MATH319 Slides 100 Laplace transform calculation 102 Proof of realization

101 Realization of a linear system

Consider a SISO linear system $Y=LU$ where $L$ is a linear operator, and such that all the input $U$ and output $Y$ satisfy $(E)$ . Let $\hat{Y}$ and $\hat{U}$ be the Laplace transforms of $Y$ and $U$ . Suppose that $T(s)$ is a function such that

\hat{Y}(s)=T(s)\hat{U}(s)\qquad(s>\beta).

Then $T(s)$ is called the transfer function of the linear system.

Theorem

Let $T$ be a complex rational function. Then there exists a SISO linear system $\Sigma$ , possibly with feedback, composed of taps, amplifiers, summing junctions, integrators, and differentiators, such that the transfer function of $\Sigma$ is $T$ .