Zero of a transfer function
High school
Consider a fraction of polynomials, after canceling common factors. A zero of the fraction is a value where the numerator is zero, so that the function is zero.
Undergraduate
Consider a transfer function $G(s)$, a function of the complex variable $s$. A zero of $G$ is a value of $s$ at which $G(s) = 0$: for a rational function, a root of the numerator after common factors are canceled.
Examples
Worked examples
- $G(s) = \frac{1}{s + 2}$ : None High school
-
$G(s) = \frac{s - 1}{(s + 2)(s + 3)}$
: $s = 1$
High school
The zero of $G$ is $1$, where its numerator is zero. The zero in the right half-plane, which makes the system non-minimum phase.
-
$G(s) = \frac{s + 1}{(s + 1)(s + 2)}$
: None
High school
$G(s) = \frac{s + 1}{(s + 1)(s + 2)}$ has no transmission zeros, because the factor $s + 1$ cancels.
Properties
- Suppose $z$ is a zero of $G$ such that $z$ is not a pole of $G$. Then there is an initial state such that the input $u(t) = e^{z t}$ produces the output $y(t) = 0$ for all $t \geq 0$. In this sense, the system blocks the input $e^{z t}$ completely.
- A system is minimum phase if all of its zeros have negative real part. Zeros in the right half-plane limit the performance that any controller can achieve.
Notes
For state-space systems, the invariant zeros are the values of $s$ where the Rosenbrock system matrix loses rank. They include the transmission zeros and also any zeros canceled by uncontrollable or unobservable modes.