๐ Luenberger Observer – Complete Theory & Worked Examples
In practical systems, all states cannot be measured. Observer is used to estimate internal states using input and output. Very important topic in Modern Control.
๐น 1. Why Observer Needed?
- Not all state variables measurable
- Sensors expensive or unavailable
- Need full state for state feedback
๐น 2. Original System
ẋ = Ax + Bu y = Cx
๐น 3. Observer Model
x̂̇ = Ax̂ + Bu + L(y − Cx̂)
Where:- x̂ = Estimated state
- L = Observer gain matrix
๐น 4. Error Dynamics
Define error:e = x − x̂
Then:ė = (A − LC)e
Observer poles = Eigenvalues of (A − LC)๐น 5. Condition for Observer Design
System must be:Observable
Check rank of observability matrix.๐น 6. Worked Example – Observer Gain Calculation
Given:
A = [ 0 1 -2 -3 ] C = [1 0]
Step 1: Check Observability
Compute:CA = [0 1]
Observability matrix:O = [ 1 0 0 1 ]
Rank = 2 → Observable.Step 2: Desired Observer Poles
Choose faster poles:s = -8, -9
Desired polynomial:(s + 8)(s + 9) = s² + 17s + 72
Step 3: Compute L
Let:L = [ l₁ l₂ ]
Characteristic equation of (A − LC): After solving:L = [14 43]
Observer poles placed at -8 and -9.๐น 7. Important Design Rule
Observer poles should be:- 2 to 5 times faster than system poles
- Too fast → Noise amplification
๐น 8. Duality Principle
Observer design is dual of pole placement. Replace:- A → Aแต
- B → Cแต
๐ฏ GATE Important Points
- Check observability first
- Error dynamics stability important
- Observer poles can be placed freely
- Dual concept frequently asked
Observer = Estimator of Hidden States
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