SYSTEMS_AND_CONTROL.PDF

7/27/2019 SYSTEMS_AND_CONTROL.PDF

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1540 Introduction To Mechatronics: Slide 1Stefan Williams

Introduction to MechatronicsMech-1540

Systems and Control


Systems and Control

A System is a device or process that takes a

given input and produces some output:

A DC motor takes as input a voltage and producesas output rotary motion

A chemical plant takes in raw chemicals andproduces a required chemical product

SystemInput Output


What is a Control System

A Process that needs to be controlled:

To achieve a desired output

By regulating inputs

A Controller: a mechanism, circuit or algorithm Provides required input

For a desired output

Required

Input

Desired

OutputProcess

Output

Controller


Closed Loop Control

Closed-loop control takes account of actualoutput and compares this to desired output

Measurement

DesiredOutput

+-

Process

Dynamics

Controller/

Amplifier

OutputInput

Open-loop control is blind to actual output


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Feed-Back Control

A Measurement of actual output is taken A comparison between measured and desired

output is made

A controller uses this comparison to provideinput to the process

The input is designed to make desired andmeasured values the same

Called a Feedback Controller


What is a Control System ?

A process to be controlled

A measurement of process output

A comparison between desired and actual output

A controllerthat generates inputs from comparison

Measurement

+ -

ProcessController OutputDesiredOutput

Comparison


Example: DC Motor Speed Control

Desired speed d Actual speed Tachometer measurements plus noise

Control signal is a voltage Variations in Load Torque

Actual Speed Measurement

+

-Load Torque

Power

AmplifierController

Motor

Tacho

d


Example: DC Motor Speed Control

Feedback provides insurance against:

Motor Non-linearities

Changes in Load Torque

Actual Speed Measurement

+

-Load Torque

Power

AmplifierController

Motor

Tacho

d


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Example: Batch Reactor

Temperature Control

Goal: Keep Temperature at desired value Td

If T is too large, exothermic reaction may cause explosion

If T is too low, poor productivity may result

Feedback is essential because process dynamics are notwell known

ControllerSteam

Water

Measured Temperature

Coolant

ReactantsDesired

Temperature


Example: Aircraft Autopilot

Standard components in modern aircraft

Goal: Keep aircraft on desired path

Disturbances due to wind gust, air density, etc.

Feedback used to reject disturbances

GPS/Inertial

Path controller

RudderElevons

Measured pathRoute

SensorsActuators

Disturbances


Block Diagrams

Formalise control systems as pictures

Ascribes mathematical models to systemcomponents

Components can be combined to produce an

overall mathematical description of systems

Interaction between elements is well defined


Block Diagrams: Summation

Ideal, no delay or dynamics

Two inputs: ( ) ( ) ( )z t d t y t=

Three ormore: ( ) ( ) ( ) ( )z t f t g t y t= +

( )z t( )z t

( )y t( )y t

( )d t( )f t

( )g t+

++


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Block Diagrams: Transfer

Functions

Transfers input to output:

( )y t( )x t G

.y G x=

A constant gain (K)

A differential equation

A look-up table


Time Out: The Laplace Transform

In this course, the Laplace operator or

Laplace variable is a useful notation:

ds

dt

22

2

ds

dt etc

And logically: 1.dts


For Example I

( ) ( ) ( ) ( ) ( )p pdy

mc y t u t sY s mc Y s U sdt

+ = + =

pm c

( )Q u t=&

( )T y t=

Physical model

( ) ( ) ( )

( ) ( ) ( )

1( ) ( )

p

p

p

sY s mc Y s U s

s mc Y s U s

Y s U ss mc

+ =+ =

=

+


For Example I

( ) ( ) ( ) ( ) ( )p pdy

mc y t u t sY s mc Y s U sdt

+ = + =

pm c

( )Q u t=&

( )T y t=

Physical model

1

ps mc+( )U s ( )Y s

Block Diagram

model


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For Example I

( ) ( ) ( ) ( ) ( )p pdy

mc y t u t sY s mc Y s U s

dt

+ = + =

pm c

( )Q u t=&

( )T y t=

Physical model

( )G s( )U s ( )Y s

Transfer Function

1( )

p

G ss mc

=+


For Example II

( )x t

( )u tM

C

K

2

2[ ( ) ( )]

d x dxM C K x t u t

dt dt + =

22 2

22 ( ) ( )

d x dxx t u t

dt dt + + =

2 C

M=

2 K

M =

K

M=

2

C

KM=


For Example II

( )2 2 22 ( ) ( )s s X s U s + + =

22 2

22 ( ) ( )

d x dxx t u t

dt dt + + =

2 2 2. ( ) 2 . ( ) . ( ) ( )s X s s X s X s U s + + =

Laplace Transform

2

2 2( ) ( )

2X s U s

s s

= + +


For Example II

2

2 22s s

+ +

( )X s( )U s

( )x t

( )u tM

C

K

Physical Model

Block Diagram model


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Block Diagrams:

Transfer Functions

Transfer FunctionG(s) describes system

component

An operatorthat transfers input to output

Described as a Laplace transform because

( )Y s( )X s ( )G s

( ) ( ) ( )Y s G s U s= ( ) ( ) ( )y t g t u t


Single-Loop Feedback System

DesiredValue

Output

Transducer

+-

Feedback

Signal

error

Controller Plant

ControlSignal

( )C s ( )G s

K

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

( )f t

Error Signal

The goal of the Controller C(s) is:To produce a control signal u(t)

Which drives the error e(t) to zero

( ) ( ) ( ) ( ) ( )e t d t f t d t Ky t = =


Controller Objectives

Controller cannot drive error to zero

instantaneouslyas the plantG(s) has dynamics

Clearly a large control signal will move the plant

more quickly The gain of the controller should be large so that

even small values of e(t) will produce largevalues of u(t)

However, large values of gain will causeinstability


Why Use Feedback ?

Perfect feed-forward controller: Gn

( )y t( )d tG1/ nG

( )l t

+

Output and error: nGy d GlG

=

1n

Ge d y d Gl

G

= = +

Only zero when:n

G G=

0l=

(Perfect Knowledge)

(No load or disturbance)


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Why Use Feedback ?


Why Use Feedback ?

ud yl

e++

K G

unity feedback

controller plant

demand output

A sensor

load

, ( ),e d y y G u l u Ke= = =

No dynamics

Here !

( ) ( )e d y d G u l d G Ke l = = =

So:


Closed Loop Equations

e d GKe Gl = +

( )1 KG e d Gl + = +Collect terms:

1

1 1

Ge d l

KG KG= +

+ +Or:

Demand to Error

Transfer Function

Load to Error

Transfer Function


Closed Loop Equations

1

G

KG+K

dl

y+

+

Equivalent Open Loop Block Diagram

1 1

GK Gd l

KG KG= +

+ +Similarly

Demand to Output

Transfer Function

Load to Output

Transfer Function


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Rejection of Loads and

Disturbances

1

Gy l

KG= +

Load to OutputTransfer Function

If Kis big: 1KG>>

10

Gy l l

KG K =

If Kis big:

Perfect Disturbance Rejection

Independent of knowing G

Regardless of l


Tracking Performance

1

GKy d

KG= +

Demand to OutputTransfer Function

If Kis big: 1KG>>

GKy d d

KG =

IfKis big:

Perfect Tracking of Demand

Independent of knowing G

Regardless of l


Two Key Problems

ud yl

e++

K G

( )u K d y= Power: LargeKrequires largeactuator power u


Two Key Problems

ud yl

e++

K G

( )u K d y n= +Noise:

LargeKamplifies

sensor noiseu Kd

In practise a Compromise K is required

+n


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Control Criteria

Speed of Response

Robustness to unknownplant and load

Stability


Response of a First-Order

System

1( ) ( ) ( ) ( )

dyay t x t Y s X s

dt s a+ = =

+

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Output

Response of First Order Lag to Impulse Input

0( ) aty t y e

=

General Solution:


Step Response

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Output

Time

Response of First Order Lag to Step Input

( ) (1 )at

fy t y e=


Speed of Response

ux ye+

K1

s a+

, ( )dy

ay u u K x ydt

+ = = Equations:

( )dy

ay K x ydt

+ = ( )dy

a K y Kxdt

+ + =


10/141 0


System Descriptions

( ) ( )( )

KY s X ss a K= + +

( )

K

s a K+ +( )X s ( )Y s

( )dy a K y Kxdt + + =

( )

0( ) a k ty t y e +=


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Output

Speed of Response

( )

0( ) a k ty t y e +=

0( ) aty t y e=

Increasing K increases Speed of Response


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Output

Time (s)

Speed of Response to Step

( )( )0( ) 1 a k t

y t y e +=

0( ) (1 )at

y t y e=

Increasing K increases Speed of Response


The Time Constant of a System

The time constant of a system is defined as the

time taken to reach 1/e of the final value

( )

0( ) a k ty t y e +=

0( ) at

y t y e

=

1dyy x

dt T+ = The time-constant of a system

characterises its speed of response

Reach 1/e when

1t T

a K=

+ @

1

t Ta= @


11/141 1


Tracking Response

A system with dynamics takes time to respond

(like e-at for example)

If the demand also changes, the output will lag.

For example a ramp demand:


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

Time (s)

Tracking Error

Steady-State Error

Initial Response

Input

Output


Characterising

Tracking Errors

Tracking Errors are characterised byconsidering the response of the system to

sine waves of different frequencies


Low Frequency Response

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

Input/Output

Input

Output

Lag


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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Input/Output

Time (s)

High Frequency Response

Input

Output

Lag

Bigger lag

Diminished

Amplitude


Frequency Response

For most physical systems:

When input varies slowly, output tracks inputclosely (small lag, similar amplitude)

When input varies quickly, output does not trackinput well (large lag, reduced amplitude)

The speed of input changes that can be trackedis a universal measure of system performance


Example I

sindy

ay tdt

+ = 1( ) sin( )y t ta

= ++

1tana

=

Plot amplitude of output

Against input frequency

Plot phase lag of output

Against input frequency


10-2

10-1

100

101

10-2

10-1

100

Frequency (rad/s)

Amplitude

Amplitude

Amplitude Falls by half

Bandwidth


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Bandwidth Calculation

Definition of Bandwidth:

Frequency at which input

Amplitude is half DC value

1A

a=

+

0=

a=

1A

a=

1

2A

a=

Bandwidth here is

a rads/second


Phase Lag

10-2

10-1

100

101

-1.5

-1

-0.5

0

Phase(rads)

Frequency (rad/s)


Increased Gain Increases

Bandwidth

10-2

10-1

100

101

10-2

10-1

100

Bandwidth

is (a+k)


Bandwidth and Time-constants

The Bandwidth and Time -constant of a

system are the inverse of each other:

1 1

bw

Ta

= = 1bw

aT

= =

1bw a K

T = + =

1 1

bw

Ta K

= =+


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A Word About Stability

0 1 0 20 30 4 0 5 0 6 0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 0 20 30 4 0 5 0 60

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Start here G 180ophaseinversion

0 1 0 20 30 4 0 5 0 60

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Bigger hereKAnother phase inversion

If G is such that input is phase reversed

(180o out of phase) for any frequency,

then input will be back in phase

If loop gain >1 then

system will be unstable

BANG !

If System is unstable for one input,

it will be unstable for all inputs

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