Flatness (systems theory)

Flatness in the Systems Theory is a system property, which extends the concept of controllability of linear systems to nonlinear systems. A system that has the flatness characteristic is flat system. Flat systems have an output so that all state and input variables can be fully described by using this flat output and a finite number of its time derivatives. The flatness concept of system theory is based on the mathematical concept of flatness of commutative algebra and finds application in control engineering.

Definition

A non-linear dynamic system

Is called flat if there is a (virtual) output

Is that satisfies the following conditions:

• The sizes can be expressed as a function of states and inputs, and a finite number of time derivatives.

• The states and input variables can be expressed as a function of a finite number of people and time derivatives.

• The components of are differentially independent, ie they do not fulfill any differential equation of the form.

If these conditions are fulfilled at least locally, is the name of the potentially fictitious output flat output and the system is called ( differentially ) flat.

Note: If this is not the third condition is always fulfilled.

Respect to controllability of linear systems

A linear system with respect to the non -linear system is exactly the same as defined dimensions for then flat when it is controlled. For linear systems, both properties are equivalent and interchangeable.

Importance

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