# 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 LinearTimeInvariantSystem 2013-03-22 14:22:25 2013-03-22 14:22:25 Mathprof (13753) Mathprof (13753) 11 Mathprof (13753) Definition msc 93A10 LTI Controllability Observability SystemDefinitions