# Re: Basic Question (fwd)

Feijun Song (fsong@oe.fau.edu)
Fri, 25 Jun 1999 12:48:34 +0200 (MET DST)

I would like to have a summary of the discussion. Please excuse me
if I am wrong.

for highly nonlinear system, an optimal control policy is a hingly
nonliner function of controller's inputs, a FLC can outperform any
conventional linear controller because it can approximate the nonlinear
control policy.

The apporximation ability of a FLC depends on its number of rules, rule
antecedents,...., for a thorough proof of the conclusion that a FLC is
an universal approximator, please refer to a serial of papers written
by Dr. Hao Ying and Dr.Kosko, Dr. Li Xin Wang. The lasters one is in

IEEE International Conference on System, Man and Cybernetics 1998.

there is a pretty good bibliograph inside too.

---------- Forwarded message ----------
Date: Thu, 24 Jun 1999 11:54:43 PDT
To: fsong@oe.fau.edu
Subject: Re: Basic Question

Dear Feijun Hello!!
you are quite right in saying........
>Generally, when the control objective is near the set point, one
>would say a FLC would function equally to a conventional P(I)D
>controller since the objective is fairly a linear system now.
>
>Where the above statement is true for 2D systems, based on my limited
>experience on 4D systems, I would like to point out that even near
>the set point, a FLC could outperform a P(I)D controller for 4D systems.
>The reason is below:
>
>We know for 4D system, a system trajectory oscillates a lot with a P(I)D
>controller, and a trajectory goes to the set point after several
>oscillations. This kind of trajectory is energy-comsuming, time-consuming.
>There are possibilities that a better trajectory exist within the
>objective's ability, and all the control commands along this better
>trajectory is a nonlinear function of system states. A FLC can approximate
>this nonlinear function fairly well whereas a P(I)D could never do.
>So even when system is near the set point, a P(I)D controller still
>cause lots of oscillation, by contrast, a FLC can directly drive the
>objective to the set point without much oscillation.
>
>I have designed a LQR and a TS type FLC for a 4D inverted pendulum, lots
>of trajecotries tell me that the FLC drive the pole and cart directly to
>the set point, whereas the LQR always drive the pole and cart in an
>oscillating way, which costs lots of time and energy.
>

I have no experience of working on FLC. Whatever I posted earlier regarding
my perception about fuzzy logic but in the light of my perception after
doing a lot of work in dynamic system modeling I like to add few words in
your posting about non-linear PI FLC. Basically in TS type FLC if the
premise are e(t) and ce(t) the consequent is either PI, PD or PI(D)
depending upon what form of consequent one has to choose. During aggregation
of fired rules FLC interpolates between consequent(which is linear function
of e(t) and ce(t) in TS type FLC) of rule. This interpolation results in
nonliner function of e(t) and ce(t) which termed as non-linear PI PD or
list, you can do that.
Yours

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