Gains, poles and zeros

We are going to look at systems a lot, so we need a nice common set of things to fully characterise any LTI system. These three magical properties are:

• The gain / steady state gain / DC gain: defined as the value of $G(0)$,
• The poles: the set of $s$-values for which $G(s)$ is not defined, i.e. when $(sI -A)$ is not invertible or a division by 0 happens in the transfer function,
• The zeros: the set of $s$-values for which $G(s) = 0$. (zeros are underrated and you'll see why later)