Luenberger observer

A Luenberger observer is a deterministic state estimator for linear systems that reconstructs the full state vector from available output measurements. Introduced by David Luenberger in 1964, the observer is a dynamic system that runs in parallel with the plant, using a copy of the plant's dynamics and a correction term driven by the discrepancy between the measured output and the output predicted by the observer model. For a system \(\dot{x} = Ax + Bu\) with output \(y = Cx\), the observer takes the form \(\dot{\hat{x}} = A\hat{x} + Bu + L(y - C\hat{x})\), where \(L\) is the observer gain matrix designed to place the observer's eigenvalues at desired locations.

The observer solves a fundamental problem in control theory: most state-feedback designs — including pole placement via Ackermann's formula — require knowledge of the full state vector, but sensors rarely measure every state variable directly. The Luenberger observer provides a theoretical guarantee that if the system is observable, the estimate \(\hat{x}\) converges to the true state \(x\) asymptotically, and the convergence rate can be made arbitrarily fast by appropriate choice of \(L\). When combined with state feedback, the observer and controller form a complete output-feedback architecture governed by the separation principle: the controller poles and observer poles can be placed independently, and the combined system retains closed-loop stability.