Re: Basic Question

Feijun Song (fsong@oe.fau.edu)
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.
U.S.A

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