- Flatness (systems theory)
Flatness in
systems theory is a system property that extends the notion ofControllability from linear systems to nonlineardynamical system s. 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 incommutative algebra and is applied incontrol theory .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, ext{Rang} frac{partialmathbf{f}(mathbf{x},mathbf{u})}{partialmathbf{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 y_i,i=1,...,m are representable as functions of the states x_i,i=1,...,n and inputs u_i,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 x_i,i=1,...,n and inputs u_i,i=1,...,m are representable as functions of the outputs y_i,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".
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}_0with 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.
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.
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 [http://cas.ensmp.fr/~rouchon/publications/PR1995/IJC95.pdf]
See also
*
Control theory
*Control engineering
*Controller
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