Differential Flatness Control of Nonlinear Systems
Summary
Differential flatness is a structural property of certain nonlinear dynamical systems that permits the representation of all system states and inputs in terms of a set of outputs—known as flat outputs—and a finite number of their derivatives. In practice, flatness enables the formulation of motion planning and feedback control tasks as algebraic trajectory design followed by direct feedforward and feedback synthesis, bypassing the need for local linearisation or successive approximations. For a differentially flat system, one selects a flat output whose derivatives fully parameterise the system evolution. The control law is then constructed in two stages: an open-loop component that inverts the system mapping from flat outputs to states and inputs, and a closed-loop stabilising correction, often via state feedback linearisation. This approach unifies trajectory generation and tracking under a single framework and has been applied to robotic manipulators, vehicle dynamics, crane systems and process control. The global significance lies in the ability to handle nonlinearities explicitly, ensuring high-performance operation over wide working envelopes and facilitating the inclusion of constraints in a systematic manner.
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Differential Flatness Control of Nonlinear Systems publication trend
The graph below shows the total number of articles in differential flatness control of nonlinear systems across all publications each year (not limited to Nature Index journals).
Technical terms
Differential flatness: A property of a control system whereby all its states and inputs can be expressed as functions of a chosen output and a finite number of its derivatives.
Flat output: An output variable whose derivatives parametrise the full state and input trajectories of a flat system.
Feedback linearisation: A control technique that algebraically cancels nonlinearities in a system model to yield an equivalent linear input–output behaviour.
Triangular form: A system representation in which the dynamics are arranged in a hierarchical chain structure, facilitating the identification of flat outputs and inversion.
Discrete-time flatness: The extension of differential flatness concepts to systems evolving in discrete time, often requiring tailored discretisation and state transformations to preserve the flatness property.
References
- Trajectory tracking for robot manipulators using differential flatness. Ingeniería e Investigación (2011).
- Discrete-time flatness-based control of a gantry crane. Control Engineering Practice (2022).
- A normal form for two-input forward-flat nonlinear discrete-time systems. International Journal of Systems Science (2021).
- A structurally flat triangular form based on the extended chained form. International Journal of Control (2020).
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