02 - First Order Systems

Original ODE: \[\tau_p \frac{dy}{dt} + y = K_px\] ODE in deviation variable: \[\tau_p \frac{dY}{dt} + Y = K_pX\] where \(Y=y-y_s\) is deviation variable of output; \(X=x-s_s\) is deviation variable of input.

Laplace transform: \[G_p(s) = \frac{Y(s)}{X(s)} = \frac{K_p}{\tau_p s + 1}\]