Re: Basic Question

Feijun Song (
Sat, 3 Jul 1999 15:51:07 +0200 (MET DST)

> On Tue, 22 Jun 1999, Feijun Song wrote:
> >
> > 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.
> >
> The nearness to setpoint/equilibrium allows many nonlinear systems to be
> well-defined by a linear one - but only the *system*, not the controllers.
> If the FLC is non-adaptive (say), or fixed, then it will remain exactly
> as nonlinear near or far from equilibrium. Much analysis shows simply
> that the FLC-PI is like a gain-scheduled PI. Anyway, a good nonlinear
> controller (FLC) will control even a linear process equal or better than a
> a linear one. The emphasis on this resemblance is more theoretical, and
> tells very little to nothing about the performance.

Performance of any controller depends on plants, when plants is
linear, you get a performance, when it is nonlinear, you get another
performance for the same controller. It is difficult to compare
controllers without talking about control objectives. "near set point" is
a good place to compare a nonliner and a linear controller. If there is
no way to linearize a nonliner plant, it would be difficult for us to
compare a nonliner controller and a liner controller theorectically ( Cell Mapping
is a computational approach to do this, that's why I mentioned the
comparison of a LQR and a FLC ). I agree that this kind of comparisons
are mostly theoretical. The only reason a FLC can overperform a linear
controller is its nonliearity.

> > 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:
> >

for 2D system, the comparsion of a nonliner and a liner
controllers holds true for a pretty large area in state space. When more
states involved, this area should shrink. Although a PD controller can
easily performance as well as a FLC for a 2D system over a large region of state
space, it is very difficult for it to achieve the same well performance
for a 4D system, espeicially when all states are coupled. That's why I
emphersize the importance of a FLC for high order systems ( above 2 ).

> > 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.
> >
> The nonlinear function mapped by the FLC is removed from the system
> states. The FLC(e, \delta e) depends only on "output feedback" error,
> not the directly the states of the system. The inverted pendulum has 4
> states but *only* 2 outputs (theta, omega) hence only 2 SISO (TISO)
> FLC's. The (e, \delta e) map is not a state phase-space map, but an
> input/output map only. This is nominally independent of the no. of
> states.

Sorry I was not clear in my last post. If the 4D inverted pendulum
I mentioned before could be decoupled, all my postings would
not hold. The 4D inverted pendulum I mentioned is fully observable, all
states are observable, " pole angle", " angular velocity", "cart
position", and " cart speed " are all observable ( of couse it is an
assumption ), and the set points is [0 0 0 0]. For this system, the
input/output map is equal to a state space/output map since all states
are fedback. for a detailed state equation, please see:

[Smith, 1990] Smith, S.M., and Comer, D.J., Self-Tuning of a Fuzzy Logic
Controller Using a Cell State Space Algorithm, Proceedings of the 1990
IEEE In ternational Conference on Systems, Man, and Cybernetics,
pp.445-450, 1990.

> > 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.
> >
> LQR is a linear structure. FLC is nonlinear. The plant is strongly
> nonlinear. Ergo.

What's is Ergo?!, This is a academic mailing list, I appreciated
any of your posting, as long as it is within academic discussion.

Feijun Song
Ph.D candidate
Ocean Engineering Dept.
Florida Atlantic Univ.

This message was posted through the fuzzy mailing list.
(1) To subscribe to this mailing list, send a message body of
"SUB FUZZY-MAIL myFirstName mySurname" to
(2) To unsubscribe from this mailing list, send a message body of
(3) To reach the human who maintains the list, send mail to
(4) WWW access and other information on Fuzzy Sets and Logic see
(5) WWW archive: