Stability of FuzzyLogic Controller UAV

[AIStudent]

Hello,
I've designed a FLC controller for an UAV and I want to analyze its dynamic stability.
In all "Flight dynamics and control" books I've read, the analysis is based on transfer functions of the aircraft (and exemplified on a specific aircraft like Cessna 172) and of the pilot (human or automatic).
On the other hand, I've found an article that is using Lyapunov stability theorem to prove whether a FLC is stable or not.

1. Is there such a thing as "transfer function for fuzzy logic controllers"?
2. Is it possible to analyze the stability of an aircraft in the sense of Lyapunov stability?
3. Do you have any other ideas to analyze the dynamic stability of an FLC?

Thanks!

 

[milesyoung]

1. Is there such a thing as "transfer function for fuzzy logic controllers"?

In general, no. A fuzzy control system is nonlinear in general and the notion of a transfer function is only applicable to linear time-invariant systems. You could design the fuzzy controller to have the same response as some linear controller, but that would defeat the purpose of using a fuzzy control system.

2. Is it possible to analyze the stability of an aircraft in the sense of Lyapunov stability?

This question is extremely broad, but yes, it's possible.

3. Do you have any other ideas to analyze the dynamic stability of an FLC?

This is one of the pitfalls of a fuzzy control system - stability proofs are hard to come by. You often see engineers "bruteforce" their way to a sense of the systems stability limits by simulating the system response to a wide array of inputs and disturbances far beyond what the system is designed to handle.

 

[AIStudent]

Thanks for the answer!

I found another article that gives a theorem for stability analysis of FLC.
I came up with the following steps:
1. write the longitudinal (short period and phugoid) and lateral (duch roll and spiral) modes equations;
2. for each of the 4 sets of equations, use the variable gradient method to determine Lyapunov functions V(x);
3. prove that V(x) > 0 and V_dot(x) <= 0, for a given aircraft and flight condition and based on the rules in FLC;
4. use second article to state that the fuzzy logic control system (described by the article) is globally asymptotically stable in the origin/equilibrium point.

Am I on the right path?

Thanks!

 

[milesyoung]

I don't know enough about your system to verify your procedure (would probably also be a bit more work than I'm willing to put in), but I can say for certain that this:

3. prove that V(x) > 0 and V_dot(x) <= 0, for a given aircraft and flight condition and based on the rules in FLC;

is not enough to prove global or local asymptotical stability of the equilibrium at the origin, even for an autonomous system. I assume you got the idea from the first article you posted, which I skimmed, and their claim of stability on the basis of a negative semi-definite Lyapunov function derivative stands out as extremely dubious, at best.

The second article you posted makes more sense, as they further include LaSalle's invariance principle, but again - I skimmed it.

If you really want a good resource on nonlinear control, I can recommend 'Applied Nonlinear Control' by Slotine and Li.

Edit: Typo

 

[AIStudent]

Thanks for the reply!
The idea for the procedure came from both articles.
The first pointed to a resource 'Nonlinear Control Systems Analysis and Design' - Horacio J. Marquez where the variable gradient method is defined and how to determine the V(x) based on that gradient - step 2.
From the second, having V(X), I can find P > 0 and satisfy all the conditions of the theorem (from the design of the FLC and a specific aircraft and flight condition) - steps 3 & 4.

In any case, thanks for your feedback! I'll look into the resource you pointed out.