Frequency-Domain Descriptions

In Definition 4.3.5, we introduced the notion of a transfer function for a state linear system Σ(A, B, C. D) and showed that it was equal to D + C(sI – A)-1B. In this section, we study the input-output relationship directly in the frequency domain without reference to any state-space descriptions. More specifically, we suppose that we have a scalar input function of time u: $$0,\infty ) \mapsto$$ ℂ and a scalar output function of time y: $$0,\infty ) \mapsto $$ ℂ, which arc Laplace transformable and we suppose that their Laplace transforms û(.) and ŷ(.) are related by 7.1$$\hat y\left( s \right) = g\left( s \right)\hat u\left( s \right),$$ where g(s) is an irrational function of the complex variable s. We call the g(s) the transfer function.