transfer function models of dynamical processes · process control time domain transfer function...

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Transfer Function Modelsof Dynamical Processes

Process Dynamics and ControlProcess Dynamics and Control

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Linear SISOLinear SISO Control SystemsControl Systems

General form of a linear SISO control system:General form of a linear SISO control system:

this is a underdeterminedthis is a underdetermined higher order differentialhigher order differentialequationequation

thethe functionfunction must be specified for this ODE tomust be specified for this ODE toadmit a well defined solutionadmit a well defined solution

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Transfer FunctionTransfer Function

Heated stirred-tank model (constant flow, )Heated stirred-tank model (constant flow, )

Taking the Taking the Laplace Laplace transform yields:transform yields:

or lettingor letting

Transfer functions

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Heated stirred tank exampleHeated stirred tank example

e.g. e.g. The block is called the transfer function relating Q(s) to T(s)

+

++

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Process ControlProcess Control

Time DomainTime Domain

Transfer function Modeling, Controller Design and Analysis

Process Modeling,Experimentation and

Implementation

Laplace Laplace DomainDomain

Ability to understand dynamics in Laplace and time domains is extremely important in thestudy of process control

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Transfer functionsTransfer functions

Transfer functions are generallyTransfer functions are generally expressed as a ratio ofexpressed as a ratio ofpolynomialspolynomials

WhereWhere

The polynomial is called the The polynomial is called the characteristic polynomialcharacteristic polynomial of of

Roots ofRoots of are theare the zeroeszeroes of of Roots ofRoots of are the are the polespoles of of

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Transfer functionTransfer function

Order of underlying ODE is given by degree ofcharacteristic polynomial

e.g. First order processes

Second order processes

Order of the process is the degree of the characteristic(denominator) polynomial

The relative order is the difference between the degree of thedenominator polynomial and the degree of the numeratorpolynomial

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Steady state behavior of the process obtained form theSteady state behavior of the process obtained form thefinal value theoremfinal value theoreme.g. First ordere.g. First order processprocess

For a unit-step input,For a unit-step input,

From the finalFrom the final value theorem, the ultimate value ofvalue theorem, the ultimate value of isis

This implies that the limit exists,This implies that the limit exists, i.e.i.e. that the system is that the system is stable.stable.

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Transfer functionTransfer function

Transfer function is the unit impulse responseTransfer function is the unit impulse response

e.g.e.g. First order process,First order process,

Unit impulse responseUnit impulse response is given byis given by

In the time domain,In the time domain,

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Unit impulse response of a 1st order processUnit impulse response of a 1st order process

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Deviation VariablesDeviation Variables

To remove dependence on initial conditione.g.

Compute equilibrium condition for a given and

Define deviation variables

Rewrite linear ODE

or

0

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Deviation VariablesDeviation Variables

Assume that we start at equilibrium

Transfer functions express extent of deviation from a givensteady-state

Procedure Find steady-stateWrite steady-state equation Subtract from linear ODEDefine deviation variables and their derivatives if required Substitute to re-express ODE in terms of deviation variables

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Process ModelingProcess Modeling

Gravity tank

Objectives:Objectives: height of liquid in tankFundamental quantity:Fundamental quantity: Mass, momentumAssumptions:Assumptions:

Outlet flow is driven by head of liquid in the tank Incompressible flow Plug flow in outlet pipe Turbulent flow

h

L

F

Fo

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Transfer FunctionsTransfer Functions

From mass balance and Newton’s law,

A system of simultaneous ordinary differential equations results

Linear or nonlinear?

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Nonlinear Nonlinear ODEsODEs

Q: If the model of the process is nonlinear, how do weexpress it in terms of a transfer function?

A: We have to approximate it by a linear one (i.e.Linearize)in order to take the Laplace.

f(x0)

f(x)

!!fxx( )0

xx0

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Nonlinear systemsNonlinear systems

First order Taylor series expansion

1. Function of one variable

2. Function of two variables

3. ODEs

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Transfer Transfer FFunctionunction

Procedure to obtain transfer function from nonlinearprocess models Find an equilibrium point of the system Linearize about the steady-state Express in terms of deviations variables about the steady-state Take Laplace transform Isolate outputs in Laplace domain

Express effect of inputs in terms of transfer functions

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Block DiagramsBlock Diagrams

Transfer functions of complex systems can be representedTransfer functions of complex systems can be representedin block diagram form.in block diagram form.

3 basic arrangements of transfer functions:3 basic arrangements of transfer functions:

1.1. Transfer functions in seriesTransfer functions in series

2.2. Transfer functions in parallelTransfer functions in parallel

3.3. Transfer functions in feedback formTransfer functions in feedback form

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Transfer functions in seriesTransfer functions in series

Overall operation is the multiplication of transfer functionsOverall operation is the multiplication of transfer functions

Resulting overall transfer functionResulting overall transfer function

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Transfer functions in series (two first order systems)Transfer functions in series (two first order systems)

Overall operation is the multiplication of transfer functionsOverall operation is the multiplication of transfer functions

Resulting overall transfer functionResulting overall transfer function

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DCDC Motor example:Motor example: In terms of angular velocityIn terms of angular velocity

In terms of the angleIn terms of the angle

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Transfer function in parallelTransfer function in parallel

OverallOverall transfer function is the addition oftransfer function is the addition of TFs TFs in parallelin parallel

+

+

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Transfer function in parallelTransfer function in parallel

OverallOverall transfer function is the addition oftransfer function is the addition of TFs TFs in parallelin parallel

+

+

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Transfer functions in (negative) feedback formTransfer functions in (negative) feedback form

Overall transfer functionOverall transfer function

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Transfer functions in (positive) feedback formTransfer functions in (positive) feedback form

Overall transfer functionOverall transfer function

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Example 3.20Example 3.20

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ExampleExample

A positive feedback loopA positive feedback loop

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Two systems in parallelTwo systems in parallel ReplaceReplace byby

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Two systems in parallelTwo systems in parallel

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A negative feedback loopA negative feedback loop

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Two process in seriesTwo process in series

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First Order SystemsFirst Order Systems

First order systemsFirst order systems are systems whose dynamics areare systems whose dynamics aredescribed by the transfer functiondescribed by the transfer function

wherewhere

is the systemis the system’’s s (steady-state) gain(steady-state) gain

is the is the time constanttime constant

First orderFirst order systems are the most common systems are the most common behaviourbehaviourencountered in practiceencountered in practice

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ExamplesExamples, Liquid storage

Assume: Incompressible flow Outlet flow due to gravity

Balance equation: Total Flow In Flow Out

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Balance equation:Balance equation:

Deviation variables about the equilibriumDeviation variables about the equilibrium

Laplace Laplace transformtransform

First order system withFirst order system with

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ExamplesExamples: Cruise control

DC Motor

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Liquid Storage Tank

Speed of a car

DC Motor

First order processes are characterized by:

1. Their capacity to store material, momentum and energy2. The resistance associated with the flow of mass, momentum or energy in reaching theircapacity

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Step response of first order process

Step input signal of magnitude M

The ultimate change in is given by

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Step responseStep response

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What do we look for?What do we look for? SystemSystem’’s Gain: Steady-State Responses Gain: Steady-State Response

Process Time Constant:Process Time Constant:

What do we need?What do we need? System initially atSystem initially at equilibriumequilibrium Step input of magnitude MStep input of magnitude M Measure process gain from new steady-stateMeasure process gain from new steady-state Measure time constantMeasure time constant

Time Required to Reach 63.2% of final value

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First order systems areFirst order systems are also called systems with finitealso called systems with finitesettling timesettling time The settling time is the time required for the system comesThe settling time is the time required for the system comes

within 5% of the total change and stays 5%within 5% of the total change and stays 5% for all timesfor all times

Consider the step responseConsider the step response

The overall change isThe overall change is

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Settling timeSettling time

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Process initially at equilibrium subject to a step ofProcess initially at equilibrium subject to a step ofmagnitude 1magnitude 1

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First order processFirst order process

Ramp response:Ramp response:

Ramp input of slope a

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

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First OrderFirst Order SystemsSystems

Sinusoidal responseSinusoidal response

Sinusoidal input Asin(ωt)

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

2

AR

φ

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10 -2 10 -1 10 0 10 1 10 210 -2

10 -1

10 0Bode Plots

10 -2 10 -1 10 0 10 1 10 2-100

-80-60

-40-20

0

High Frequency

AsymptoteCorner Frequency

Amplitude Ratio Phase Shift

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Integrating SystemsIntegrating Systems

Example: Liquid storage tankExample: Liquid storage tank

Laplace Laplace domaindomain dynamicsdynamics

If there is no outlet flow,If there is no outlet flow,

h

F

Fi

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ExampleExample CapacitorCapacitor

Dynamics of both systems is equivalentDynamics of both systems is equivalent

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Step input of magnitude MStep input of magnitude M

Out

put

Time

Inpu

t

Time

Slope =

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Unit impulse responseUnit impulse response

Out

put

Time

Inpu

t

Time

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Rectangular pulse responseRectangular pulse response

Out

put

Time

Inpu

t

Time

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Second order SystemsSecond order Systems

Second order process:Second order process: Assume the general formAssume the general form

wherewhere = Process steady-state gain= Process steady-state gain= Process time constant= Process time constant= Damping Coefficient= Damping Coefficient

Three families of processesThree families of processes

UnderdampedUnderdampedCritically DampedCritically DampedOverdampedOverdamped

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Second Order SystemsSecond Order Systems

Three types of second order process:

1. Two First Order Systems in series or in parallele.g. Two holding tanks in series

2. Inherently second order processes: Mechanical systemspossessing inertia and subjected to some external forcee.g. A pneumatic valve

3. Processing system with a controller: Presence of acontroller induces oscillatory behaviore.g. Feedback control system

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Multicapacity Multicapacity Second Order ProcessesSecond Order Processes Naturally arise from two first order processes in seriesNaturally arise from two first order processes in series

By multiplicative property of transfer functionsBy multiplicative property of transfer functions

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First order systems in parallelFirst order systems in parallel

OverallOverall transfer function a second order process (with one zero)transfer function a second order process (with one zero)

+

+

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Inherently second order process:Inherently second order process:e.g. Pneumatic Valve

x

p

By Newton’s law

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Feedback Control SystemsFeedback Control Systems

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Second order process:Second order process: Assume the general formAssume the general form

wherewhere = Process steady-state gain= Process steady-state gain= Process time constant= Process time constant= Damping Coefficient= Damping Coefficient

Three families of processesThree families of processes

UnderdampedUnderdampedCritically DampedCritically DampedOverdampedOverdamped

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Roots of the characteristic polynomial

Case 1) Two distinct real rootsSystem has an exponential behavior

Case 2) One multiple real rootExponential behavior

Case 3) Two complex rootsSystem has an oscillatory behavior

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Step response of magnitude MStep response of magnitude M

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ξ=2

ξ=0

ξ=0.2

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Observations

Responses exhibit overshoot when

Large yield a slow sluggish response

Systems with yield the fastest response without overshoot

As (with ) becomes smaller, system becomes moreoscillatory

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Characteristics of underdamped second order process

1. Rise time,2. Time to first peak,3. Settling time,4. Overshoot:

5. Decay ratio:

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Step responseStep response

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Sinusoidal ResponseSinusoidal Response

wherewhere

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More Complicated SystemsMore Complicated Systems

Transfer function typically written as rational function of polynomials

where and can be factored following the discussion on partialfraction expansion

s.t.

where and appear as real numbers or complex conjuguate pairs

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Poles and zeroesPoles and zeroes

Definitions: the roots of are called the zeroszeros of G(s)

the roots of are called the polespoles of G(s)

Poles: Directly related to the underlying differential equation

If , then there are terms of the form in vanishes to a unique point

If any then there is at least one term of the form does not vanish

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PolesPoles

e.g. A transfer function of the form

with can factored to a sum of

A constant term from A from the term A function that includes terms of the form

Poles can help us to describe the qualitative behavior of a complex system(degree>2)

The sign of the poles gives an idea of the stability of the system

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Poles

Calculation performed easily in MATLAB Function POLE, PZMAP

e.g.» s=tf(‘s’);» sys=1/s/(s+1)/(4*s^2+2*s+1);Transfer function:Transfer function: 1 1--------------------------------------------------4 s^4 + 6 s^3 + 3 s^2 + s4 s^4 + 6 s^3 + 3 s^2 + s» pole(sys)ans ans == 0 0 -1.0000 -1.0000 -0.2500 + 0.4330i -0.2500 + 0.4330i -0.2500 - 0.4330i -0.2500 - 0.4330i MATLAB

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PolesPoles

Function PZMAPFunction PZMAP» pzmap(sys) MATLAB

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PolesPoles

One constant poleOne constant pole integrating featureintegrating feature

One real negative poleOne real negative pole decaying exponentialdecaying exponential

A pair of complex roots with negative real partA pair of complex roots with negative real part decaying sinusoidaldecaying sinusoidal

What is the dominant feature?What is the dominant feature?

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PolesPoles

Step ResponseStep Response

Integrating factorIntegrating factor dominates thedominates the dynamic dynamic behaviourbehaviour

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PolesPoles

ExampleExample

PolesPoles -1.0000-1.0000

-0.0000 + 1.0000i-0.0000 + 1.0000i -0.0000 - 1.0000i-0.0000 - 1.0000i

One negative real poleOne negative real pole Two purely complex polesTwo purely complex poles

What is the dominant feature?What is the dominant feature?

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PolesPoles


Purely complex poles dominate the dynamicsPurely complex poles dominate the dynamics

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PolesPoles

ExampleExample


-0.6564-0.6564 -0.1868-0.1868

Three negative real polesThree negative real poles

What is the dominant dynamic feature?What is the dominant dynamic feature?

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PolesPoles


Slowest exponential (Slowest exponential ( ) dominates the dynamic) dominates the dynamicfeature (yields a time constant of 1/0.1868= 5.35 sfeature (yields a time constant of 1/0.1868= 5.35 s

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PolesPoles

ExampleExample

PolesPoles -4.6858-4.6858 0.3429 + 0.3096i0.3429 + 0.3096i 0.3429 - 0.3096i0.3429 - 0.3096i

One negative real rootOne negative real root Complex Complex conjuguate conjuguate roots withroots with positive realpositive real part part

The system is unstable (grows without bound)The system is unstable (grows without bound)

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PolesPoles


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PolesPoles

Two types of polesTwo types of poles

Slow (or dominant) polesSlow (or dominant) poles Poles that are closer toPoles that are closer to the imaginary axisthe imaginary axis

Smaller real part means Smaller real part means smaller exponential termsmaller exponential term andandslower decayslower decay

Fast polesFast poles Poles that are further from thePoles that are further from the imaginary axisimaginary axis

LargerLarger real part means larger exponentialreal part means larger exponential term and fasterterm and fasterdecaydecay

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PolesPoles

PolesPoles dictate the stability of the processdictate the stability of the process

Stable polesStable poles Poles with negative real partsPoles with negative real parts Decaying exponentialDecaying exponential

Unstable polesUnstable poles Poles withPoles with positive real partspositive real parts Increasing exponentialIncreasing exponential

Marginally stable polesMarginally stable poles Purely complexPurely complex polespoles Pure sinusoidalPure sinusoidal

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PolesPoles

Stable Unstable

MarginallyStable

Pure Exponential

OscillatoryBehaviour

IntegratingBehaviour

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PolesPoles

ExampleExample

Slow Poles

Unstable Poles

Stable Poles

Fast Poles

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Poles and zeroesPoles and zeroes

Definitions: the roots of are called the zeroszeros of G(s)

the roots of are called the polespoles of G(s)

Zeros: Do not affect the stability of the system but can modify the dynamical

behaviour

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ZerosZeros

Types of zeros:Types of zeros:

Slow zerosSlow zeros are are closer to the imaginary axis than the dominant poles ofcloser to the imaginary axis than the dominant poles of thethetransfer functiontransfer function Affect the Affect the behaviourbehaviour Results in overshoot (or undershoot)Results in overshoot (or undershoot)

Fast zerosFast zeros are further away to the imaginary axis are further away to the imaginary axis have a negligible impact on have a negligible impact on dynamicsdynamics

Zeros with negative real parts,Zeros with negative real parts, , are , are stable zerosstable zeros Slow stable zeros lead to overshoot Slow stable zeros lead to overshoot

Zeros withZeros with positive real parts,positive real parts, , are , are unstable zerosunstable zeros Slow unstable zeros lead to undershoot (or inverse response) Slow unstable zeros lead to undershoot (or inverse response)

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ZerosZeros

Can result from two processes in parallel

If gains are of opposite signs and time constants are different then a right halfplane zero occurs

+

+

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ZerosZeros

ExampleExample


-0.6564-0.6564 -0.1868-0.1868

ZerosZeros -10.000-10.000

What is the effect of the zero on the dynamic What is the effect of the zero on the dynamic behaviourbehaviour??

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ZerosZeros

Poles and zerosPoles and zeros

DominantPole

Fast, StableZero

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ZerosZeros

Unit step response:Unit step response:

Effect of zero isEffect of zero is negligiblenegligible

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ZerosZeros

ExampleExample


-0.6564-0.6564 -0.1868-0.1868

ZerosZeros -0.1000-0.1000


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ZerosZeros

Poles and zeros:Poles and zeros:

Slow (dominant)Stable zero

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ZerosZeros

UnitUnit step response:step response:

Slow dominant zeroSlow dominant zero yields an overshoot.yields an overshoot.

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ZerosZeros

ExampleExample


-0.6564-0.6564 -0.1868-0.1868

ZerosZeros 10.00010.000


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ZerosZeros

Poles and ZerosPoles and Zeros

Fast UnstableZero

DominantPole

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ZerosZeros

Unit Step Response:Unit Step Response:

Effect of unstableEffect of unstable zero is negligiblezero is negligible

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ZerosZeros

ExampleExample


-0.6564-0.6564 -0.1868-0.1868

ZerosZeros 0.10000.1000


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ZerosZeros

Poles and zeros:Poles and zeros:

Slow Unstable Zero

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ZerosZeros


Slow unstable zero causesSlow unstable zero causes undershoot (inverse undershoot (inverse reponsereponse))

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ZerosZeros

Observations:

Adding a stable zero to an overdamped system yields overshoot

Adding an unstable zero to an overdamped system yieldsundershoot (inverse response)

Inverse response is observed when the zeros lie in right halfcomplex plane,

Overshoot or undershoot are observed when the zero is dominant(closer to the imaginary axis than dominant poles)

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ZerosZeros

Example: System with complex zerosExample: System with complex zeros

DominantPole

SlowStable Zeros

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ZerosZeros

Poles and zerosPoles and zeros

PolesPoles havehave negative real parts: The system is stablenegative real parts: The system is stable

Dominant poles are real: yieldsDominant poles are real: yields an an overdamped behaviouroverdamped behaviour

A pair of slow complex stable poles: yields overshootA pair of slow complex stable poles: yields overshoot

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ZerosZeros


Effect ofEffect of slow zero is significant and yields oscillatory slow zero is significant and yields oscillatory behaviourbehaviour

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ZerosZeros

Example: System with zero at theExample: System with zero at the originorigin

Zero atOrigin

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ZerosZeros

Unit step responseUnit step response

Zero at the origin:Zero at the origin: eliminates the effect of the stepeliminates the effect of the step

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ZerosZeros

Observations:Observations:

ComplexComplex (stable/unstable) zero that is dominant yields an(stable/unstable) zero that is dominant yields an(overshoot/undershoot)(overshoot/undershoot)

ComplexComplex slow zero can introduceslow zero can introduce oscillatory oscillatory behaviourbehaviour

Zero at the origin eliminates (or zeroes out) the system responseZero at the origin eliminates (or zeroes out) the system response

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DelayDelay

Time required for the fluid to reach the valve

usually approximated as dead time

h

Fi

Control loop

Manipulation of valve does not lead to immediate change in level

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DelayDelay

Delayed transfer functions

e.g. First order plus dead-time

Second order plus dead-time

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DelayDelay

Dead time (delay)

Most processes will display some type of lag timeDead time is the moment that lapses between input changes and

process response

Step response of a first order plus dead time process

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DelayDelay Delayed step response:Delayed step response:

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DelayDelay Problem

use of the dead time approximation makes analysis (poles andzeros) more difficult

Approximate dead-time by a rational (polynomial)functionMost common is Pade approximation

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Pade Pade ApproximationsApproximations In general Pade approximations do not approximate dead-time very

well

Pade approximations are better when one approximates a first orderplus dead time process

Pade approximations introduce inverse response (right half planezeros) in the transfer function

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Pade Pade ApproximationsApproximations Step response of Step response of Pade Pade ApproximationApproximation

Pade Pade approximation introduces inverse responseapproximation introduces inverse response

transfer function models of dynamical processes · process control time domain transfer function...

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