Nonlinear State Observation and Control Techniques

Nonlinear state observation and control constitute a fundamental pillar of modern systems engineering where the underlying process dynamics cannot be faithfully captured by linear models. Observers are algorithms that reconstruct unmeasured internal states from accessible outputs, while control strategies apply computed inputs to steer the system towards desired trajectories or equilibria. The inherent challenges of nonlinearity – including multiple equilibria, bifurcations and sensitivity to disturbances – demand advanced theoretical frameworks and robust design methods. Over the past decade, research has focused on high-gain observers, sliding-mode and extended-state observers, adaptive schemes and optimisation-based approaches to ensure rapid convergence, disturbance rejection and noise attenuation. Integrating estimation with feedback control has enabled applications across robotics, aerospace, renewable energy systems and environmental monitoring. Recent progress has also embraced global observers via state-space embedding and convex optimisation techniques to achieve performance guarantees under broad classes of nonlinearities.

Research from Nature Portfolio

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Research from all publishers

Recent studies have advanced high-gain observer design by embedding the original polynomial system into a higher-dimensional Euclidean space. This embedding avoids singularities in the observation matrix, enabling the construction of global observers that ensure accurate state reconstruction across the entire operating range. Projection techniques then retrieve the original state estimates with proven convergence properties.

A novel cascade extended-state observer utilises a dynamic dead-zone structure to trim high-frequency noise and drive disturbance estimation errors to zero. By organising the observer into a two-stage cascade and bounding the high-gain parameter growth, this method achieves input-to-state stability while maintaining robustness against measurement noise and external perturbations.

An optimisation-driven observer design exploits sum-of-squares optimisation tools to perform simultaneous state and disturbance estimation for a permanent-magnet synchronous machine model. By formulating the observer synthesis as a semidefinite programming problem, the approach decomposes the estimation task into lower-dimensional subproblems, enabling efficient online computation and ensuring asymptotic convergence under bounded uncertainties.

Nonlinear State Observation and Control Techniques publication trend

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

Technical terms

State observer: An algorithm that estimates unmeasured internal variables of a dynamical system using available output measurements and a model of system dynamics.

High-gain observer: A nonlinear observer that employs large observer gains to achieve rapid convergence of the estimation error, often at the expense of sensitivity to measurement noise.

Extended state observer: A technique that augments the system model with additional states to estimate disturbances and unmodelled dynamics alongside the original state variables.

Sum-of-squares optimisation: A convex optimisation framework that represents polynomial positivity constraints as sum-of-square decompositions, solvable via semidefinite programming.

Input-to-state stability (ISS): A property ensuring that the system state remains bounded and converges when subject to bounded inputs or disturbances.

References

1. A High-Gain Observer for Embedded Polynomial Dynamical Systems. Machines (2023).
2. A cascade dead-zone extended state observer for a class of systems with measurement noise. AIMS Mathematics (2023).
3. SOS-Based Nonlinear Observer Design for Simultaneous State and Disturbance Estimation Designed for a PMSM Model. Sustainability (2022).