Dynamic Response

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Dynamic Response

• Unit step signal
• Step response y(s)H(s)/s, y(t)L-1H(s)/s
• Unit impulse signal d(t)??1
• Impulse response h(t) L-1 H(s)
• In Matlab use step, impulse, lsim, etc

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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

Time domain response specifications
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By final value theorem
In MATLAB num .. .. .. .. b0
num(length(num)), or num(end) a0
den(length(den)), or den(end) yssb0/a0
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If numerical values of y(t) available, abs(y
yss) lt 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.
tst_out(end)
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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
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• If ymax is reached as t?8, 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
• t50t(ylt0.5yss)
• tdt50(end)

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• 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

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If t50 t(y gt 0.5yss), this contains all
time points when y(t) is 50 of yss so
the first such point is td. tdt50(1) Simila
rly, t10 t(y gt 0.1yss) t90 t(y gt
0.9yss) can be used to find tr. trt90(1)-t10
(1)
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90yss
tr0.45
10yss
ts
ts
tp0.9sec
td0.35
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5 ts0.45
yss1 ess0 O.S.0 Mp0 tp8 td0.2
tr0.35
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tp0.35
O.S.0.4 Mp40 yss1 es0
ts0.92
td0.2
tr0.1
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Steady-state tracking sys. types

• Unity feedback control

plant

e
r(s)
y(s)
G(s)
C(s)
-

e
r(s)
Go.l.(s)
y(s)
-
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r(t)
t
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r(t)
0
t
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ref. input
sys. type
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Example of tank

H
C
-
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r(s)

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example
e(s)
y(s)
r(s)
G(s)
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