📘 Pole Placement & Ackermann’s Formula – Complete Explanation
Pole Placement is used to design state feedback controller to place closed-loop poles at desired locations. Only possible if system is controllable.
🔹 1. State Feedback Concept
Original system:
ẋ = Ax + Bu
Apply state feedback:u = -Kx
Closed-loop system becomes:ẋ = (A - BK)x
Poles = Eigenvalues of (A - BK)🔹 2. Condition for Pole Placement
System must be:Controllable
Check rank of controllability matrix.🔹 3. Ackermann’s Formula
For nth order system:K = [0 0 0 ... 1] Q⁻¹ φ(A)
Where:- Q = Controllability matrix
- φ(A) = Desired characteristic polynomial evaluated at A
🔹 4. Worked Example 1 (2nd Order System)
Given:
A = [ 0 1 -2 -3 ] B = [ 0 1 ]
Step 1: Check Controllability
Compute:AB = [ 1 -3 ]
Controllability matrix:Q = [ 0 1 1 -3 ]
Determinant ≠ 0 → Controllable.Step 2: Desired Poles
Let desired poles:s = -4, -5
Desired characteristic polynomial:(s + 4)(s + 5) = s² + 9s + 20
Step 3: Compare with Current Polynomial
Original:s² + 3s + 2
Step 4: Find K = [k₁ k₂]
After solving:K = [18 6]
Closed-loop poles placed at -4 and -5.🔹 5. Important Insights
- Pole placement changes system dynamics
- Faster poles → Faster response
- Too large poles → High control effort
- Not possible if system not controllable
🔹 6. Conceptual GATE Question Type
If system not controllable:- Pole placement impossible
- Some poles cannot be moved
🎯 Modern Control Summary
- Check controllability first
- Choose desired poles
- Use Ackermann formula
- Compute K
- Verify closed-loop eigenvalues
Pole Placement = Direct Control of System Dynamics
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