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Luenberger Observer

Overview & Motivation

State-feedback controllers require the complete state vector, but physical systems typically expose only a few measured outputs. The Luenberger observer reconstructs unmeasured states from input and output measurements using a copy of the plant model continuously corrected by the difference between predicted and actual outputs.

Unlike the Kalman filter, the Luenberger observer is deterministic — it requires no noise statistics and no covariance propagation. For embedded systems with well-characterized models and deterministic environments, it offers the same state reconstruction at a fraction of the computational cost.

Mathematical Theory

Discrete-Time State-Space System

$$x[k+1] = A x[k] + B u[k]$$ $$y[k] = C x[k] + D u[k]$$

Observer Update Law

The observer maintains a state estimate $\hat{x}[k]$ updated by:

$$\hat{y}[k] = C \hat{x}[k] + D u[k]$$ $$\hat{x}[k+1] = A \hat{x}[k] + B u[k] + L (y[k] - \hat{y}[k])$$

The term $y[k] - \hat{y}[k]$ is the innovation (output prediction error). The gain matrix $L \in \mathbb{R}^{n \times p}$ scales how aggressively the estimate is corrected.

Error Dynamics

Defining the estimation error $e[k] = x[k] - \hat{x}[k]$:

$$e[k+1] = (A - LC), e[k]$$

The error decays to zero if and only if all eigenvalues of $(A - LC)$ lie strictly inside the unit circle. Choosing $L$ to place those eigenvalues at desired locations is the pole placement problem for observers.

Ackermann's Formula (SISO Output)

For single-output systems ($p = 1$), the observer gain that places eigenvalues of $(A - LC)$ at ${\mu_1, \dots, \mu_n}$ is:

$$L = \varphi_d(A), \mathcal{O}^{-1}, e_n$$

where:

  • $\varphi_d(z) = \prod_{i=1}^n (z - \mu_i)$ is the desired characteristic polynomial, evaluated at $A$
  • $\mathcal{O} = \begin{bmatrix} C \ CA \ \vdots \ CA^{n-1} \end{bmatrix}$ is the observability matrix
  • $e_n = [0, \dots, 0, 1]^T$ is the last standard basis vector

This is the dual of Ackermann's controller placement formula.

Observability Condition

Ackermann's formula requires $\mathcal{O}$ to be invertible, which holds if and only if the pair $(A, C)$ is observable: every state affects the output through some combination of shifts. An unobservable pair makes the formula degenerate — the gain cannot force arbitrary error convergence.

Complexity Analysis

Phase Time Space Notes
Design (Ackermann) $O(n^3)$ $O(n^2)$ Observability matrix build + linear solve
Update (per step) $O(n^2 + np)$ $O(n^2)$ Matrix-vector products; dominant cost is $A\hat{x}$

The design phase is offline. The real-time cost per sample is dominated by the $n \times n$ state-transition multiply.

Step-by-Step Walkthrough

System: Double integrator, $n=2$, $p=1$, desired observer poles ${\mu_1, \mu_2} = {0.2,, 0.3}$

$$A = \begin{bmatrix}1 & 1\0 & 1\end{bmatrix}, \quad B = \begin{bmatrix}0\1\end{bmatrix}, \quad C = \begin{bmatrix}1 & 0\end{bmatrix}$$

Step 1 — Build observability matrix:

$$\mathcal{O} = \begin{bmatrix}C\CA\end{bmatrix} = \begin{bmatrix}1 & 0\1 & 1\end{bmatrix}$$

Step 2 — Evaluate desired polynomial at $A$:

$$\varphi_d(z) = (z - 0.2)(z - 0.3) = z^2 - 0.5z + 0.06$$

$$\varphi_d(A) = A^2 - 0.5A + 0.06I = \begin{bmatrix}0.56 & 1.5\0 & 0.56\end{bmatrix}$$

Step 3 — Solve for gain:

$$\mathcal{O}^{-1} = \begin{bmatrix}1 & 0\-1 & 1\end{bmatrix}, \quad \mathcal{O}^{-1} e_2 = \begin{bmatrix}0\1\end{bmatrix}$$

$$L = \varphi_d(A) \cdot \begin{bmatrix}0\1\end{bmatrix} = \begin{bmatrix}1.5\0.56\end{bmatrix}$$

Verification: eigenvalues of $A - LC = \begin{bmatrix}-0.5 & 1\-0.56 & 1\end{bmatrix}$ are ${0.2, 0.3}$. ✓

Pitfalls & Edge Cases

  • Unobservable pair. If $\mathcal{O}$ is rank-deficient, the linear solve in Ackermann's formula fails. Verify observability with a rank test before design.
  • Pole placement too aggressive. Observer poles much faster than the controller poles amplify measurement noise, since every output error is fed back through $L$. Typical practice: observer poles 2–5× faster than controller poles.
  • Deadbeat design (all poles at zero) converges in exactly $n$ steps but maximizes noise sensitivity and requires large $L$ entries, risking numerical overflow in fixed-point.
  • MIMO output limitation. Ackermann's formula applies to SISO output ($p=1$). Multi-output systems require alternative pole-placement methods (e.g., Brogan's formula or numerical optimization).
  • Model mismatch. If the observer plant differs from the true plant, the error dynamics no longer satisfy $e[k+1] = (A-LC)e[k]$ exactly; a bounded steady-state error results rather than zero convergence.

Variants & Generalizations

Variant Key Difference
Kalman Filter Stochastic design — minimizes covariance rather than placing poles; handles noise statistics
Extended Luenberger Observer Linearizes a nonlinear plant around the estimate for quasi-linear operation
Unknown-Input Observer Estimates states in the presence of unmeasured disturbances
Reduced-Order Observer Estimates only the unmeasured states, using measured outputs directly
Continuous-time observer Uses $\dot{\hat{x}} = A\hat{x} + Bu + L(y - C\hat{x})$; same structure, continuous pole placement

Applications

  • Motor control — estimating velocity and back-EMF from position and current sensors.
  • Automotive suspension — reconstructing unsprung mass velocity from chassis accelerometers.
  • Satellite attitude estimation — inferring angular rates from gyros and star trackers.
  • Observer-based compensator — pairing with LQR to form the deterministic equivalent of LQG when noise statistics are unavailable.

Connections to Other Algorithms

graph LR
    LO["Luenberger Observer"]
    LTI["LinearTimeInvariant"]
    GE["Gaussian Elimination"]
    LQR["LQR Controller"]
    KF["Kalman Filter"]
    LTI -->|"plant model"| LO
    GE -->|"solve O^{-1}"| LO
    LO -->|"state estimate"| LQR
    KF -.->|"stochastic counterpart"| LO
Loading
Algorithm Relationship
Linear Time-Invariant Model Supplies the $(A, B, C, D)$ matrices used in both design and update steps
Gaussian Elimination Solves $\mathcal{O} x = e_n$ inside Ackermann's formula
LQR Controller Primary consumer of the observer's state estimate
Kalman Filter Stochastic counterpart; adds noise covariance propagation

References & Further Reading

  • Luenberger, D.G., "An Introduction to Observers," IEEE Transactions on Automatic Control, 16(6), 1971.
  • Franklin, G.F., Powell, J.D. and Emami-Naeini, A., Feedback Control of Dynamic Systems, 8th ed., Pearson, 2019 — Chapter 7.
  • Chen, C.-T., Linear System Theory and Design, 4th ed., Oxford University Press, 2013 — Chapter 8.