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| '''Flatness''' in [[systems theory]] is a system property that extends the notion of [[Controllability]] from [[LTI system theory|linear systems]] to [[Nonlinearity|nonlinear]] [[dynamical system]]s. A system that has the flatness property is called a ''flat system''. Flat systems have a (fictitious) ''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 [[Flat morphism|flatness]] in [[commutative algebra]] and is applied in [[control theory]].
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| == Definition ==
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| A nonlinear system
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| <math>\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{Rank} \frac{\partial\mathbf{f}(\mathbf{x},\mathbf{u})}{\partial\mathbf{u}} = m</math>
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| Is flat, if there exists an output
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| <math>\mathbf{y}(t) = (y_1(t),...,y_m(t))</math>
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| that satisfies the following conditions:
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| * The signals <math>y_i,i=1,...,m</math> are representable as functions of the states <math>x_i,i=1,...,n</math> and inputs <math>u_i,i=1,...,m</math> and a finite number of derivatives with respect to time <math>u_i^{(k)}, k=1,...,\alpha_i</math>: <math>\mathbf{y} = \Phi(\mathbf{x},\mathbf{u},\dot{\mathbf{u}},...,\mathbf{u}^{(\alpha)})</math>.
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| * The states <math>x_i,i=1,...,n</math> and inputs <math>u_i,i=1,...,m</math> are representable as functions of the outputs <math>y_i,i=1,...,m</math> and of its derivatives with respect to time <math>y_i^{(k)}, i=1,...,m</math>.
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| * The components of <math>\mathbf{y}</math> are differentially independent, that is, they satisfy no differential equation of the form <math>\phi(\mathbf{y},\dot{\mathbf{y}},\mathbf{y}^{(\gamma)}) = \mathbf{0}</math>.
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| If these conditions are satisfied at least locally, then the (possibly fictitious) output is called ''flat output'', and the system is ''flat''.
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| == Relation to controllability of linear systems ==
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| A [[LTI system theory|linear system]]
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| <math>\dot{\mathbf{x}}(t) = \mathbf{A}\mathbf{x}(t) + \mathbf{B}\mathbf{u}(t), \quad \mathbf{x}(0) = \mathbf{x}_0</math> | |
| with the same signal dimensions for <math>\mathbf{x},\mathbf{u}</math> as the nonlinear system is flat, if and only if it is [[Controllability|controllable]]. For [[LTI system theory|linear systems]] both properties are equivalent, hence exchangeable.
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| == Significance ==
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| 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.
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| == Literature ==
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| * 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 [http://cas.ensmp.fr/~rouchon/publications/PR1995/IJC95.pdf]
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| * A. Isidori, C.H. Moog et A. De Luca. A Sufficient Condition for Full Linearization via Dynamic State Feedback. 25th CDC IEEE, Athens, Greece, pp.203 - 208, 1986 [http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=4048739&contentType=Conference+Publications&matchBoolean%3Dtrue%26searchField%3DSearch_All%26queryText%3D%28%28p_Authors%3Aisidori%29+AND+p_Authors%3Amoog%29]
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| == See also ==
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| * [[Control theory]]
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| * [[Control engineering]]
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| * [[Controller (control theory)]]
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| * [[Flat pseudospectral method]]
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| [[Category:Control theory]]
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