Transfer Function $H(s)$

A transfer function is the Laplace transform of the Impulse Response of a LTI with a single input and single output when you set the initial conditions to zero.

Informally

The transfer function of a system is the ratio of the Laplace transform of its output and its input, assuming zero initial conditions.

Transfer functions allow us to connect several systems in series by performing convolution through simple multiplication.

We often write the transfer function as $H(s)=\frac{1}{s^n+a_{n-1}{s}^{n-1}+\cdots +a_0}$

Abstract

Transfer functions are a Laplace domain representation of the system.

Definition 1: Zero-State and Zero-Input Responses and the Transfer function

Given a linear DE with constant coefficients of the form $\frac{d^n}{d{t}^n}y+a_{n-1}\frac{d^{n-1}}{d^{n-1}}y+\cdots +a_{0}y=f(t)$ along with the needed initial conditions $y(0),y^{\prime }(0),\dots ,y^{(n-1)}(0)$, the Laplace transform of the equation will be of the form $\mathbf{Y}(\mathbf{s})=\frac{\mathbf{F}(\mathbf{s})}{{\mathbf{s}}^{\mathbf{n}}+{\mathbf{a}}_{\mathbf{n}-1}{\mathbf{s}}^{\mathbf{n}-1}+\cdots +{\mathbf{a}}_0}+\frac{\mathbf{g}({\mathbf{a}}_0,...,{\mathbf{a}}_{\mathbf{n}-1},\mathbf{y}(0),...,\mathbf{y}(\mathbf{n}-1)(0),\mathbf{s})}{{\mathbf{s}}^{\mathbf{n}}+{\mathbf{a}}_{\mathbf{n}-1}{\mathbf{s}}^{\mathbf{n}-1}+\cdots +{\mathbf{a}}_0}$ where $g$ is some function of the ICs and coefficients.

• The zero-input response is $\frac{g(a_0,...,a_{n-1},y(0),...,y(n-1)(0),s)}{s^n+a_{n-1}{s}^{n-1}+\cdots +a_0}$ . This is the response of the system the DE models to the initial conditions (i.e. when the forcing term is 0).
• The zero-state response is $\frac{F(s)}{s^n+a_{n-1}{s}^{n-1}+\cdots +a_0}$ . This is the response of the system the DE models to the forcing term (i.e. when the ICs are all $0$).
• The transfer function is $\frac{1}{s^n+a_{n-1}{s}^{n-1}+\cdots +a_0}$. This function determines the zero-input and the effect of the forcing term.

Theorem

The zero-state response of a linear DE is the convolution of the input (i.e. forcing term) with the system’s Impulse Response.

Transfer functions are super convenient because they enable us to compose multiple LTI Systems together!

In my own words, Motivation

I swear, I went through MATH213 without understanding how transfer functions work. I did pretty bad in that course.

• Looking at this reddit thread: https://www.reddit.com/r/DSP/comments/8k1fdt/can_someone_explain_how_and_why_transfer/

Transfer functions are just functions that map an input to an input.

for any input X, you get output 2X: X * 2 = 2X

• the transfer function is 2 in this case

Input * Transfer function = Output

• its the convolution in the time domain, but whatever, in the Laplace Domain its much simpler

So for an LTI system, the transfer function is just the output over the input

• Set the input to be 1 (the impulse function) and the output is your transfer function. AHH.

Why is the transfer function useful? Because you’ll end up with a polynomial as a transfer function and if you factor the polynomial, you can find the zeroes and poles of the function super easy. You can determine whether the system is stable or not, what the gain is. Why you can even plot the frequency response.

SE380

For a SISO system, the transfer function is given by output / control input, i.e. $G(s)=\frac{Y(s)}{U(s)}$

• we work in frequency domain to be able to write these as fractions. Else, in the time-domain, it’s some convolution?

A transfer function $G(s)$ is real rational if it can be written as $G(s)=\frac{b_m{s}^m+\cdots +b_{1}s+b_0}{s^n+a_{n-1}{s}^{n-1}+\cdots +a_0}{a}_i,b_i\in \mathbb{R}$

• $G(s)$ is proper if ${\lim }_{s\to \infty }G(s)$ exists. $\to$ $n\ge m$
• $G(s)$ is strictly proper if ${\lim }_{s\to \infty }G(s)=0$ $\to$ $n>m$

$p\in \mathbb{C}$ is a pole of $G(s)$ if ${\lim }_{s\to p}|G(s)|=\infty$ $\to$ root of denominator $z\in \mathbb{C}$ is a zero of $G(s)$ if ${\lim }_{s\to z}|G(s)|=0$ $\to$ root of numerator

let’s consider this differential equation $\{\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$

Then we can write G(s) as a matrix form $G(s)=C(sI-A{)}^{-1}B+D$

Related

• LTI System
• Standard Second Order System