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<span style="color:blue">end</span>h = findobj(gcf, <span style="color:#B20000">'type'</span>, <span style="color:#B20000">'text'</span>);set(h, <span style="color:#B20000">'rot'</span>, 90, <span style="color:#B20000">'fontsize'</span>, 11, <span style="color:#B20000">'hori'</span>, <span style="color:#B20000">'right'</span>);drawnow<span style="color:green">% ====== Generate input_index for bjtrain.m</span>[a b] = min(trn_error);input_index = index(b,:);title(<span style="color:#B20000">'Training (Circles) and Checking (Asterisks) Errors'</span>);ylabel(<span style="color:#B20000">'RMSE'</span>);</pre><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:gray; font-style:italic;">Train 36 ANFIS models, each with 3 inputs selected from 10 candidates...ANFIS model = 1: y(k-1) y(k-2) u(k-1) --> trn=0.0990, chk=0.0962ANFIS model = 2: y(k-1) y(k-2) u(k-2) --> trn=0.0852, chk=0.0862ANFIS model = 3: y(k-1) y(k-2) u(k-3) --> trn=0.0474, chk=0.0485ANFIS model = 4: y(k-1) y(k-2) u(k-4) --> trn=0.0808, chk=0.0822ANFIS model = 5: y(k-1) y(k-2) u(k-5) --> trn=0.1023, chk=0.0991ANFIS model = 6: y(k-1) y(k-2) u(k-6) --> trn=0.1021, chk=0.0974ANFIS model = 7: y(k-1) y(k-3) u(k-1) --> trn=0.1231, chk=0.1206ANFIS model = 8: y(k-1) y(k-3) u(k-2) --> trn=0.1047, chk=0.1085ANFIS model = 9: y(k-1) y(k-3) u(k-3) --> trn=0.0587, chk=0.0626ANFIS model = 10: y(k-1) y(k-3) u(k-4) --> trn=0.0806, chk=0.0836ANFIS model = 11: y(k-1) y(k-3) u(k-5) --> trn=0.1261, chk=0.1311ANFIS model = 12: y(k-1) y(k-3) u(k-6) --> trn=0.1210, chk=0.1151ANFIS model = 13: y(k-1) y(k-4) u(k-1) --> trn=0.1420, chk=0.1353ANFIS model = 14: y(k-1) y(k-4) u(k-2) --> trn=0.1224, chk=0.1229ANFIS model = 15: y(k-1) y(k-4) u(k-3) --> trn=0.0700, chk=0.0765ANFIS model = 16: y(k-1) y(k-4) u(k-4) --> trn=0.0817, chk=0.0855ANFIS model = 17: y(k-1) y(k-4) u(k-5) --> trn=0.1337, chk=0.1405ANFIS model = 18: y(k-1) y(k-4) u(k-6) --> trn=0.1421, chk=0.1333ANFIS model = 19: y(k-2) y(k-3) u(k-1) --> trn=0.2393, chk=0.2264ANFIS model = 20: y(k-2) y(k-3) u(k-2) --> trn=0.2104, chk=0.2077ANFIS model = 21: y(k-2) y(k-3) u(k-3) --> trn=0.1452, chk=0.1497ANFIS model = 22: y(k-2) y(k-3) u(k-4) --> trn=0.0958, chk=0.1047ANFIS model = 23: y(k-2) y(k-3) u(k-5) --> trn=0.2048, chk=0.2135ANFIS model = 24: y(k-2) y(k-3) u(k-6) --> trn=0.2388, chk=0.2326ANFIS model = 25: y(k-2) y(k-4) u(k-1) --> trn=0.2756, chk=0.2574ANFIS model = 26: y(k-2) y(k-4) u(k-2) --> trn=0.2455, chk=0.2400ANFIS model = 27: y(k-2) y(k-4) u(k-3) --> trn=0.1726, chk=0.1797ANFIS model = 28: y(k-2) y(k-4) u(k-4) --> trn=0.1074, chk=0.1157ANFIS model = 29: y(k-2) y(k-4) u(k-5) --> trn=0.2061, chk=0.2133ANFIS model = 30: y(k-2) y(k-4) u(k-6) --> trn=0.2737, chk=0.2836ANFIS model = 31: y(k-3) y(k-4) u(k-1) --> trn=0.3842, chk=0.3605ANFIS model = 32: y(k-3) y(k-4) u(k-2) --> trn=0.3561, chk=0.3358ANFIS model = 33: y(k-3) y(k-4) u(k-3) --> trn=0.2719, chk=0.2714ANFIS model = 34: y(k-3) y(k-4) u(k-4) --> trn=0.1763, chk=0.1808ANFIS model = 35: y(k-3) y(k-4) u(k-5) --> trn=0.2132, chk=0.2240ANFIS model = 36: y(k-3) y(k-4) u(k-6) --> trn=0.3460, chk=0.3601</pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" src="drydemo_img08.gif"><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:#990000; font-weight:bold; font-size:medium; page-break-before: auto;"><a name=""></a></p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">The popped window shows ANFIS predictions on both training and checkingdata sets. Obviously the performance is better than those of the ARXmodel.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px"><span style="color:blue">if</span> ishandle(winH1), delete(winH1); <span style="color:blue">end</span>ss = 0.01;ss_dec_rate = 0.5;ss_inc_rate = 1.5;trn_data = data(1:trn_data_n, [input_index, size(data,2)]);chk_data = data(trn_data_n+1:600, [input_index, size(data,2)]);<span style="color:green">% generate FIS matrix</span>in_fismat = genfis1(trn_data);[trn_out_fismat trn_error step_size chk_out_fismat chk_error] = <span style="color:blue">...</span> anfis(trn_data, in_fismat, [1 nan ss ss_dec_rate ss_inc_rate], <span style="color:blue">...</span> nan, chk_data, 1);subplot(2,1,1);index = 1:trn_data_n;plot(index, y(index), index, yp(index), <span style="color:#B20000">'.'</span>);rmse = norm(y(index)-yp(index))/sqrt(length(index));title([<span style="color:#B20000">'(a) Training Data (Solid Line) and ARX Prediction (Dots) with RMSE = '</span> num2str(rmse)]);disp([<span style="color:#B20000">'[na nb d] = '</span> num2str(nn)]);xlabel(<span style="color:#B20000">'Time Steps'</span>);subplot(2,1,2);index = (trn_data_n+1):(total_data_n);plot(index, y(index), index, yp(index), <span style="color:#B20000">'.'</span>);rmse = norm(y(index)-yp(index))/sqrt(length(index));title([<span style="color:#B20000">'(b) Checking Data (Solid Line) and ARX Prediction (Dots) with RMSE = '</span> num2str(rmse)]);xlabel(<span style="color:#B20000">'Time Steps'</span>);</pre><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:gray; font-style:italic;">ANFIS info: Number of nodes: 34 Number of linear parameters: 32 Number of nonlinear parameters: 18 Total number of parameters: 50 Number of training data pairs: 300 Number of checking data pairs: 300 Number of fuzzy rules: 8Start training ANFIS ... 1 0.0474113 0.0485325Designated epoch number reached --> ANFIS training completed at epoch 1.[na nb d] = 5 10 2</pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" src="drydemo_img09.gif"><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:#990000; font-weight:bold; font-size:medium; page-break-before: auto;"><a name=""></a></p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px">y_hat = evalfis(data(1:600,input_index), chk_out_fismat);subplot(2,1,1);index = 1:trn_data_n;plot(index, data(index, size(data,2)), <span style="color:#B20000">'-'</span>, <span style="color:blue">...</span> index, y_hat(index), <span style="color:#B20000">'.'</span>);rmse = norm(y_hat(index)-data(index,size(data,2)))/sqrt(length(index));title([<span style="color:#B20000">'Training Data (Solid Line) and ANFIS Prediction (Dots) with RMSE = '</span> num2str(rmse)]);xlabel(<span style="color:#B20000">'Time Index'</span>); ylabel(<span style="color:#B20000">''</span>);subplot(2,1,2);index = trn_data_n+1:600;plot(index, data(index, size(data,2)), <span style="color:#B20000">'-'</span>, index, y_hat(index), <span style="color:#B20000">'.'</span>);rmse = norm(y_hat(index)-data(index,size(data,2)))/sqrt(length(index));title([<span style="color:#B20000">'Checking Data (Solid Line) and ANFIS Prediction (Dots) with RMSE = '</span> num2str(rmse)]);xlabel(<span style="color:#B20000">'Time Index'</span>); ylabel(<span style="color:#B20000">''</span>);</pre><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:gray; font-style:italic;">Warning: Some input values are outside of the specified input range.</pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" src="drydemo_img10.gif"><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="color:#990000; font-weight:bold; font-size:medium; page-break-before: auto;"><a name=""></a></p><p xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd">The above table is a comparison among various modeling approaches. TheARX modeling spends the least amount of time to reach the worseprecision, which the ANFIS modeling via exhaustive search takes thelargest amount of time to reach the best percision. In other words, iffast modeling is the goal, then ARX is the right choice. But ifprecision is the utmost concern, then we can go for ANFIS that isdesigned for nonlinear modeling and higher precision.</p><pre xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" style="position: relative; left:30px">subplot(1,1,1)a=imread(<span style="color:#B20000">'drytable.jpg'</span>, <span style="color:#B20000">'jpg'</span>);image(a); axis image;axis off;</pre><img xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" src="drydemo_img11.gif"><originalCode xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd" code="%% Nonlinear system identification
% This demo addresses the use of ANFIS function in the Fuzzy Logic
% Toolbox for nonlinear dynamical system identification. This demo also
% requires the System Identification Toolbox, as a comparison is made
% between a nonlinear ANFIS and a linear ARX model.
%
% Copyright 1994-2002 The MathWorks, Inc. 
% $Revision: 1.9 $

% Exit if the IDENT toolbox is not on the path
if exist('arx.m','file') == 0
 errordlg('DRYDEMO requires the System Identification Toolbox.');
 return;
end


%%
% The data set for ANFIS and ARX modeling was obtained from a laboratory
% device called Feedback's Process Trainer PT 326, as described in Chapter
% 17 of Prof. Lennart Ljung's book "System Identification, Theory for the
% User", Prentice-Hall, 1987. The device's function is like a hair dryer:
% air is fanned through a tube and heated at the inlet. The air temperature
% is measure by a thermocouple at the outlet. The input u(k) is the voltage
% over a mesh of resistor wires to heat incoming air; the output y(k) is
% the outlet air temperature. Here is a the system model

a=imread('dryblock.jpg', 'jpg');
image(a); 
axis image;
axis off;

%%
% Here are the results of the test.

load dryer2;
data_n = length(y2);
output = y2;
input = [[0; y2(1:data_n-1)] ...
 [0; 0; y2(1:data_n-2)] ...
 [0; 0; 0; y2(1:data_n-3)] ...
 [0; 0; 0; 0; y2(1:data_n-4)] ...
 [0; u2(1:data_n-1)] ...
 [0; 0; u2(1:data_n-2)] ...
 [0; 0; 0; u2(1:data_n-3)] ...
 [0; 0; 0; 0; u2(1:data_n-4)] ...
 [0; 0; 0; 0; 0; u2(1:data_n-5)] ...
 [0; 0; 0; 0; 0; 0; u2(1:data_n-6)]];
data = [input output];
data(1:6, :) = [];
input_name = str2mat('y(k-1)','y(k-2)','y(k-3)','y(k-4)','u(k-1)','u(k-2)','u(k-3)','u(k-4)','u(k-5)','u(k-6)');
trn_data_n = 300;
index = 1:100;
subplot(2,1,1);
plot(index, y2(index), '-', index, y2(index), 'o');
ylabel('y(k)');
subplot(2,1,2);
plot(index, u2(index), '-', index, u2(index), 'o');
ylabel('u(k)');

%%
% The data points was collected at a sampling time of 0.08 second. One
% thousand input-output data points were collected from the process as the
% input u(k) was chosen to be a binary random signal shifting between 3.41
% and 6.41 V. The probability of shifting the input at each sample was 0.2.
% The data set is available from the System Identification Toolbox; and the
% above plots show the output temperature y(k) and input voltage u(t) for
% the first 100 time steps.

%%
% A conventional method is to remove the means from the data and assume a
% linear model of the form:
%
% y(k)+a1*y(k-1)+...+am*y(k-m)=b1*u(k-d)+...+bn*u(k-d-n+1)
%
% where ai (i = 1 to m) and bj (j = 1 to n) are linear parameters to be
% determined by least-squares methods. This structure is called the ARX
% model and it is exactly specified by three integers [m, n, d]. To find an
% ARX model for the dryer device, the data set was divided into a training
% (k = 1 to 300) and a checking (k = 301 to 600) set. An exhaustive search
% was performed to find the best combination of [m, n, d], where each of
% the integer is allowed to changed from 1 to 10 independently. The best
% ARX model thus found is specified by [m, n, d] = [5, 10, 2], with a
% training RMSE of 0.1122 and a checking RMSE of 0.0749. The above figure
% demonstrates the fitting results of the best ARX model.

trn_data_n = 300;
total_data_n = 600;
z = [y2 u2];
z = dtrend(z);
ave = mean(y2);
ze = z(1:trn_data_n, :);
zv = z(trn_data_n+1:total_data_n, :);
T = 0.08;

% Run through all different models
V = arxstruc(ze, zv, struc(1:10, 1:10, 1:10));
% Find the best model
nn = selstruc(V, 0);
% Time domain plot
th = arx(ze, nn);
th = sett(th, 0.08);
u = z(:, 2);
y = z(:, 1)+ave;
yp = idsim(u, th)+ave;

xlbl = 'Time Steps';

subplot(2,1,1); 
index = 1:trn_data_n;
plot(index, y(index), index, yp(index), '.');
rmse = norm(y(index)-yp(index))/sqrt(length(index));
title(['(a) Training Data (Solid Line) and ARX Prediction (Dots) with RMSE = ' num2str(rmse)]);
disp(['[na nb d] = ' num2str(nn)]);
xlabel(xlbl);

subplot(2,1,2); 
index = (trn_data_n+1):(total_data_n);
plot(index, y(index), index, yp(index), '.');
rmse = norm(y(index)-yp(index))/sqrt(length(index));
title(['(b) Checking Data (Solid Line) and ARX Prediction (Dots) with RMSE = ' num2str(rmse)]);
xlabel(xlbl);

%%
% The ARX model is inherently linear and the most significant advantage is
% that we can perform model structure and parameter identification rapidly.
% The performance in the above plots appear to be satisfactory. However, if
% a better performance level is desired, we might want to resort to a
% nonlinear model. In particular, we are going to use a neuro-fuzzy
% modeling approach, ANFIS, to see if we can push the performance level by
% using a fuzzy inference system.

%%
% To use ANFIS for system identification, the first thing we need to do is
% input selection. That is, to determine which variables should be the
% input arguments to an ANFIS model. For simplicity, we suppose that there
% are 10 input candidates (y(k-1), y(k-2), y(k-3), y(k-4), u(k-1), u(k-2),
% u(k-3), u(k-4), u(k-5), u(k-6)), and the output to be predicted is y(k).
% A heuristic approach to input selection is called sequential forward
% search, in which each input is selected sequentially to optimize the
% total squared error. This can be done by the function seqsrch; the result
% is shown in the above plot, where 3 inputs (y(k-1), u(k-3), and u(k-4))
% are selected with a training RMSE of 0.0609 and checking RMSE of 0.0604.

trn_data_n = 300;
trn_data = data(1:trn_data_n, :);
chk_data = data(trn_data_n+1:trn_data_n+300, :);
[input_index, elapsed_time]=seqsrch(3, trn_data, chk_data, input_name);
fprintf('\nElapsed time = %f\n', elapsed_time);
winH1 = gcf;

%%
% For input selection, another more computation intensive approach is to do
% an exhaustive search on all possible combinations of the input
% candidates. The function that performs exhaustive search is exhsrch,
% which selects 3 inputs from 10 candidates. However, exhsrch usually
% involves a significant amount of computation if all combinations are
% tried. For instance, if 3 is selected out of 10, the total number of
% ANFIS models is C(10, 3) = 120.
%
% Fortunately, for dynamical system identification, we do know that the inputs should not come from either of the following two sets of input candidates exclusively:
%
% Y = {y(k-1), y(k-2), y(k-3), y(k-4)}
%
% U = {u(k-1), u(k-2), u(k-3), u(k-4), u(k-5), u(k-6)}
%
% A reasonable guess would be to take two inputs from Y and one from U to
% form the inputs to ANFIS; the total number of ANFIS models is then
% C(4,2)*6=36, which is much less. The above plot shows that the selected
% inputs are y(k-1), y(k-2) and u(k-3), with a training RMSE of 0.0474 and
% checking RMSE of 0.0485, which are better than ARX models and ANFIS via
% sequential forward search.

group1 = [1 2 3 4]; % y(k-1), y(k-2), y(k-3), y(k-4)
group2 = [1 2 3 4]; % y(k-1), y(k-2), y(k-3), y(k-4)
group3 = [5 6 7 8 9 10]; % u(k-1) through y(k-6)

anfis_n = 6*length(group3);
index = zeros(anfis_n, 3);
trn_error = zeros(anfis_n, 1);
chk_error = zeros(anfis_n, 1);
% ======= Training options 
mf_n = 2;
mf_type = 'gbellmf';
epoch_n = 1;
ss = 0.1;
ss_dec_rate = 0.5;
ss_inc_rate = 1.5;
% ====== Train ANFIS with different input variables
fprintf('\nTrain %d ANFIS models, each with 3 inputs selected from 10 candidates...\n\n',...
 anfis_n);
model = 1;
for i=1:length(group1),
 for j=i+1:length(group2),
 for k=1:length(group3),
 in1 = deblank(input_name(group1(i), :));
 in2 = deblank(input_name(group2(j), :));
 in3 = deblank(input_name(group3(k), :));
 index(model, :) = [group1(i) group2(j) group3(k)];
 trn_data = data(1:trn_data_n, [group1(i) group2(j) group3(k) size(data,2)]);
 chk_data = data(trn_data_n+1:trn_data_n+300, [group1(i) group2(j) group3(k) size(data,2)]);
 in_fismat = genfis1(trn_data, mf_n, mf_type);
 [trn_out_fismat t_err step_size chk_out_fismat c_err] = ...
 anfis(trn_data, in_fismat, ...
 [epoch_n nan ss ss_dec_rate ss_inc_rate], ...
 [0 0 0 0], chk_data, 1);
 trn_error(model) = min(t_err);
 chk_error(model) = min(c_err);
 fprintf('ANFIS model = %d: %s %s %s', model, in1, in2, in3);
 fprintf(' --> trn=%.4f,', trn_error(model));
 fprintf(' chk=%.4f', chk_error(model));
 fprintf('\n');
 model = model+1;
 end
 end
end

% ====== Reordering according to training error
[a b] = sort(trn_error);
b = flipud(b); % List according to decreasing trn error
trn_error = trn_error(b);
chk_error = chk_error(b);
index = index(b, :);

% ====== Display training and checking errors
x = (1:anfis_n)';
subplot(2,1,1);
plot(x, trn_error, '-', x, chk_error, '-', ...
 x, trn_error, 'o', x, chk_error, '*');
tmp = x(:, ones(1, 3))';
X = tmp(:);
tmp = [zeros(anfis_n, 1) max(trn_error, chk_error) nan*ones(anfis_n, 1)]';
Y = tmp(:);
hold on; 
plot(X, Y, 'g'); 
hold off;
axis([1 anfis_n -inf inf]);
set(gca, 'xticklabel', []);

% ====== Add text of input variables
for k = 1:anfis_n,
 text(x(k), 0, ...
 [input_name(index(k,1), :) ' ' ...
 input_name(index(k,2), :) ' ' ...
 input_name(index(k,3), :)]);
end
h = findobj(gcf, 'type', 'text');
set(h, 'rot', 90, 'fontsize', 11, 'hori', 'right');
drawnow

% ====== Generate input_index for bjtrain.m
[a b] = min(trn_error);
input_index = index(b,:);
title('Training (Circles) and Checking (Asterisks) Errors');
ylabel('RMSE');

%%
% The popped window shows ANFIS predictions on both training and checking
% data sets. Obviously the performance is better than those of the ARX
% model.

if ishandle(winH1), delete(winH1); end

ss = 0.01;
ss_dec_rate = 0.5;
ss_inc_rate = 1.5;

trn_data = data(1:trn_data_n, [input_index, size(data,2)]);
chk_data = data(trn_data_n+1:600, [input_index, size(data,2)]);

% generate FIS matrix
in_fismat = genfis1(trn_data);

[trn_out_fismat trn_error step_size chk_out_fismat chk_error] = ...
 anfis(trn_data, in_fismat, [1 nan ss ss_dec_rate ss_inc_rate], ...
 nan, chk_data, 1);

subplot(2,1,1);
index = 1:trn_data_n;
plot(index, y(index), index, yp(index), '.');
rmse = norm(y(index)-yp(index))/sqrt(length(index));
title(['(a) Training Data (Solid Line) and ARX Prediction (Dots) with RMSE = ' num2str(rmse)]);
disp(['[na nb d] = ' num2str(nn)]);
xlabel('Time Steps');
subplot(2,1,2);
index = (trn_data_n+1):(total_data_n);
plot(index, y(index), index, yp(index), '.');
rmse = norm(y(index)-yp(index))/sqrt(length(index));
title(['(b) Checking Data (Solid Line) and ARX Prediction (Dots) with RMSE = ' num2str(rmse)]);
xlabel('Time Steps');

%%
y_hat = evalfis(data(1:600,input_index), chk_out_fismat);

subplot(2,1,1);
index = 1:trn_data_n;
plot(index, data(index, size(data,2)), '-', ...
 index, y_hat(index), '.');
rmse = norm(y_hat(index)-data(index,size(data,2)))/sqrt(length(index));
title(['Training Data (Solid Line) and ANFIS Prediction (Dots) with RMSE = ' num2str(rmse)]);
xlabel('Time Index'); ylabel('');

subplot(2,1,2);
index = trn_data_n+1:600;
plot(index, data(index, size(data,2)), '-', index, y_hat(index), '.');
rmse = norm(y_hat(index)-data(index,size(data,2)))/sqrt(length(index));
title(['Checking Data (Solid Line) and ANFIS Prediction (Dots) with RMSE = ' num2str(rmse)]);
xlabel('Time Index'); ylabel('');

%%
% The above table is a comparison among various modeling approaches. The
% ARX modeling spends the least amount of time to reach the worse
% precision, which the ANFIS modeling via exhaustive search takes the
% largest amount of time to reach the best percision. In other words, if
% fast modeling is the goal, then ARX is the right choice. But if
% precision is the utmost concern, then we can go for ANFIS that is
% designed for nonlinear modeling and higher precision.

subplot(1,1,1)
a=imread('drytable.jpg', 'jpg');
image(a); 
axis image;
axis off;"></originalCode>⌨️ 快捷键说明
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