# Re: Basic Question

Pramit 'Jake' Sarma (psarma@che.iitb.ernet.in)
Sat, 3 Jul 1999 05:15:28 +0200 (MET DST)

These questions overlap into control design 'philosophy' ...

That's not just the only point. It is OK to assume that FLC will perform
better than PID in general. However, finally if it's a REAL system, the
control designer must choose using a 'cost/benefit' approach: is the 2-3
floatg-pt ops (flops) calculation of the PID *that* much worse than the
O[100+] flops FLC? Sometimes only! To play "devil's advocate" for the PID,
I could always add a few 10's of flops with an adaptive gain!

It has nothing to do with a decoupled [or not] system: full-blown MIMO
FLC's with NxN, N >= 3 are rarely used, as the FRB is so hard to construct
that the utility of the FLC dies, due to the "curse of dimensionality".
SISO controllers are normally used, (TISO or {e,\delta e} for FLC). For
linear PID/LQR an interaction analysis is done ... it can be very useful
to design the SISO/TISO FLC too.

Perhaps you are concentrating on the i. pend. (IP) problem, but there are
many many plants which are simply not analysable thus,
theoretically/semi-theoretically. A case I'm looking at right now, has
110 differential/algebraic equations. No simple rules are good
here. I am otherwise reasonably familiar with the test-case of the IP. It
is generally erroneous to use inductive logic to generalise the results
from a single case. One could always use results from, say, exact
linearising control (ELC) from diff geom, and show that it
equals/outperforms FLC! But that would be specific only to the IP, as ELC
needs a formal mathematical model ... which is where FLC scores over ELC
(it doesn't need explicit knowledge of the physics).

All this is fine, from a chapter on FLC analysis. A host of papers can be
hunted down from IEEE SMC/a/b/c, FUZZ on such comparisons. None of them,
really contribute all that much: The FLC was structured by Prof Mamdani
like a PI to begin with, and to improve on it. So (other than a lot of
papers) where's the mystery?!

Near-equilibrium operations (for any nonlinear plant) are useful, but not
paramount. While the plant is linearisable there and all sorts of nice
looking Lyapunov functions for stability etc. can be checked, in the end
even for a simple plant like the IP, closed-loop control is most useful
only *far* from equilibrium. Also, in general, the FLC's are used most for
a plant with poor/no models, so the only way to compare the performance
then is by *simulations*. This is not a problem with todays computing
powers. So, it makes perfect sense even to compare a linear (PID/LQR/MAC
..) controller with a nonlinear (FLC/NNC/ELC ...) where ||x-x_{eq}|| is
quite large. Or "theory of control" only gets you so far.

The point is whether it (a) works (b) works very well and (c) is worth the
effort to replace a simpler reasonable structure. That is the point of all
control, including fuzzy control. Microfinessing on one process alone will
never teach enough about either FLC or nonlinear dynamical systems.

('Ergo' is Latin for 'therefore')

On Wed, 30 Jun 1999, Feijun Song wrote:

>
> > 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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Pramit "Jake" Sarma
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