TransferFunction 2003-10-16 22:20:32 2006-10-08 16:28:20 Definition frequency domain stable unstable % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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. Consider a canonical dynamical system \begin{eqnarray*} \dot x(t) &=& A x(t) + B u(t) \\ y (t) &=& C x(t) + D u(t) \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. The frequency domain representation is $$ y(s) = (D + C(sI - A)^{-1}B)u(s), $$ and thus the transfer function matrix is $D + C(sI - A)^{-1}B$. In the case of single-input-single-output systems ($m=n=1$), the transfer function is commonly expressed as a rational function of $s$: $$ H(s) = \frac{\prod_{i=0}^Z (s - z_i)}{\prod_{i=0}^P (s - p_i)}. $$ 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}.