PI controller

In control engineering, a PI Controller (proportional-integral controller) is a feedback controller which drives the plant to be controlled with a weighted sum of the error (difference between the output and desired set-point) and the integral of that value. It is a special case of the common PID controller in which the derivative (D) of the error is not used.

Controller output = Constant K_I x Summation (Error x Time)

Advantages Of a Proportional Plus Integral Controller

A PI Controller is used to remove steady-state errors.

PI Controller Model

A PI Controller can be modelled easily in software such as Simulink using a "flow chart" box involving laplace operators : -

Failed to parse (syntax error): {\displaystyle C=\frac{G(1+sτ)}{sτ}}

Where

G = Proportional Gain

τ = Time Integral

Finding a value for G

Setting a value for G is a trade off between decreasing rise time and increasing settling time, the optimal range of values for G is between 4 and 8.

Finding a value for τ

Finding a proper value for τ is an iterative process.

1) Set a value for G from the optimal range.

2) View the Nichols Plot for the open-loop response of the system. Observe where the response curve crosses the 0dB line. This frequency is known as the cross-over frequency ($f_{\mathrm {c} }$).

3) The value of τ can be calculated by doing:

$1/f_{\mathrm {c} }$

4) Decreasing τ decreases the phase margin, however it eliminates a greater proportion of the steady-state errors.

Disadvantages Of A Proportional Plus Integral Controller

The problem with using a PI controller is that it introduces a phase-lag. This means that on a Nichols Plot, the stability margin (the phase margin) decreases. So careful design considerations with respect to the gain must be considered.

Rectifying The Phase Lag

The phase-lag can be countered/rectified by introduces a Proportional plus Differential Controller (PD Controller).