| Version current |
Version 7 |
| A \emph{linear time invariant system} (LTI) is a linear dynamical system $T(p)$, |
A \emph{linear time invariant system} (LTI) is a linear dynamical system $T(p)$, |
|
|
| \begin{align*} |
\begin{align*} |
| y(k) &= T(p) \; u(k), |
y(k) &= T(p) \; u(k), |
| \end{align*} |
\end{align*} |
|
|
| with parameter $p$ that is time independent. $y(k)$ denotes the |
with parameter $p$ that is time independent. $y(k)$ denotes the |
| system output and $u(k)$ denotes the input. The independent variable |
system output and $u(k)$ denotes the input. The independent variable |
| $k$ can be denoted as time, index for a discrete sequences or |
$k$ can be denoted as time, index for a discrete sequences or |
| differential operaters (e.g. such as $s$ in Laplace domain or $\omega$ |
differential operaters (e.g. such as $s$ in Laplace domain or $\omega$ |
| in frequency domain). |
in frequency domain). |
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| For example, for a simple mass-spring-dashpot system, the system |
For example, for a simple mass-spring-dashpot system, the system |
| parameter $p$ can be selected as the mass $m$, spring constant $k$ and |
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 |
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 |
as the force applied to the mass and the output $y$ can be chosen as the |
| mass's displacement. |
mass's displacement. |
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|
| LTI system has the following properties. |
LTI system has the following properties. |
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|
| \begin{description} |
\begin{description} |
| \item[Linearity:] |
\item[Linearity:] |
| If $y_1 = T x_1$ and $y_2 = T x_2$, then |
If $y_1 = T x_1$ and $y_2 = T x_2$, then |
| $$T \{\alpha x_1 + \beta x_2 \} = \alpha y_1 + \beta y_2 $$ |
$$T \{\alpha x_1 + \beta x_2 \} = \alpha y_1 + \beta y_2 $$ |
| \item[Time Invariance:] |
\item[Time Invariance:] |
| If $y(k) = T x(k)$, then |
If $y(k) = T x(k)$, then |
| $$ y(k+\delta_k) = T x(k + \delta_k) $$ |
$$ y(k+\delta_k) = T x(k + \delta_k) $$ |
| \item[Associative:] |
\item[Associative:] |
| $$ T_1 \cdot ( T_2 \cdot T_3 ) = (T_1 \cdot T_2) \cdot T_3 $$ |
$$ T_1 \cdot ( T_2 \cdot T_3 ) = (T_1 \cdot T_2) \cdot T_3 $$ |
| \item[Commutative:] |
\item[Commutative:] |
| $$ T_1 \cdot T_2 = T_2 \cdot T_1 $$ |
$$ T_1 \cdot T_2 = T_2 \cdot T_1 $$ |
| \end{description} |
\end{description} |
|
|
| A LTI system can be represented with the following: |
A LTI system can be represented with the following: |
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|
| \begin{itemize} |
\begin{itemize} |
| \item Transfer function of Laplace transform variable $s$, which is commonly |
\item Transfer function of Laplace transform variable $s$, which is commonly |
| used in control systems design. |
used in control systems design. |
| \item Transfer function of Fourier transform variable $\omega$, which is |
\item Transfer function of Fourier transform variable $\omega$, which is |
| commonly used in communication theory and signal processing. |
commonly used in communication theory and signal processing. |
|
\item Transfer function of z-transform variable $z^{-1}$, which is
|
\item Transfer function of z-tranform variable $z^{-1}$, which is
|
| commonly used in digital signal processing (DSP). |
commonly used in digital signal processing (DSP). |
| \item State-space equations, which is commonly used in modern control |
\item State-space equations, which is commonly used in modern control |
| theory and mechanical systems. |
theory and mechanical systems. |
| \end{itemize} |
\end{itemize} |
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| Note that all transfer functions are LTI systems, but not all |
Note that all transfer functions are LTI systems, but not all |
| state-space equations are LTI systems. |
state-space equations are LTI systems. |
