Differential flatness

Differential flatness is a property of certain nonlinear systems that makes trajectory generation and feed-forward control significantly easier. A system is differentially flat if there exists a set of so-called flat outputs from which all states and all inputs can be recovered without integrating the dynamics — purely algebraic combinations of the flat outputs and a finite number of their time derivatives.

The concept is due to Fliess, Lévine, Martin, and Rouchon [1][2].

Definition

Consider a nonlinear control-affine system

(8)\[\dot x = f(x, u), \qquad x \in \mathbb{R}^n,\; u \in \mathbb{R}^m.\]

The system is differentially flat if there exists an \(m\)-vector of smooth functions — the flat output —

\[y = \phi(x,\, u,\, \dot u,\, \ddot u,\, \dots,\, u^{(p)}) \;\in\; \mathbb{R}^m,\]

such that the state and input can be expressed as smooth functions of the flat output and its derivatives up to some finite order \(q\):

(9)\[\begin{split}\begin{aligned} x &= \phi_x\bigl(y,\, \dot y,\, \ddot y,\, \dots,\, y^{(q)}\bigr), \\ u &= \phi_u\bigl(y,\, \dot y,\, \ddot y,\, \dots,\, y^{(q)}\bigr). \end{aligned}\end{split}\]

A few consequences follow immediately:

  • Dimension matching. The number of flat outputs equals the number of inputs, \(\dim y = \dim u = m\).

  • Integration-free trajectory generation. Any smooth curve \(y(\cdot)\) (with enough derivatives) defines a valid reference trajectory in the full state/input space — just apply (9). No ODE solver needed.

  • Feed-forward control. The resulting \(\bigl(x(t),\, u(t)\bigr)\) is the nominal state and input pair for tracking \(y(\cdot)\). A closed- loop controller then only has to reject deviations from it.

A small example — double integrator

Consider the system

\[\ddot x = u, \qquad x, u \in \mathbb{R}.\]

The choice \(y = x\) is a flat output: the state and input fall out as

\[x = y, \qquad \dot x = \dot y, \qquad u = \ddot y.\]

Any smooth reference \(y(t)\) that’s at least \(C^2\) yields a valid \((x(t), \dot x(t), u(t))\). Trivial in this case, but the same structure extends to the systems we care about in udaan.

With the tuple \((y,\, \dot y,\, \ddot y,\, \dots,\, y^{(q)})\) written as a \(q\)-th order jet \(J^q_t\, y\) of the flat output, the flat-to-state map (9) reads compactly as

(10)\[(x, u) \;=\; \Phi\!\bigl(J^q_t\, y\bigr),\]

a single map from the jet space \(J^q \mathbb{R}^m\) to the state-input space \(\mathbb{R}^n \times \mathbb{R}^m\). See Jets of flat outputs for the formal definition, the udaan.flatness.Jet implementation, and the operations (differentiation, truncation, prolongation) the abstraction makes available.

References