| Version current |
Version 9 |
| The \emph{transfer function} of a linear dynamical system is the ratio of the Laplace transform of its output to the Laplace transform of its input. In systems theory, the Laplace transform is called the ``frequency domain'' representation of the system. |
The \emph{transfer function} of a linear dynamical system is the ratio of the Laplace transform of its output to the Laplace transform of its input. In systems theory, the Laplace transform is called the ``frequency domain'' representation of the system. |
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| Consider a canonical dynamical system |
Consider a canonical dynamical system |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \dot x(t) &=& A x(t) + B u(t) \\ |
\dot x(t) &=& A x(t) + B u(t) \\ |
| y (t) &=& C x(t) + D u(t) |
y (t) &=& C x(t) + D u(t) |
| \end{eqnarray*} |
\end{eqnarray*} |
| with input $u: R \mapsto R^n$, output $y: R \mapsto R^m$ and state $x:R \mapsto R^p$, and $(A,B,C,D)$ are constant matrices of conformable sizes. |
with input $u: R \mapsto R^n$, output $y: R \mapsto R^m$ and state $x:R \mapsto R^p$, and $(A,B,C,D)$ are constant matrices of conformable sizes. |
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| The frequency domain representation is |
The frequency domain representation is |
| $$ |
$$ |
| y(s) = (D + C(sI - A)^{-1}B)u(s), |
y(s) = (D + C(sI - A)^{-1}B)u(s), |
| $$ |
$$ |
| and thus the transfer function matrix is $D + C(sI - A)^{-1}B$. |
and thus the transfer function matrix is $D + C(sI - A)^{-1}B$. |
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| In the case of single-input-single-output systems ($m=n=1$), the transfer function is commonly expressed as a rational function of $s$: |
In the case of single-input-single-output systems ($m=n=1$), the transfer function is commonly expressed as a rational function of $s$: |
| $$ |
$$ |
| H(s) = \frac{\prod_{i=0}^Z (s - z_i)}{\prod_{i=0}^P (s - p_i)}. |
H(s) = \frac{\prod_{i=0}^Z (s - z_i)}{\prod_{i=0}^P (s - p_i)}. |
| $$ |
$$ |
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The values $z_i$ are called the zeros of $H(s)$, and the values $p_i$ are called the poles. If any of the poles has positive real part, then the transfer function is termed \emph{unstable}; if all of the poles have strictly negative real part, it is \emph{stable}.
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The values $z_i$ are called the zeros of $H(s)$, and the values $p_i$ are called the poles. If any of the poles has positive real part, then the transfer function is termed unstable; if all of the poles have strictly negative real part, it is stable.
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