MECH466 L5 ModelDCmotorLinearization

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2008/09 MECH466 : Automatic Control 1

MECH466: Automatic Control

Dr. Ryozo Nagamune

Department of Mechanical Engineering

University of British Columbia

University of British Columbia Lecture 5

Lecture 5

Modeling of DC motors

linearization, time delays

Course roadmap

Laplace transform Laplace transform Transfer function Transfer function

Models for systems

•

•electricalelectrical •

•mechanicalmechanical •

•electromechanicalelectromechanical

Linearization

Modeling

Modeling AnalysisAnalysis DesignDesign

Time response

•

•TransientTransient •

•Steady stateSteady state

Frequency response

•

•Bode plotBode plot

Stability

•

•RouthRouth--HurwitzHurwitz

•

•NyquistNyquist

Design specs Design specs Root locus Root locus Frequency domain Frequency domain

PID & Lead

PID & Lead--laglag

Design examples

Matlab

Matlabsimulations simulations & laboratories& laboratories

What is DC motor?

An actuator, converting electrical energy into

rotational mechanical energy

(You will see DC motor during Lab #3 (You will see DC motor during Lab #3--#4.)#4.)

Why DC motor?

Advantages Advantages

high torquehigh torque

speed controllabilityspeed controllability

portability, etc.portability, etc.

Widely used in control applications: robot, tape Widely used in control applications: robot, tape drives, printers, machine tool industries, radar drives, printers, machine tool industries, radar tracking system, etc.

tracking system, etc.

Used for moving loads whenUsed for moving loads when

Rapid (microseconds) response is not requiredRapid (microseconds) response is not required

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How does DC motor work?

Armature

Armature

(from Dorf and Bishop book)

Model of DC motor

e

eaa(t(t))

i

iaa(t(t)) RRaa LLaa

e

ebb(t(t)) Jm

B

Bmm

Armature circuit

Armature circuit Mechanical loadMechanical load

Input

Output

Armature circuitArmature circuit

Connection between mechanical/electrical partsConnection between mechanical/electrical parts

Motor torqueMotor torque

Back EMFBack EMF

Mechanical loadMechanical load

Modeling of DC motor: time domain

Load torque

Armature circuitArmature circuit

Connection between mechanical/electrical partsConnection between mechanical/electrical parts

Motor torqueMotor torque

Back EMFBack EMF

Mechanical loadMechanical load

Modeling of DC motor: s

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DC motor: Block diagram

Fact:

Fact:KiKi=Kb.=Kb. _{Feedback system}_{Feedback system} Tachometer Tachometer

(Sec.4 (Sec.4--55--2)2)

Encoder Encoder (Sec.4 (Sec.4--55--3)3)

Transfer functions with feedback

NegativeNegativefeedback systemfeedback system

E

Eaa

Ex: Derivation of transfer functions

E

Eaa EEaa

Compute transfer functions from

Compute transfer functions from R(sR(s) to ) to Y(sY(s).).

E

Eaa

E

Eaa

DC motor: Transfer functions (TF)

2

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DC motor: Transfer functions (cont

’

d)

Note:

Note:In many cases La<<Ra. Then, an In many cases La<<Ra. Then, an approximated TF is obtained by setting La=0. approximated TF is obtained by setting La=0.

1

1stst_{order system}_{order system} 2

2ndnd_{order system}_{order system}

Course roadmap

Laplace transform

Transfer function

Models for systems

•

•electricalelectrical •

•mechanicalmechanical •

•electromechanicalelectromechanical

Linearization

Modeling

Modeling AnalysisAnalysis DesignDesign

Time response

•

•TransientTransient •

•Steady stateSteady state

Frequency response

•

•Bode plotBode plot

Stability

•

•RouthRouth--HurwitzHurwitz

•

•NyquistNyquist

Design specs

Root locus

Frequency domain

PID & Lead

PID & Lead--laglag

Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

What is a linear system?

A system having A system having Principle of SuperpositionPrinciple of Superposition

System

Why linearization?

Real systems are inherently nonlinear. (Linear Real systems are inherently nonlinear. (Linear systems do not exist!)

systems do not exist!) Ex.Ex.f(t)=f(t)=Kx(tKx(t), ), v(tv(t)=)=Ri(tRi(t))

TF models are only for linear time-TF models are only for linear time-invariant (LTI, invariant (LTI, see next slide) systems.

see next slide) systems.

Many control analysis/design techniques are Many control analysis/design techniques are available for linear systems.

available for linear systems.

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Time

-

invariant & time

-

varying

A system is called A system is called timetime--invariant (timeinvariant (time--varying) varying) if system parameters do not (do) change in time. if system parameters do not (do) change in time.

Example: Example: MxMx’’’’(t(t)=)=f(tf(t) & ) & M(t)xM(t)x’’’’(t(t)=)=f(tf(t))

For timeFor time--invariant systems:invariant systems:

This course deals with timeThis course deals with time--invariant systems.invariant systems.

Sys

Sys Time shift

Time shift Time shiftTime shift

Linearization of water tank model

(Laboratory 1)

Mass flow rate equation Mass flow rate equation

Q. How to

Q. How to linearizelinearize??

Linearization of water tank model

(cont

’

d)

Why is the system linear when Why is the system linear when α=1?α=1?

System

System

Linearization of water tank model

(cont

’

d)

1.

1. Select an operating pointSelect an operating pointhh00, around which you , around which you

want to

want to linearizelinearizethe equation.the equation. 2.

2. Consider a small perturbationConsider a small perturbation from hfrom h0 0

3.

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Linearization of water tank model

(cont

’

d)

RHS : RHS :

Linearized

Linearizedmodelmodel

This model is accurate enough ONLY around h This model is accurate enough ONLY around h0 0 !!!!!!

Some remark

Actually, this “Actually, this “linearizedlinearized””model is model is notnotlinear, but linear, but affine, due to the constant term.

affine, due to the constant term.

However, if we define a new inputHowever, if we define a new input

then the system is linear. then the system is linear.

You don’You don’t need this fact in Laboratory #1.t need this fact in Laboratory #1.

Linearization of a pendulum model

Motion of the pendulum Motion of the pendulum

Q. How to

Q. How to linearizelinearize??

For a small For a small

Linearization of a pendulum model around

general operating point

Taylor series expansionTaylor series expansionof f(xof f(x) around x=x) around x=xoo

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Time delay examples (p.115)

Delay time

Time delay transfer function

TF derivationTF derivation

The more time delay is, the more difficult to The more time delay is, the more difficult to control (Imagine that you are controlling the control (Imagine that you are controlling the temperature of your shower with a very long temperature of your shower with a very long hose. You will either get burned or frozen!) hose. You will either get burned or frozen!)

(Memorize this!)

Summary and Exercises

Modeling of, DC motor, nonlinear systems and Modeling of, DC motor, nonlinear systems and systems with time delay

systems with time delay

NextNext

StabilityStabilityof linear control systems, one of the most of linear control systems, one of the most important topics in feedback control

important topics in feedback control

ExercisesExercises

Read Sections 2.8, 2.10, 2.11 & Lab #1 manual Read Sections 2.8, 2.10, 2.11 & Lab #1 manual “

“Review of the modeling of fluidReview of the modeling of fluid--flow through a flow through a constriction

constriction””..

Problems 2.45, 2.49, 2.50 in page 100.Problems 2.45, 2.49, 2.50 in page 100.

Main message until this point

Many systems

can be represented

as

transfer functions

!

Using the transfer functions,

…

.

(to be continued)