It *can* be controlled perfectly well with a PI controller as long as you
keep away from the gain reversal. Before you embark on your maths, you need
to specify exactly what it is that you want to prove.

On Fri, 25 Nov 2011 13:30:50 +0800, Bruce Varley wrote:

Well, as long as you don't have to control across the gain reversal, and
as long as your bandwidth is high enough at the low gain end of the
operating range, and as long as your damping is low enough at the high
gain end of the operating range.
After that point, consider that gain scheduling can make you a hero, but
it can also make you look like a buffoon. (Nothing, but nothing, equals
the impression you can make by spending an hour in the conference room
explaining why your approach is superior, then walking out into the lab
or the factory floor and blowing something up).

--
Tim Wescott
Control system and signal processing consulting

Yep, anyone who works in the process industries has had more experience with
this than they may realise. Heaps of flow controllers utilise equal
percentage control valves, whose installed characteristic is often just as,
or more bendy than a quadratic. The limited operating range of the plant
usually protects you from the nonlinearity, as long as you don't make the
mistake of doing the tuning at a low operating point.

Your problem is still a semantic one. If you want to prove that conditions
can exist where positive feedback will cause the loop response to be
unbounded, then that's trivial. If you want more than this, then what do you
mean by 'problem'?
Sorry to appear difficult, but I can assure you that many real world control
projects are derailed by failure to clarify situations like this.

Lets start here with the trivial. Under which conditions is the loop
response unbound?

No problem. As you may assume, I found "the problem" in a complex
control loop. Don't know where the instability results from. So I try
to divide the system or the problem into smaller pieces. The quadratic
behaviour is one of those, as I see that the derivate of the process
function changes its sign. Meaning, the process function is not
monotonous (?). I hope to get a better understanding if I can put it
into mathmatics and would be happy if someone can give me a hint how
to setup the mathematical prove.

That may not be so trivial after all. But if you have an element in the
system whose instantaneous gain is changing from positive to negative,
then it is very doubtful that a system whose linearized model is stable
for one case will be stable for the other, unless you have a very strange
(and very nonlinear) controller.

You mean monotonic (you should be able to look up "monotonic function" in
Wikipedia). A monotonous process function would be boring:
malfunctioning equipment is, unfortunately, not often boring.

The most direct way that I could see to do this would be to take my
system model and do a root locus plot around the changing gain of the
part, over the expected gain range of the part. If you've got just one
pole at the origin (from the controller integrator) then the system will
either be stable for some values of positive gain and not at all for
negative, or the converse.
In fact, the only way that I could see that you might have a system with
overall stability in the face of an element whose gain changes sign would
be if that portion of the system were loaded down somehow with an element
(or elements) that were so very dissipative that they would swamp out the
quadratic element. I can't even begin to think of a concrete example of
this outside of electronic circuits, and within that discipline the
"swamping out" is generally pretty wasteful in terms of energy and often
space.

--
Tim Wescott
Control system and signal processing consulting

This sort of thing does happen occasionally. My experience is that
you won't get expected results :) Since real world systems have
limits (say saturation) they can occasionally be made to work;
sometimes in odd ways.
Why not explicitly explicitly describe the system and the part that
you think has quadratic part and gain reversal? Give the physical
reason you believe that this is happening. There are plenty of
people here who are quite knowledgeable about a wide range of real
world applications. I don't think this is a problem that can be
treated in the abstract.
BTW: Scilab is a free program that has a graphical interface allowing
a variety of non-linear blocks and allowing the execution the block
diagram.
A physical description and a accurate executable block diagram would
probably lead to really effective answers. In engineering
mathematical answers typically require an approach from the bottom
up. Understanding the physical system and identifying a mathematical
models that describe it.
Ray

If you mean "transfer function" in the sense of a Laplace transfer
function or a z-domain transfer function, that's because the whole
concept of a transfer function with quadratic gain is entirely
meaningless: transfer functions are a way of modeling system behavior in
the Laplace or z domain, and modeling in those domains only makes sense
for linear, time invariant (or shift-invariant, for z domain) systems.
If you want to use transfer function analysis then you need to linearize
the system around an operating point, then find the transfer function of
the resulting linear model. Doing so and finding that the resulting
linearized system is unstable means that your overall system is unstable
around that operating point. In a wide sense this may or may not mean
that you're in trouble: some systems work perfectly well under these
circumstances, but usually only if the nonlinearity is well behaved and
makes them oscillate around the desired set point. In your case, there's
a good change that getting on the wrong side of that quadratic gain curve
will mean that the system will go berserk.
If the system, once you cross over the zero-gain part of the quadratic
characteristic, tends to go further in that direction (instead of
breaking into a sinusoidal oscillation, for instance), then you can be
pretty sure that it'll go berserk. Even if it does something more benign
but still out of the correct operating range, then you have a problem.
Why can't you just restrict the system from driving that part into the
negative gain region?

--
Tim Wescott
Control system and signal processing consulting

I have started with a simple example. With the transfer function of a
PI controller G_C(s)=K_C*(1+T_C*s)/T_C*s and the transfer function of
a PT1 process G_P(s)=K_P/(1+T_P*s), I build the closed loop and
calculate the characteristic equation
1+(K_P*K_C*(1+T_C*s))/((1+T_P*s)*T_C*s)=0
Now, I take the part between "1+" and "=0", name it G_CP and figure
out the zeros and poles. These should be
s_z1 = -1/T_C (zero 1)
s_p1= 0 (pole 1)
s_p2=-1/T_P (pole 2).
Finally the curve parameter K_0=K_P*K_C/T_P.
With this I could play with the time constants and gains of process
and controller. As you suggested, I can try to minimize K_P like it
would be in the area of the gain reversal. I can also define a process
with negative gain and examine the stability on the other side of the
gain reversal. I have the feeling that both, the gain reversal and the
small gain cause instability. I think you wrote about the small gain
problem before but I wasn't sure what you mean.
I chose some values for the time constants and gains and used a root
locus java applet on some web site. My first question is, can you
suggest a web page with root locus applet? My second question is, how
do I evaluate the stability with the root locus plot? I have
literature about this but after reading I still have no clue.
Example values: T_P=T_C=500ms, K_C=K_P=1
Applet Page: http://users.ece.gatech.edu/bonnie/book/OnlineDemos/InteractiveRootLocus/applet.html
numerator setting: "0.0 0.5 1.0"
denominator setting: "0.3 0.5 0.0"
gain: 0 to 5

Some stupid designed it in the middle of the operating range. A design
change is my first choice. But still I would like to understand what
is happening in that region.
Best regards,
pt

Get a copy of Scilab, and use the "Evans" function (because the _full_
name of the root locus plot is the _Evans_ root locus plot).

The root locus shows the paths (the loci) of the roots as the gain
changes. If you have a root that falls outside of the stability region
(for the Laplace domain that's for any root with a non-negative real
part) then that root is unstable. Roots close to the stability boundary
indicate a system that will take a long time to settle, and that may not
be robust to changes in system parameters.

InteractiveRootLocus/applet.html
Oh Joy!!

Depending on the plant you may not be able to stabilize it at all without
going to wacky extremes -- sign changes in your gains are Very Bad Things.
But like everything, it depends.

--
Tim Wescott
Control system and signal processing consulting

I have severe problems with my scilab installation. For some reason I
can't get the scilab plot working. There seems to be a problem with
the interface java to openGL which stopped functioning after a kernel
security update. This may take a while. But I'm eager to try out your
proposal.
pt

Ouch! I know nothing!
Except that you could try scicoslab. There used to be one Scilab group,
and one Scilab. Then they (apparently) underwent miosis, and now there's
two. Switching from one to another is -- in the extreme -- a way to
dodge bugs.

--
Tim Wescott
Control system and signal processing consulting

[...]
After a long time I found a bugfix for scilab 5.3 on ubuntu 11.10 here
(http://www.equalis.com/forums/posts.asp?topic=321201 ). The solution
proposed by R. Rivière on 11/22/2011 worked fine.
And I plotted the root locus for the linearized quadratic function. We
discussed that the two problems are the small gain left and right of
the gain reversal and the gain reversal itself. The root locus plots I
saved at

http://imageshack.us/photo/my-images/687/gainreversal.png/.
As far as I understood the roots for gains less the -0.1 move towards
the non-negative area and would be unstable.
pt

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