Sistemi di controllo - 01(Eng)

POLITECNICO -LESSON

DI MILANO NOTES

CONTROL SYSTEMS

Casella

2023 F

2024 PROF

a -

. y

. . .

CONTENTS

0 .

1 SYSTEMS

DYNAMICAL

. SYSTEM MODEL

DYNAMICAL

A

2 MOTION

. IN METHOD

TRANSFER

.

3 FUNCTION

TRANSFORM

INVERSE LAPLACE

4

. DETERMINATION

STABILITY FOR

.

5 LT I

STABILITY NON-LINEAR SYSTEMS

.

6 IN

STEP RESPONSES

7 .

8 BLOCK DIAGRAMS

.

9 BLOCK STABILITY

DIAGRAMS AND

.

10 RESPONSE

HARMONIC

. SYSTEMS

CONTROL

11 . FEEDBACK DESIGN

CONTROLLER

12 . DISTURBANCE

DIRECT COMPENSATION

13 . TWO-DEGREES FREEDOM CONTROLLERS

OF

14 . SATURATION

15 ACTUATORS

. CASCADED CONTROL

.

16

17 MULTIVARIABLE CONTROL

. PYRAMID

18 . AUTOMATION

19 ON/OFF CONTROLLERS

. DYNAMICAL SYSTEMS

1

. da

letts idealized

consider representation real S

of focusing ...

system some

an a on

. U ,

variables

specific by the

described such /causes) (from

and

aspects world Y

INDUTS >

as g i

bothi

disturbanced

manipolated variables

divided u and which

the in

to system) , > "Yn

variables

the these

letterts)

will of

All

system at not

act cuTPUTs 1 ores are

are un

on , . BLOCK-DIAGRAM

the

time to time

dynamic E me

t

The take

if

key make

point that

system is

. a

we = DOCK-DIAGRAM

a

from

dipendence

have time scale time)

large I past present

a

on

a :

,

FASE

the

depends

(E) inputs d(t)

ult) CINERTIA)

Y on . different

for of

lets

An types

the

abstract such

theory examples

approach us use as

same :

,

temperation

single

(well-mixedain impresentative T

TEMPERATURE

ROOM

· heat by

dissipated

Golt) occupants Qinm( + /

windows Qult)

immadiated through

(t)

Qian power vv T(t)

fan fore temperative contro

heating g(t) ↳

git) S

FtzE

dIt)

<ult)

have

situation y(E)

this

in <

we ,

consider relacity

just vI

(let's

LONGITUDINAL A

DYNAMICS CAR

OF

· Vw(t) x(t)

flow

fuel mate

971) vv

(t)

+

9 V(t)

the

of breaking

position pedal

bit) i

b(t) L

with

face n(t)

speed

aire/wind

resistance

VwIt/ -

transmission

of

number the

nit) gean road

the

of

stope

It)

< N ~ &

thorst

to T

cicle produce

(Brayton

TURBOJET ENGINE

· ~ ~

C T

the

of

speed

wit) rotation engine

fuel value

the

of

angle

t

6 opening Pert rity

Tex t

conditions

external

ten

Te x t Pert

, v

v - T(

(t)

velocity <

(

(intare

velocity

air/wind

vit) S

S cw(t

(

GDP/PIL

pot

Leconamy's at

ITALIAN ECONOMY

· ↓ ↓

population ...

NIt) elt) at

numer price i(t) GDP(t)

<

of

awit/ natio working people &

mit) price N(t)

gas S

+ >

(t) , , >

mate

interest

ilt) "

(t) change

commency

n i

taxation

It) mate

t abstract

the

So contrad

the control to problem

solve

We

problem want

in most

approach

we a

way .

variables

of IS these

with and

time dit) variables

ult) want

interesting yit)

system we

generic

a , , I

condition

be lovesuef/desidened the

want

specific

connected JET-PANT)

at

to to yout) pot

a on ·

dit) to

the disturbances

be possible of

despite (unknown

the

to

to set-point

output as cose on

as behaviour

be uncertain despite the

determined if and of

the

which of

measurcable) recentanty

are

S' of

the values variables control

the

to manipulated variables)

(ore

Our aim ult)

is compute

. .

be automatic

solution

The operative determines

human

manual controd directly

(a u)

can an

a on

artificial determines

controd (an Let's

u)

system CONTROL STRATEGIES

same

see :

. information of

1) (controter the

has

LOOP/FEED-FORWARD

CONTROL

. OPEN result)

output

no

the controbur and about

dynamical the

which has information

determining set point

C is system u

a be

distinbances (dotted determine

measured line)

could

about C

We despite

to

that ulty

and use

some . measured

of

unuentanties the

the be with

of well

Works

and disturbances that cannot

system .

electrical heater

limited ununtainty the of

take

Let's example an

.

themmal balance

with first-principal

Let's make

inventia

same di

a

. ac

to the resistence vit)

: youts Il

EYR c

Q

BAL VYQ

EN CHMLAW R

+

. =

. = E R

Qo

if >

In

there's andertocated

known

V the

well

M

inuntainty are

no . .

first and

Qo value

the

fixed evaluate try

I

output compute

a

we can

, , need

do know

to

determined We not

Q0)

Unawn

of R always

(which is . r

but of

controd

the the

real ander to in

Q

at in system

put cale

paven O

blind)

lit

contridur

loop cannot

unurtainty is

significant guarante you

an open y .

LOOP/FEED-BACK

CONTROL

.) CLOSED

2 -To d ( t )

it

blind

the but

controdur not ult)

you)

act knowing

is Y(t)

can S

C o c

(

feed-back

the thanks to

what at autputy

is going a

on M

with

deal

loop better

much ununtainty

This system can :

. the

fare

for but

need desired

distinbance

do the

not output

just campaning

sensor

we a :

one yo

every

with something

do

controller

the

the

and Until

measuredy uncertainty

helps The

caping can

y = yo

. .

downside between unstable

feed-back behavior that

that lead

delays to is

strong

is + can

usy an delays

with

deal

the to

controllen

difficult It order

design

control

to to well in

really is recessary

. bad

with behavior

otherwise be

end about

must stability

(oscillations) worried

We and

up

we a .

behaviour

unstable

understand the BACK (ECRENLOOD)

FEED PRESERVE STABILITY

LOOPS DO NOT

. .

of

describe S'

behaviours Let's

to the dynamical consider

system

We DYNAMICAL SYSTEMS MODELS

use a

. :

S'

dynamical then of dynamical

model behavior

S another Ms

system whose

system approximated

is

. a

,

of

behaviors behavioure real

the Swell of

the

that the

make predictions system

enough we

so can on .

types models

of

two

There are : and

time-consuming

electrical

modds

(scale analogies)

PHYSICAL poicey

1

.) MODELS very

,

made by equations (ar

) MODELS

MATHEMATICAL

2 case

. d(t)

wit

have functions

then of that

We time quantities

physical

represent : Y(t)

Ms <

3

(dit) by

Ikea le" analogy

y(t)

ust) : ,

time consider

description could (Tn)

LIRe) usually

Considering DISCRETE-TIME

CONTINUOUS-TIME

we ,

first described

model by

The

spaceh spaced)

Inon-even is

equally EVENT-BASED TIME

on one

even .

differential the

consider

functions that time

The dit)

equations real

ult) yit)

on we arre

more . ,

,

functions Let's examples

see some :

. (t)

Te x t

L

atdocos

Selectorical heatin known

placed

THERMAL Te x t

MODEL -

· T

Di .

EH

C

with distributed

Tu n i fo r m l y

experiment

let's thought

make Dext

a balance

write

transfer

heat dynamic

and We

constant can

. :

a energy Dext

-Dext Q-hSIT-Text

Gi

AE Mc

BAL c ri

=

= >

-

. R T

&T Q

the -h

egration

mondering IT-TAt ,

i

= Q

if

the

what

understand to Let's

happens

want manually cal

to change

system

we j

we .

that variable =1

derivative

quantity under state the

the is in

T have

example) We

= :

.

& Q h

T Imathematical mol

(T-TAt) STATE EQUATION

= -

T X VARIABLE

STATE

OUTPUT EQUATION I

Y = =

=

. I i

to

I

hS(T OUTPUT EQUATION

Te x t ]

yz = - initial

if the

values

have at

states TIto)

point X(to)

NB same +

same given >

we

. Qi(t)

of

behaviouse for

the [to that knaw

inputs salve

(t)

5) Te x t

so

- we

we can

, ,

and problem

the salve to

equation

state It)

Carchy's compute

TC) 2

compute >

a cm

the

the equations

yilt)

atput atput

using .

because the depends

model of the

the also evolution

dynamical

Why output

is past

system

a now on

:

depend [to E)

the

the

The depend

which

statesX

system atpots te

inpots in

on on

. ,

, .

↓

(flid cross-section A)

PRESCRIBED FLOS Wolt

TANK WITH

· , wilt)

balance

fundamental

the dynamic equation is mass l(t)

a 1 T <

A

BAL wi-w

wi-Woc

. = =

dt Wi

-

l

where E

-In wo

= =

y 00 Wo

>

Wi

(fid cross-section A)

VALUE

TA N K OTLET

WITH

· , *

becomes

but Av

the input

contand

langer Wi

we can now

no 00 Wa

AMWi-Wo M

BAL Wi-Ar Ar

S

zg

=

. Wilt

=A ↓

where

Wi-Ave Av(t) l(t)

T <

>

rumember with

flow

confuse

to not input/output inputs ult)

system

model

(t) The bandamy

system conditions

depends

atpots strngly

on .

on

y . T

G)

e-h To

WisTi

( fid el"-

cu <p

ELECTRICAL

WATER HEATER Wo

=

=

· ,

,

, ,

Qu

leak

insulated themmal

(no of

surface

perfectly

we assume a -

capacity

heat

and (we

walls

thin the

of wall)

the

power) neglect Til ↓w

AM-Wi-W

MASSBAL WiWow Qic

I

. IT

HE

dE Q Mc Di

Wihi-wcho welTi-To)

ENERGYBAL + i +

< =

=

. dt its

controd behavioure

dynamic model of that it's

to

when doing system assume

oss can

a we

a ,

been do

that

designed semplifications

(good-design principle)

preoporty same

so we can

AT

obtain the ITi-To) Q

equations where

W

system =T

x

we +

=

It's variable

the

that the

variable that

to state under

state

important just

is

I appears

the

of model

the

doesn't actual

to

and always which

equals

derivative output (t)

is y

sign .

(WATER

PRESSURE (

BOILER

HIGH HEATER

· Ti To

Wi Wo T

=

, ,

the internal

since particularly high

is we can an

.

no

pressure the

have

thin We approximations

walls

longer past

same

assume . ↑

but uniferm

external wall

with heating temperature Tw Q

sarece +

an :

MASS BALANCE Wi Wo W

= =

& McAwalTi-T)

Qic

E Wihi-Woho USIT T

BALANCE

ENERGY FLUID + +

= Q-Qi

NEW Q-hSIT

MwCwAT T

BALANCE

ENERGY <

WALLS = =

&T

let's the explicit

equations

make h (Tw

-T) -T)

Ti (1)

+

= I ↓Ti

~

-IT

AT Te

= Q HPB T

> 1

variable

manipulated

,

is u

I

T

y = with

have different that two

differential have

equations state

two

system

we so

a we lineare

have states)

variables second (we

Tw andem two

The is

T Xz

X system non .

a one

= =

1 . ,

. oscillator

Charmonic (

SPRING-MASS OSCILLATOR

· E

motion

ht's and

officienth

foriction horizontal

only

assurce .

a F

balance >

ned

We write

to momentum

a 7

: O O > P

d

BALANCE F- Up-hv

Mr)

MOMENTUM = MF-K-

that &P have

considering v we :

= ↑

Ad

F p

be

to

condition

the

F-p-tr and output

= - is

(non-linean mechanical

PENDULUM System)

SIMPLE

· · le

Let's and

mass-less

consider and

infinitly real M

rigid plaudati m

mass

a a

of

the the

end force

to horizontal *

We have

We

F

apply

assume

mope an :

. . "sing

ALMEW)

ANGULAR BALANCE

MOMENTUM FIOR-Mglsint-hw

=

AR.

with form

the write the

relaity explicit

Let's

being

angular w= F

: sp ?

F-glesint-1w

AW lint

where S Ex w

co y = =

= ,

. Mez

linears

this of that non-linear

example of

equation contains

nice type

is a

a non every

functions states)

rulation (trigonometrical and

between

product

mathematical inputs .

, ↓ ↓

I

↓ I

(electrical (

CIRCUITS

RLC systems

· O

%

v

different

three with

consider -you

We comparents

can of

being the result

with

linear each

equations one a CEl LI

law

conservation of the also E

ERI

have

change We generators

. =

.

if circuit write

consider Kinchhoft's

Mareover ↓

can

we

a

we I Ia

, =

⑳

for

and

far

comment law closed

node

law voltage

2 I a

a

0

: = EEY

of dectrical this

the

Carration field)

mesh For

IEI 0 -

= . .

linear differential

write

of algebraica

RIC systems

type can

we - ⑧

Let's consider circuit

simple

equations a :

. Y

m =

,

CVladV/dt (n-ES/Rc

-E

CAPACITOR =

= ↑ V

uy

(U -E)"R

E I

RI

KIRCHHOFF

RESISTOR + U c

+ =

= #

and In-E)/R

obtained

balances

constitutive laws I

combining we = =

y u

. El

>

lineare but the "

onder equal

have to

in general

We system get

we an

a

of

number for

(except

active constraints)

particular

components same .

SIR MODEL

EPIDEMIOLOGICAL MODEL

· : sX

of

idea down

that

the models

write compartimental

is : 2

by

fran

isdated composed

N components

population starting [

became infected

then

which

people

supsceptible contagios

+ R

dead)

third of removed (healed

and then stage

people an

a .

balance

write for compartiment

equations

Let's :

every B(t) 2(t)

SH > 3

A

BALANCE BICt

S = - . N B(t) I(t)

AZ SH-

BICt

BALANCE =+

I . ?

> >

↑

VI

BALANCE

R don't contad

how to

know

really BLA

= we

of

if good

this

to exam