If the alignment of the output variables to the input variables in a transfer element is described by a system of differential equations or differential equations and standard equations, by introducing state variables x 1(t)...x n(t) as intermediate variables, this can always be converted into a system of differential equations or 1st order differential equations and a system of standard equations. These equations are called the system's state equations (DIN 19 229):
As a shorthand, input variables (u i), output variables (y i) and state variables (x i) can be pooled into vectors.
In the case of linear, time-variable systems, the state equations can be represented as follows:
The system matrix A describes the inner dynamic performance of the system: It specifies the damped movements and provides information on the stability of the system (stability criteria).
The state variables enable a particularly clear description of transfer elements. In addition, when looking according to a mathematical model of systems on the basis of the laws of nature, they generally turn out far more natural than differential equations of the nth order.