5. MechatronicsWorkshop KCCDay2 Session1-2.pdf

8/19/2019 5. MechatronicsWorkshop KCCDay2 Session1-2.pdf

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K. Craig 1

Mechatronics for the 21st Century

Agreement?

Control System

Design

Agreement?

Expected

Closed-Loop

System

Response

Predicted

Closed-Loop

System

Response

System

Design

Concept

System Design

and Performance

Specifications

Concept

Physical

Model

Concept

Mathematical

Model

Predicted

Open-Loop

System

Response

ExpectedComponent

and Open-

Loop System

Response

Is predicted

response acceptable

with respect to

specifications?no

yes

no

Mechatronic System

Design ProcessSTART

HERE

Simplifying

Assumptions

no

yes

yes

no

Build and Test Physical SystemCheck That System Meets Specifications

Evaluate, Iterate, and Improve As Needed

Apply

Laws of Nature

Solve Equations:

Analytical &

Numerical

Past

Experience &Experiments

Engineering

JudgmentRe-evaluate

Physical Model

Assumptions &

Parameters

Identify

Model

Parameters

Improve Control Design:

Feedback, Feedforward,

Observers, Filters

Improve System Design:

Parameters and/or

Configuration / Concept

Design &

Simulate

R e a l W o r l d

S i m ul a t i onW or l d Re-evaluate

Past

Experience &

Experiments


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K. Craig 2

Mechatronics Master Class

Schedule Day 1 Day 2 Day 3

Session 1Mechatronics

and

Innovation

Modeling &

Analysis of

Dynamic Physical

Systems

High-Performance

Mechatronic

Motion Systems

Session 2Human-Centered

Design

Automotive

Mechatronics

Session 3

Model-Based

Design

Control System

Design: Feedback,Feedforward, &

Observers

Web-Handling

Mechatronic Applications

Session 4Mechatronic

System Design

Fluid Power

Mechatronic

Applications


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Modeling Engineering Systems K. Craig 1

Modeling Engineering Systems


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Engineering System Investigation Process

Physical

System

System

Measurement

Measurement Analysis

Physical

Model

Mathematical

Model

Parameter

Identification

Mathematical Analysis

Comparison:

Predicted vs.

Measured

Design

Changes

Is The

Comparison

Adequate ?

NO

YES

START HERE

The cornerstone of

modern engineering

practice !

Engineering

SystemInvestigation

ProcessModeling


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• Apply the steps in the process when:

– An actual physical system exists and one desires to

understand and predict its behavior.

– The physical system is a concept in the design process

that needs to be analyzed and evaluated.

• After recognizing a need for a new product or service, onegenerates design concepts by using: past experience

(personal and vicarious), awareness of existing hardware,

understanding of physical laws, and creativity.• The importance of modeling and analysis in the design

process has never been more important. Design concepts

can no longer be evaluated by the build-and-test approach

because it is too costly and time consuming. Validating thepredicted dynamic behavior in this case, when no actual

physical system exists, then becomes even more dependent

on one's past hardware and experimental experience.


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• What is a Physical Model?

– In general, a physical model is an imaginary physical system

which resembles an actual system in its most significant

features, but which is simpler, more ideal, and is thereby

more amenable to analytical studies.

– There is a hierarchy of physical models of varying complexitypossible.

– The difference between a physical system and a physical

model is analogous to the difference between the actualphysical terrain and a map.

– Not oversimplified, not overly complicated - a slice of reality.

– The very crux of engineering analysis and the hallmark ofevery successful engineer is the ability to make shrewd and

viable approximations which greatly simplify the system and

still lead to a rapid, reasonably accurate prediction of its

behavior.


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• What is a Mathematical Model?

– We apply the Laws of Nature to the Physical Model,NOT to the Physical System, to obtain the

Mathematical Model.

– What Laws of Nature?• Newton’s Laws of Motion

• Maxwell’s Equations of Electromagnetism

• Laws of Thermodynamics – Once we have the Mathematical Model of the

Physical Model, we then solve the equations, either

analytically or numerically, or both to get the greatestinsight, to predict how the Physical Model behaves.

This predicted behavior must then be compared to

the actual measured behavior of the Physical System.


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Balance: The Key to Success

Computer Simulation Without Experimental Verification

Is At Best Questionable, And At Worst Useless!


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Balance in Engineering is the Key!

• The essential characteristic of an engineer and

the key to success is a balance between thefollowing sets of skills:

– modeling (physical and mathematical), analysis

(closed-form and numerical simulation), and controldesign (analog and digital) of dynamic physical

systems

– experimental validation of models and analysis andunderstanding the key issues in hardware

implementation of designs


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• Dynamic Physical System

– Any collection of interacting elements for which thereare cause-and-effect relationships among the time-

dependent variables. The present output of the system

depends on past inputs.

• Analysis of the Dynamic Behavior of Physical

Systems

– Cornerstone of modern technology – More than any other field links the engineering

disciplines

• Purpose of a Dynamic System Investigation – Understand & predict the dynamic behavior of a system

– Modify and/or control the system, if necessary


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• Essential Features of the Study of Dynamic

Systems – Deals with entire operating machines and

processes rather than just isolated components.

– Treats dynamic behavior of mechanical, electrical,electromechanical, fluid, thermal, chemical, and

mixed systems.

– Emphasizes the behavioral similarity betweensystems that differ physically and develops general

analysis and design tools useful for all kinds of

physical systems. – Sacrifices detail in component descriptions so as to

enable understanding of the behavior of complex

systems made from many components.


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– Uses methods which accommodate componentdescriptions in terms of experimental

measurements, when accurate theory is lacking or

is not cost-effective, and develops universal lab

test methods for characterizing component

behavior.

– Serves as a unifying foundation for many practical

application areas, e.g., vibrations, measurementsystems, control systems, acoustics, vehicle

dynamics, etc.

– Offers a wide variety of computer software toimplement its methods of analysis and design.


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Electro-Dynamic Vibration Exciter

Physical System vs. Physical Model


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• This "moving coil" type of device converts an electrical

command signal into a mechanical force and/or motion and

is very common, e.g., vibration shakers, loudspeakers,linear motors for positioning heads on computer disk

memories, and optical mirror scanners.

• In all these cases, a current-carrying coil is located in asteady magnetic field provided by permanent magnets in

small devices and electrically-excited wound coils in large

ones.• Two electromechanical effects are observed in such

configurations:

– Generator Effect: Motion of the coil through the magnetic fieldcauses a voltage proportional to velocity to be induced into the coil.

– Motor Effect: Passage of current through the coil causes it toexperience a magnetic force proportional to the current.


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• Flexure Kf is an intentional soft spring (stiff, however, in

the radial direction) that serves to guide the axial motion

of the coil and table.

• Flexure damping Bf is usually intentional, fairly strong,

and obtained by laminated construction of the flexure

spring, using layers of metal, elastomer, plastic, and soon.

• The coupling of the coil to the shaker table would ideally

be rigid so that magnetic force is transmitted undistortedto the mechanical load. Thus Kt (generally large) and Bt(quite small) represent parasitic effects rather than

intentional spring and damper elements.• R and L are the total circuit resistance and inductance,

including contributions from both the shaker coil and the

amplifier output circuit.


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t t f t f t t c t t c t

c c t c t t c t

i c

M x K x B x B (x x ) K (x x )

M x B (x x ) K (x x ) Ki

Li e Ri Kx

= − − + − + −

= − − − − +

= − −

Equationsof Motion

f t f t t tt t

t t t tt t

ic c

t t t tc c

c c c c c

0 1 0 0 0

0K K B B K Bx x0M M M M 0x x

0 0 0 1 0 0ex x

K B K B 0K x xM M M M M 1

i iLK R

0 0 0L L

⎡ ⎤⎢ ⎥− − − − ⎡ ⎤⎢ ⎥⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

− −⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎣ ⎦− −⎢ ⎥

⎢ ⎥⎣ ⎦

State-Space

Representation

Mathematical Modeling: Laws of Nature applied to Physical Model


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Analytical Frequency Response Plots

Typical

Parameter

Values

(SI Units)L = 0.0012

R = 3.0

K = 190

Kt

= 8.16E8

Bt = 3850

Kf = 6.3E5

Bf = 1120

Mc = 1.815

Mt = 6.12

resonance due to flexures

parasitic resonance

due to coil-table

coupling


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Experimental Frequency Response Plot

B&K

PM Vibration Exciter

Type 4809


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Electro-PneumaticTransducer

Σ Σ


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This system can be

collapsed into a

simplified

approximate overall

model when

numerical valuesare properly

chosen:


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• It is interesting to note here that while the blockdiagram shows one input for the system, i.e.,

command voltage ein , there are possible undesired

inputs that must also be considered.

• For example, the ambient temperature will affect the

electric coil resistance, the permanent magnet

strength, the leaf-spring stiffness, the damper-oil

viscosity, the air density, and the dimensions of the

mechanical parts. All these changes will affect the

system output pressure po in some way, and the

cumulative effects may not be negligible.


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Temperature FeedbackControl System:

A Larger-Scale

Engineering System

Bridge

Circuit Amplifier Controller

Electro-

Pneumatic

Transducer

ValveChemical

Process

Thermistor

RV

eE

RC

DesiredTemperature

(set with RV)

Block Diagram of an Temperature Control System

Σ

Actual

Temperature

(measured with

RC)

eM pM xVTC


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• Note that in the block diagram of this system, the

detailed operation of the electropneumatic transducer

is not made apparent; only its overall input/output

relation is included.

• The designer of the temperature feedback-controldynamic system would consider the electropneumatic

transducer an off-the-shelf component with certain

desirable operating characteristics.• The methods of system dynamics are used by both

the electropneumatic transducer designer and the

designer of the larger temperature feedback-controlsystem.


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Physical Modeling

Less Real, Less Complex, More Easily Solved

Truth Model Design Model

More Real, More Complex, Less Easily Solved

Hierarchy Of Models

Always Ask: Why Am I Modeling?


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• Physical System

– Define the physical system to be studied, along withthe system boundaries, input variables, and output

variables.

• Physical System to Physical Model – In general, a physical model is an imaginary physical

system which resembles an actual system in its

salient features, but which is simpler, more ideal, andis thereby more amenable to analytical studies.

There is a hierarchy of physical models of varying

complexity possible. – Not oversimplified, not overly complicated - a slice of

reality.


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– The astuteness with which approximations are

made at the onset of an investigation is the verycrux of engineering analysis.

– The ability to make shrewd and viable

approximations which greatly simplify the systemand still lead to a rapid, reasonably accurate

prediction of its behavior is the hallmark of every

successful engineer. – What is the purpose of the model? Develop a set

of performance specifications for the model based

on the specific purpose of the model. Whatfeatures must be included? How accurately do

they need to be represented?


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– The challenges to physical modeling are formidable:• Dynamic behavior of many physical processes is

complex.

• Cause and effect relationships are not easilydiscernible.

• Many important variables are not readily identified.

• Interactions among the variables are hard tocapture.

• Engineering Judgment is the Key!


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– In modeling dynamic systems, we consider matter

and energy as being continuously, though not

necessarily uniformly, distributed over the space

within the system boundaries.

– This is the macroscopic or continuum point of view.We consider the system variables as quantities which

change continuously from point to point in the system

as well as with time and this always leads to adistributed-parameter physical model which results in

a partial differential equation mathematical model.

– Models in this most general form behave most like thereal systems at the macroscopic level.


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– Because of the mathematical complexity of these

models, engineers find it necessary and desirable towork with less exact models in many cases.

– Simpler models which concentrate matter and energy

into discrete lumps are called lumped-parameterphysical models and lead to ordinary differential

equation mathematical models.

– An understanding of the difference between

distributed-parameter and lumped-parameter models

is vital to the intelligent formulation and use of lumped

models.

– The time-variation of the system parameters can be

random or deterministic, and if deterministic, variable

or constant.


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• Comments on Truth Model vs. Design Model

– In modeling dynamic systems, we use engineering judgment and simplifying assumptions to develop a

physical model. The complexity of the physical model

depends on the particular need, and the intelligent

use of simple physical models requires that we have

some understanding of what we are missing when we

choose the simpler model over the more complex

model. – The truth model is the model that includes all the

known relevant characteristics of the real system.

This model is often too complicated for use inengineering design, but is most useful in verifying

design changes or control designs prior to hardware

implementation.


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– The design model captures the important features

of the process for which a control system is to be

designed or design iterations are to be performed,

but omits the details which you believe are not

significant.

– In practice, you may need a hierarchy of models of

varying complexity:

• A very detailed truth model for final

performance evaluation before hardwareimplementation

• Several less complex truth models for use in

evaluating particular effects

• One or more design models


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Approximation Mathematical

Simplification

Neglect small effects Reduces the number and

complexity of the equations of

motion

Assume the environment isindependent of system motions

Reduces the number andcomplexity of the equations of

motion

Replace distributed

characteristics with appropriatelumped elements

Leads to ordinary (rather than

partial) differential equations

Assume linear relationships Makes equations linear; allows

superposition of solutions Assume constant parameters Leads to constant coefficients in

the differential equations

Neglect uncertainty and noise Avoids statistical treatment


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• Neglect Small Effects

– Small effects are neglected on a relative basis. In

analyzing the motion of an airplane, we are unlikely to

consider the effects of solar pressure, the earth's

magnetic field, or gravity gradient. To ignore these

effects in a space vehicle problem would lead to

grossly incorrect results!

• Independent Environment

– In analyzing the vibration of an instrument panel in a

vehicle, we assume that the vehicle motion is

independent of the motion of the instrument panel.


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• Lumped Characteristics

– In a lumped-parameter model, system dependentvariables are assumed uniform over finite regions of

space rather than over infinitesimal elements, as in a

distributed-parameter model. Time is the onlyindependent variable and the mathematical model is

an ordinary differential equation.

– In a distributed-parameter model, time and spatialvariables are independent variables and the

mathematical model is a partial differential equation.

– Note that elements in a lumped-parameter model donot necessarily correspond to separate physical parts

of the actual system.


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– A long electrical transmission line has resistance,

inductance, and capacitance distributed continuouslyalong its length. These distributed properties are

approximated by lumped elements at discrete points

along the line. – It is important to note that a lumped-parameter model,

which may be valid in low-frequency operations, may

not be valid at higher frequencies, since the neglected

property of distributed parameters may become

important. For example, the mass of a spring may be

neglected in low-frequency operations, but it becomes

an important property of the system at highfrequencies.


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• Linear Relationships

– Nearly all physical elements or systems are inherentlynonlinear if there are no restrictions at all placed on

the allowable values of the inputs, e.g., saturation,

dead-zone, square-law nonlinearities. – If the values of the inputs are confined to a sufficiently

small range, the original nonlinear model of the

system may often be replaced by a linear model

whose response closely approximates that of the

nonlinear model.

– When a linear equation has been solved once, the

solution is general, holding for all magnitudes of

motion.


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– Linear systems also satisfy conditions for

superposition.

– The superposition property states that for a system

initially at rest with zero energy: (1) multiplying the

inputs by any constant multiplies the outputs by the

same constant, and (2) the response to several inputsapplied simultaneously is the sum of the individual

responses to each input applied separately.

• Constant Parameters – Time-varying systems are ones whose characteristics

change with time.

– Physical problems are simplified by the adoption of a

model in which all the physical parameters are

constant.


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• Neglect Uncertainty and Noise

– In real systems we are uncertain, in varying degrees,about values of parameters, about measurements,

and about expected inputs and disturbances.

Disturbances contain random inputs, called noise,

which can influence system behavior.

– It is common to neglect such uncertainties and noise

and proceed as if all quantities have definite values

that are known precisely.


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• Summary

– The most realistic physical model of a dynamicsystem leads to differential equations of motion that

are:

• nonlinear, partial differential equations with time-varying and space-varying parameters

• These equations are the most difficult to solve.

– The simplifying assumptions discussed lead to aphysical model of a dynamic system that is less

realistic and to equations of motion that are:

• linear, ordinary differential equations with constantcoefficients

• These equations are easier to solve and design

with.


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Classes of Systems


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Pure and Ideal Elements vs. Real Devices

• A pure element refers to an element (spring, damper,

inertia, resistor, capacitor, inductor, etc.) which has only the

named attribute.• For example, a pure spring element has no inertia or friction

and is thus a mathematical model (approximation), not a

real device.• The term ideal, as applied to elements, means linear , that

is, the input/output relationship of the element is linear, or

straight-line. The output is perfectly proportional to theinput.

• A device can be pure without being ideal and ideal without

being pure.


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• From a functional engineering viewpoint, nonlinear

behavior may often be preferable, even though it leadsto difficult equations.

• Why do we choose to define and use pure and ideal

elements when we know that they do not behave like thereal devices used in designing systems?

– Once we have defined these pure and ideal elements,

we can use these as building blocks to model realdevices more accurately.

• For example, if a real spring has significant friction and

mass, we model it as a combination of pure/ideal spring,mass, and damper elements, which may come quite

close in behavior to the real spring.


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Physical Model of a Real Spring


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• There are three basic types of building blocks,two that can store energy and one that

dissipates energy.

– All Mechanical Systems have three attributes - mass,springiness, and energy dissipation. The basic

elements corresponding to these attributes are inertia

(mass), damper (friction), and spring (compliance). – All Electrical Systems have three attributes -

inductance, resistance, and capacitance. The

corresponding basic elements are inductor, resistor,and capacitor.


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Mechanical System Elements

• It is hard to imagine any engineering system that does

not have mechanical components.

• Motion and force are concepts used to describe the

behavior of engineering systems that employ mechanical

components.

• The three basic mechanical building block elements are:

– Spring (elasticity or compliance) element

– Damper (friction or energy dissipation) element

– Mass (inertia) element

• There are both translational and rotational versions of

these basic building blocks.


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• These are passive (non-energy producing) devices.

• Driving Inputs are force and motion sources which causeelements to respond.

• Each of the elements has one of two possible energy

behaviors: – stores all the energy supplied to it

– dissipates all energy into heat by some kind of

“frictional” effect• Spring stores energy as potential energy.

• Mass stores energy as kinetic energy.

• Damper dissipates energy into heat.

• The Dynamic Response of each element is important.

S i El t


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• Spring Element

– A spring is a fundamental mechanical component found

intentionally or unintentionally in almost everymechanical system.

– Real-world spring is neither pure nor ideal.

– Real-world spring has inertia and friction.

– Pure spring has only elasticity - it is a mathematical

model, not a real device.

– Some dynamic operation requires that spring inertia

and/or damping not be neglected.

– Ideal spring is linear. However, nonlinear behavior may

often be preferable and give significant performanceadvantages.

2 1

F K(x x )= −

Pure & Ideal


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0

s

2 2x

s 0 s 0s

0

Differential Work Done

(f )dx (K x)dx

Total Work Done

K x C f (K x)dx

2 2

= =

= = =∫

Pure & IdealSpring Element


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• Damping Element

– A damper is a mechanical component often found inengineering systems.

– A pure damper dissipates all the energy supplied to it,

i.e., converts the mechanical energy to thermalenergy.

– Various physical mechanisms, usually associated

with some form of friction, can provide this dissipativeaction, e.g.,

• Coulomb (dry friction) damping

• Material (solid) damping

• Viscous (fluid) damping


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Pure & IdealDamper Element

Step Input

Forcecauses instantly

(a pure damper

has no inertia) aStep of dx/dt

and a

Ramp of x


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• Mass or Inertia Element

– All real mechanical components used in engineeringsystems have mass.

– A designer rarely inserts a component for the purpose

of adding inertia; the mass or inertia element oftenrepresents an undesirable effect which is unavoidable

since all materials have mass.

– There are some applications in which mass itselfserves a useful function, e.g., flywheels – a flywheel

is an energy-storage device and can be used as a

means of smoothing out speed fluctuations in enginesor other machines.

N t ’ L d fi th b h i f l t


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– Newton’s Law defines the behavior of mass elements

and refers basically to an idealized “point mass”:

– The concept of rigid body is introduced to deal with

practical situations. For pure translatory motion,every point in a rigid body has identical motion.

– Real physical bodies never display ideal rigid

behavior when being accelerated.

– The pure and ideal inertia element is a model, not a

real object. Real inertias may be impure (have some

springiness and friction) but are very close to ideal.

forces (mass)(acceleration)=∑

F Ma=

Pur & Id l


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Pure & IdealInertia Element

Real inertias may be

impure (have some

springiness and friction) but

are very close to ideal.

2 2

x 1 1(D) (D)

f MD T JD

θ= =

Inertia Element stores energy

as kinetic energy:

2 2Mv J or

2 2

ω


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Ideal vs. Real Sources

• External driving agencies are physical quantities which

pass from the environment, through the interface into the

system, and cause the system to respond.• In practical situations, there may be interactions between

the environment and the system; however, we often use

the concept of ideal source.• An ideal source (force, motion, voltage, current, etc.) is

totally unaffected by being coupled to the system it is

driving.• For example, a “real” 6-volt battery will not supply 6 volts

to a circuit! The circuit will draw some current from the

battery and the battery’s voltage will drop.


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Physical Model to Mathematical Model

• We derive a mathematical model to represent the physical

model, i.e., apply the Laws of Nature to the physical model and

write down the differential equations of motion.• The goal is a generalized treatment of dynamic systems,

including mechanical, electrical, electromechanical, fluid,

thermal, chemical, and mixed systems.

– Define System: Boundary, Inputs, Outputs

– Define Variables: Through and Across Variables

– Write System Relations: Dynamic Equilibrium Relations and

Compatibility Relations

– Write Constitutive Relations: Physical Relations for Each

Element

– Combine: Generate State Equations


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• Define System

– A system must be defined before equilibrium and/orcompatibility relations can be written.

– Unless physical boundaries of a system are clearly

specified, any equilibrium and/or compatibilityrelations we may write are meaningless.

• Define Variables

– Physical Variables

• Select precise physical variables (velocity, voltage,

pressure, flow rate, etc.) with which to describe the

instantaneous state of a system, and in terms ofwhich to study its behavior.


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– Through Variables

• Through variables (one-point variables) measure the

transmission of something through an element, e.g.,

– electric current through a resistor

– fluid flow through a duct – force through a spring

– Across Variables

• Across variables (two-point) variables measure adifference in state between the ends of an element,

e.g.,

– voltage drop across a resistor – pressure drop between the ends of a duct

– difference in velocity between the ends of a

damper


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– In addition to through and across variables, integrated

through variables (e.g., momentum) and integratedacross variables (e.g., displacement) are important.

• Write System Relations

– Dynamic Equilibrium Relations• Write dynamic equilibrium relations to describe the

balance - of forces, of flow rates, of energy - which

must exist for the system and its subsystems.• Equilibrium relations are always relations among

through variables, e.g.,

– Kirchhoff’s Current Law (at an electrical node) – continuity of fluid flow

– equilibrium of forces meeting at a point


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– Compatibility Relations

• Write system compatibility relations to describehow motions of the system elements are

interrelated because of the way they are

interconnected.• These are inter-element or system relations.

• Compatibility relations are always relations among

across variables, e.g., – Kirchhoff’s Voltage Law around a circuit

– pressure drop across all the interconnected

stages of a fluid system

– geometric compatibility in a mechanical system


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• Write Physical Relations for Each Element

– These relations are called constitutive physicalrelations as they concern only individual elements or

constituents of the system.

– They are natural physical laws which the individualelements of the system obey, e.g.,

• mechanical relations between force and motion

• electrical relations between current and voltage• electromechanical relations between force and

magnetic field

• thermodynamic relations between temperature,pressure, etc.


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– They are relations between through and across

variables of each individual physical element. – They may be algebraic, differential, integral, linear or

nonlinear, constant or time-varying.

– They are purely empirical relations observed byexperiment and not deduced from any basic

principles.

• Combine System Relations and Physical

Relations to Generate System Differential

Equations


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• Study Dynamic Behavior – Study the dynamic behavior of the mathematicalmodel by solving the differential equations of

motion either through mathematical analysis or

computer simulation.

– Dynamic behavior is a consequence of the system

structure - don’t blame the input!

– Seek a relationship between physical model

structure and behavior.

– Develop insight into system behavior.

• Comparison: Actual vs Predicted


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Comparison: Actual vs. Predicted

– Compare the predicted dynamic behavior to the

measured dynamic behavior from tests on the actualphysical system; make physical model corrections, if

necessary.

• Make Design Decisions

– Make design decisions so that the system will behave

as desired:• modify the system (e.g., change the physical

parameters of the system, add a sensor, change

the actuator or its location)• control the system (e.g., augment the system,

typically by adding a dynamic system called a

compensator or controller)


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Dynamic System Response K. Craig 1

Dynamic System Response


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Dynamic System Response

• LTI Behavior vs. Non-LTI Behavior

• Solution of Linear, Constant-Coefficient, Ordinary

Differential Equations

– Classical Operator Method

– Laplace Transform Method

• Laplace Transform Properties

• 1st-Order Dynamic System Time and Frequency

Response• 2nd-Order Dynamic System Time and Frequency

Response

• Dynamic System Response Example Problems


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LTI Behavior vs. Non-LTI Behavior

• Linear Time-Invariant (LTI) Systems

– A frequency-domain transfer function is limited todescribing elements that are linear and time invariant

– severe restrictions! No real-world system meets

them!

– Linear Time-Invariant System Properties

• Homogeneity: If r(t) → c(t), then kr(t) → kc(t)

• Superposition: If r 1(t) → c1(t) and r 2(t) → c2(t), thenr 1(t) + r 2(t) → c1(t) + c2(t)

• Time Invariance: If r(t) → c(t), then r(t-t1) → c(t-t1)

C f S


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– Comments on the Principle of Superposition

• The principle of superposition states that if the

system has an input that can be expressed as the

sum of signals, then the response of the system can

be expressed as the sum of the individual

responses to the respective signals. Superpositionapplies if and only if the system is linear .

• Using the principle of superposition, we can solve

for the system responses to a set of elementarysignals. We are then able to solve for the response

to a general signal simply by decomposing the

given signal into a sum of the elementarycomponents and, by superposition, concluding that

the response to the general signal is sum of the

responses to the elementary signals.

• In order for this process to work, the


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p ,

elementary signals need to be sufficiently rich

that any reasonable signal can be expressedas a sum of them, and their responses have to

be easy to find. The most common candidate

for elementary signals for use in linear systems

are the impulse and the exponential.

• The unit impulse δ(t) is a pulse of zero durationand infinite height. The area under the unit

impulse (its strength) is equal to one.

• The impulse δ(t) is defined by δ(t-τ) = 0 for all t≠ τ. It has the property that if f(t) is continuousat t = τ, then

f ( ) (t )d f (t)

∞

−∞ τ δ − τ τ =∫

(t )dt 1+

−

τ

τδ − τ =∫

Th i l i h t d i t th t


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• The impulse is so short and so intense that no

value of f matters except over the short range

where the δ occurs. The integral can be viewed asrepresenting the function f as a sum of impulses.

• To find the response to an arbitrary input, the

principle of superposition tells us that we need onlyfind the response to a unit impulse.

• For a linear, time-invariant system, the impulse

response, i.e., the response at time t to an impulseapplied at time τ, can be expressed as h(t – τ)because the response at time t to an input applied

at time τ depends only on the difference betweenthe time the impulse is applied and the time we areobserving the response.


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• For linear, time-invariant systems with input

u(t), the superposition integral, called the

convolution integral, is:

• Here u(t) is the input to the system and h(t-τ) is

the impulse response of the system.

y(t) u( )h(t )d u(t )h( )d u h

∞ ∞

−∞ −∞= τ − τ τ = − τ τ τ = ∗∫ ∫

Convolution Example y ky u (t) y(0 ) 0−+ = = δ =


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– Convolution Example0 0 0

0 0 0

y y ( ) y( )

ydt k ydt (t)dt

y(0 ) y(0 ) k(0) 1

y(0 ) 1

y ky 0 y(0 ) 1

+ + +

− − −

+ −

+

+

+ = δ⎡ ⎤− + =⎣ ⎦

=+ = =

∫ ∫ ∫

st

st st

kt

Assume y Ae

Ase kAe 0

s k A 1

y(t) h(t) e for t 0−

=

+ =

= − == = >

0 t 01 t 0

<≥

1(t)

kty(t) h(t) e 1(t)−= =

The response to a general input u(t) is

given by the convolution of this impulseresponse and the input:

kt

kt

0

y(t) e 1(t)u(t )d

e u(t )d

∞ −

−∞

∞ −

= − τ τ

= − τ τ

∫

∫

unit step function

impulse response

A consequence of convolution is the concept of transfer


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– A consequence of convolution is the concept of transfer

function.

– Note that both the input and output are exponential timefunctions and that the output differs from the input only in

the amplitude H(s).

– The function H(s) is the transfer function from the inputto output of the system. It is the ratio of the Laplace

transform of the output to the Laplace transform of the

input assuming all initial conditions are zero.

( )

s( t ) st

st s st s

st s

y t h( )u(t )d Convolution Integral

h( )e d Input e

h( )e e d e h( )e d

e H(s) where H(s) h( )e d

∞

−∞∞ −τ

−∞

∞ ∞− τ − τ

−∞ −∞

∞ − τ

−∞

= τ − τ τ

= τ τ =

= τ τ = τ τ= = τ τ

∫∫

∫ ∫∫

– If the input is the unit impulse function δ(t), then y(t) is


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If the input is the unit impulse function δ(t), then y(t) isthe unit impulse response. The Laplace transform of

δ(t) is 1 and the Laplace transform of y(t) is Y(s), soY(s) = H(s).

– The transfer function H(s) is the Laplace transform of

the unit impulse response h(t) where the Laplacetransform is defined by:

– Thus, if one wishes to characterize a linear time-invariant system, one applies a unit impulse, and the

resulting response is a description (inverse Laplace

transform) of the transform function.

st

st

0

H(s) h(t)e dt

h(t)e dt since h(t) 0 for t 0

∞ −

−∞∞ −

=

= =


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a linear time-invariant system is in finding the

frequency response. – Euler’s relation is:

– Let s = jω and, by superposition, the response tothe sum of these two exponentials which make up

the cosine signal, is the sum of the responses:

– The transfer function H(jω) is a complex number

that can be expressed in polar form (magnitudeand phase form) as H(jω) = M(ω)e jΦ(ω).

j t j tAAcos( t) (e e )2

ω − ωω = +

j t j tAy(t) H( j )e H( j )e2

ω − ω⎡ ⎤= ω + − ω⎣ ⎦

j( t ) j( t )Ay(t) M e e AM cos( t )2

ω +φ − ω +φ⎡ ⎤= + = ω + φ⎣ ⎦

– This means that if a system represented by the


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– This means that if a system represented by the

transfer function H(s) has a sinusoidal input with

magnitude A, the output will be sinusoidal at thesame frequency with magnitude AM and will be

shifted in phase by the angle Φ.

– The response of a linear time-invariant system to asinusoid of frequency ω is a sinusoid with the samefrequency and with an amplitude ratio equal to the

magnitude of the transfer function evaluated at theinput frequency. The phase difference between input

and output signals is given by the phase of the

transfer function evaluated at the input frequency.

The magnitude ratio and phase difference can be

computed from the transfer function or measured

experimentally.

T f F i h b i f l i l l


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– Transfer Functions, the basis of classical control

theory, require LTI systems, but no real-world system

is completely LTI. There are regions of nonlinear

operation and often significant parameter variation.

– Examples of Linear Behavior

• addition, subtraction, scaling by a fixed constant,

integration, differentiation, time delay, and

sampling

– No practical control system is completely linear and

most vary over time.

– LTI systems can be represented completely in the

frequency domain. Non-LTI systems can have

frequency response plots, however, these plots

change depending on system operating conditions.

• Non-LTI Behavior


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Non LTI Behavior

– Non-LTI behavior is any behavior that violates one

or more of the three criteria for an LTI system.

– Within nonlinear behavior, an important distinction

is whether the variation is slow or fast with respect

to the loop dynamics.

• Slow Variation

– When the variation is slow, the nonlinear behavior

may be viewed as a linear system with parametersthat vary during operation.

– The dynamics can still be characterized effectively

with a transfer function. However, the frequencyresponse plots will change at different operating

points.

• Fast Response


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Fast Response

– If the variation of the loop parameter is fast with

respect to the loop dynamics, the situation becomesmore complicated. Transfer functions cannot be

relied upon for analysis.

– The definition of fast depends on the systemdynamics.

• Fast vs. Slow

– The line between fast and slow is determined bycomparing the rate at which the parameter changes

to the bandwidth of the control system. If the

parameter variation occurs over a period of time nofaster than 10 times the control loop settling time, the

effect can be considered slow for most applications.


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• Dealing with Nonlinear Behavior

– Nonlinear behavior can usually be ignored if the

changes in parameter values effect the loop gain

by no more than about 25%. A variation this small

will be tolerated by systems with reasonablemargins of stability.

– If a parameter varies more than that, there are at

least three courses of action:• Modify the plant

• Tune for the worst-case conditions

• Compensate for the nonlinearity of the controlloop

• Modify the Plant


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• Modify the Plant

– Modifying the plant to reduce the parameter

variation is the most straightforward solution to

nonlinear behavior. It cures the problem without

adding complexity to the controller or

compromising system performance.

– This solution is commonly employed in the sense

that components used in control systems are

generally better (closer to LTI) than componentsused in open-loop systems.

– Enhancing the LTI behavior of a loop component

can increase its cost significantly. Componentsfor control systems are often more expensive than

open-loop components.

• Tune for the Worst-Case Conditions


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Tune for the Worst Case Conditions

– Assuming that the variation from the non-LTI

behavior is slow with respect to the control loop, itseffect is to change gains in the control loop. In this

case, the operating conditions can be varied to

produce the worst-case gains while tuning the controlsystem. Doing so will ensure stability for all

operating conditions.

– Tuning the system for worst-case operatingconditions generally implies tuning the proportional

gain of the inner loop when the plant gain is at its

maximum. This ensures that the inner loop will bestable in all conditions; parameter variation will only

lower the loop gain, which will reduce

responsiveness but will not cause instability.

– The other loop gains (inner loop integral and the outer

loops) should be stabilized when the plant gain is


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loops) should be stabilized when the plant gain is

minimized. This is because minimizing the plant gain

reduces the inner loop response; this will provide the

maximum phase lag to the outer loops and again

provides the worst case for stability.

– So tune the proportional gain with a high plant gain

and tune the other gains with a low plant gain to

ensure stability in all conditions.

– The penalty for tuning for worst case is the reduction inresponsiveness. Consider the proportional gain.

Because the proportional term is tuned with the highest

plant gain, the loop gain will be reduced at operatingpoints where the plant gain is low.

– In general, you should expect to lose responsiveness

in proportion to plant variation.

• Compensate in the Control Loop


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Compensate in the Control Loop

– Compensating for the nonlinear behavior in the

controller requires that a gain equal to the inverseof the non-LTI behavior be placed in the loop.

– This is called gain scheduling. By using gain

scheduling, the impact of the non-LTI behavior iseliminated from the control loop.

– Gain scheduling requires that the non-LTI

behavior be slow with respect to any transferfunctions between the non-LTI component and the

scheduled gain.

– This is a less onerous requirement than beingslow with respect to the loop bandwidth because

the loop components are always much faster than

the loop itself.

– Gain scheduling assumes that the non-LTI behavior

b di t d t bl ( ll


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can be predicted to reasonable accuracy (generally ±

25%) based on information available to the controller.This is often the case.

– Many times, a dedicated control loop will be placed

under the direction of a larger system controller. Themore flexible system controller can be used to

accumulate information on a changing gain and then

modify gains inside dedicated controllers to affect the

compensation.

– The chief shortcoming of gain scheduling via the

system controller is limited speed. The system

controller may be unable to keep up with the fastestplant variations. Still this solution is commonly

employed because of the controller’s higher level of

flexibility and broader access to information.

Solution of Linear, Constant-Coefficient


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,

Ordinary Differential Equations• A basic mathematical model used in many areas of

engineering is the linear, ordinary differential

equation with constant coefficients:

• qo is the output (response) variable of the system

• qi is the input (excitation) variable of the system

• an and bm are the physical parameters of the system

n n 1

o o on n 1 1 0 on n 1

m m 1

i i im m 1 1 0 im m 1

d q d q dqa a a a q

dt dt dt

d q d q dq b b b b q

dt dt dt

−

− −

−

− −

+ + + + =

+ + + +

• Straightforward analytical solutions are available no


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g y

matter how high the order n of the equation.

• Review of the classical operator method for solving lineardifferential equations with constant coefficients will be

useful. When the input qi(t) is specified, the right hand

side of the equation becomes a known function of time,f(t).

• The classical operator method of solution is a three-step

procedure: – Find the complimentary (homogeneous) solution qoc for

the equation with f(t) = 0.

– Find the particular solution qop with f(t) present. – Get the complete solution qo = qoc + qop and evaluate

the constants of integration by applying known initial

conditions.

• Step 1


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Step 1

– To find qoc

, rewrite the differential equation using

the differential operator notation D = d/dt, treat the

equation as if it were algebraic, and write the

system characteristic equation as:

– Treat this equation as an algebraic equation in the

unknown D and solve for the n roots (eigenvalues)

s1, s2, ..., sn. Since root finding is a rapid

computerized operation, we assume all the roots

are available and now we state rules that allowone to immediately write down qoc.

n n 1

n n 1 1 0a D a D a D a 0−

−+ + + + =

– Real unrepeated root s1:1s t

oc 1q c e=


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Real, unrepeated root s1:

– Real root s2

repeated m times:

– When the a0 to an in the differential equation are

real numbers, then any complex roots that mightappear always come in pairs a ± ib:

– For repeated root pairs a ± ib, a ± ib, and soforth, we have:

– The c’s and φ ’s are constants of integration whose

numerical values cannot be found until the last

step.

oc 1q c e

2 2 2 2s t s t s t s t2 m

oc 0 1 2 mq c e c te c t e c t e= + + + +

at

ocq ce sin(bt )= + φ

at at

oc 0 0 1 1q c e sin(bt ) c te sin(bt )= + φ + + φ +


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• Step 2

– The particular solution qop takes into account the"forcing function" f(t) and methods for getting the

particular solution depend on the form of f(t).

– The method of undetermined coefficients providesa simple method of getting particular solutions for

most f(t)'s of practical interest.

– To check whether this approach will work,differentiate f(t) over and over. If repeated

differentiation ultimately leads to zeros, or else to

repetition of a finite number of different timefunctions, then the method will work.


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– The particular solution will then be a sum of terms

made up of each different type of function found inf(t) and all its derivatives, each term multiplied by

an unknown constant (undetermined coefficient).

– If f(t) or one of its derivatives contains a termidentical to a term in qoc, the corresponding terms

should be multiplied by t.

– This particular solution is then substituted into thedifferential equation making it an identity. Gather

like terms on each side, equate their coefficients,

and obtain a set of simultaneous algebraic

equations that can be solved for all the

undetermined coefficients.


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• Step 3 – We now have qoc (with n unknown constants) and

qop (with no unknown constants).

– The complete solution qo = qoc + qop. – The initial conditions are then applied to find the n

unknown constants.

• Certain advanced analysis methods are most easily

d l d h h h f h L l T f


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developed through the use of the Laplace Transform.

• A transformation is a technique in which a function istransformed from dependence on one variable to

dependence on another variable. Here we will

transform relationships specified in the time domaininto a new domain wherein the axioms of algebra can

be applied rather than the axioms of differential or

difference equations.

• The transformations used are the Laplace

transformation (differential equations) and the Z

transformation (difference equations).

• The Laplace transformation results in functions of the

time variable t being transformed into functions of the

frequency-related variable s.

• The Z transformation is a direct outgrowth of the


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g

Laplace transformation and the use of a modulated

train of impulses to represent a sampled functionmathematically.

• The Z transformation allows us to apply the

frequency-domain analysis and design techniques ofcontinuous control theory to discrete control systems.

• One use of the Laplace Transform is as an

alternative method for solving linear differentialequations with constant coefficients. Although this

method will not solve any equations that cannot be

solved also by the classical operator method, itpresents certain advantages.

– Separate steps to find the complementary


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p p p y

solution, particular solution, and constants of

integration are not used. The complete solution,including initial conditions, is obtained at once.

– There is never any question about which initial

conditions are needed. In the classical operatormethod, the initial conditions are evaluated at t =

0+, a time just after the input is applied. For some

kinds of systems and inputs, these initialconditions are not the same as those before the

input is applied, so extra work is required to find

them. The Laplace Transform method uses theconditions before the input is applied; these are

generally physically known and are often zero,

simplifying the work.

– For inputs that cannot be described by a single

f l f th i ti b t t b d fi d


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formula for their entire course, but must be defined

over segments of time, the classical operator methodrequires a piecewise solution with tedious matching of

final conditions of one piece with initial conditions of

the next. The Laplace Transform method handles

such discontinuous inputs very neatly.

– The Laplace Transform method allows the use of

graphical techniques for predicting system

performance without actually solving system

differential equations.

• All theorems and techniques of the Laplace Transform

derive from the fundamental definition for the direct

Laplace Transform F(s) of the time function f(t):

[ ] st0

L f (t) F(s) f (t)e dt t > 0 s complex variable i∞ −= = = = σ + ω

∫

• This integral cannot be evaluated for all f(t)'s, but

h it it t bli h i i f f ti


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when it can, it establishes a unique pair of functions,

f(t) in the time domain and its companion F(s) in the sdomain. Comprehensive tables of Laplace

Transform pairs are available. Signals we can

physically generate always have corresponding

Laplace transforms. When we use the Laplace

Transform to solve differential equations, we must

transform entire equations, not just isolated f(t)

functions, so several theorems are necessary for this.

• Linearity (Superposition and Amplitude Scaling)

Theorem:

[ ] [ ] [ ]1 1 2 2 1 1 2 2 1 1 2 2L a f (t) a f (t) L a f (t) L a f (t) a F (s) a F (s)+ = + = +


100/175

• Integration Theorem:


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– Again, the initial values of f(t) and its integrals areevaluated numerically at a time instant before the

driving input is applied.

• Convolution Theorem:

– Convolution in the time domain corresponds to

multiplication in the frequency domain.

( )

( )( )

( ) ( )

1

k nn

n n k 1

k 1n -0n

F(s) f (0)

L f (t)dt s s

F(s) f (0)L f (t)

s s

where f (t) f (t)(dt) and f (t) f (t)

−

−−

− +

=−

⎡ ⎤ = +⎣ ⎦

⎡ ⎤ = +⎣ ⎦

= ∫ ∫ =

∫∑

[ ]1 2 1 2L f (t) f (t) F (s)F (s)∗ =

• Time Delay Theorem:


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y

– The Laplace Transform provides a theorem useful

for the dynamic system element called dead time

(transport lag) and for dealing efficiently with

discontinuous inputs.

• Time Scaling Theorem: – If the time t is scaled by a factor a, then the

Laplace transform of the time-scaled signal is

u(t) 1.0 t 0

u(t) 0 t 0

u(t a) 1.0 t a

u(t a) 0 t a

= ⇒ >

= ⇒ <

− = ⇒ >− = ⇒ <

[ ] asL f (t a)u(t a) e F(s)−− − =

1 sF

a a

⎛ ⎞⎜ ⎟⎝ ⎠

• Final Value Theorem:

If we know Q (s) q ( ) can be found quickly without


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– If we know Q0(s), q0(∞) can be found quickly without

doing the complete inverse transform by use of the finalvalue theorem.

– This is true if the system and input are such that the

output approaches a constant value as t approaches ∞. – The DC gain of a system, the steady-state value of theunit-step response, is given by:

• Initial Value Theorem:

– This theorem is helpful for finding the value of f(t) just

after the input has been applied, i.e., at t = 0+. In gettingthe F(s) needed to apply this theorem, our usual

definition of initial conditions as those before the input is

applied must be used.

t s 0lim f (t) lim sF(s)→∞ →=

t 0 slim f (t) lim sF(s)→ →∞=

s 0 s 0

1DC Gain limsG(s) limG(s)

s→ →= =

• Impulse Function (t)δ


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– The step function is the integral of the impulse

function, or conversely, the impulse function is thederivative of the step function.

1/b

b

Approximating function for

the unit impulse function

p(t)

t

[ ]

b 0(t) lim p(t)

du 1L (t) L sU(s) s 1.0

dt s

→δ =

⎡ ⎤δ = = = =⎢ ⎥

⎣ ⎦

(t) 0 t 0

(t)dt 1 0+ε

−ε

δ = ⇒ ≠δ = ⇒ ε >∫

– When we multiply the impulse function by some


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When we multiply the impulse function by some

number, we increase the “strength of the impulse”,but “strength” now means area, not height as it

does for “ordinary” functions.

• An impulse that has an infinite magnitude and zero

duration is mathematical fiction and does not occur in

physical systems.

• If, however, the magnitude of a pulse input to a

system is very large and its duration is very short

compared to the system time constants, then we can

approximate the pulse input by an impulse function.

• The impulse input supplies energy to the system in

an infinitesimal time.


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Approximate and Exact

Impulse Functions

If es

=1.0 (unit step function),

its derivative is the unit

impulse function with a

strength (or area) of one unit.

This “non-rigorous” approach

does produce the correct result.

• Inverse Laplace Transformation


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– A convenient method for obtaining the inverse

Laplace transform is to use a table of Laplace

transforms. In this case, the Laplace transform must

be in a form immediately recognizable in such a table.

– If a particular transform F(s) cannot be found in atable, then we may expand it into partial fractions and

write F(s) in terms of simple functions of s for which

inverse Laplace transforms are already known. – These methods for finding inverse Laplace transforms

are based on the fact that the unique correspondence

of a time function and its inverse Laplace transformholds for any continuous time function.


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Time Response & Frequency Response


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1st-Order Dynamic System

Example: RC Low-Pass Filter

in out

in out

Dynamic System Investigation

of the

RC Low-Pass Filter

Zero-Order Dynamic System Model


110/175


Zero Order Dynamic System Model

Step ResponseFrequency Response

Validation of a Zero-Order


111/175


Dynamic System Model


112/175

• How would you determine if an experimentally-

d t i d t f t ld b


113/175


determined step response of a system could be

represented by a first-order system step response?

t

o is

t

o is

is

t

o

is

o10 10

is

q (t) Kq 1 e

q (t) Kqe

Kqq (t)

1 eKq

q (t) t tlog 1 log e 0.4343

Kq

−τ

−τ

−τ

⎛ ⎞

= −⎜ ⎟⎝ ⎠

−= −

− =

⎡ ⎤− = − = −⎢ ⎥ τ τ⎣ ⎦

Straight-Line Plot:

o10

is

q (t)log 1 vs. t

Kq

⎡ ⎤−⎢ ⎥

⎣ ⎦

Slope = -0.4343/τ

– This approach gives a more accurate value of τ sinceth b t li th h ll th d t i t i d th


114/175


the best line through all the data points is used rather

than just two points, as in the 63.2% method.Furthermore, if the data points fall nearly on a straight

line, we are assured that the instrument is behaving as

a first-order type. If the data deviate considerably froma straight line, we know the system is not truly first order

and a τ value obtained by the 63.2% method would be

quite misleading. – An even stronger verification (or refutation) of first-order

dynamic characteristics is available from frequency-

response testing. If the system is truly first-order, the

amplitude ratio follows the typical low- and high-

frequency asymptotes (slope 0 and –20 dB/decade) and

the phase angle approaches -90° asymptotically.


115/175


– If these characteristics are present, the numerical

value of τ is found by determining ω (rad/sec) atthe breakpoint and using τ = 1/ωbreak. Deviationsfrom the above amplitude and/or phase

characteristics indicate non-first-order behavior.

• What is the relationship between the unit-step


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What is the relationship between the unit step

response and the unit-ramp response and betweenthe unit-impulse response and the unit-step

response?

– For a linear time-invariant system, the response tothe derivative of an input signal can be obtained

by differentiating the response of the system to the

original signal.

– For a linear time-invariant system, the response to

the integral of an input signal can be obtained by

integrating the response of the system to the

original signal and by determining the integrationconstants from the zero-output initial condition.

• Unit-Step Input is the derivative of the Unit-Ramp


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p p p

Input.• Unit-Impulse Input is the derivative of the Unit-Step

Input.

• Once you know the unit-step response, take thederivative to get the unit-impulse response and

integrate to get the unit-ramp response.

System Frequency Response


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System Frequency Response

Bode Plotting of


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dB = 20 log10 (amplitude ratio)

decade = 10 to 1 frequency change

octave = 2 to 1 frequency change

g

1st

-Order Frequency

Response

0.01 40 dB0.1 20 dB

0.5 6 dB

1.0 0 dB2.0 6 dB

10.0 20 dB

100.0 40 dB

= −= −

= −

==

=

=

A l El t i


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Analog Electronics:

RC Low-Pass Filter

Time Response &

Frequency Response

outin

outin

outout

in

ee RCs 1 R

ii Cs 1

e 1 1 when i 0

e RCs 1 s 1

+ − ⎡ ⎤⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦⎣ ⎦ ⎣ ⎦

= = =+ τ +

Time Response to Unit Step Input


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0 1 2 3 4 5 6 7 8

x 10-4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (sec)

A m p l i t u d e

Time Constant τ = RC

R = 15 K Ω

C = 0.01 μF


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• Time Constant τ

– Time it takes the step response to reach 63% of

the steady-state value

• Rise Time Tr

= 2.2 τ

– Time it takes the step response to go from 10% to

90% of the steady-state value

• Delay Time Td = 0.69 τ – Time it takes the step response to reach 50% ofthe steady-state value

Frequency

R


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Bandwidth = 1/τ

Response

1out

2 2 1 2 2in

e K K 0 K (i ) tan

e i 1 ( ) 1 tan ( ) 1

−

−

∠ω = = = ∠ − ωτ

ωτ + ωτ + ∠ ωτ ωτ +

R = 15 K Ω

C = 0.01 μF

• Bandwidth


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– The bandwidth is the frequency where the amplituderatio drops by a factor of 0.707 = -3dB of its gain at

zero or low-frequency.

– For a 1

st

-order system, the bandwidth is equal to1/τ.

– The larger (smaller) the bandwidth, the faster

(slower) the step response.

– Bandwidth is a direct measure of system

susceptibility to noise, as well as an indicator of the

system speed of response.

1Response to Input 1061 Hz Sine Wave

Amplitude Ratio = 0.707 = -3 dB Phase Angle = -45°


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0 0.5 1 1.5 2 2.5 3 3.5 4

x 10-3

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (sec)

a m p l i t u d e

Input

Output


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Physical Model Ideal Transfer Function

⎛ ⎞⎛ ⎞


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7

6 1 3 2 5out

2in

3 2 1 2 4 2 3 4 2 5

R 1

R R R C Ce(s)

e 1 1 1 1s s

R C R C R C R R C C

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠=

⎛ ⎞+ + + +⎜ ⎟

⎝ ⎠

2nd-Order Dynamic

System Model

0n

2

aundamped natural frequency

aω =


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System Model

2

0 02 1 0 0 0 i2

20 0

0 i2 2

n n

d q dqa a a q b q

dt dt

d q dq1 2q Kq

dt dt

+ + =

ζ+ + =ω ω

1

2 0

0

0

a damping ratio2 a a

b

K steady-state gaina

ζ =

=

Step Response

of a2nd-Order System

2

0 00 i2 2

d q dq1 2q Kq

dt dt

ς+ ζ + =

ω ω

Step Response

of ad d


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n ndt dtω ω

( )n t 2 1 2

o is n

2

1q Kq 1 e sin 1 t sin 1 1

1

−ζω −⎡ ⎤

= − ω − ζ + − ζ ζ ⎢ ⎥ζ − ζ −⎢ ⎥+⎢ ⎥ζ −

⎣ ⎦

( ) n to is nq Kq 1 1 t e 1−ω⎡ ⎤= − + ω ζ =⎣ ⎦

Over-

damped

Critically Damped

Underdamped

Frequency Response

of a


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of a

2nd-Order System

( )o 2i

2

n n

Q K s

s 2 sQ 1

=

ζ+ +ω ω

( ) 1o22i 2 2 n

2 n

n n

Q K 2i tan

Q 41

− ζω = ∠⎛ ⎞ω ω⎡ ⎤ −⎛ ⎞ω ζ ω ⎜ ⎟− +⎢ ⎥ ω ω⎜ ⎟ ⎝ ⎠ω ω⎢ ⎥⎝ ⎠⎣ ⎦

Laplace Transfer Function

Sinusoidal Transfer Function


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Frequency Response

of a

2nd-Order System


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Frequency Response

of a

2nd-Order System

-40 dB per decade slope

• Some Observations


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• When a physical system exhibits a natural oscillatorybehavior, a 1st-order model (or even a cascade of

several 1st-order models) cannot provide the desired

response. The simplest model that does possess

that possibility is the 2nd-order dynamic system

model.

• This system is very important in control design.

– System specifications are often given assuming

that the system is 2nd order.

– For higher-order systems, we can often use

dominant pole techniques to approximate the

system with a 2nd-order transfer function.

• Damping ratio ζ clearly controls oscillation; ζ < 1 isrequired for oscillatory behavior.


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• The undamped case (ζ = 0) is not physically realizable(total absence of energy loss effects) but gives us,mathematically, a sustained oscillation at frequency ωn.

• Natural oscillations of damped systems are at thedamped natural frequency ωd, and not at ωn.

• In hardware design, an optimum value of ζ = 0.64 isoften used to give maximum response speed withoutexcessive oscillation.

• Undamped natural frequency ωn is the major factor inresponse speed. For a given ζ response speed isdirectly proportional to ωn.

2

d n 1ω = ω − ζ

• Thus, when 2nd-order components are used in

feedback system design, large values of ωn (small


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lags) are desirable since they allow the use of largerloop gain before stability limits are encountered.

• For frequency response, a resonant peak occurs for ζ

< 0.707. The peak frequency is ωp and the peakamplitude ratio depends only on ζ.

• Bandwidth

– The bandwidth is the frequency where theamplitude ratio drops by a factor of 0.707 = -3dB of

its gain at zero or low-frequency.

2

p n 2

K 1 2 peak amplitude ratio

2 1ω = ω − ζ =

ζ − ζ

– For a 1st -order system, the bandwidth is equal to

1/τ.

The larger (smaller) the bandwidth the faster


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Documents

5. MechatronicsWorkshop KCCDay2 Session1-2.pdf