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

sstytee

1lim

Time domain response specifications

H s Y s U s 1U ss

1 0

1 0

mmn

b s b s b s bH s

a s s a s a

1Y s H ss

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0 00

lim lim limss t s s

by y t sY s H sa

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%ssp

ss

y yMy

max 1 100%1

y

If ymax is reached as t→∞, there is no peak time and there is no overshoot.

Delay time td = the time when y(t) first reaches

50% of yss

– Not frequently used

– Some people use a percentage different from 50%

t50=t(y<=0.5*yss);td=t50(end)

Rise time tr = the time it takes for y(t) to go from 0.1yss to 0.9yss for the first time.

• Rise time captures how fast a system responds to changes in a reference input

• td, tp has similar effect

If t50 = t(y >= 0.5·yss),this contains all time points wheny(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(1)

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90%yss

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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

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