How much additional positive phase should we request from a

How much additional positive phase should we request from a DT PID controller with a standard selection of the sampling rate at 6 samples-per-rise-time (assuming ZOH introduces a half-sample delay, and bandwidth is 1.5 times the cutoff frequency):

Solution

P-I-D controller has the optimum control dynamics including zero steady state error, fast response (short rise time), no oscillations and higher stability. The necessity of using a derivative gain component in addition to the PI controller is to eliminate the overshoot and the oscillations occurring in the output response of the system. One of the main advantages of the P-I-D controller is that it can be used with higher order processes including more than single energy storage. In order to observe the basic impacts, described above, of the proportional, integrative and derivative gain to the system response, see the simulations below prepared on MATLAB in continuous time with a transfer function and unit step input. The results will lead to tuning methods

Transient Response of P-I-D Controller:

P-I-D controller is the optimal controller for high order plants. It has zero steady state error together with acceptable transient response. The only problem with P-I-D control is tuning. Fortunately, MATLAB has automatic tuning option. However, automatic tuning does not usually provide the best results, it only provides optimal results. P-I-D tuning is an engineering art and should be manually done by control engineers. The following simulations were done on MATLAB-Simulink to illustrate the performance of P-I-D control on first and second order plants.