# PI controller

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.

The controller output is given by :$K_P Delta + K_I int Delta,dt$where $Delta$ is the set-point error.

Advantages of a Proportional Plus Integral Controller

The integral term in a PI controller causes the steady-state error to be zero.

PI Controller Model

A PI controller can be modelled easily in software such as Simulink using a "flow chart" box involving Laplace operators::$C=frac\left\{G\left(1+s au\right)\right\}\left\{s au\right\}$where:$G = K_P$ = proportional gain:$G/ au = K_I$ = integral gain

Finding a value for G

Setting a value for $G$ is often a trade off between decreasing overshoot and increasing settling time.

Finding a value for $au$

Finding a proper value for $au$ 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 ($f_\left\{mathrm\left\{c$).

3) The value of $au$ can be calculated as:

:$au = 1/f_\left\{mathrm\left\{c$

4) Decreasing $au$ 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.

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