# Top Ten Research Problems in Nonlinear Control

## January 1996

It's been a while since I have updated this. Probably no one is bothering
to look here anymore anyway. But just in case (and because I have 15
minutes to burn before going to a thesis defense)...
Here is my personal list of the biggest research problems in nonlinear
control theory (including some relevant links, where appropriate). If
you don't agree with these (which is likely), feel free to send me e-mail. This is more or
less a way for me to think online, so I wouldn't take any of this too
seriously.

## 10.
Building representative 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 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.

## 9.
Convincing industry to invest in new nonlinear methodologies

This used to be much higher on my list until I went out and visited some
aerospace companies and found out that dynamic inversion is one of the
standard approaches that people use for flight control. So it seems clear
that the problem is not that industry is not interested in trying out new
techniques, but rather that researchers are not presenting convincing
arguments for why someone should invest time in trying out a new technique.
Writing better software is a good start.

## 8.
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. Nonlinear theory directed at stabilization about a single
operating point is a waste of effort: linear controllers do a great and
typically have great domains of attraction for stabilization of a single
(nonlinear) plant to an equilibrium.

## 7.
Exploiting special structure to analyze and design 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.

## 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

## 5.
Recognizing the difference between performance and operability

One of the things that nonlinear control can do is increase the range
over which a system can run without catastrophic failure. This is
different than providing good performance and is a particularly hard
problem because you have to know about the global behavior of the
system in order to define something like operability. An example that
has motivated me is active control rotating stall and surge in
compression systems, where the main issue is to keep the system from
getting stuck in deep stall in the presence of disturbances. Good
performance is only required in normal operating conditions, so the
real issue is dealing with system nonlinearities that appear when
operating near the (uncontrolled) stability limits of the system.

## 4.
Finding nonlinear normal systems for control

Most of the research in nonlinear control to date has concentrated on
extending linear methodologies to nonlinear problems. In essence, 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. Even more nonlinear approaches
like backstepping really only apply to problems that are absolutely equivalent to
linear systems.

## 3.
Global robust stabilization and local robust performance

Here's what I think you want to be able to tell a person about your control
method: when the plant is near the design conditions, this controller
provides gauranteed performance in the presence of (relatively small) noise,
unmodeled dynamics and parameter uncertainty. If the system gets into a
region which is far from the design condition, the performance could be
terrible but the global dynamics of the system have been designed to be
stable in the presence of (bounded) noise, unmodeled dynamics, and parameter
uncertainty. I think that a really good example is some of the work that Blaise Morton has done at Honeywell on the use of
dynamic inversion for pitch control of an F14. Design issues also come up
in that work, in particular there are certain inequalities that have to hold
in the aerodynamic coefficients if you want to guarantee global convergence
of the trajectories to a (nice) positively invariant set.
(Yes, I know, this contradicts what I just said about point
stabilization. I think the dynamical systems view of the problem
combined with the goal of local performance appeals to me.)

## 2.
Magnitude and rate saturation

I am sometimes amazed at how little we know about how to do good design or
analysis of nonlinear control systems in the presence of saturation
(magnitude and rate). Every interesting nonlinear control
system that I know of is limited by saturations. It's the old actuator
bandwidth vs. performance limitations that Gunter Stein described so
eloquently in his Bode lecture at the 1989 (?) CDC. Except everything is
nonlinear, so limited bandwidth is replaced by magnitude and rate limits.

## 1.
Writing numerical software for implementing nonlinear theory

In this day and age, the only way anyone is going to use your personal
technique for designing 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. In particular, we need to write software that can handle 10s and 100s
of state space equations in a reasonable way and doesn't rely on symbolic
representations of the dynamics (so that we can deal with lookup tables for
things like aerodynamic coefficients). People in linear control systems and
dynamical systems have already solved this problem via programs like Matlab,
Matrixx, and Dstool.

Richard Murray (murray@indra.caltech.edu)
Last modified: Mon Feb 12 18:51:26 1996