The compensator required to improve the steady state response of a system is

Lag compensator:

Transfer function:

If it is in the form of \(\frac{{1 + aTs}}{{1 + Ts}}\), then a < 1

If it is in the form of \(\frac{{s + a}}{{s + b}}\), then a > b

Maximum phase lag frequency:

\({\omega _m} = \frac{1}{{T\sqrt a }}\)

Maximum phase lag::

\({\phi _m} = {\sin ^{ - 1}}\left( {\frac{{a - 1}}{{a + 1}}} \right)\)

ϕm is negative

Pole zero plot:

The pole is nearer to the origin.

Filter: It is a low pass filter (LPF)

Effect on the system:

• Rise time and settling time increases and Bandwidth decreases
• The transient response becomes slower
• The steady-state response is improved
• Stability decreases

Lead compensator:

Transfer function:

If it is in the form of \(\frac{{1 + aTs}}{{1 + Ts}}\), then a > 1

If it is in the form of \(\frac{{s + a}}{{s + b}}\), then a < b

Maximum phase lead frequency:

\({\omega _m} = \frac{1}{{T\sqrt a }}\)

Maximum phase lead:

\({\phi _m} = {\sin ^{ - 1}}\left( {\frac{{a - 1}}{{a + 1}}} \right)\)

ϕm is positive

Pole zero plot:

The zero is nearer to the origin.

Filter: It is a high pass filter (HPF)

Effect on the system:

• Rise time and settling time decreases and Bandwidth increases
• The transient response becomes faster
• The steady-state response is not affected
• Improves the stability