📘 Steady State Error & Static Error Constants – Complete Explanation
Steady State Error (SSE) is the difference between input and output as time approaches infinity. This is a very important concept in Control Systems.
🔹 1. Definition of Steady State Error
ess = lim (t → ∞) e(t)
Using Final Value Theorem:ess = lim (s → 0) sE(s)
🔹 2. Unity Feedback System
For unity feedback:E(s) = R(s) / (1 + G(s))
🔹 3. Type of System
System type = Number of poles at origin in G(s).- Type 0 → No pole at origin
- Type 1 → One pole at origin
- Type 2 → Two poles at origin
🔹 4. Static Error Constants
Position Error Constant (Kp)
Kp = lim (s → 0) G(s)
Step input error:ess = 1 / (1 + Kp)
Velocity Error Constant (Kv)
Kv = lim (s → 0) sG(s)
Ramp input error:ess = 1 / Kv
Acceleration Error Constant (Ka)
Ka = lim (s → 0) s²G(s)
Parabolic input error:ess = 1 / Ka
🔹 5. Error Table Summary
| System Type | Step | Ramp | Parabolic |
|---|---|---|---|
| Type 0 | Finite | ∞ | ∞ |
| Type 1 | 0 | Finite | ∞ |
| Type 2 | 0 | 0 | Finite |
🔹 6. Worked Example 1
Given:
G(s) = 10 / (s + 5)
Step 1: Identify Type
No pole at origin → Type 0Step 2: Find Kp
Kp = 10 / 5 = 2
Step 3: Step Input Error
ess = 1 / (1 + 2) = 1/3
Ramp input error → Infinite.🔹 7. Worked Example 2
Given:
G(s) = 20 / [s(s + 4)]
Step 1: Type
One pole at origin → Type 1Step 2: Find Kv
Kv = lim s→0 [ s × 20 / (s(s+4)) ] = 20 / 4 = 5
Step 3: Ramp Error
ess = 1/5 = 0.2
Step error = 0🎯 Important GATE Points
- System type determined only by open-loop G(s)
- Use Final Value Theorem carefully
- Ramp and parabolic errors frequently asked
- Increasing system type reduces steady state error
Higher System Type = Lower Steady State Error
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