SME
Control Systems · Problem type
PI control: gains for a damping ratio
Use this for “A plant 1/(s + 1) under PI control with Kₚ = 3 and Kᵢ = 8: find ωₙ and ζ of the closed loop.”
Chart
Step response with a zero at s = −2.67: y = y₀ + y₀′ ÷ 2.67, ζ = 0.7071, ωₙ = 2.828 rad/s
Peak 1.12 at Tₚ = 0.946 s: OS = 11.9% with the zero, 4.32% without it.
Tₛ = 1.78 s, read where the curve stays in the ±2% band.
This is an example. Type your own values to start a new problem.
G(s)
=
K
s
+
1
C(s)
=
Kₚ
+
Kᵢ
/s
ωₙ
Natural frequency
calculated · rad/s
ζ
Damping ratio
calculated
Formulas
ωₙ
=
√(
K
×
Kᵢ
τ
)
2.8284
=
√(1
×
8
1
)
ζ
=
1 +
K
× Kₚ
2 ×
τ
× ωₙ
0.7071
=
1 + 1 × 3
2 × 1 × 2.8284
Assumptions
•
Unity feedback around the plant K ÷ (τs + 1); the controller is Kₚ + Kᵢ ÷ s.
•
The closed-loop denominator is τs² + (1 + KKₚ)s + KKᵢ; dividing by τ matches it to s² + 2ζωₙ s + ωₙ².
•
The integral term removes the offset, so the output settles at the unit step.
Step-by-step
WE KNOW
τ = 1 s, K = 1, Kₚ = 3, Kᵢ = 8 s⁻¹
FIND
natural frequency (ωₙ), damping ratio (ζ)
Step 1 · Find natural frequency (ωₙ)
Use ωₙ = √(K × Kᵢ ÷ τ)
The
constant
term
of
s
2
+
(1
+
KKₚ)
s
τ
+
KKᵢ
τ
is
ωₙ
2
.
ωₙ
=
√(1
×
8
1
)
ωₙ = 2.8284 rad/s
Step 2 · Find damping ratio (ζ)
Use ζ = (1 + K × Kₚ) ÷ (2 × τ × ωₙ)
The
middle
coefficient
1 + KKₚ
τ
is
2ζωₙ.
ζ
=
1 + 1 × 3
2 × 1 × 2.8284
ζ
=
4
5.6568
ζ
=
0.7071
Check
Put the numbers back in. Both sides should match.
2.8284
=
√(1
×
8
1
)
✓
0.7071
=
1 + 1 × 3
2 × 1 × 2.8284
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.