Sistemi di controllo - 02 (Eng)

STEP RESPONSES -

.

7

unit

the first

Let's function

ander

of transfer

step

see response a : fix

G the

Let's

model)

(accumulatore

known integrator

GIS) the

M/s system

as

= .

unit %

the t Y

input Step

step M/s

u(t) (t) namp(t)

(S) M/sa <y(t)

S M

as . =

= = = .

.

. .

initial

study final

and value

the

We can : ya

(ndegz1

G then strictly

(5)

(0 ) system (

+

y preopen

0

=

= him positive fore

y (0 t-

SG(s)

+ stope

) of

M M

= = d)

S st

+

+ 0

lim

(0) depending the

G(s) o

7 sign

=I o

on

= M

② = -

YIS) the

=

then get time

and

Ms

4(s) in

we = -

st

IST

danain m[1-expl-t/ )] step

y(H) Values

(t) .

+ :

= +

co

Ya

linh(s) Indegu

(0 )

+ 0

y =

= limshis) positive

MIT sop

y(07) per

=

= #

lit blows fer

it

410) T(0)

(0) Tso

Y up

M

=

= considere

t concluded

transient [SETTLINGTIME)

attere the phase

5T

47

os can

: we

= Mi-T/T)

=

↑

e the

③ and

Ms

Y

have in

4(s) we = -

= 1/T

s +

M[1 T

domain exp(-te)]

time

the y(t) Values

Step (t)

- :

= . Ya

= TsT

lin immediate

(0 his)

) Indeg

+ ( ⑧

↑

y response

-

=

= -

s[MIT)

l()) T

tim OLT <

o

(ot)

↑ = = =

S +

- 0 t

>

I

O -

him ST

o 0

= if value

initial with

the

connect

we

I TO

G(0)

(8) STATIC GAIN

M

y = stope

get

= the

the (T

point M) we

. pdes/zenos term

by

the transient

then contribute the white

phase

to

only

When the

given

is

g 0

= by finish the

behavioure time

the needed

characterized IST

to transient

asymptotic The Ts)

is .

M =

by

determined pdes transient

the depends

the

white the

initial the

of

part (St o)

is Zeros

on + :

if have initial

effect undershoot (the

white

it has anticipator

>0 to we response

an

an , final

first the value)

overeshoot latput above

starts

it

reapidly)

drops then T

increases Ta an

an .

1fst

Q ist

= Es Se

have

then

al) Y(S) = and

↑ M

we

= = .

damain

the time Values

#exp1

y(t) )

in + / ya

:

+

= · uoudenchoot

response

= Limmediate

tim (1)

y(0 )

+ = S + + 0

Sim stsYs) t

PI · >

10

i )

+ - T

=

= final

initial value

and

connecting at

the

17

G(0)

y(x) T30

0 initial stope

the

t get

=

= T we

=

if the of

end the

at

the the

the transient

then step

origin

in

is

os zero is

response 0

.

second-order

of

switch to

Let's systems

examples

same :

M

⑤ distinct

two then

like with

poles and G(s) Ti

Tz

temos O

>

no = (1 ) STz)

(1

ST

+ +

i

M (MF

-

Y(s) = (PI

2)/(s 1/TC)

)/(s

/T ) +

+

·

= - -

.

(1 +

) sta)

(

ST T

+ + .

=

. - I expl-/Tc)Jstepit

M[1-l- (since the

and e

y(t) exp1-tT ) system

.

= Initial

exp) final

two

have of

the

ander values

second sommation and

is we

a one :

. fact

fast guides

dynamics 5T

limh(s) =

Indeg

(0+) 0

y =

= effect)

lif Tic Tz

> no

n

Y

Hat)

Starts

(response

(ndeg 2)

(0t)

y d

0 =

= M

=.. - -

line if

yj(0 MITIT2

524(s)

)

+ ps

=

= s + + o ⑧ to it

final value

(5) G(x)

y M /

=

= in t

only

10)

y <

0

0 =

= the slowest Clargest

constant

figure

matters the night

at

to term

what time time

is

It's vietul to compute

constant) the this

orle max(TI in

Tz) 5T

5

Ts

is case

.

=

: ..

; =

local

y(t) and the maximum/minimum

compute

0

= .

fasten approximate

first

than the

second pole the

it (TICLT2)

the then

much

is

oss can

one we

behaviour the fast)

of first

with (initial imentia

onder System is very

a

one .

M(1 ST)

⑥ distinct like

two +

poles and then

G(s) and

zero

one = )(1

(1 STz)

ST

+ +

,

= [])

-

have =

YIS) IPI)12 4/Th)

we 17 ) +

+

- -

.

Texpl-/Tc) Jstepit)

M[1-lexp1-t/Ti)

and Values

y(t) -

= . stationary points

yn O

him Irdeg Ti

e

(0 = G11)

)

y 0 t

=

= MIT

y(0 )

+ =

=... to st

y(5) M

=... =

it the .

Stationary then

t (t)

points

compute The

plot

.

make

S can

we a

we

0

. =

initial to

compared

the

accelerates behavioure

of the

adding

effect previous

zero case

a .

=

⑦ (0)

tw then e e

like have

(SMS

pos his)

0)

h

+ we

my =

M 5 m

YIS) mi-clut]stes

function and

Ms ylt)

.

= =

13 =

-

1 + Wi

Initial final values

and Ya 5/

: ST Zwn =+ o

=

Y

limh(s) ⑧

~

Inde 1 1

(04)

y 0

=

= Indeg 2)

y(0 + ) 0 =

=... = ⑧

io Mrn

+ ) ↓

↓

=... = =

period ocillation

of T

F = oscillation

linfinite

7) (

* ...

↓ but

because

the IRelp)

theoum (daba)

IRe(p)

final have

value

cannot co

we

use o

we no =

1 -

⑧ poes have Y

of complex then

↑1

G(s)

paine we )5

= +

3

+ +

2

(1 +

wegis

Stun W time

the

= the

domain got

and in

↑ we

- zwn)

(stzwul" wa

+ +

+ 3

M[e-e-Sant)-coslut) sin(it))]Step Initial final

function values

and

(t)

y(t) + :

.

= 32

1 - damped

a oscillations

y

lin 6(s) Indez

10 )

+

y 0

=

= i

y(0 Indeg 2)

)

+ 0 =

=

=...

yj(ot) MW

=... = ⑧ > t

g PJEDO-Period

im final value it

7(s) only

his) M

= 2/weit/we

= =

time fastest

the the

settling exponential the

that

term to

part 0

is gas so we can

: and of

number

compute 5/5wn

exp1-5Ts) 5/-1Re(p)

expl-Junt)

impose Ts The

= = = ·

pseudo-periods /

-

N Ts /

is .

see

we +

= -

D M

⑨ it function

consider but with

the same

we :

3

3 ant)

m[ M

sin (wt))]Step (t)

y(7) (s(wt)

- +

e

- -

= 32

1 - but

the value

initial derivative

still have and

same

we > t

exponential diverging

envelope blows

the Toscillation up)

is diverging scillations

-

GIS)

TIME/GAIN

RESPONSES FOR

OBSERVATIONS STEP

ON behavior M

(steadystate)

represents

static the

G(0) static g =

We

gain

Ms

= can

· .

Possible

final theorem

value results 0

the box

with

it the ga

compute in

are

. : gia

amplification S

basically factor between atput

the and

It steady state.

input in

is stability

stability

about (

(IRe(p)

tell information asymptatic

615)

of

poles us < 0

· by lendogenous

poles phenomenon) when

oscillations complex

step E20

given

in 7

response are

· .

the of (TS

rulated position

to

settling poes

time for

the (Ti) real poles on

is Smax

· =

(1/

Ts poles)

for

) just complex

5 max (IRe(P)

(Ti " iwni)

To

= . 5 max

5

:

wn max

;

, =

= . ,

the

anticipate time

if (acullerated

constant

the

zeros can response <=>

Tso zo

· delaying

initial the it Idelayed

response) have initial

zo

iso

an =>

response we

of If

the large

and if

ovenshoot

have

system) enough

is

response I io

is 0 on

we an

. 0)

assumed

for undershoot example

value (in above

of get the Ma

we

T

every we an BLOCK DIAGRAMS-

8 . ↓ d

Let's components

several

have of system each

assume a

we , ya

by function

described want

trausten Y

to

61s) We 21 52

one a > -

>

-

. -

describe of

the need

autput System We

LTI

response a :

. M

① G(s)U(s)

t Y (1)

BLOCK Y(S)

V(s)

S G(s)

= >

. >

-

. - ① Y(1)

Vill)

② =U Uz

t (1)

Y(S) >

(s)

SUMMATION >

=

NODES S -

-

,

=

. . # VIS

combine signals)

to

ander

( them in

use

we Y (s)

Vi(S) >

③ ,

Y

t

BRANCHING V

(S) Y2(s) (s)

NODE S = i

.

.

. Do

, boxes

distributes

Cit the multiple

signal to Yz(s)

same < -

function

figure

to 4

transfer

the all

need also rules system

at

We aver . St >

-

basic diagrams

Let's configurations (asculare)

block

for siso

see T

: VISI

① VIS)

have

St

CASCADED CONNECTION We an

. H(s)

G(s) Y(S)

<

> >

intermediate variable GIS)V(S)

VIS) and

= ⑭

variable

atput (G

H U)

H(s)

Y(S) V(S)

an . .

=

= U(s) YISS

1 (HG)U

(4U) HisS

G(6)

only)

for

H We

SISO > >

= can nunge

= .

·

black computing

the into ((5) H(S)

system one . Vils) Y (S)

,

4())

>

V(s)

② t have Vi()

((S)

PARALLELED CONNECTION S We Y(S)

=

= <

. . 30

and simplify

Y(s)

Uc(s) [ Y2

=Y We

(5)

(s) can

=> = , · Yz/s)

H(S)

>

and write (EGIH)U (summing

Y(s) GU HUz

= Uz

. (s)

= ⑭

the functions)

transfer V(1) Y(s)

IG(1) [H(S)

< <

③ VIS)

the VIS)

t compute

FEEDBACK CONNECTION S We can Y(S)

4())0

. . 10 >

<

⑦

functions N

Y(S) G(S) and

WIS)

VIS) HISSY(S)

= =

, H(s) <

by

simplify writing

VIS) [U(s) We WIS

#WIS) can

= . G(s)(U() ⑭

#HISSYISS)

U(S)

VIS) [ H(s) Y (S)

Y(s) <

= = 41)

limplicitegration Y(S) Gis

Y1)

in VISI

< = > Yi

G(S)H(S)

1 = 61/His)

1

U(s) = /

G(S) the

where TRANSFER

LOOP FUNCTION

is

G(S)H(S)

1 =

It's of

examples DIAGRAMS

COMPLEX

see some : VIS) Y(1)

① Als) B(S)03

the <

diagram >

· <

reduce

Let's <

system

siso &

. by

to recognising

simplem patterns

a one > ?

that know

we : C(S)

function

transfer

with 1

block

like a

UIS) U(S)

e Be

>

then 1

+

AB Y(s)

final following

and result the

the 1

VIS

is + <

>

: BC

1 + Vis

↓

VIS

② that the

know

systems for

We systems

MISO LTI 4

tut

100

co

.

superposition applies consider

(we

principle on

time)

at

input B(S)

a : <

Y(s)

Als) o

30 >

< A

V(x) V(s)

0 *

0

+ Y(S)

VIS E

= was >

, AB

1 +

By VIS)

Vis

↓ ·

Bus

VK) Vk) > j

0 cos e

0

n

= , E

by VIS

final

draw the the

system combining single A

we can >

transfer functions above

computed AB

each

in in

case 1

a Y

+

summation node : VIS) ~

Ab

there to single

possibility compute

oss is a

no ABU +

between

function two

transtin (they

the inputs VIS

Y(S) = AB

+

for

different units

have

cold instance) SHALT

THE

.

FOR SYSTEMS

TRANSFER FAB

SUM FUNCTIONS +

THE MISO

NOT Y(s) AB)

1 WIS)

+

. =

NOOO

8 dumb to the

salve

system The

SIMO way

. Y (S)

< ,

it

simplify

to following

the

system is in

Ismant

atput time

at is

way

one a

way UI) BIss

Als) Yels)

< >

<

>@ o

considering Y)

B

Yz :

.

= ⑦

V(S) Y (S)

Als)

g >

> oc .

just Y A

LS) O

, <Y(s)

E UIs) < AB

1 +

BISS A Y (s)

S < ,

AB AB

1

Y2(S)

simularly

just VIS

Yzis) +

compute U(I)

we = 30

final fellowing

the

write the results

St in AB

can

we way :

. Yels)

S c

AB

1 +

VIS)

that w(s)

Odd

⑪ sove the

with

cannot roles z(s)

cases we Y(s)

//2 BIS)

All) <

& <

0

In always

these

know apply VISI

cases can

we we

. ②

the mathematical t

operations S :

. R(S) V

Os C(s) <

S(S)

S(s) W(s)

R(s)

A(s) VIS)

N(s) +

=

= (1) [Ar CY]

Y

BISSWIs) BAV

VIS) -R(s) U

V

VIS)

z(s) AND +

+ <

= =

=

=

S(s) ((s) (s)

z (1

A(s) V (s)

( B(s)((s))

V(s) 1) U(S)

+

= < +

-

=

final

lineare

solving figure

equations

the solution

of the

compute simplem

at

system We

we can a

can

.

branch solution

loops

feedback sequencially the

break

if write

the in using

we can

way one

we

: brooken loop

for

tearing (2

be .

(to explicitly) equation

expressed system)

variable VIS) every

a BLOCK STABILITY

AND

DIAGRAMS -

9

. block diagram

two-system

have

Let's where

assume a

we Biss

Ull) Als) <

<

di(s)

da Knowing that

and nB(s)

na(c) (1) Y(S)

co-prime

are

,

, . ⑭

asymptetically what

stable about

and BIS)

Als) say

can we

are <MA UB

VIS) YIS)

.

of

the stability the <

CASCADED CONNECTION : da dB

MASSUBS

Y/S)

let's Als)BLS(VII) possible

compute VIEISV) cas.

= =

① (pdes <poes Gl5

WApdes of

B(s))

co-price) of

all

cancellations of

di

(da Acs)]

no na uo :

i =

, , stable

asymptatically

of

poles the

of

also and

then the system

GIS)

set initial

is so .

asymptatically

asymptatically stable

Gls)

stable

A B

We >

say

can : , .

Al30ypdes of

Clpdes

② (pds BISS)

of his)]

cancellation

there t of of

kind 415)

in

is S

same . . asymptatically

asymptatically stable (E).

stable Gls)

A B

We =>

say

can : ,

I consider

instead PARALLELED A(S)

CONNECTION >

a ↓

we : S

U(s)

Bud

T ↳

=

1A

B(s)

=A(s)

4(s) = =

= B(S)

>

of

the the

possible the previous so

cases same ⑭

we

are ore

that BOTH PARALLELED CONNECTION

CASCADED AND VIl

say

can Y(s)

G(s)

> <

rustable

can't

PRESERVE (we make

STABILITY

ASYMDICTIC an

by

component .

it with

stable component)

connecting As

an

I consider >Y (S)

FEEDBACK-LOOP CONNECTION U() <Als)

we a >

: o

⑦

dB(s)

na(s)

na/da

FAB

4()) (InB)

= B(S)

dakdBIs) Funksul

1 ⑭

da(s)dg(s) the

known

nals)np/s)

where XIS) VIl

is

= as

= Y(s)

G(s)

> <

Possible

fed-back

polynoncial of the

characteristic cases

. : Im a

correlation

there betw