linear time invariant system

A linear time invariant system (LTI) is a linear dynamical system $T(p)$ ,

$\displaystyle y(k)$

$\displaystyle=T(p)\;u(k),$

with parameter $p$ that is time independent. $y(k)$ denotes the system output and $u(k)$ denotes the input. The independent variable $k$ can be denoted as time, index for a discrete sequences or differential operaters (e.g. such as $s$ in Laplace domain or $\omega$ in frequency domain).

For example, for a simple mass-spring-dashpot system, the system parameter $p$ can be selected as the mass $m$ , spring constant $k$ and damping coefficient $d$ . The input $u$ to the said system can be chosen as the force applied to the mass and the output $y$ can be chosen as the mass’s displacement.

LTI system has the following properties.

Linearity:

If $y_{1}=Tx_{1}$ and $y_{2}=Tx_{2}$ , then

$T\{\alpha x_{1}+\beta x_{2}\}=\alpha y_{1}+\beta y_{2}$

Time Invariance:

If $y(k)=Tx(k)$ , then

$y(k+\delta_{k})=Tx(k+\delta_{k})$

Associative:

$T_{1}\cdot(T_{2}\cdot T_{3})=(T_{1}\cdot T_{2})\cdot T_{3}$

Commutative:

$T_{1}\cdot T_{2}=T_{2}\cdot T_{1}$

A LTI system can be represented with the following:

•

Transfer function of Laplace transform variable $s$ , which is commonly used in control systems design.
•

Transfer function of Fourier transform variable $\omega$ , which is commonly used in communication theory and signal processing.
•

Transfer function of z-transform variable $z^{-1}$ , which is commonly used in digital signal processing (DSP).
•

State-space equations, which is commonly used in modern control theory and mechanical systems.

Note that all transfer functions are LTI systems, but not all state-space equations are LTI systems.

Title	linear time invariant system
Canonical name	LinearTimeInvariantSystem
Date of creation	2013-03-22 14:22:25
Last modified on	2013-03-22 14:22:25
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	11
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 93A10
Synonym	LTI
Related topic	Controllability
Related topic	Observability
Related topic	SystemDefinitions