Observer Design

Observer design addresses a fundamental limitation of state feedback control: the assumption that all state variables are directly measurable. In real systems, only outputs y(t) are measured, and the complete state vector x(t) is often inaccessible. A state observer reconstructs the state vector from the available input u(t) and output y(t) measurements, making full-state feedback practically implementable.

Core Concept Explanation

The Luenberger observer is a deterministic state estimator for linear systems. It consists of a copy of the plant model with an added correction term. The observer state equation is:

dx̂/dt = Ax̂ + Bu + L(y - Cx̂)

Here x̂ is the estimated state, and the term L(y - Cx̂) is the correction. The matrix L is the observer gain, an n×p matrix. The quantity (y - Cx̂) is the output prediction error: the difference between the actual measured output y and the output predicted by the observer model Cx̂. This error drives the observer to correct its estimate.

Define the estimation error e = x - x̂. Subtracting the observer equation from the plant equation:

de/dt = dx/dt - dx̂/dt = (Ax + Bu) - (Ax̂ + Bu + L(Cx - Cx̂)) = A(x-x̂) - LC(x-x̂) = (A - LC)e

The error dynamics are governed solely by the matrix (A - LC). If the eigenvalues of (A - LC) all have negative real parts, the estimation error decays to zero regardless of the initial error. The designer chooses L to place the eigenvalues of (A - LC) at desired locations.

Mathematical Expression

The observer pole placement problem is the dual of state feedback pole placement. Just as state feedback places eigenvalues of (A - BK) by choosing K, observer design places eigenvalues of (A - LC) by choosing L. By duality, the system (A, C) must be completely observable for arbitrary observer pole placement to be possible.

The dual Ackermann formula for observer gain is:

L = φ(A) · Mo⁻¹ · eₙ

where φ(A) is the desired observer characteristic polynomial evaluated at A, Mo is the observability matrix, and eₙ is the last standard basis column vector. Alternatively, coefficients of L can be found by expanding det(sI - A + LC), setting it equal to the desired observer polynomial, and matching coefficients.

The separation principle states that when the system is both controllable and observable, the state feedback gain K and the observer gain L can be designed completely independently. The combined closed-loop eigenvalues are the union of the controller poles (eigenvalues of A - BK) and the observer poles (eigenvalues of A - LC). This decoupling greatly simplifies design.

Practical Understanding

Observer poles must typically be placed faster (more to the left in the s-plane) than the controller poles. A common rule of thumb is to place observer poles 3 to 5 times faster than the desired closed-loop poles. This ensures the estimation error decays quickly and the observer tracks the true state before the control loop acts on inaccurate estimates.

In the presence of measurement noise and process disturbances, the optimal observer is the Kalman filter, which is the stochastic generalization of the Luenberger observer. The Kalman filter computes the optimal L that minimizes the mean-square estimation error. For GATE purposes, the Luenberger deterministic observer is the primary topic.

Numerical Example

Consider a system with A = [[0, 1], [-2, -3]], C = [[1, 0]]. Design a Luenberger observer with desired observer poles at s = -5 and s = -6. The desired observer characteristic polynomial is (s+5)(s+6) = s² + 11s + 30. Set L = [l₁, l₂]ᵀ and compute (A - LC).

Given:
A = [[0,1],[-2,-3]],  C = [[1,0]],  n = 2
Desired observer poles: s = -5, s = -6

Why this formula applies:
Verify observability: Mo = [C; CA] = [[1,0],[0,1]], rank = 2 (observable).
Error dynamics governed by (A - LC). Choose L to place poles.

Formula:
det(sI - (A - LC)) = s² + 11s + 30

Substitution:
L = [[l₁],[l₂]],   LC = [[l₁,0],[l₂,0]]
A - LC = [[0-l₁, 1],[−2-l₂, -3]]

Calculation:
det(sI-(A-LC)) = det([[s+l₁, -1],[2+l₂, s+3]])
= (s+l₁)(s+3) + (2+l₂)
= s² + (3+l₁)s + (3l₁ + 2 + l₂)

Matching coefficients:
3 + l₁ = 11  →  l₁ = 8
3(8) + 2 + l₂ = 30  →  l₂ = 4

Final Answer:
L = [[8],[4]]
Observer matrix A-LC = [[-8, 1],[-6, -3]]
Eigenvalues: s = -5, -6 (verified)

Exam Tip: Observer design is the dual of state feedback design. Replace A with Aᵀ, B with Cᵀ, and K with Lᵀ in any state feedback formula to get the corresponding observer formula. This duality shortcut saves significant time in GATE.

Mechanism Points

• The observer runs a copy of the plant model and uses the output prediction error (y - Cx̂) multiplied by gain L to continuously correct the state estimate.
• Error dynamics: de/dt = (A - LC)e. Choosing L to make (A - LC) stable ensures estimation error decays to zero.
• Observer pole placement requires complete observability: rank(Mo) = n. Without observability, some modes cannot be corrected by any L.
• Observer poles should be 3 to 5 times faster than controller poles to ensure estimation converges before control actions rely on the estimate.
• Separation principle: the combined system eigenvalues are the union of controller and observer poles, independently designed. This is valid only when both controllability and observability conditions are met.
• The observer-based output feedback controller replaces the full-state feedback requirement: the control is u = -Kx̂ rather than u = -Kx, using the estimated state.

Quick Revision

• Luenberger observer equation: dx̂/dt = Ax̂ + Bu + L(y - Cx̂).
• Estimation error dynamics: de/dt = (A - LC)e. Observer poles are eigenvalues of (A - LC).
• Observability condition: rank(Mo) = n is required for arbitrary observer pole placement.
• Observer gain L found by: expand det(sI - A + LC), match with desired observer polynomial, solve for L elements.
• Duality: observer design for (A, C) is equivalent to state feedback for (Aᵀ, Cᵀ). Transpose and swap to convert between the two problems.
• Separation principle: design K (controller) and L (observer) independently. Combined eigenvalues = {A-BK poles} + {A-LC poles}.
• Exam trap: Observer poles must be placed in the left half plane and typically faster than controller poles. Placing them too slow leads to poor state estimates and degraded control performance.