Post on 19-Mar-2016
description
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
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• 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
By final value theorem 0
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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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
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