PI controller

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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 Δ is the set-point error.

Contents

[edit] Advantages of a Proportional Plus Integral Controller

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

[edit] PI Controller Model

A PI controller can be modelled easily in software such as Simulink using a "flow chart" box involving [Laplace transform|Laplace] operators:

C=\frac{G(1+s\tau)}{s\tau}

where

G = KP = proportional gain
G / τ = KI = integral gain

[edit] Finding a value for G

Setting a value for G is a trade off between decreasing rise time and increasing settling time. One rule of thumb for G is between 4 and 8.

[edit] 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 (fc).

3) The value of τ can be calculated as:

τ = 1 / fc

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

[edit] 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.

[edit] Rectifying the Phase Lag

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

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