Linear system (systems theory)

In systems theory , a linear system is a model for a sufficiently well isolated part of nature, in which all functions that occur are linear images .

A linear system consists of internal state variables and a dynamic that describes the development of these state variables over time. Furthermore, there are observable quantities that are only functions of the internal state variables and do not clearly characterize the internal state. From outside the isolated area there are interactions that are assumed to be weak, but still modify the internal dynamics.

For example, a linear differential system (i.e. a system with continuous time, infinite value ranges and continuous system operators) with the internal state , external influences and externally observable signals can be represented as ${\displaystyle x(t)}$ ${\displaystyle u(t)}$ ${\displaystyle y(t)}$

${\displaystyle {\begin{aligned}{\dot {x}}(t)&=A(t)x(t)+B(t)u(t)\\y(t)&=C(t)x(t)+D(t)u(t),\end{aligned}}}$

wherein , , , time-dependent matrices are suitable dimension, in particular must be square. The matrices can be combined into a block matrix , which is then called the system matrix . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle A}$

A linear system is called a linear time-invariant system (LZI system) if the system matrix does not depend on time . ${\displaystyle t}$

However, systems with discrete time and finite value ranges can also be linear if corresponding linear mappings are defined on the sets and operators. A typical example are linear automata with the non-equivalence as a linear operation, e.g. B. a linear feedback shift register .

See also

• State space representation
• Non-linear system

literature

• Rolf Unbehauen : Systems Theory 1 . Oldenbourg Verlag, Munich Vienna 2002, ISBN 978-3-486-25999-5 .
• Gerhard Wunsch : Handbook of Systems Theory . R. Oldenbourg Verlag, Munich Vienna 1986, ISBN 3-486-20017-8 .
• Michael Gössel: Applied automata theory II. Linear automata and shift registers . Akademie-Verlag, Berlin 1972.