Observer Design for Nonlinear and Linear Dynamic Systems

Observer design underpins the estimation of internal states in dynamic systems where direct measurement is impractical or impossible. In linear systems, observers such as the Luenberger observer exploit system matrices and measurement outputs to reconstruct the full state vector with guaranteed convergence properties. Nonlinear observer design extends these principles to systems exhibiting complex behaviours, including Lipschitz continuity, one-sided Lipschitz constraints and descriptor forms. Key challenges include handling model uncertainties, time-varying delays, unknown inputs and faults while ensuring stability and robustness. Modern approaches employ Lyapunov-based analyses, linear matrix inequalities and H∞ or quadratic boundedness techniques to derive observer gains that bound estimation errors under disturbances. Applications span from robotic manipulators and autonomous vehicles to fault diagnosis in power networks and aerospace guidance. The interplay between linear and nonlinear methods has led to hybrid schemes that combine the analytical tractability of linear observers with the flexibility required for real-world nonlinear dynamics.

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Research from all publishers

Recent work has advanced the reconstruction of faults in descriptor-style nonlinear systems by devising observers that transform the original dynamics into augmented forms, facilitating the use of linear matrix inequalities to compute gains that guarantee bounded fault estimation despite disturbances. Applications demonstrate successful fault recovery in simulation examples, illustrating resilience against both state and output perturbations. Complementary research introduces robust adaptive unknown input observers that incorporate online parameter adaptation to track exogenous disturbances and faults without prior knowledge of their bounds. This adaptive law, combined with LMI-based stability conditions, ensures convergence of estimation errors and has been validated through numerical case studies. Another strand of research offers a unified framework for simultaneous actuator and sensor fault estimation in nonlinear systems by extending the state vector to absorb sensor faults and leveraging both H∞ and quadratic boundedness methods. Comparative analyses reveal trade-offs between robustness margins and computational complexity, with illustrative examples on multi-rotor and multi-tank setups demonstrating practical fault diagnosis and tolerance capabilities.

Observer Design for Nonlinear and Linear Dynamic Systems publication trend

The graph below shows the total number of articles in observer design for nonlinear and linear dynamic systems across all publications each year (not limited to Nature Index journals).

Technical terms

Observer: An algorithmic construct that uses system inputs and outputs to estimate unmeasured internal states.

Nonlinear system: A dynamic system whose behaviour cannot be expressed as a linear combination of its states and inputs.

Lipschitz condition: A constraint on nonlinearity growth ensuring the difference between outputs is bounded by a constant times the difference between inputs.

Linear matrix inequality (LMI): A convex constraint expressed as a matrix inequality, widely used for observer gain computation.

Unknown input observer (UIO): An observer designed to estimate system states and certain exogenous signals or faults without direct measurement of those inputs.

References

1. A Nonlinear Observer for Robust Fault Reconstruction in One-Sided Lipschitz and Quadratically Inner-Bounded Nonlinear Descriptor Systems. IEEE Access (2021).
2. Robust adaptive unknown input observer design for a class of disturbed systems. Systems Science & Control Engineering (2020).
3. Towards Robust Simultaneous Actuator and Sensor Fault Estimation for a Class of Nonlinear Systems: Design and Comparison. IEEE Access (2019).