Flatness (systems theory)

Flatness in systems theory is a system property that extends the notion of controllability of linear systems to nonlinear systems . A system which has the flatness property is called a flat system. Flat systems have an output, so that all state and input variables can be fully described using this flat output and a finite number of its time derivatives.

definition

A non-linear dynamical system

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

is called flat if there is a (virtual) output

${\displaystyle \mathbf {y} (t)=(y_{1}(t),...,y_{m}(t))}$

that meets the following conditions:

• The sizes can be as a function of states and inputs and a finite number of time derivatives expressed: . ${\displaystyle y_{i},i=1,...,m}$ ${\displaystyle x_{i},i=1,...,n}$ ${\displaystyle u_{i},i=1,...,m}$ ${\displaystyle u_{i}^{(k)},k=1,...,\alpha _{i}}$ ${\displaystyle \mathbf {y} =\Phi (\mathbf {x} ,\mathbf {u} ,{\dot {\mathbf {u} }},...,\mathbf {u} ^{(\alpha )})}$

• The states or input variables can be expressed as a function of and a finite number of their time derivatives . ${\displaystyle x_{i},i=1,...,n}$ ${\displaystyle u_{i},i=1,...,m}$ ${\displaystyle y_{i},i=1,...,m}$ ${\displaystyle y_{i}^{(k)},i=1,...,m}$

• The components of are differentially independent; that is, they do not satisfy a differential equation of form . ${\displaystyle \mathbf {y} }$ ${\displaystyle \phi (\mathbf {y} ,{\dot {\mathbf {y} }},\mathbf {y} ^{(\gamma )})=\mathbf {0} }$

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

Note: If applies, the third condition is always met. ${\displaystyle n=m}$

Relation to the controllability of linear systems

A linear system is flat if and only if it can be controlled . For linear systems , both properties are therefore equivalent and interchangeable. ${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t),\quad \mathbf {x} (0)=\mathbf {x} _{0}}$

meaning

The flatness property is useful for the analysis and synthesis of nonlinear dynamical systems. It is particularly advantageous for trajectory planning and asymptotic follow-up control of non-linear systems.

literature

• M. Fliess, JL 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
• Hebertt Sira-Ramírez, Sunil K. Agrawal: Differentially Flat Systems (Control Engineering) . CRC: 2004. ISBN 0-824-75470-0
• Rudolph, Joachim: Contributions to the flatness-based follow-up control of linear and non-linear systems of finite and infinite dimensions . Shaker Verlag , Aachen 2003. ISBN 3-8322-1765-7

See also

• Control loop
• Regulator