## Solution

Three major classifications of PID controller algorithms are: ideal, parallel, and series.

As shown in the equations below,

*K*_{p},

*K*_{c} are gain parameters;

*T*_{i} and

*T*_{d }are integral and derivative time respectively.

where:

**Proportional term or Gain**

The proportional term make changes to the output that is proportional to the current error value.

The proportional gain may also be expressed as proportional band (PB).

So a proportional band of 50% is equivalent to a proportional gain of 2.

The integral term is proportional to both the magnitude and the duration of the error. Increasing integral time makes the output respond slower to an error, which is opposite of the effect of increasing integral gain. The integral gain in the parallel PID is equal to the overall gain divided by integral time in the ideal PID.

The derivative term demonstrates the rate of change of the process error by determining the slope of the error over time. The derivative gain in the parallel PID is equal to the overall gain times derivative time in the ideal formulation.