State Estimation Techniques for Stochastic Systems

State estimation in stochastic systems involves the reconstruction of unobserved or partially observed variables from noisy and uncertain measurements. Central to this endeavour is the Kalman filter, which provides optimal linear estimation for systems subject to Gaussian noise. Extensions to nonlinear and non-Gaussian contexts include the Extended Kalman Filter, Unscented Kalman Filter and particle-based approaches, each offering different trade-offs between computational cost and statistical fidelity. More recent advances have introduced cubature and sigma-point methods to approximate complex integrals in high dimensions, as well as robust and H∞ filters designed to tolerate model mismatches and adversarial disturbances. Innovations in simultaneous input and state estimation address scenarios with unknown or time-varying exogenous signals by coupling filter design with adaptive or minimax optimisation. Applications span aerospace navigation, autonomous vehicles, environmental monitoring, biomedical signal processing and financial forecasting. Practical challenges such as missing data, packet drops and real-time constraints have motivated hybrid schemes that integrate model predictive control with filtering, sliding-mode observers for fault detection and multi-stage transformations to reduce computational burden. The interconnection between theory and experiment is evident in laboratory demonstrations of torque estimation in marine propulsion, bias-compensated tracking in robotics and sensorless control of electric motors, underscoring global relevance across engineering domains.

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Recent work has derived a unified Kalman-filtering framework for linear discrete-time systems with direct feedthrough, demonstrating how input and state estimation converge to classical Kalman solutions as input variance grows. This approach offers an unbiased estimator resilient to unknown inputs and packet drops, thus broadening applicability to networked control systems. Another study has introduced an augmented Kalman filter for torque estimation in marine propulsion drives, modelling excitations as quasi-stationary signals with bounded spectra. Bench tests and full-scale thruster experiments confirm high-accuracy estimation of torque excitations and responses under varying load and environmental conditions. A further development employs a two-stage cubature Kalman filter based on a nonlinear T-transform, achieving block-diagonal covariance updates to reduce computational load without sacrificing estimation accuracy. Equivalence to the standard cubature filter is established analytically, and wheeled-robot trajectory experiments validate robust state and bias tracking in the presence of unknown disturbances.

State Estimation Techniques for Stochastic Systems publication trend

The graph below shows the total number of articles in state estimation techniques for stochastic systems across all publications each year (not limited to Nature Index journals).

Technical terms

Stochastic system: A dynamical model influenced by random processes or noise.

Kalman filter: An algorithm providing optimal linear state estimation under Gaussian noise assumptions.

Cubature Kalman filter: A sigma-point method that approximates multidimensional integrals via cubature rules for nonlinear estimation.

Direct feedthrough: A system configuration where unknown inputs appear directly in the measurement equation.

Quasi-stationary signal: A random process whose statistical properties vary slowly over time.

Block-diagonalisation: A matrix transformation that partitions a covariance matrix into independent sub-blocks for computational efficiency.

References

1. A Kalman-filtering derivation of input and state estimation for linear discrete-time systems with direct feedthrough. Automatica (2024).
2. Torque estimation in marine propulsion systems. Mechanical Systems and Signal Processing (2022).
3. Two‐Stage Cubature Kalman Filtering Based on T‐Transform and Its Application. Complexity (2021).