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.

Current Previous
rank Research problem rank
10 Building representative experiments for evaluating controllers
9 Convincing industry to invest in new nonlinear methodologies
8 Recognizing the difference between regulation and tracking
7 Exploiting special structure to analyze and design controllers
6 Integrating good linear techniques into nonlinear methodologies
5 Recognizing the difference between performance and operability
4 Finding nonlinear normal systems for control
3 Global robust stabilation and local robust performance
2 Magnitude and rate saturation
1 Writing numerical software for implementing nonlinear theory

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 (
Last modified: Mon Feb 12 18:51:26 1996