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linear time invariant system (Definition)

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 = T x_1$ and $ y_2 = T x_2$, then
$\displaystyle T \{\alpha x_1 + \beta x_2 \} = \alpha y_1 + \beta y_2 $
Time Invariance:
If $ y(k) = T x(k)$, then
$\displaystyle y(k+\delta_k) = T x(k + \delta_k) $
Associative:
$\displaystyle T_1 \cdot ( T_2 \cdot T_3 ) = (T_1 \cdot T_2) \cdot T_3 $
Commutative:
$\displaystyle T_1 \cdot T_2 = T_2 \cdot T_1 $

A LTI system can be represented with the following:

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



"linear time invariant system" is owned by Mathprof. [ full author list (2) | owner history (1) ]
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See Also: controllability, observability, system definitions

Other names:  LTI
Keywords:  LTI
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Cross-references: equations, theory, Fourier transform, design, control systems, Laplace transform, transfer function, properties, force, coefficient, mass, simple, frequency domain, domain, sequences, discrete, index, variable, system, independent, parameter, dynamical system
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This is version 8 of linear time invariant system, born on 2004-05-20, modified 2006-09-16.
Object id is 5864, canonical name is LinearTimeInvariantLTISystems.
Accessed 11566 times total.

Classification:
AMS MSC: 93A10 (Systems theory; control :: General :: General systems)
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