Top Ten Research Problems in Nonlinear Control
Here is my personal list of the biggest research problems in nonlinear
control theory (including links to what my group is doing about them).
If you don't agree with these (which is likely), feel free to send me e-mail.
10.
Using experiments and applications to motivate new theory
If you want to have an impact on engineering problems, you need to start
with engineering problems as a basis for reseach. This has been the
starting point for my own research on
nonholonomic motion planning,
flight control, and
active control of fluids.
9.
Building good university experiments for evaluating controllers
One of the hardest parts about doing controls research in a university is
figuring out how to validate your results on an experiments that are
representative of real engineering systems while at the same time being
simple enough to be built, maintained, and used by faculty and graduate
students (as opposed to a full-time, technical support staff). Two
experiments that we have built at Caltech that I am reasonable happy with
are the ducted fan and a
low-speed compressor system.
8.
Integrating algorithmic control with dynamical control
Modern controllers are implemented on computers and often consist of a lot
of logic surrounding a core of feedback control algorithms. Figuring out
how to integrate the logic with the controllers and how to design
controllers which are compatible with higher level algorithms is basically
an unsolved problem. We aren't doing a lot of work on this problem in my
group right now, mainly because it hasn't yet come up in any of the problems
we are working on (but it will...).
7.
Recognizing the difference between regulation and tracking
For linear control systems, regulation and tracking are essentially
identical. For nonlinear systems, and particularly motion control systems, the
problem of tracking is significantly different and considerably harder. The
role of trajectory
generation is very important in nonlinear problems and is the motivation
for much of our work in
differential flatness,
nonholonomic motion planning,
and
mechanical systems with
symmetries.
6.
Integrating good linear techniques into nonlinear methodologies
People who work in nonlinear control need to figure out how to make use of
all of the latest advances in linear control techniques when they apply.
The fact is that for a lot of control problems, the dynamic, error
correction (feedback) portion of the controller can be made linear. And in
that case, you may as well use a good linear controller with gauranteed
robustness and performance rather than just using static, linear or
nonlinear feedback (like pole placement). This is what we are trying to do
on the ducted fan and is the basic idea underlying two degree of freedom design
techniques.
5.
Recognizing the difference between linear and nonlinear stabilization
For some classes of problems, looking at regulation to a single equilibrium
point is not the right problem to study. The example that we are working on
is active control of rotating
stall, where you are usually much more interested in getting rid of
hysteris behavior or guaranteeing a large domain of attraction in the
presence of significant noise.
4.
Finding nonlinear normal forms for control
Most of the research in nonlinear control to date has concentrated on
extending linear methodologies to nonlinear problems. In essense, we
convert or approximate nonlinear systems by linear ones and then applying
traditional ideas. It is often very expensive (in terms of control energy)
to convert a nonlinear system to a linear one and linear approximations are
becoming increasingly inaccurate as we push the envelope of controller
performance.
I envision a time when there is a big (online) catalog of nonlinear normal
forms for control, with software for determining how close a given system is
to each normal form listed, and methods and techniques for control of that
normal form. All of this in some consitent format, so that an engineer can
get a first cut design by combining existing results to attack their
problems (kind of like a set operating system interface functions or a
programming library in the world of computers).
3.
Writing software for implementing theory
In this day and age, the only way anyone is going to use your personal
technique for synthesizing controllers is if you write software to implement
it. There is a strong need for a software protocol for nonlinear control
which allows easy integration of modules from a variety of sources. Our
initial work in this area has so-far been limited to
Sparrow,
RobotLinks, and
EDSpack.
A lot more needs to be done.
2.
Exploiting special structure to synthesize controllers
You can't build a theory for nonlinear control that works for everything.
Nonlinear systems are a lot more complicated than that.
Concentrating on special classes of systems, like mechanical systems and propulsion systems, is the most likely way
make significant progress in synthesizing nonlinear controllers.
1.
Transferring successful nonlinear control methods to science and industry
The biggest research problem in nonlinear control is figuring out how to get
people to use it. Most of the theoretical work in nonlinear control is just
that: theory. In order to make control theory useful, we need to spend more
energy on solving practical problems and applying techniques to physical
examples (like the ducted fan, here at Caltech).
Richard Murray (murray@indra.caltech.edu)
Last modified: Fri Jan 12 09:44:05 1996