An example of ringing occurs when a digital proportional-only controller is applied to a process that is so fast that it reaches steady state within one execution period, ∆t. The following calculations, which are simple enough to be carried out by hand, will help explain ringing. (a) Calculate several steps of the response of a control system with a steady-state process with Kp = 1.0(θ = τ = 0) and a proportional only controller, Kc = 0.8. Assume that the system is initially at steady state and a set point change of 5 units is made. (b) Repeat the calculation for an integral-only controller. Find a value of the parameter (Kc∆t/ Ti) by trial and error that gives good dynamic performance for the controlled and manipulated variables. (c) Generalize the results in (a) and (b) and give a tuning rule for integral only, digital control of a fast process.