Hi Elio Stefano Brizio Valdez

Your rules affect the values of the PID gains. Can you explain how you scientifically determine the rules in Fuzzy Associative Matrix? I asked this, because I think that determining the gains using the rules (53 = 125) is way much more tedious than tuning the only 3 control parameters (23 = 8) via increase/decrease method. Did you really test 125 outcomes to set the rules?

Hello Yew-Chung Chak

Well, I don't know if I misunderstood, but I believed that with 5 membership functions in each variable y will have 25 rules for each gain, getting a total of 75 rules. The way I tuned the rules was considering part of the criteria exposed in Jingwen, C. (2010). Re-winder Tension Control Based on Fuzzy Adaptive PID Algorithm, he explained:

1. When the absolute value of the error (e) is great, no matter how big is de/dt, you must take a big or low value of Kp, and Kd and Ki should be small.

2. When e*de/dt>0, Kp should be big, Kd must not be big as Kp, and Ki still be small.

3.When e*de/dt

Currently, with a linear model of the plant and the "pidtune" command in MATLAB I've found new PID gains (Kp=3.72, Ki=52.2 and Kd=0.008, they work better with r0=0.65 m), they don't provide a better response than (Kp=27, Ki=20 and Kd=0.01), but I'm trying to show that you can improve the control performance using tuned gains that you previously found with a conventional method, so I taking the normalization:

Kp=Gp*dKp+Kp0

Ki=Gi*dKi+Ki0

Kd=Gd*dKi+Ki0

From -1 to 1 with the membership functions Negative Big, Negative Small, Zero, Positive Small and Positive Big in dKp, dKi and dKd.

Notice that the linear and the non linear plant have almost the same behavior. For r0=0.44 m, Kp=3.75, Ki=35.6 and Kd=0.011, and for r0=1 m, Kp=3.71, Ki=79.9 and Kd=0.011. With the default operation fixed in r0=0.65 m, Gp=(3.75-3.71)=0.04, Gi=(79.9-35.6)=44.3 and Gd=(0.011-0.005)=0.006, or Gp=(3.75-3.71)/2=0.02, Gi=(79.9-35.6)/2=22.15 and Gd=(0.011-0.005)/2=0.003. I haven't programmed the rules for this type of control yet, so at the end I will find out which gains are more suitable.

Hi Elio Stefano Brizio Valdez

Ya I think you are correct on the number of rules. Nevertheless, having 75 rules is not easy to tune. Moreover, the guidelines in (Jingwen, 2010) may be not perfectly applicable to your case.

Since you already have the conventional PID control gain, it is relatively easier to design a Fuzzy Controller in the Sugeno-form. Technically, it can be Fuzzy PD+I controller, which means that the PD control parts are fuzzified and the I control part remains. This method shall at least guarantee that the Fuzzy PD+I performs as good as the designed PID controller.

Once the performance is guaranteed, then you can move to the next phase to improve the fuzzy controller performance by reshaping the input membership functions.

Hello Yew-Chung Chak

Thanks for the suggestion, I may apply this method in a future investigation, but unfortunately I can't change method in a drastic way since I'm currently in an advance step of my work and if I had to change it I have to explain it based in papers that apply the same method. I'm so sorry.

Currently, I decided to apply only 3 membership functions, as you suggested, in e, ec, dKp, dKi and dKd [Negative, Zero and Positive] so I could tune easily each gain. The method for normalize the output of the FLC that I'm using is:

Kp=Gp*dKp+Kp0

Ki=Gi*dKi+Ki0

Kd=Gd*dKi+Ki0

Where Kp0, Ki0 and Kd0 are tuned by an conventional method. I observed that even in a simple plant, the Fuzzy PID controller doesn't response well as you can see in the figure "Response". The way that rules has according to my experience about the effects of each individual PID gains, also each case is illustrated in the "Error and Fuzzy Control" (The graph is a generic error, but I will explain how it is reflected to the output):

According to my FAM, my question is: What is incorrect in the rules that I tuned? How can I fix it?

Notice that the output curve recovers better when it is below if the set value, but not the same when it is above (See "Responsex2").

Sorry for bother and thanks for your help. My best wishes.

Hi Elio Stefano Brizio Valdez

Switching to Sugeno-type FIS (with the stabilizing PID gains) is actually way easier than you think. If you are reluctant to do so, Mamdani-type shall work too. However, you need to be an expert of the system because only you understand how it works (on the dynamical level) and this knowledge must be properly translated to the fuzzy rules.

From the comparison of the responses, I guess the problem lies in the rules for [Z, Z]. You stated that "at the point (Z, Z) the output reached the set value, so Kp, Ki and Kd must be medium". From your rules, the output is Z. Does output Z mean medium? If the input MFs {Z} are wide and the output MF {Z} is centered around Zero (the number 0), then the fuzzy controller will produce a low value around this region.

You can check this in the control surface.

Hello Yew-Chung Chak

I think that I didn't explain it well, in the point (Z,Z) dKp, dKi, dKd are zero, that means that the final constants Kp, Ki and Kd will tend to de Kp0, Ki0 and Kd0 gains original tuned.

I understood that the purpose of the FLC that tunes de PID is to optimize the original PID in some points to give the optimal gains in order to get a better response.

So, in order to follow to suggestion but considering what I said before, should I still change the form of the MF's, or with that I have to reconsider my fuzzy rules?

Thanks for the attention.

Hi Elio Stefano Brizio Valdez

Can you plot the MFs and show them? If you don't want to switch to Sugeno-type, then the first thing is to modify the MFs until a desired result is achieved around the equilibrium.