What effect does the proportional parameter of control response have on the rise time in a closed-loop control system?

Concept:

The controller is a device that is used to alter or maintain the transient state & steady-state region performance parameter as per our requirement.

Proportional Controller- 

The standard Proportional Controller as shown:

In space-form - 

\({G_C}\left( s \right) = \frac{{U\left( s \right)}}{{E\left( s \right)}} = \frac{{{K_p}}}{s(s+1)}\)

In time-domain form - 

p(t) = Kp e(t) + po

Where,

po = controller output with zero error  

Kp = proportional gain constant.

Some effects of the proportional controller are as follows:

• The P-controller can stabilize a first-order system, can give a near-zero error, and improves the settling time by increasing the bandwidth.
• It can also destabilize the system by using very high gains because it reduces the gain margin.
• Even though the system can be stable by the use of a small gain proportional controller, the performance of the system are generally not so good.

Analysis:

Rise time (tr):  It is defined as the time required for a pulse or signal to increase from one specified value (as 10 %) of its amplitude to another (as 90%).

3-dB Bandwidth: The frequency at which the power level of the signal decreases by 3 dB from its maximum value is called the 3 dB bandwidth, The 3 dB bandwidth is the frequency at which the signal amplitude reduces by 3 dB i.e. becomes half its value.

The relationship between the rise time(tr) and bandwidth is given by:

\(B.W. = \frac{{0.35}}{{{t_r}}}\)

As the rise time is inversely proportional to the bandwidth, with the proportional controller the rise time decreases.