Control Theory for Two-Dimensional Dynamical Systems

Control theory for two-dimensional dynamical systems addresses the analysis and design of controllers for processes characterised by two independent evolution variables, often representing spatial and temporal dimensions. Such systems extend classical one‐dimensional state‐space models to capture the dynamics of image frames, heat diffusion across a plate, distributed parameter processes and repetitive manufacturing lines. Two principal model formulations prevail: the Roesser model, which partitions states into horizontal and vertical components, and the Fornasini–Marchesini model, which integrates interactions along both dimensions. Central objectives include the assessment of stability, performance under uncertainty and disturbance rejection. Stability criteria frequently hinge on the construction of Lyapunov functions defined over a two‐dimensional index set, while controller synthesis commonly employs convex optimisation over linear matrix inequalities to guarantee robustness. Advanced methods also address H∞ performance, fault detection and isolation, as well as event‐triggered mechanisms to manage communication constraints in networked implementations. Practical realisations span thermal regulation in industrial reactors, image enhancement in signal processing and trajectory planning in grid‐based robotic systems, emphasising the global significance of rigorous two‐dimensional control strategies.

Research from Nature Portfolio

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Research from all publishers

Recent advances in non‐Nature publications have expanded both theoretical underpinnings and practical schemes for two‐dimensional controllers. One effort proposed an event‐triggered framework for integrated fault detection, isolation and control in discrete Roesser systems operating over constrained networks. By formulating robustness conditions for H∞, H2 and L1 objectives as linear matrix inequalities, the scheme ensures reliable operation under stochastic and bounded disturbances, with simulation studies demonstrating effective thermal and gas‐absorption regulation. Another strand established robust stability criteria for uncertain two‐dimensional discrete‐time systems described by a polytopic parameterisation. A parameter‐dependent Lyapunov function was constructed and feasibility conditions were derived at polytope vertices, resulting in tractable linear matrix inequality tests that guarantee stability despite parameter variations. Complementing these contributions, a polynomial‐parameter‐dependent approach to robust H∞ filtering for discrete Roesser models has been developed. By extending filter structures to higher‐order polynomial dependency, conservatism in estimation error norms is reduced, affording improved noise attenuation in numerical examples while preserving asymptotic stability across admissible uncertainties.

Control Theory for Two-Dimensional Dynamical Systems publication trend

The graph below shows the total number of articles in control theory for two-dimensional dynamical systems across all publications each year (not limited to Nature Index journals).

Technical terms

Two‐dimensional dynamical system: A system whose state evolution depends on two independent discrete or continuous indices, such as time and space.

Roesser model: A state‐space representation that separates system states into horizontal and vertical direction components for two‐dimensional processes.

Fornasini–Marchesini model: A two‐dimensional state‐space framework that captures coupled dynamics along both coordinate directions in a unified form.

Linear matrix inequality (LMI): A convex constraint of the form L(X)=A0+∑AiXi<0, used to encode stability and performance requirements in controller synthesis.

Lyapunov function: A scalar function of system states that decreases along system trajectories, used to certify stability.

H∞ control: A robust control paradigm that minimises the worst‐case gain from disturbance inputs to regulated outputs.

Event‐triggered control: A strategy that updates control actions only when specified state or output conditions are met, reducing communication or computation load.

References

1. Event-triggered robust fault diagnosis and control of linear Roesser systems: A unified framework. Automatica (2021).
2. LMI Conditions for Robust Stability of 2D Linear Discrete‐Time Systems. Mathematical Problems in Engineering (2008).
3. Robust H∞ Filtering of 2D Roesser Discrete Systems: A Polynomial Approach. Mathematical Problems in Engineering (2012).