Nonlinear Stochastic Control and Filtering Systems

Nonlinear stochastic control and filtering systems constitute a branch of control theory and signal processing concerned with the behaviour, regulation and estimation of systems whose dynamics combine nonlinear deterministic laws with random perturbations. Such systems are typically modelled by stochastic differential equations driven by processes that represent environmental uncertainty or intrinsic noise. The control objective is to design feedback laws that stabilise the system, attenuate disturbances and achieve performance measures under worst-case conditions. Complementarily, filtering aims to reconstruct or estimate the system state from noisy measurements, optimising accuracy in the face of stochastic perturbation. Mathematical tools central to this field include Lyapunov-based stability theory, Hamilton-Jacobi inequalities, convex optimisation via linear matrix inequalities and approximation techniques such as Takagi-Sugeno fuzzy models or deep neural network architectures. Advances in computational methods have enabled real-time implementation of robust H∞ controllers and mixed H₂/H∞ filters on applications ranging from unmanned aerial vehicles and robotic formations to power systems and financial networks, underscoring the global significance of ensuring reliable operation under uncertainty.

Research from Nature Portfolio

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Research from all publishers

Recent work has introduced a deep neural network H∞ control scheme that embeds Hamilton-Jacobi-Isaacs equations within a learning framework to design robust controllers for nonlinear time-varying systems with external disturbances, demonstrating effective stabilisation and tracking in unmanned aerial vehicles. Another advance presents a decentralised H∞ PID team formation tracking strategy for large-scale stochastic quadrotor formations, converting complex Hamilton-Jacobi inequalities into bilinear matrix inequalities via Takagi-Sugeno fuzzy interpolation and enabling independent convex synthesis of each controller. Foundational research on robust mixed H₂/H∞ filtering for nonlinear stochastic systems with state-dependent noise has established linear matrix inequality formulations that balance disturbance attenuation and estimation accuracy, validated through numerical examples.

Nonlinear Stochastic Control and Filtering Systems publication trend

The graph below shows the total number of articles in nonlinear stochastic control and filtering systems across all publications each year (not limited to Nature Index journals).

Technical terms

Nonlinear Stochastic Differential Equation: A differential equation in which the rate of change of the state depends nonlinearly on the state and control and includes random noise terms modelled by stochastic processes.

H∞ Control: A robust control methodology aiming to design a feedback law that minimises the worst-case gain from exogenous disturbances to predefined performance outputs.

Hamilton-Jacobi Inequality: A partial differential inequality whose solution characterises stability and performance conditions for nonlinear control or filtering problems under uncertainty.

Linear Matrix Inequality (LMI): A convex constraint expressed as a linear function of decision matrices, widely used for synthesising controllers and filters via efficient numerical solvers.

Takagi-Sugeno Fuzzy Model: An approach to approximate complex nonlinear dynamics by a weighted combination of linear subsystems, facilitating controller and observer design through interpolation.

Deep Neural Network: A multilayer function approximator trained on data to represent control laws or value functions, capable of handling high-dimensional nonlinearities in stochastic control problems.

References

1. DNN-Based H∞ Control Scheme of Nonlinear Time-Varying Dynamic Systems With External Disturbance and its Application to UAV Tracking Design. IEEE Access (2021).
2. Decentralized H PID Team Formation Tracking Control of Large-Scale Quadrotor UAVs Under External Disturbance and Vortex Coupling. IEEE Access (2022).
3. Robust H2/H∞ Filter Design for a Class of Nonlinear Stochastic Systems with State‐Dependent Noise. Mathematical Problems in Engineering (2012).