Time invariance

In systems theory, time invariance is the property of a system to show the same behavior at all times with the same input - it is invariant over time . The parameters of its mathematical description are invariable with time, and the matrices of the state space representation are constant.

A system is a structure of several elements that form a unit, e.g. B. an electronic circuit or a pendulum. The parameters of a system are then the parameters of the electronic components or geometric dimensions.

Together with the linearity , the system description is simplified to the linear, time-invariant systems .

Time invariance

From the system property time invariance it follows that the time shift of the input signal of the system leads to a similar shift of the output signal without influencing its time course in any other way.

That is, the system

{\ displaystyle y (t) = H \ {x (t) \}}

supplies an input signal that has been delayed by the time , an identical, correspondingly delayed output signal : ${\ displaystyle x}$ ${\ displaystyle t_ {0}}$ ${\ displaystyle y}$

{\ displaystyle y (t-t_ {0}) = H \ {x (t-t_ {0}) \}.}

A system that does not have the property described above is called time variant .

Energy conservation

According to Noether's theorem , every continuous symmetry in physics also has a conservation quantity . The conservation of energy belongs to the time invariance ( homogeneity of time) .

Examples of time-invariant systems are closed systems , e.g. B. an ideal pendulum without taking friction into account . In this case, its kinetic energy changes along with the speed of the pendulum (i.e. the system) and its potential energy changes with its position in space , but their sum, the total energy, remains constant. It doesn't matter at what point in time the pendulum is viewed; its energy E is always the same:

{\ displaystyle {\ frac {\ mathrm {d} E} {\ mathrm {d} t}} = 0 \ quad \ Leftrightarrow \ quad E = {\ text {const.}}}

Examples

1st example An electrical resistance R is time invariant. If a constant current I flows through it, then a voltage U from drops across it . Several minutes later, the same tension is applied to it. ${\ displaystyle U = R \ cdot I}$

On closer inspection, the voltage is slightly higher because the resistor has heated up due to the flow of current. This heating is not directly dependent on the time, but on the current input signal, the heat dissipation and the output temperature. Under the same starting conditions, it will deliver the same voltage at all times.

Example 2 Imagine the following two systems:

System A: ${\ displaystyle y (t) = t \ cdot x (t)}$
System B: ${\ displaystyle y (t) = 10 \ cdot x (t)}$

Since system A uniquely depends on t , it is time-variant. System B is not directly dependent on t and is therefore time in variant.