# PID Controller

Cuthbert Nyack

The control system is represented by the following diagram.
*G*_{1}(s) is the transfer function of the
controller and *G*_{2}(s)
is the transfer function of the system.
Consider first the case with a PD controller. For the PD controller the transfer
functions for *G*_{1}(s) is given by the
following equation with *K* being the proportional
gain and *T*_{d} the Derivative time. This
transfer function has a zero at *-1/T*_{d}.
The system is a second order type 0 system consisting of
2 first orders in series. Its transfer function
*G*_{2}(s) has 2 poles at *-1/T*_{1} and
*-1/T*_{2} and
is given by the following expression.
Combining the above equations gives the following overall
transfer function.

The applet below shows the step response (red curve) of the system with a PD
controller. The response has oscillations and is offset from the input
(magenta curve). Increasing *T*_{d} damps the oscillations
and increasing *K* reduces the offset but cannot remove it. From the above
equations the steady state response is *K/(1 + K)* which is offset from
the input by *1/(K + 1)*. In the applet, the horizontal axis goes from 0 to
4 when Hg = 1.

For PI control only the overall transfer function is.
and the response is shown below. I control removes the offset but introduces
additional phase lag which may make the loop unstable if T_{i} is
made too small.

The transfer function for a PID controller is.
and the following overall loop transfer function.
Examining the above expression(set s to 0) shows that
the steady state response is one with zero offset. The step
response of the system with a PID controller is shown in the
applet below. The input is in magenta and the response is in red.
With Hg = 1 the horizontal axis goes from 0 to 4. To simplify the
usage of PID controllers, it is common to start with *T*_{i} =
*4 T*_{d}. In this case the open loop system has 3 poles at
0, *-1/T*_{1}, *-1/T*_{2} and 2 zeros
at *-1/(2T*_{d}).

## In summary, the rise time can be
reduced with P control, D control can be used to damp out any transient
oscillations and I control can be used to remove any steady
state error between input and output.

The above applet shows the root locus(interface of red and blue regions) of the system. If Ti > 4Td then both zeros are on the real axis and if Ti < 4Td the zeros become complex conjugate off the real axis. A suitable set of parameters might be T1 = T2 = 0.5, Ti = 0.8, Td = 0.2, K = 5. These parameters can be set in both applets to show the step response and closed loop poles. However depending on the values chosen the system can become unstable. An example is T1 = T2 = 1.0, Ti = Td = 0.2. For these parameters the applet shows that a portion of the root locus lies to the right of the imaginary axis. Changing K( the loci now start at the closed loop poles) shows that the system is stable for K < 0.8 and > 13 (approximately). These numbers can be verified with the applet for the step response.

The above applet shows the Bode plot of the PID system shown above. Changing the parameters shows how the Bode plot corresponds with the transient and root locus information shown above. The Integrator increases the low frequency gain to reduce the offset but introduces a 90º phase lag which can make the system unstable. The D controller is added to reduce the phase lag and improve the stability and transient behaviour of the system.

The same information is shown in the Nyquist plot above for P, PD, PI and PID controllers.

The above applet shows the Routh Hurwitz criterion applied to the PID system. As for the Bode plot, this is consistent with the other analyses.

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COPYRIGHT © 2005 Cuthbert A. Nyack.
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