How to tune the PID parameters using Fuzzy Logic?

Question

I previously used the Ziegler method to tune the parameters of my PID controller to control my robot's position. I then implemented fuzzy logic for self-tuning the parameters. I have two inputs to the fuzzy logic controller; one is the position error and the error rate.

I know that my problem might be due to not understanding the effect of each parameter very well.

The problem is that I am confused in setting up the fuzzy rules. When do I need to use high and low values for Kp, Kd and Ki to achieve the best tuning? Is it that Kp must be very low when the error is almost zero (hence, the robot is at the desired position)? The same question applies for all of the three parameters.

Answer 1

The paper Controlling of Quadrotor UAV Using a Fuzzy System for Tuning the PID Gains in Hovering Mode by E. Abbasi, M. J. Mahjoob explains how to tune PID gains with fuzzy logic. You can find many papers about singleton tuning but this paper shows totally fuzzy control

1. find PID gains with ziegler-nichols (or another technique)
2. Create a fuzzy PID gain changer which has inputs error (e) and change in error(de)

3. Define fuzzification graphs for inputs and outputs. Define limits (also you can change the shape) like

   name [min,peak,max]

   very small [-1,-1,-0.6], small [-1,-0.6,0], medium [-0.6,0,0.6], big [0,0.6,1], very big [0.6,1,1]

4. create rules like

   if **e** and/or **de** *fuzzyname* (small,big etc.) than KI is fuzzyname (small,big etc.)

5. Defuzzyfy the result.

You can use tools like matlab fuzzy toolbox or python skfuzzy

The tipping problem can be used as Fuzzy-PID just change qualtiy as e and service as de and lastly you can change tip output as KP/ KI/ KD (there is example about tipping problem: python scikit fuzzy - Fuzzy Control Systems: The Tipping Problem)

Note 1: Error ranges should be well defined so you must log the error and change in error. The limits must be in max and min values of these values

Note 2: The output value range is good between -1 and 1.

An example code for Fuzzy-PID in python is here:

# -*- coding: utf-8 -*-
"""
@author: acs
"""

import skfuzzy as fuzz
from skfuzzy import control as ctrl
import acspid
import numpy as np
from matplotlib import pyplot as plt

plt.ion()
fig=plt.figure()

ferr = ctrl.Antecedent(np.arange(-150, 150, 1), 'ferr')
fder = ctrl.Antecedent(np.arange(-150, 150, 1), 'fder')
fout = ctrl.Consequent(np.arange(-1, 1, 0.01), 'fout')

ferr.automf(5)
fder.automf(5)
fout.automf(5)
fout['poor'] = fuzz.trimf(fout.universe, [-1, -1, -0.5])
fout['mediocre'] = fuzz.trimf(fout.universe, [-1, -0.5, 0])
fout['average'] = fuzz.trimf(fout.universe, [-0.1, 0, 0.1])
fout['decent'] = fuzz.trimf(fout.universe, [0, 0.5, 2])
fout['good'] = fuzz.trimf(fout.universe, [0.5, 1, 1])
fout.view()
ferr.view()
fder.view()
plt.show()
plt.pause(0.0001)

#'poor'; 'mediocre'; 'average'; 'decent', or 'good'
rules=[]
rules.append(ctrl.Rule(ferr['average'] | fder['average'] , fout['average']))
rules.append(ctrl.Rule(ferr['decent'] | fder['decent'] , fout['decent']))
rules.append(ctrl.Rule(ferr['good'] | fder['good'] , fout['good']))
rules.append(ctrl.Rule(ferr['mediocre'] | fder['mediocre'] , fout['mediocre']))
rules.append(ctrl.Rule(ferr['poor'] | fder['poor'] , fout['poor']))

fctrl = ctrl.ControlSystem(rules)
fpid = ctrl.ControlSystemSimulation(fctrl)

pid=acspid.pidcont(1.2,0.02,0.01,5,-5)

pid2=acspid.pidcont(1.2,0.02,0.01,5,-5)

d=np.zeros(10)
for i in range(10):
    d=np.append(d,np.ones(10)*np.random.uniform(-100,100,1))

print len(d)
m=[]
m.append(0.0)
m2=[]
m2.append(0.0)
e=[]
de=[]
e2=[]
de2=[]

kp=pid.kp
kd=pid.kd
ki=pid.ki
for i in range(len(d)):
    pid.setDesired(d[i])
    print "e:",pid.error ,"\t de:", pid.ed
    fpid.input['ferr'] = pid.error
    fpid.input['fder'] = pid.ed
    fpid.compute()
    newpid=np.abs(fpid.output['fout'])
    print "PID:", newpid*pid.kp,"\t",newpid*pid.ki,"\t",newpid*pid.kd
    pid.setGains(newpid*kp,newpid*ki,newpid*kd)
    newm=pid.update(m[-1])
    newm=m[-1]+newm
    print i,m[-1],newm
    m.append(newm)
    e.append(pid.error)
    de.append(pid.ed)

    pid2.setDesired(d[i])
    newm2=pid2.update(m2[-1])
    newm2=m2[-1]+newm2
    m2.append(newm2)
    e2.append(pid2.error)
    de2.append(pid2.ed)

    ax1 =plt.subplot(2,1,1)
    ax1.set_xlim([0, len(d)])
    ax1.set_ylim([-200, 200])
    plt.grid()
    plt.plot(range(len(m)),m,linewidth=5.0)
    plt.plot(range(len(m2)),m2,linewidth=2.0)
    plt.plot(range(len(d)),d,'g--')

    plt.title('Status')
    ax2=plt.subplot(2,1,2)
    ax2.set_xlim([0, 50])
    ax2.set_ylim([-100, 100])
    plt.plot(range(len(e)),e,'r-',range(len(de)),de,'g-')
    plt.grid()
    plt.title('e and ed')
    #plt.draw()
    plt.show()
    plt.pause(0.0001)

Fuzzy input membership functions:

Fuzzy output Membership function:

Status: In the status plot dashed line is target value, red line is PID and green line is Fuzzy-PID

Here the acspid class

class pidcont():
    def __init__(self,P,I,D,pmax,pmin):
        self.kp=P
        self.kd=D
        self.ki=I
        self.pidmax=pmax
        self.pidmin=pmin
        self.desired=0.0
        self.error=0.0
        self.elast=0.0
        self.esum=0.0
        self.eder=0.0
    def update(self,current):
        self.error=self.desired-current
        self.eder=self.error-self.elast
        self.elast=self.error
        self.esum=self.esum+self.error
        if self.esum>self.pidmax:
            self.esum=self.pidmax
        elif self.esum<self.pidmin:
            self.esum=self.pidmin

        self.P=self.kp*self.error
        self.D=self.kd*self.eder
        self.I=self.ki*self.esum
        pid=self.P+self.I+self.D
        return pid
    def setDesired(self,d):
        self.desired=d
    def setGains(self,P,I,D):
        self.kp=P
        self.kd=D
        self.ki=I
    def setLimits(self,pmax,pmin):
        self.pidmax=pmax
        self.pidmin=pmin