Pid Fuzzy

DTuning Of Fuzzy PID Controllers
Jan Jantzen

jj@iau.dtu.dk:1

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Since fuzzy controllers are nonlinear, it is more difficult to set the controller gains compared to proportional-integral-derivative (PID) controllers. This research paper proposes
a design procedure and a tuning procedure that carries tuning rules from the PID domain
over to fuzzy single-loop controllers. The idea isto start with a tuned, conventional PID
controller, replace it with an equivalent linear fuzzy controller, make the fuzzy controller
nonlinear, and eventually fine-tune the nonlinear fuzzy controller. This is relevant whenever a PID controller is possible or already implemented.



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When the control problem is to regulate the process output around a setpoint, it is naturalto consider HUURU as an input, even to a fuzzy controller, and it follows that the integral
of the error and the derivative of the error may be useful inputs as well. In a fuzzified
PID controller, however, it is difficult to tell the effect of each gain factor on the rise time,
overshoot, and settling time, since it is most often nonlinear and has more tuning gains
than a PID controller. Theobjective in this paper is to find a systematic tuning procedure
by carrying PID tuning rules over to the fuzzy domain. A systematic tuning procedure
would make it easier to install fuzzy controllers, and it might pave the way for auto-tuning
of fuzzy controllers.
PID controllers may be tuned in a variety of ways, including hand-tuning, ZieglerNichols tuning, loop shaping, analytical methods,by optimisation, pole placement, or autotuning (Smith, 1979; Åström & Hägglund, 1995). Furthermore, fuzzy controllers show
similarities with PID controllers under certain assumptions (Siler & Ying, 1989; Mizumoto, 1992; Qiao & Mizumoto, 1996; Tso & Fung, 1997). But there is still a gap, it seems,
between the PID tuning methods and a design strategy for fuzzy controllers of the PID type.
Thispaper proposes a design strategy, which makes use of known PID design techniques, before implementing the fuzzy controller:
1. Tune a PID controller
2. Replace it with an equivalent linear fuzzy controller
3. Make the fuzzy controller nonlinear
 Technical University of Denmark, Department of Automation, Bldg 326, DK-2800 Lyngby, DENMARK.
Tech. report no 98-H 871 (fpid), 30 Sep 1998.

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Figure 1: Proportional control with load o and noise q.
4. Fine-tune it
It seems sensible to start the controller design with a crisp PID controller, maybe even
just a P controller, and get the system stabilised. From there it is easier to go to fuzzy
control. Each step will be investigated in thefollowing.



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The first step in the design strategy is to install and tune a PID controller. The ideal continuous PID controller


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(1)
h g . Wg
x @ Ns h .
Wl 3
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returns the controller output x, the constant Ns is the SURSRUWLRQDO JDLQ, Wl is the LQWHJUDO
WLPH, Wg the GHULYDWLYH WLPH, and h is the HUURU between the reference and the processoutput
| +h @ Uhi |,. We are concerned with digital control, and for small sampling periods Wv ,
the equation may be approximated by a discrete approximation. Replacing the derivative
term by a backward difference and the integral by a sum using rectangular integration, an
approximation is
3
4
q
4[
hq hq 4 D
xq @ Ns Chq .
hm Wv . Wg
(2)
Wl m @4
Wv
Index q refers to the time instant.By WXQLQJ we shall mean the activity of adjusting the
parameters Ns > Wl > and Wg =
Several tuning aspects may be illustrated by static considerations (Åström & Hägglund,
1995). For purely proportional control +Wg @ 3 and 4@Wl @ 3,, the control law (2) reduces
to
xq @ Ns hq
(3)
Consider the feedback loop in Fig. 1, where the controller has the proportional gain Ns and
the process has the...