State Observers and the Luenberger Observer
An observer reconstructs unmeasured states by running a model of the plant corrected by the difference between predicted and actual outputs.
Estimating What You Cannot Measure
State feedback needs the full state, but sensors usually measure only some outputs. A state observer reconstructs the missing states by simulating a model of the plant in parallel and correcting that simulation using the measured output. The Luenberger observer is the classic deterministic form.
The observer equation
The observer runs x-hat' = A*x-hat + B*u + L*(y - C*x-hat). The first two terms replicate the plant model; the third is the correction, where L is the observer gain and (y - C*x-hat) is the output error between what is measured and what the model predicts. When the estimate is correct, the error is zero and no correction is applied.
Error dynamics
Define the estimation error e = x - x-hat. Subtracting the observer equation from the plant equation gives e' = (A - L*C)*e. The error decays to zero if the eigenvalues of A - L*C lie in the left half-plane. Choosing L to place those eigenvalues is exactly the dual of choosing a feedback gain to place controller poles, and it is possible whenever the system is observable.
Choosing the observer speed
- The observer poles are usually placed faster than the controller poles, so the estimate converges before the controller acts on it.
- Placing them too fast makes the observer sensitive to measurement noise, since fast correction amplifies noise.
- This speed-versus-noise trade is the deterministic analogue of the Kalman filter's gain balance.
Relation to the Kalman filter
The Luenberger observer and the Kalman filter share the same structure of model plus output correction. The difference is how L is chosen: the Luenberger observer places error poles by hand, while the Kalman filter computes the gain optimally from noise statistics. In a noise-free deterministic setting the Luenberger observer is simpler and sufficient.
Combined with state feedback under the separation principle, an observer yields a complete output-feedback controller, letting state-based design methods run on systems where the full state is never directly measured.