SME
Control Systems · Problem type
DC gain from poles and zeros
Use this for “Find the DC gain of G(s) = 5(s + 2)/((s + 1)(s + 4)).”
Diagram
Poles (×) at s = −1, −4; zeros (○) at s = −2.
G(0)
=
5
×
2
1 × 4
=
2.5.
This is an example. Type your own values to start a new problem.
G(s)
=
(s
+
)
(s
+
)(s
+
)
G(0)
=
Formulas
G(0)
=
K
×
z
p₁
×
p₂
2.5
=
5
×
2
1 × 4
Assumptions
•
G(s) = K(s + z) ÷ ((s + p₁)(s + p₂)): a zero at −z and two real poles at −p₁ and −p₂.
•
The DC gain is G(0): put s = 0, so each factor (s + r) becomes r.
•
Both poles are in the left half-plane, so a unit step response settles to G(0).
Step-by-step
WE KNOW
K = 5, z = 2, p₁ = 1, p₂ = 4
FIND
DC gain (G(0))
Step 1 · Find DC gain (G(0))
Use G(0) = K × z ÷ (p₁ × p₂)
Put
s
=
0:
the
zero’s
factor
s
+
z
becomes
z
and
each
pole’s
factor
s
+
p
becomes
p.
G(0)
=
5
×
2
1 × 4
G(0)
=
10
4
G(0) = 2.5
Check
Put the numbers back in. Both sides should match.
2.5
=
5
×
2
1 × 4
✓
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Steady-state error to step inputs
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