dynamic response
DESCRIPTION
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” - PowerPoint PPT PresentationTRANSCRIPT
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
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ssss
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s
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1 0
mmn
b s b s b s bH s
a s s a s a
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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
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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
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