Observer-Based Estimation Methods

Updated 11 December 2025

Observer-based estimation is a framework that uses mathematical observers to reconstruct unmeasured states and unknown parameters from available outputs under noise and uncertainty.
Techniques range from classical Luenberger observers for linear systems to advanced adaptive, nonlinear, distributed, and neural approaches, ensuring convergence through model-predictive updates.
Recent advances include multi-observer fusion, consensus-based designs, and data-driven neural observers, which enhance performance in high-dimensional and infinite-dimensional system settings.

Observer-based estimation refers to a broad set of methodologies in control, signal processing, and system identification, where a dynamical model (the observer) is used to reconstruct unmeasured states or unknown parameters from available output measurements, often in real time and under noise or uncertainty. Rooted in the notion of observer theory for linear systems, the field now encompasses highly nonlinear, infinite-dimensional, distributed, and data-driven systems. Observer-based approaches are characterized by recursive assimilation of measurements and model predictions, aiming to achieve provable stability, robustness, convergence, and optimality properties.

1. Classical Observer-Based Estimation in Linear Systems

The fundamental framework is the Luenberger observer, designed for linear time-invariant (LTI) systems: $\dot x = A x + B u, \quad y = C x + \nu(t)$ with the observer dynamics

$\dot{\hat x} = A\hat x + B u + L(y - C\hat x)$

where $L$ denotes the observer gain matrix. The error $e = x - \hat x$ evolves as $\dot e = (A - L C)e$ , and a suitable choice of $L$ ensures asymptotic convergence of $\hat x$ to $x$ under detectability assumptions. Extensions for output feedback and noise (Kalman filtering) are standard.

For simultaneous state and parameter estimation, iterative observer schemes integrate parameter updating steps with state reconstruction. Aalto (Aalto, 2016) introduces a joint observer-based state and parameter estimation method for linear and bilinear systems using a Back-and-Forth Nudging (BFN) approach and an interleaved Gauss–Newton parameter step. The algorithm runs a forward observer (with parameter guess), then a backward observer (with updated parameter), passing initial states between legs, and applies the Gauss–Newton update using sensitivity operators to minimize an output-error cost functional: $J(x_0,\theta) = \frac12\int_0^T \|y(t) - Cx_{x_0,\theta}(t)\|^2 dt + \frac12(\theta-\theta_0)^T W (\theta - \theta_0)$ This results in provably attractive fixed points for the optimal state/parameter pair minimizing $J$ , under mild assumptions such as skew-adjointness and exact observability.

2. Observer Design and Convergence in Nonlinear and High-Order Systems

For nonlinear or uncertain systems, observer construction is nontrivial due to structural and stability constraints. Observers for such systems often require design strategies beyond direct state feedback, including:

Adaptive Observers: Joint state and parameter estimation is achieved by embedding adaptation laws in the observer and exploiting regressor excitation properties. For instance, parameter estimation-based observers for overparametrized linear systems are derived without the need for canonical forms, relying on canonical similarity transforms and adaptive laws ensuring algebraic instead of differential state reconstruction, with Lyapunov proofs under finite-excitation conditions (Glushchenko et al., 2023).
Nonlinear Observers with Riccati/LMI Design: For epidemic processes modeled by polynomial (mass-action) nonlinearities, the observer architecture leverages additional feedback terms to suppress nonlinearity in the estimation error dynamics. Convergence is certified by algebraic Riccati inequalities (ARI) or equivalent Linear Matrix Inequalities (LMIs), incorporating Lipschitz or generalized Lipschitz bounds on the nonlinearity. The ARI admits an explicit expression:

$(A-LCA-JC)^TP + P(A-LCA-JC) + PN GG^T N^T P + \ell^2(H-KC)^T(H-KC) < 0$

guaranteeing exponential estimation error decay for appropriate gains $L, J, K$ (Niazi et al., 2022).

Observers for Infinite-Dimensional Systems: For PDEs such as the wave equation or Cosserat rod models, state and source estimation is implemented with boundary-injected observers or fully discrete adaptive schemes. For instance, an adaptive observer for the discretized wave equation combines state and source estimation with recursive update rules on a discrete grid, converging under verification of suitable matrix perturbations and persistent excitation in sensor placement (Asiri et al., 2014). In continuum robotics, a boundary observer injects tip-velocity measurements through a dissipation term at the boundary, yielding local input-to-state stability of the estimation error for the full infinite-dimensional PDE system and is robust to spatial and parametric uncertainties (Zheng et al., 2023).

3. Advanced Observer Architectures: Switching, Distributed, and Data-Driven Design

Recent work expands observer-based estimation beyond centralized, fixed-gain designs:

Observer Switching and Fusion: In nonlinear and partially observable networks (e.g., CSTR cascades), multi-observer frameworks run a bank of advanced estimators (EKF, UKF, QKF, PF) in parallel. At every time instant, a composite cost function (combining $L_1$ output error and KL divergence) selects the best observer for the current step, yielding a modular, robust solution that reacts to changing regimes, process nonlinearities, and sensor failures (Bárzaga-Martell et al., 16 Jun 2025).
Distributed Consensus-Based Observers: For multi-agent systems tracking a moving target using bearing measurements, distributed consensus-based observers combine local innovation (measurement residual adjustment) and consensus (state averaging with neighbors) terms. Only position estimates are exchanged over the network, minimizing communication. Uniform global exponential stability (UGES) is proved using matrix inequalities involving time-averaged projection matrices, with geometric conditions relating to agent–target configurations (Jacinto et al., 18 Jul 2025).
Neural Observer Frameworks: In high-dimensional, nonlinear contexts (e.g., fluid flows), observer-based estimation is generalized to loops where each component (prediction, measurement, correction) is implemented via neural networks. Training involves closed-loop unrolling and direct minimization of output and state estimation error over finite horizons under noisy, partial observations. The neural observer subsumes classical gains as learned mappings, providing improved robustness and performance relative to naive open-loop surrogates or super-resolution networks, though without general convergence proofs (Déda et al., 4 Nov 2025).

4. Observer-Based Estimation for System Identification and Inverse Problems

Observer-based estimation forms the backbone for online system identification, inverse source problems, and reinforcement learning:

Fixed-Time and Sliding-Mode Adaptations: For frequency estimation in oscillatory signals, observer-based methods combine fixed-time high-order sliding-mode differentiators with adaptation laws to achieve estimation times independent of the initial error, with explicit Lyapunov-based guarantees (Shi et al., 2020). Similarly, super-twisting sliding-mode and high-gain observers are rigorously analyzed for derivative estimation of noisy signals, presenting explicit error bounds in terms of noise and system smoothness parameters (Huynh et al., 23 Oct 2025).
Observer-Based Inverse Reinforcement Learning: For linear-quadratic optimal control agents, IRL is posed as an augmented state-parameter observer design problem: an augmented Luenberger observer estimates both the agent’s state and the cost function parameters online; convergence is ensured under persistent excitation and detectability, with further enhancements via history-stack (concurrent learning) approaches (Self et al., 2020).
Dictionary-Based High-Dimensional Estimators: Observer-based inference principles extend to high-dimensional settings using observable dictionary learning: a basis is learned to be both accurate for posterior mean reconstruction and “observable” (sensed by available sensors), ensuring that state recovery avoids unobservable subspaces and yields tighter posteriors than standard PCA or K-SVD dictionaries (Mathelin et al., 2017).

5. Practical Guidelines, Limitations, and Extensions

The practice of observer-based estimation is governed by a range of considerations:

Model and Measurement Structure: Exact observability (or suitable detectability notions) is a non-negotiable requirement for guarantees. For high-dimensional models, observability structure directly impacts the design and performance of observer gains and learned dictionaries.
Robustness, Noise, and Excitation: Observers’ convergence rates and noise rejection performance depend on explicit tuning (gain selection, adaptation scaling, regularization), choice of feedback architecture (sliding-mode, high-gain, neural), and excitation conditions (persistent, finite, or concurrent learning for parameters).
Computational Feasibility: Advanced observer frameworks (multi-observer banks, neural or dictionary learning loops) present significant computational challenges. Bank approaches scale linearly with the number of observers; neural observers suffer from training and long-horizon simulation costs. For embedded applications, achieving real-time capability may dictate the complexity of the chosen strategy.
Application-Specific Choices: For battery SOC estimation, PI and PID observers offer a trade-off among accuracy, convergence, robustness to uncertainty, and computation. Flow control and high-dimensional inference favor data-driven (dictionary or neural observer) approaches for flexible fusion of sensor data and model prediction. In distributed systems and sensor networks, communication constraints drive design decisions toward consensus architectures.
Research Directions: The field is advancing toward fully integrated estimation–control architectures (e.g., observer-based RL, observer–controller neural loops), rigorously extending observer-based estimation to stochastic, hybrid, or networked systems, and developing systematic design tools for nonlinear, infinite-dimensional, or data-driven settings.

6. Notable Variants: Nonclassical Systems and Mathematical Structures

Observers generalize to unconventional settings:

Idempotent Semirings and Event-Graphs: In systems governed by tropical algebra (max-plus linear), observer design uses residuation theory to construct estimators in the idempotent semiring, ensuring estimates are lower bounds matching measured outputs. This has applications in manufacturing systems, scheduling, and networked event-timing (Hardouin et al., 2013).
Quantum Systems: In quantum state estimation by continuous measurement, the BFN approach is adapted to the stochastic master equation, alternating forward and backward observer integration to recover the initial density matrix from time series data, with Lyapunov-based convergence proven in the qubit case under observability and suitable control richness (Leghtas et al., 2010).

Observer-based estimation has evolved from classical theory to encompass modern, distributed, high-dimensional, and learning-based paradigms. It provides a flexible, mathematically principled backbone for fusing dynamical models and measurements under noise, nonlinearity, and partial observability. Developments across control, signal processing, physics, and machine learning continually expand its reach and technical foundations.