Dynamic Response

• Steady State Response: the part of resp. when t→∞

• Transient response: the part of resp right after the input is being applied.

• Both are part of the total resp.

total resp = z.i. resp + z.s.resp.z.i. resp = “Output due to i.c. when input ≡ 0”z.s. resp = “Output due to input excitation when all i.c.

are set=0 at t=0”

Typical test signal

• Unit step signal:

• Unit impulse:δ(t)

0

us(t)

1

stu

tutu

s

s

1))((

)()(

L

t

s

b

a

dtttut

b

adtt

tt

)()(,1))((

0

01)(

0,0)(

any

L

δ(t)

t

• Unit ramp:

• Unit acc. signal:

r(t)

2

1))((

)()()(

00

0)(

str

dttututtr

t

tttr

t

ss

L

3

2

1))((

)(

00

02

1)(

sta

dttr

t

ttta

t

L

a(t)

t

t

0.5

10

• Exponential signal:

• sinusoidal:as

tue

a

t

tetue

sat

at

sat

1))((

0

00

0)(

L

22))()(sin(

00

0sin)()sin(

stut

t

tttut

s

s

L

0

1

t

• Unit step response:

In Matlab: step

• Unit impulse resp:

Matlab: impulse

H(s)u(s)=1

s

y(s)=H(s)1

s

0))((1

i.c.

input whenoutputu.i.r

(t)sHL

0

1)(

i.c.

or input whenoutputu.s.r s

tus

H(s)u(s)=1 y(s)=H(s)

Dynamic Response

• Unit step signal:

• Step response: y(s)=H(s)/s, y(t)=L-1{H(s)/s}

• Unit impulse signal: δ(t)1

• Impulse response: h(t)= L-1 {H(s)}

• In Matlab: use “step”, “impulse”, “lsim”, etc

stutu s

1)()(

• Defined based on unit step response• Defined for closed-loop system

• Steady-state value yss

• Steady-state error ess

• Settling time ts

= time when y(t) last enters a tolerance band

tutyy st

input,lim

sst

ytee

1lim

Time domain response specifications

%1001

1%100

:overshoot percentage

:Overshoot

)( :hence

);max(

);( :Peak time

valuemaximum its reaches )( when time Peak time

maxmax

max

max

max

max

y

y

yyM

yyM

tyy

yy

yytt

tyt

ss

ssp

ssp

p

p

p

50% from different percentage a use people some

used freq. not

of 50%

reaches first whentime the timeDelay

overshoot no is there &

time peak no is there as reached is If

ss

d

y

tyt

ty

)(

,max

effectsimilar has ,

input referencein changes to

responds system afast how captures timerise

.first time for the 90 to10

from go to)(for it takes timethe timerise

pd

ssss

r

tt

y.y.

tyt

H s Y s U s 1U s

s

1 0

1 0

mmn

b s b s b s bH s

a s s a s a

1Y s H s

s

By final value theorem

0

0 00

lim lim limsst s s

by y t sY s H s

a

In MATLAB: num = [ .. .. .. .. ]

b0 = num(length(num)), or num(end)

a0 = den(length(den)), or den(end)

yss=b0/a0

1ss sse y

If numerical values of y(t) available,

abs(y – yss) < tol means inside band

abs(y – yss) ≥ tol not inside

e.g. t_out = t(abs(y – yss) ≥ tol) contains all those time points when y is not inside the band.

Therefore, the last value in t_out will be the settling time.

ts=t_out(end)

Peak time tp = time when y(t) reaches its maximum value.

Peak value ymax = y(tp)

Hence: ymax = max(y);

tp = t(y = ymax);

Overshoot: OS = ymax - yss

Percentage overshoot:max 100%ss

pss

y yM

y

max 1

100%1

y

If t50 = t(y >= 0.5·yss),

this contains all time points when

y(t) is ≥ 50% of yss

so the first such point is td.

td=t50(1);

Similarly, t10 = t(y <= 0.1*yss)

& t90 = t(y >= 0.9*yss)

can be used to find tr.

tr=t90(1)-t10(end)

%158.0

12.0

12.08.092.0,92.0

2.0

1,8.0)(

max

.

...

o.s. percentage

overshoot

0i.c.

stepu resp. step on defined are specs

y

yye

yyy

ssdss

ssdss

tp≈0.9sec

10%yss

90%yss

tr≈0.45

td≈0.35

ts ts

tr≈0.35

±5% ts=0.45

yss=1

ess=0

O.S.=0

Mp=0

tp=∞

td≈0.2

tr≈0.1

td≈0.2

ts≈0.92

tp=0.35O.S.=0.4

Mp=40%

yss=1

es=0

Steady-state tracking & sys. types

• Unity feedback control:

G(s) C(s)+

-r(s) e y(s)

plant

Go.l.(s)+

-r(s) e

y(s)ol

ol

G

G

sr

sy

1)(

)(

T.F. get & open, loop cut i.e.

yto e from T.F. loop open the is )()(

)(.. sG

se

sylo

s. an cancel can otherwise , need , If

but

:into factored be always can

00

0

0

0,0

)1()1)(1(

)1()1)(1(

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011

011

11

1

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21..

..

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ba

bK

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asasasasas

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sTsTsTKG

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mN

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nn

n

mm

pN

mbalo

lo

psloss

slos

ss

lo

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KsGe

ssr

sG

ssrsseete

srsG

sysrse

srsG

sGsy

1

1

)(1

1

1)(

)(1

)()(lim)(

)()(1

1)()()(

)()(1

)()(

0..

0..0

..

..

..

step to

:input step For

:tracking state-steady

:error tracking

:loop-closed

finite

r, to respect with0" type" called is system the If

K control alproportion withconfused be to not p, small use here

step to Then

const. error positionstatic called

denote

P

0

0..

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)(

1

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)0()(lim

a

bGK

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