SME
Control Systems · Problem type
Steady-state error to step inputs
Use this for “Find the steady-state step error of G(s) = 30/((s + 1)(s + 5)) under unity feedback.”
Chart
Underdamped step response: ζ = 0.5071, ωₙ = 5.916 rad/s
Peak
0.992
at
Tₚ
=
π
ωd
=
0.616
s:
OS
=
exp(−ζπ/
√
1 −
ζ
2
)
=
15.7%
Tₛ
=
4
ζωₙ
=
1.33
s:
the
envelope
enters
the
±2%
band.
This is an example. Type your own values to start a new problem.
G(s)
=
K
(s
+
)(s
+
)
e
ss
=
1
1
+
=
eₛₛ
c(∞)
Final output
worked out (not typed)
ωₙ
Natural frequency
worked out (not typed) · rad/s
ζ
Damping ratio
worked out (not typed)
Formulas
Kₚ
=
K
a
×
b
6
=
30
1 × 5
eₛₛ
=
1
1 + Kₚ
0.1429
=
1
1 + 6
c(∞)
=
Kₚ
1 + Kₚ
0.8571
=
6
1 + 6
ωₙ
=
√
a
×
b
+
K
5.9161
=
√
1 × 5 + 30
ζ
=
a
+
b
2 × ωₙ
0.5071
=
1 + 5
2 × 5.9161
Assumptions
•
Unity feedback around G(s) = K ÷ ((s + a)(s + b)): type 0, with no pole at s = 0.
•
The input is one unit step; eₛₛ = 1 ÷ (1 + Kₚ) needs the closed loop stable, as it is for positive K, a and b.
•
The closed loop is K ÷ (s² + (a + b)s + ab + K), so ωₙ² = ab + K and 2ζωₙ = a + b; ζ above 1 means no overshoot.
•
K, a and b are plain numbers, read with s in s⁻¹, so ωₙ comes out in rad/s.
Step-by-step
WE KNOW
K = 30, a = 1, b = 5
FIND
position constant (Kₚ), steady-state error (eₛₛ), final output (c(∞)), natural frequency (ωₙ), damping ratio (ζ)
Step 1 · Find position constant (Kₚ)
Use Kₚ = K ÷ (a × b)
Kₚ
is
G(0):
put
s
=
0,
so
(s
+
a
)(s
+
b
)
becomes
ab.
Kₚ
=
30
1 × 5
Kₚ = 6
Step 2 · Find steady-state error (eₛₛ)
Use eₛₛ = 1 ÷ (1 + Kₚ)
The
output
settles
where
c
=
Kₚ(1
−
c),
so
the
error
1
−
c
is
1
1 + Kₚ
.
eₛₛ
=
1
1 + 6
eₛₛ
=
1
7
eₛₛ = 0.1429
Step 3 · Find final output (c(∞))
Use c(∞) = Kₚ ÷ (1 + Kₚ)
Where the output settles: the step less the error.
c(∞)
=
6
1 + 6
c(∞)
=
6
7
c(∞) = 0.8571
Step 4 · Find natural frequency (ωₙ)
Use ωₙ = √(a × b + K)
The
closed
loop’s
constant
term
is
ab
+
K
,
which
is
ωₙ
2
.
ωₙ
=
√
1 × 5 + 30
ωₙ
=
√
35
ωₙ = 5.9161 rad/s
Step 5 · Find damping ratio (ζ)
Use ζ = (a + b) ÷ (2 × ωₙ)
The
closed
loop’s
s
term
is
a
+
b
,
which
is
2ζωₙ.
ζ
=
1 + 5
2 × 5.9161
ζ
=
6
11.8322
ζ
=
0.5071
Check
Put the numbers back in. Both sides should match.
6
=
30
1 × 5
✓
0.1429
=
1
1 + 6
✓
0.8571
=
6
1 + 6
✓
5.9161
=
√
1 × 5 + 30
✓
0.5071
=
1 + 5
2 × 5.9161
✓
Topic check
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