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A reviewer of the last paper I sent replied me that the it is very dangerous to update a PID with next kind of formula (paper is about quadrotor control):

$$ K_p (t + 1) = K_p (t)+e(t) (μ_1 (Pe(t)) + μ_4 (Pe(t))) $$

$Pe(t)$ is the % relationship between the desired angles and the real angles, and $e(t)$ is the difference between those angles. $μ_1$ and $μ_4$ are the membership functions of a fuzzy function. I think that the reviewer is talking about the time increment update rather than the fuzzy usage and specific formula.

How can stability of this formula be tested, please?

EDIT:

membership functions are represented in following graph: enter image description here

$e(t)$ is not the absolute difference between angles, just the difference. It can be negative

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  • $\begingroup$ A time step of 1 second is rather large and less general. The stability is mainly dangerous because you basically use forward Euler integration. What would be the update formula for a time step of $dt$? $\endgroup$ – fibonatic May 13 '16 at 15:54
  • $\begingroup$ My $ dt=0.1segs. $ I do use Euler integration for calculating displacement and velocity out of acceleration. How to proof the stability in that case? Is transfer function and Bode diagram or something similar necessary? $\endgroup$ – galtor May 13 '16 at 19:56
  • $\begingroup$ Can you give us more information? To analyze stability, we need to know more about $\mu_1$ and $\mu_4$. Also, if the system is discretized, then $dt$ might need to be in the expression for $K_p(t+1)$. $\endgroup$ – NBCKLY May 13 '16 at 19:56
  • $\begingroup$ Done it: triangular shape functions. I've seen Gaussian too, but this aspect think doesn't matter $\endgroup$ – galtor May 16 '16 at 17:38
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I think I may have reason for your reviewer's caution. You are updating the proportional gain of a PID controller, and you are concerned with the stability of your update law. Restating your update: $$K_p(t+1)=K_p+e(t)(\mu_1(Pe(t))+\mu_4(Pe(t))).$$ You have re-stated that $e(t)$ is the difference, which means that the error is not positive semi-definite ($e(t)\geq 0$). However, the membership functions $\mu_1$ and $\mu_4$ are positive semi-definite; this implies that $E(t)=e(t)(\mu_1(Pe(t))+\mu_4(Pe(t)))$ is either positive or negative semi-definite, which means that $K_p$ grows unbounded in some region $D$.

At this point, it is clear that the update law is not stable in some region (in the sense of Lyapunov) for $e(t)\neq 0$. We can substantiate this with discrete-time Lyapunov analysis.

$Proof:$ Dropping the time dependence for clarity, let $$V(K_p)=\frac{1}{2}K_p^2$$ be a candidate Lyapunov function. The rate of change along solutions is given by $$\begin{align}\nabla V(K_p)&=V(K_p+E)-V(K_p)\\&=\frac{1}{2}(K_p+E)^2-\frac{1}{2}K_p^2\\&=K_pE+\frac{1}{2}E^2\end{align}.$$ For stability of the system, we must have $\nabla V(K_p)<0$. This implies $$K_pE+\frac{1}{2}E^2<0$$$$\to K_p<-\frac{1}{2}E.$$ We can conclude from converse Lyapunov results that the system is unstable for at least $K_p>-\frac{1}{2}E$, which is an obvious reason for concern. There may be a better Lyapunov candidate function to demonstrate this result, but we can be sure that the system is locally unstable.

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  • $\begingroup$ ok, I wrongly used "absolute difference" expression. It was just to differentiate from $Pe$, which is % based difference. $ e(t) $ is just the difference, but it can take negative values. Does this change your arguments? $\endgroup$ – galtor May 16 '16 at 20:30
  • $\begingroup$ This absolutely changes the argument. I will edit my response, but you should include your edit in the question. $\endgroup$ – NBCKLY May 16 '16 at 20:45
  • $\begingroup$ @NBCKLY - Do you have time to make the edit? Your current answer is terrific for the absolute difference case; I'd love to see your response! $\endgroup$ – Chuck Jun 16 '16 at 14:05
  • $\begingroup$ I gave it a shot. The result was clear cut for the original post, but I think my response highlights at least one concern. The more I think about it, the more I am wondering what the general effect of controlling $K_p$ based on an error function will have on the stability of a system--I suspect that this is not a good practice, but if somebody can find a paper proposing this, I'm sure that OP would appreciate it. $\endgroup$ – NBCKLY Jun 16 '16 at 15:53
  • $\begingroup$ I posted original question.Thx, i'm not expert in lyapunov, but I see the problem. Finally, do you have any suggestion about alternative update? $\endgroup$ – galtor Jun 21 '16 at 21:16

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