Transfer Function

The transfer function of an LTI system is the ratio of the Laplace transform of the output over that of the input, i.e.

$$G(s)=\frac{Y(s)}{U(s)}$$

State-Space to Transfer Function

Consider an LTI system described by state space equation:

$$\begin{array}{rl}\dot{x}(t)& =Ax(t)+bu(t)\\ y(t)& =cx(t)+du(t)\end{array}$$

Taking the Laplace transform with zero initial conditions:

$$\begin{array}{rl}sX(s)& =AX(s)+bU(s)\\ Y(s)& =cX(s)+dU(s)\end{array}$$

which allows us to write:

$$X(s)=(sI-A{)}^{-1}bU(s)$$

Plugging into the output equation gives us:

$$G(s)=\frac{Y(s)}{U(s)}=c(sI-A{)}^{-1}b+d$$

Rational Function

The transfer function of an LTI system with a state space model is always the ratio of two polynomials:

$$G(s)=\frac{b(s)}{a(s)}$$

where $b(s)$ is called the numerator polynomial and $a(s)$ is the denominator polynomial. The ratio of two polynomials is called a rational function; thus, the transfer function of an LTI system is rational function.

$G(s)$ is called coprime if $a(s)$ and $b(s)$ don’t have common factors.

Transfer Function Forms

Zero-pole Gain Form

A transfer function or any rational function can be written in either factored or unfactored form. The factored form is also called zero-pole gain form.

Unfactored form:

$$G(s)=\frac{b_0{s}^m+b_1{s}^{m-1}+\cdots +b_m}{a_0{s}^n+a_1{s}^{n-1}+\cdots +a_n}$$

Factored (zero-pole gain form) form:

$$G(s)=K\frac{(s-z_1)(s-z_2)\dots (s-z_m)}{(s-p_1)(s-p_2)\dots (s-p_n)}$$

• The zeros of $G(s)$ are $z_1,z_2,\dots ,z_m$
• The poles of $G(s)$ are $p_1,p_2,\dots ,p_n$
• The gain (high frequency) of $G(s)$ is $K(s\to \infty )$

Transfer Function Nomenclature

• Proper: A transfer function or system $G(s)$ is proper if $\mathrm{deg}b(s)\le \mathrm{deg}a(s)$, or equivalently $|G(\infty )|<\infty$.
• Strictly proper: It is said to be strictly proper if $\mathrm{deg}b(s)<\mathrm{deg}a(s)$, or equivalently if $G(\infty )=0$.
• Bi-proper: It is said to be bi-proper if $\mathrm{deg}b(s)=\mathrm{deg}a(s)$, or equivalently $0\ne |G(\infty )|\ne \infty$.
• Degree: The difference $\mathrm{deg}a(s)-\mathrm{deg}b(s)$ is called the relative degree of $G(s)$, $\mathrm{deg}a(s)$ is called the order or degree of $G(s)$.
• DC gain: $G(0)$ is called the DC gain of $G(s)$.

$$\frac{\left(C_1{R}_{1}s+1\right)\left(C_2{R}_{2}s+1\right)}{C_1{R}_{1}s+C_2{R}_{1}s+C_2{R}_{2}s+C_1{C}_2{R}_1{R}_2{s}^2+1}$$