Flatness (systems theory)

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For other meanings of flatness, see Flatness.

Flatness in systems theory is a system property that extends the notion of Controllability from linear systems to nonlinear dynamical systems. A system that has the flatness property is called a flat system. Flat systems have a (fictituous) flat output, which can be used to explicitly express all states and inputs in terms of the flat output and a finite number of its derivatives. Flatness in systems theory is based on the mathematical notion of flatness in commutative algebra and is applied in control theory.

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[edit] Definition

A nonlinear system

\dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t),\mathbf{u}(t)), \quad \mathbf{x}(0) = \mathbf{x}_0, \quad \mathbf{u}(t) \in R^m, \quad \mathbf{x}(t) \in R^n, \text{Rang} \frac{\partial\mathbf{f}(\mathbf{x},\mathbf{u})}{\partial\mathbf{u}} = m

Is flat, if there exists an output

\mathbf{y}(t) = (y_1(t),...,y_m(t))

that satisfies the following conditions:

  • The signals yi,i = 1,...,m are representable as functions of the states xi,i = 1,...,n and inputs ui,i = 1,...,m and a finite number of derivatives with respect to time u_i^{(k)}, k=1,...,\alpha_i: \mathbf{y} = \Phi(\mathbf{x},\mathbf{u},\dot{\mathbf{u}},...,\mathbf{u}^{(\alpha)}).
  • The states xi,i = 1,...,n and inputs ui,i = 1,...,m are representable as functions of the outputs yi,i = 1,...,m and of its derivatives with respect to time y_i^{(k)}, i=1,...,m.
  • The components of \mathbf{y} are differentially independent, that is, they satisfy no differential equation of the form \phi(\mathbf{y},\dot{\mathbf{y}},\mathbf{y}^{(\gamma)}) = \mathbf{0}.

If these conditions are satisfied at least locally, then the (possibly fictitious) output is called flat output, and the system is flat.

[edit] Relation to controllability of linear systems

A linear system \dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t), \quad \mathbf{x}(0) = \mathbf{x}_0 with the same signal dimensions for \mathbf{x},\mathbf{u} as the nonlinear system is flat, if and only if it is controllable. For linear systems both properties are equivalent, hence exchangeable.

[edit] Significance

The flatness property is useful for both the analysis of and controller synthesis for nonlinear dynamical systems. it is particularly advantageous for solving trajectory planning problems and asymptotical setpoint following control.

[edit] Literature

  • M. Fliess, J. L. Lévine, P. Martin and P. Rouchon: Flatness and defect of non-linear systems: introductory theory and examples. International Journal of Control 61(6), pp. 1327-1361, 1995 [1]

[edit] See also