control//transfer function

A transfer function is the description of a linear time-invariant system by the ratio of its output to its input in the Laplace domain, \(G(s)=Y(s)/U(s)\) with zero initial conditions, and it is the working language of classical control: the outside view of a system, input against output, in which blocks multiply, loops close with one formula and the frequency response is read by setting \(s=j\omega\). A mass on a spring with a damper, \(m\ddot y+c\dot y+ky=u\), becomes

A transfer function is the description of a linear time-invariant system by the ratio of its output to its input in the Laplace domain, G(s)=Y(s)/U(s)G(s)=Y(s)/U(s)G(s)=Y(s)/U(s) with zero initial conditions, and it is the working language of classical control: the outside view of a system, input against output, in which blocks multiply, loops close with one formula and the frequency response is read by setting s=jωs=j\omegas=jω. A mass on a spring with a damper, my¨+cy˙+ky=um\ddot y+c\dot y+ky=umy¨​+cy˙​+ky=u, becomes

G(s)=1ms2+cs+k,G(s)=\frac{1}{ms^2+cs+k},G(s)=ms2+cs+k1​,

where every derivative has turned into a factor sss (Laplace transform). The roots of the denominator are the system's poles, and G(0)=1/kG(0)=1/kG(0)=1/k is its steady-state gain.

The state-space model is the inside view of the same building. For a linear system the two are equivalent, G(s)=C(sI−A)−1B+DG(s)=C(sI-A)^{-1}B+DG(s)=C(sI−A)−1B+D, and libraries convert one into the other (ss2tf and tf2ss in scipy, tf and ss in python-control). The transfer function is the natural choice with one input and one output, when the reasoning is in frequency, or when the model comes from a test on the real plant rather than from its physics; the state-space model wins with many inputs and outputs, with states to estimate, and with nonlinearities, which a transfer function cannot hold.

Block diagrams become algebra. Elements in series multiply, and a controller CCC closing a loop around a plant GGG gives the closed-loop transfer function CG/(1+CG)CG/(1+CG)CG/(1+CG); its denominator 1+CG1+CG1+CG holds the closed-loop characteristic polynomial, and CGCGCG is the loop gain.

Process engineers use it as a three-number model. A step test gives a gain KKK, a time constant T1T_1T1​ and a dead time θ\thetaθ, written Ke−θs/(T1s+1)Ke^{-\theta s}/(T_1s+1)Ke−θs/(T1​s+1); the factor e−θse^{-\theta s}e−θs is a pure delay, the one element whose transfer function is not a ratio of polynomials (FOPDT model, plant).

It is distinct from the frequency response, which is the transfer function evaluated along the imaginary axis, G(jω)G(j\omega)G(jω): a gain and a phase at each frequency, measurable with a sine sweep and drawn on a Bode plot.

It hides what the state space shows. A cancelled pole and zero disappear from G(s)G(s)G(s) while the state they belong to still exists, possibly unstable or unobservable (controllability and observability); the outside view is only as complete as the system is minimal.