How can I exactly identify the range of outputs of FuzzyPID-controller?

I always can't exactly find out the range of the outputs(kp,ki,kd).Does there is any way can I identify the range when I layout FuzzyPID-controller?

[Sam Chak]

Hi @乐乐

In the image, you designed the Mamdani fuzzy system to map the input spaces of E and EC to an output space of Kp​, where the output range [0, 3] defines all possible output values of Kp​ that the fuzzy controller can produce from 0 to 3. In reality, the Mamdani fuzzy system can only generate output values in the open interval (0, 3), excluding exact 0 or exact 3 (see Kp​ Gain Surface).

To determine the fuzzy output range, you need to address several questions:

1. What is the desired Kp value when when both E and EC are negatively large?
2. What is the desired Kp value when when E is negatively large and EC is positively large?
3. What is the desired Kp value when when E is positively large and EC is negatively large?
4. What is the desired Kp value when when both E and EC are positively large?
5. What is the desired Kp value when when both E and EC are near zero?

These desired Kp​ values are not simply derived from arbitrary assumptions. You need to verify experimentally or through simulations that these Kp​ values are feasible. The true output range is determined by the minimum and maximum elements in the set of the five Kp​ values.

Kp1 = 0.5; Kp2 = 1.0; Kp3 = 1.5; Kp4 = 2.0; Kp5 = 2.5;
Kp  = [Kp1 Kp2 Kp3 Kp4 Kp5];
Kmin= min(Kp)

Kmin = 0.5000

Kmax= max(Kp)

Kmax = 2.5000

sd  = std(Kp);
output_range = [Kmin-sd Kmax+sd]    % for MamFIS only

output_range = 1×2

-0.2906 3.2906

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Additionally, you must apply common sense such that if a negative value of Kp​ can destabilize the system, then readjust the lower bound accordingly.

By the way, I am not certain how your actual Kp​ gain surface appears. However, if you allow the Fuzzy Designer App to automatically add the AND-based rules to your Mamdani FIS for all possible input combinations, you will obtain a nearly flat inclined plane for Kp gain only​, not . This design is typically not suitable for reference tracking. Why? Because it produces a very large control action when both E & EC are positively large, and a very small control action when both E & EC are negatively large.

Do you have the plant model to which you would like to apply the Fuzzy PID Controller?

%% Fuzzy Controller
fis = mamfis;

% settings
ud  = 1;
dis = 2/3;
c1  =-ud;
c2  = c1 + dis;
c3  = c2 + dis;
c4  = c3 + dis;
sig = (1/3)/sqrt(log(4));

% Fuzzy Input 1
fis = addInput(fis, [-ud +ud], 'Name', 'Err');
fis = addMF(fis, 'Err', 'gaussmf', [sig c1], 'Name', 'N2');
fis = addMF(fis, 'Err', 'gaussmf', [sig c2], 'Name', 'N1');
fis = addMF(fis, 'Err', 'gaussmf', [sig c3], 'Name', 'P1');
fis = addMF(fis, 'Err', 'gaussmf', [sig c4], 'Name', 'P2');

% Fuzzy Input 2
fis = addInput(fis, [-ud +ud], 'Name', 'dEr');
fis = addMF(fis, 'dEr', 'gaussmf', [sig c1], 'Name', 'N2');
fis = addMF(fis, 'dEr', 'gaussmf', [sig c2], 'Name', 'N1');
fis = addMF(fis, 'dEr', 'gaussmf', [sig c3], 'Name', 'P1');
fis = addMF(fis, 'dEr', 'gaussmf', [sig c4], 'Name', 'P2');

% Fuzzy Output
fis = addOutput(fis,  [0 3], 'Name', 'Kp');
fis = addMF(fis, 'Kp', 'linzmf', [0.0 0.5],     'Name', 'K1');
fis = addMF(fis, 'Kp', 'trimf',  [0.0 0.5 1.0], 'Name', 'K2');
fis = addMF(fis, 'Kp', 'trimf',  [0.5 1.0 1.5], 'Name', 'K3');
fis = addMF(fis, 'Kp', 'trimf',  [1.0 1.5 2.0], 'Name', 'K4');
fis = addMF(fis, 'Kp', 'trimf',  [1.5 2.0 2.5], 'Name', 'K5');
fis = addMF(fis, 'Kp', 'trimf',  [2.0 2.5 3.0], 'Name', 'K6');
fis = addMF(fis, 'Kp', 'linsmf', [2.5 3.0],     'Name', 'K7');

% Fuzzy Rules
rules = [
    "Err==N2 & dEr==N2 => Kp=K1"
    "Err==N2 & dEr==N1 => Kp=K2"
    "Err==N2 & dEr==P1 => Kp=K3"
    "Err==N2 & dEr==P2 => Kp=K4"
    
    "Err==N1 & dEr==N2 => Kp=K2"
    "Err==N1 & dEr==N1 => Kp=K3"
    "Err==N1 & dEr==P1 => Kp=K4"
    "Err==N1 & dEr==P2 => Kp=K5"
    
    "Err==P1 & dEr==N2 => Kp=K3"
    "Err==P1 & dEr==N1 => Kp=K4"
    "Err==P1 & dEr==P1 => Kp=K5"
    "Err==P1 & dEr==P2 => Kp=K6"
    
    "Err==P2 & dEr==N2 => Kp=K4"
    "Err==P2 & dEr==N1 => Kp=K5"
    "Err==P2 & dEr==P1 => Kp=K6"
    "Err==P2 & dEr==P2 => Kp=K7"
    ];
fis = addRule(fis, rules);

%% plot results
figure
subplot(311)
plotmf(fis, 'input', 1), grid on, 
xlabel('Error'), ylabel('\mu')
title('Input Fuzzy sets of Err')

subplot(312)
plotmf(fis, 'input', 2), grid on, 
xlabel('Change in Error'), ylabel('\mu')
title('Input Fuzzy sets of dEr')

subplot(313)
plotmf(fis, 'output', 1), grid on, 
xlabel('Kp value'), ylabel('\mu')
title('Output Fuzzy sets of Kp')

●

figure
opt = gensurfOptions('NumGridPoints', 51);
gensurf(fis, opt);
xlabel('Error'), ylabel('Change in Error'), zlabel('Kp')
title ('Kp Gain Surface')

[乐乐]

Hi,@Sam Chak

Thank you very very much for your reply!

I want to design a FuzzyPID-cotroller to track the step signal,but I find the quality of the FuzzyPID-controller is not good.The red line in last picture is the output of FuzzyPID-controller.

Whether can I utilize the Ziegler-Nichols method to determine the range of the outputs?

Answer 1

[Sam Chak]

Hi @乐乐

Hope that you're still around.

First, you must understand that a PID controller alone cannot freely manipulate the step response of a stable 3rd-order linear plant because the controller cannot influence the accelerative component or the second derivative of the plant. Nevertheless, it is still possible to slightly improve the step response through the nonlinear effects of the fuzzy PID controller. However, the performance will be limited by the coefficient of the accelerative component.

Instead of designing a fuzzy system that varies the entire space of the control gains, a more appropriate approach would be to design dimensionless control gain multipliers specifically for and ​ using fuzzy logic. The PID gains can be obtained from the autotuner. The designer can determine the desired control gain multipliers at specific operating points and then employ ANFIS to train the data and create the Sugeno fuzzy models.

The fuzzy PD+I control law is represented by this equation:

where ​​ and ​​ are the fuzzy models, and is the static multiplier for the ​ gain. A saturation function is necessary to limit the amplitude of the rate of change of the error signal, as this signal will spike initially for a step input.

Code for ANFIS to create the Sugeno fuzzy models:

%% Fuzzy Kp multiplier
% Designer's Choice: desired Kp multiplier at specific operating points, where x1 is the error
x1      = linspace(-1.5, 1.5, 15)';
yKp     = [0.6665; 0.7771; 0.9303; 1.1531; 1.4912; 2.0020; 2.6453; 3.0000; 2.6453; 2.0020; 1.4912; 1.1531; 0.9303; 0.7771; 0.6665];

% ANFIS settings for Fuzzy Kp multiplier
genOpt1 = genfisOptions('GridPartition');
genOpt1.NumMembershipFunctions       = 7;
genOpt1.InputMembershipFunctionType  = "gaussmf";
genOpt1.OutputMembershipFunctionType = "constant";
inFIS1  = genfis(x1, yKp, genOpt1);
opt1    = anfisOptions('InitialFIS', inFIS1);
opt1.DisplayANFISInformation = 0;
opt1.DisplayErrorValues      = 0;
opt1.DisplayStepSize         = 0;
opt1.DisplayFinalResults     = 0;
KpFIS   = anfis([x1 yKp], opt1);

Warning: Number of training data is smaller than number of modifiable parameters.

KpFIS.Outputs.MembershipFunctions

ans = 
  1x7 fismf array with properties:

    Type
    Parameters
    Name

  Details:
           Name          Type       Parameters
         _________    __________    __________

    1    "out1mf1"    "constant"     0.64485  
    2    "out1mf2"    "constant"     0.93056  
    3    "out1mf3"    "constant"      1.6308  
    4    "out1mf4"    "constant"      3.3073  
    5    "out1mf5"    "constant"      1.6308  
    6    "out1mf6"    "constant"     0.93056  
    7    "out1mf7"    "constant"     0.64485  

figure
tL1 = tiledlayout(2, 1, 'TileSpacing', 'Compact');
nexttile
plotmf(KpFIS, 'input', 1), grid on
nexttile
plot(x1, yKp, 'o'), hold on
X1  = linspace(-1.5, 1.5, 301)';
plot(X1, evalfis(KpFIS, X1)), grid on, 
xlabel('Error'), ylabel('Kp multiplier')
legend('Desired Kp multiplier', 'Kp ANFIS Output', 'location', 'south')
title(tL1, 'Fuzzy Kp multiplier')

%% Fuzzy Kd multiplier
% Designer's Choice: desired Kd multiplier at specific operating points, where x2 is the de/dt
x2      = linspace(-0.75, 0.75, 15)';
yKd     = [1.3333; 1.5552; 1.8646; 2.3218; 3.0427; 4.2240; 5.9281; 7.0000; 5.9281; 4.2240; 3.0427; 2.3218; 1.8646; 1.5552; 1.3333];

% ANFIS settings for Fuzzy Kd multiplier
genOpt2 = genfisOptions('GridPartition');
genOpt2.NumMembershipFunctions       = 7;
genOpt2.InputMembershipFunctionType  = "gaussmf";
genOpt2.OutputMembershipFunctionType = "constant";
inFIS2  = genfis(x2, yKd, genOpt2);
opt2    = anfisOptions('InitialFIS', inFIS2);
opt2.DisplayANFISInformation = 0;
opt2.DisplayErrorValues      = 0;
opt2.DisplayStepSize         = 0;
opt2.DisplayFinalResults     = 0;
KdFIS   = anfis([x2 yKd], opt2);

Warning: Number of training data is smaller than number of modifiable parameters.

KdFIS.Outputs.MembershipFunctions

ans = 
  1x7 fismf array with properties:

    Type
    Parameters
    Name

  Details:
           Name          Type       Parameters
         _________    __________    __________

    1    "out1mf1"    "constant"      1.2468  
    2    "out1mf2"    "constant"      1.9364  
    3    "out1mf3"    "constant"      3.1426  
    4    "out1mf4"    "constant"      7.9134  
    5    "out1mf5"    "constant"      3.1426  
    6    "out1mf6"    "constant"      1.9364  
    7    "out1mf7"    "constant"      1.2468  

figure
tL2 = tiledlayout(2, 1, 'TileSpacing', 'Compact');
nexttile
plotmf(KdFIS, 'input', 1), grid on
nexttile
plot(x2, yKd, 'o'), hold on
X2  = linspace(-0.75, 0.75, 151)';
plot(X2, evalfis(KdFIS, X2)), grid on, 
xlabel('Rate of Change of Error'), ylabel('Kd multiplier')
legend('Desired Kd multiplier', 'Kd ANFIS Output')
title(tL2, 'Fuzzy Kd multiplier')

●

%% Nominal PID control gains
Gp  = tf(1, [1 3 3 1]);
Gc  = pidtune(Gp, 'PIDF');
Kp  = Gc.Kp;
Ki  = Gc.Ki;
Kd  = Gc.Kd;
Tf  = Gc.Tf;

Control Block Diagram:

Step responses:

[乐乐]

Hi,@Sam Chak

Thank you for your patience.

I still want to use the Mamdani fuzzy system to track step signal,but the system is second order system this time.I have used the the Ziegler-Nichols method to determine the range of the outputs,but the quality(red line) still less than the classical PID controller.I think the bad quality relate to my setting of range of Kp ,Ki and Kd.

The Kp:[38.4 120]; Ki:[0.06 0.14]; Kd:[0.4 1.6].

There are any other ways can I determine the desired Kp ,Ki and Kd value when both E and EC are negatively large?

I am sorry that I don't have enough knowledge to achieve the task,thank you for your patience again.

[Sam Chak]

Hi @乐乐

Please answer a few questions below:

1. What is the operating setpoint?
2. What is the desired settling time?
3. What is the allowable maximum overshoot?
4. What are the PID gains found by Ziegler-Nichols method?

[乐乐]

Hi，@Sam Chak

1.I’m trying,so don't have setpoint.It could be 1.

2.I hope the settling time less than 2s.

3.If it can be done, I'd like to the maximum overshoo is zero.

4.I found when Critical stability，the gain Ku=200,and the period Tu=0.2,so I set maximum Kp=0.6Ku=120.

[Sam Chak]

Hi @乐乐

Here are my comments on your current design. One reason the performance of the fuzzy control (red curve) appears to be sluggish is the use of a prefilter, . In other words, it introduces unnecessary transient delay to the fuzzy controller, making it slower to respond. The three Absolute Value Function blocks, , are also unnecessary, as your fuzzy PID gains are always positive. Additionally, using two inputs for the fuzzy controller to vary each control gain may unnecessarily complicate the control design.

By the way, I first tested the performance of your classical PID controller, which you said to have tuned using the Ziegler–Nichols method. Your plant transfer function is 2nd-order, giving a stable closed-loop system. However, we can never reach the ultimate gain ​ because the in-tuning plant's output will not exhibit sustained oscillations with constant amplitude, regardless of how high the proportional gain is increased. Therefore, the Ziegler-Nichols method fails in your case.

I ran the simulation using MATLAB Online. However, the simulated step response under the classical PID controller is significantly different from the step response (yellow curve) shown in the Scope's image. I must have used the wrong PID gains. Please double-click the PID Controller block and provide a proper screenshot of the PID gains. Additionally, during the simulation stage, you should use the continuous-time PID controller, not the discrete-time PID controller.

%% plant tf
Gp  = tf(1, [0.2, 1, 0])

Gp =
 
       1
  -----------
  0.2 s^2 + s
 
Continuous-time transfer function.

Gp  = minreal(Gp);

%% classical PID Controller using the Ziegler–Nichols method
Ku  = 200;
Tu  = 0.2;
Kp  = 0.6*Ku;
Ki  = 1.2*Ku/Tu;
Kd  = 0.075*Ku*Tu;
Gc  = pid(Kp, Ki, Kd)

Gc =
 
             1          
  Kp + Ki * --- + Kd * s
             s          

  with Kp = 120, Ki = 1.2e+03, Kd = 3
 
Continuous-time PID controller in parallel form.

%% closed-loop tf
Gcl = feedback(Gc*Gp, 1)

Gcl =
 
     15 s^2 + 600 s + 6000
  ---------------------------
  s^3 + 20 s^2 + 600 s + 6000
 
Continuous-time transfer function.

step(Gcl), grid on

stepinfo(Gcl)

ans = struct with fields:
         RiseTime: 0.0423
    TransientTime: 0.8682
     SettlingTime: 0.8682
      SettlingMin: 0.6562
      SettlingMax: 1.5590
        Overshoot: 55.8974
       Undershoot: 0
             Peak: 1.5590
         PeakTime: 0.1160

[乐乐]

OK，thank you for your patience again.@Sam Chak

The quality of step response when I use the Ziegler–Nichols method is not good first,so I have already made some adjustments of the value of P、I、D. As the picture show,P=12.5;I=0.1;D=1.

If I don't use a prefilter,there are many warning will appear.And the quality of step response is bad.

You have helped me so much and if you agree, I will express my gratitude to you in my undergraduate dissertation.Thank you again.

[Sam Chak]

Hi @乐乐

Thank you for your update. Could you please post this additional problem as a new question? I now have a better understanding of the control problem for the 2nd-order system and have formulated an idea for designing the Mamdani fuzzy controller. However, this method differs from the ANFIS-based control design mentioned in my Answer above.

[乐乐]

Hi,@Sam Chak

No problem!I'd love to.

The address:https://ww2.mathworks.cn/matlabcentral/answers/2176027-how-can-i-utilize-the-mamdani-fuzzy-controller-to-get-better-quality-of-step-response-for-2nd-order