A robot arm has position sensors but no velocity sensors. An engineer says: 'We cannot implement state feedback control because velocity is a required state variable and we cannot measure it.' What does observer theory say?
AThe engineer is correct — state feedback requires direct measurement of all states, so this system cannot use state feedback
BThe engineer is wrong — we can estimate velocity from position measurements using an observer and feed back the estimated states
CThe engineer is wrong — velocity is not a state variable in a robot arm and can be ignored
DThe engineer is partially right — we can use state feedback, but only if we add velocity sensors
This is exactly what observers are designed for. An observer (Luenberger observer) runs a parallel software model of the system, using the available output (position) to continuously correct its estimate of all states including velocity. The correction term L(y − Cx̂) drives the estimation error to zero, provided the system is observable. Once the observer has reliable state estimates, those estimates are fed back to the controller exactly as if they were direct measurements. The separation principle formalizes why this works without degrading stability.
Question 2 Multiple Choice
In a Luenberger observer, what is the purpose of the correction term L(y − Cx̂)?
AIt computes the control input needed to drive the system to the desired setpoint
BIt amplifies the measured output signal to reduce sensor noise
CIt compares predicted output to actual output and nudges the state estimate toward reality
DIt replaces the system matrix A when states are unmeasurable
The term y − Cx̂ is the output prediction error: the difference between what the real system produces (y) and what the observer model predicts (Cx̂). If the state estimate is perfectly accurate, this difference is zero and no correction is needed. In practice, initial conditions are unknown and the model is imperfect, so the correction term continuously nudges the estimate toward the true state. The gain matrix L determines how aggressively the observer corrects — larger L means faster correction but greater noise amplification.
Question 3 True / False
Observer poles (the eigenvalues of A − LC) can be placed at any location in the complex plane regardless of whether the system is observable.
TTrue
FFalse
Answer: False
Observability is the necessary and sufficient condition for arbitrary observer pole placement — exactly as controllability is necessary and sufficient for arbitrary state feedback pole placement. If the system is not observable, there are directions in state space that cannot be inferred from the output, and no choice of L can make the estimation error converge in those directions. The duality between observability/observer design and controllability/state feedback design is fundamental: the mathematics of observer pole placement mirrors state feedback, with A − LC instead of A − BK.
Question 4 True / False
Placing observer poles far to the left of the imaginary axis (fast observer) generally increases the observer's sensitivity to measurement noise.
TTrue
FFalse
Answer: True
Observer speed and noise sensitivity are in fundamental tension. Placing poles far left requires a large gain matrix L, which means the observer correction term L(y − Cx̂) heavily weights the measured output. Since measurements contain noise, this amplifies high-frequency noise into the state estimate. A slow observer (poles close to the origin) produces smooth estimates but reacts sluggishly to initial errors or disturbances. The Kalman filter resolves this tradeoff optimally when noise statistics are known; for deterministic design, a common heuristic is to place observer poles 2–5 times faster than the controller poles.
Question 5 Short Answer
Why does the estimation error in a Luenberger observer converge to zero exponentially, and what system property is required for this to be achievable?
Think about your answer, then reveal below.
Model answer: The estimation error e = x − x̂ evolves according to ė = (A − LC)e. This is a linear system whose behavior is determined by the eigenvalues of A − LC (the observer poles). If L is chosen so that A − LC has all eigenvalues in the left half-plane (continuous time), the error decays exponentially to zero regardless of initial conditions. This is possible if and only if the system is observable — observability guarantees that L can be chosen to place the eigenvalues of A − LC anywhere, including in the stable left half-plane.
The key insight is that observer design is just eigenvalue placement for a different matrix (A − LC vs. A − BK for state feedback), and the enabling condition is the dual of controllability. Observability means every state variable leaves some trace in the output — so output measurements carry enough information to reconstruct all states over time. Without observability, some state directions are invisible to the output, and those unobservable modes cannot be stabilized by any observer.