SME
Control Systems · Problem type
Closed-loop bandwidth
Use this for “Find the closed-loop bandwidth of a system with ζ = 0.4 and ωₙ = 5 rad/s.”
Chart
Underdamped step response: ζ = 0.4, ωₙ = 5 rad/s
Peak
1.25
at
Tₚ
=
π
ωd
=
0.686
s:
OS
=
exp(−ζπ/
√
1 −
ζ
2
)
=
25.38%
Tₛ
=
4
ζωₙ
=
2
s:
the
envelope
enters
the
±2%
band.
This is an example. Type your own values to start a new problem.
ωBW (rad/s)
=
ωₙ (rad/s)
×
1
−
2ζ²
+
4ζ⁴
−
4ζ²
+
2
ζ
=
%OS
Percent overshoot
calculated · %
Tₛ
Settling time
calculated
Formulas
%OS
=
100
×
e
−
ζ
× π ÷
√
1 −
ζ
2
25.3827
=
100
×
e
−0.4 × π ÷
√
1 −
0.4
2
Tₛ
=
4
ζ
× ωₙ
2
=
4
0.4 × 5
ωBW
=
ωₙ
×
√(1
−
2
×
ζ
2
+
√
4 ×
ζ
4
− 4 ×
ζ
2
+ 2
)
6.8726
=
5
×
√(1
−
2
×
0.4
2
+
√
4 ×
0.4
4
− 4 ×
0.4
2
+ 2
)
Assumptions
•
The closed loop is ωₙ² ÷ (s² + 2ζωₙ s + ωₙ²), underdamped (0 < ζ < 1), with no zeros.
•
The bandwidth ωBW is where the closed-loop gain falls 3 dB below its DC value of 1.
•
Tₛ = 4 ÷ (ζωₙ) uses the 2% band; at a fixed ζ, a wider bandwidth means a faster response.
Step-by-step
WE KNOW
ζ = 0.4, ωₙ = 5 rad/s
FIND
percent overshoot (%OS), settling time (Tₛ), bandwidth (ωBW)
Step 1 · Find percent overshoot (%OS)
Use %OS = 100 × e−ζ × π ÷ √(1 − ζ²)
The
first
peak
overshoots
by
e
−ζπ ÷
√
1 −
ζ
2
of
the
final
value:
ζ
alone
sets
it.
%OS
=
100
×
e
−0.4 × π ÷
√
1 −
0.4
2
%OS
=
100
×
e
−0.4π ÷
√
1 − 0.16
%OS
=
100
×
e
−0.4π ÷
√
0.84
%OS
=
100
×
e
−0.4π ÷ 0.91652
%OS
=
100
×
e
−1.3711
%OS = 100 × 0.25383
%OS = 25.3827%
Step 2 · Find settling time (Tₛ)
Use Tₛ = 4 ÷ (ζ × ωₙ)
The
envelope
e
−ζωₙ t
falls
to
about
2%
when
ζωₙ
t
=
4.
Tₛ
=
4
0.4 × 5
Tₛ
=
4
2
Tₛ = 2 s
Step 3 · Find bandwidth (ωBW)
Use ωBW = ωₙ × √(1 − 2 × ζ² + √(4 × ζ⁴ − 4 × ζ² + 2))
Set
the
closed-loop
gain
|T(jω)|
to
1
÷
√
2
and
solve
the
quadratic
in
ω
2
.
ωBW
=
5
×
√(1
−
2
×
0.4
2
+
√
4 ×
0.4
4
− 4 ×
0.4
2
+ 2
)
ωBW
=
5
×
√
1 − 0.32 +
√
1.4624
ωBW
=
5
×
√
1 − 0.32 + 1.2093
ωBW
=
5
×
√
1.8893
ωBW = 5 × 1.3745
ωBW = 6.8726 rad/s
Check
Put the numbers back in. Both sides should match.
25.3827
=
100
×
e
−0.4 × π ÷
√
1 −
0.4
2
✓
2
=
4
0.4 × 5
✓
6.8726
=
5
×
√(1
−
2
×
0.4
2
+
√
4 ×
0.4
4
− 4 ×
0.4
2
+ 2
)
✓
Topic check
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