Questions: State Observer: Full-State and Partial Observation

Question 1

A control engineer places the observer eigenvalues at −50 ± 5j while the closed-loop controller poles are at −5 ± 2j. What is the most significant consequence of this choice?

• A. The observer will be unstable because its poles are too far into the left half-plane
• B. The observer will converge very rapidly but will heavily amplify sensor noise in the state estimates
• C. The observer convergence will be too slow to track the plant, causing estimation lag
• D. The closed-loop system will be unstable because observer poles must match controller poles

Answer: (B)

Observer eigenvalues much further left than controller poles mean the estimation error decays very quickly — which sounds desirable, but the large observer gain L needed to achieve this amplifies measurement noise and injects it into the state estimates fed to the controller. In practice, observer poles are typically placed 3–5× faster than controller poles to balance fast convergence against noise sensitivity. There is no requirement for observer and controller poles to match; the Separation Principle allows them to be placed independently.

Question 2

A student derives that the estimation error e = x − x̂ satisfies ė = (A − LC)e. What is the most important design conclusion from this equation?

• A. The error depends on the control input u, so the observer gain L must be updated at each timestep
• B. By choosing L, we can assign the eigenvalues of (A − LC) anywhere in the complex plane (given observability), controlling how fast the estimation error decays to zero
• C. The error decays to zero only if the initial state x(0) is known exactly so that e(0) = 0
• D. The observer gain must be chosen so that A − LC has the same eigenvalues as the open-loop plant A

Answer: (B)

The key insight is that the error dynamics are homogeneous and decoupled from the input u. This means the error will converge to zero regardless of initial conditions, as long as (A − LC) is stable. Because L is a free design parameter, we can perform eigenvalue placement on (A − LC) — exactly analogous to placing closed-loop poles with state feedback — to set the convergence rate. Observability is the condition that guarantees arbitrary pole placement is possible.

Question 3

The Separation Principle states that when an observer is combined with a state-feedback controller, the closed-loop eigenvalues are the union of the controller poles and the observer poles, so both can be designed independently.

• T. True
• F. False

Answer: True

The Separation Principle is one of the most practically important results in linear control theory. It allows the designer to split a complex output-feedback problem into two independent sub-problems: choose the feedback gain K to place the controller poles where desired, then choose the observer gain L to place the observer poles. The combined system has all those poles — they do not interact. Without the Separation Principle, designing output-feedback controllers would require simultaneously satisfying both objectives, which is far more complex.

Question 4

A full-order observer reconstructs mainly the unmeasured states, since there is no need to estimate states that can be directly measured.

• T. True
• F. False

Answer: False

A full-order observer reconstructs ALL n states, including those that are directly measured. This is computationally redundant but simpler to design. A reduced-order observer reconstructs only the unmeasured states (of dimension n minus the number of outputs), which is more efficient but more involved to derive. The naming is counterintuitive: 'full-order' refers to the order of the observer being the same as the order of the plant, not to the completeness of what is measured.

Question 5

Explain the core principle of the Luenberger observer: how does the correction term L(y − ŷ) cause the state estimate to converge to the true state even when initial conditions are unknown?

Model answer: The observer runs a parallel model of the plant: x̂̇ = Ax̂ + Bu. Without correction, errors from imperfect initial conditions would persist indefinitely. The correction injects the difference between the measured output y and the predicted output ŷ = Cx̂ back into the state equation, scaled by L. Subtracting the observer equation from the true plant equation gives the error dynamics ė = (A − LC)e. This is a homogeneous linear ODE whose solutions decay to zero if (A − LC) is stable — which can be guaranteed by appropriate choice of L (given observability). The correction term continuously pushes the estimate toward the true state: whenever the model diverges from reality, the output mismatch (y − ŷ) grows and the correction signal increases, pulling the estimate back.

The elegance of the Luenberger observer is that convergence is guaranteed structurally — it depends only on the eigenvalues of (A − LC), not on the specific initial error or the input trajectory. This makes observer-based control tractable and robust to uncertain initial conditions.