + 12 k h u(t)

A sH(s) h(0 +))+ k2 h H(s) Q i(s) k h (s) + 12

k h 1s

How would tank respond to a change in qHow would tank respond to a change in q ii(t)?(t)?To a change inTo a change in (t)? Need solution!(t)? Need solution!

446-05 laplace ii (n)

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