Nonlinear System Identification

This example shows how to use anfis command for nonlinear dynamical system identification.

This example requires System Identification Toolbox™, as a comparison is made between a nonlinear ANFIS and a linear ARX model.

Problem Setup

Exit if System Identification Toolbox is not available.

if ~fuzzychecktoolboxinstalled('ident')
    errordlg('DRYDEMO needs 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

Here are the results of the test.

load drydemodata
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 = char('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)');
index = 1:100;
subplot(2,1,1);
plot(index, y2(index), '-', index, y2(index), 'o');
ylabel('y(k)','fontsize',10);
subplot(2,1,2);
plot(index, u2(index), '-', index, u2(index), 'o');
ylabel('u(k)','fontsize',10);

The data points were 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.

ARX Model Identification

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 shows 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.Ts = 0.08;
u = z(:, 2);
y = z(:, 1)+ave;
yp = sim(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)],'fontsize',10);
disp(['[na nb d] = ' num2str(nn)]);
xlabel(xlbl,'fontsize',10);

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)],'fontsize',10);
xlabel(xlbl,'fontsize',10);
[na nb d] = 5  10   2

ANFIS Model Identification

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 appears 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 with a fuzzy inference system.

To use ANFIS for system identification, the first thing we need to do is select the input. 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, :);
[~, elapsed_time] = seqsrch(3, trn_data, chk_data, input_name); % #ok<*ASGLU>
fprintf('\nElapsed time = %f\n', elapsed_time);
winH1 = gcf;
Selecting input 1 ...
ANFIS model 1: y(k-1) --> trn=0.2043, chk=0.1888
ANFIS model 2: y(k-2) --> trn=0.3819, chk=0.3541
ANFIS model 3: y(k-3) --> trn=0.5245, chk=0.4903
ANFIS model 4: y(k-4) --> trn=0.6308, chk=0.5977
ANFIS model 5: u(k-1) --> trn=0.8271, chk=0.8434
ANFIS model 6: u(k-2) --> trn=0.7976, chk=0.8087
ANFIS model 7: u(k-3) --> trn=0.7266, chk=0.7349
ANFIS model 8: u(k-4) --> trn=0.6215, chk=0.6346
ANFIS model 9: u(k-5) --> trn=0.5419, chk=0.5650
ANFIS model 10: u(k-6) --> trn=0.5304, chk=0.5601
Currently selected inputs: y(k-1)

Selecting input 2 ...
ANFIS model 11: y(k-1) y(k-2) --> trn=0.1085, chk=0.1024
ANFIS model 12: y(k-1) y(k-3) --> trn=0.1339, chk=0.1283
ANFIS model 13: y(k-1) y(k-4) --> trn=0.1542, chk=0.1461
ANFIS model 14: y(k-1) u(k-1) --> trn=0.1892, chk=0.1734
ANFIS model 15: y(k-1) u(k-2) --> trn=0.1663, chk=0.1574
ANFIS model 16: y(k-1) u(k-3) --> trn=0.1082, chk=0.1077
ANFIS model 17: y(k-1) u(k-4) --> trn=0.0925, chk=0.0948
ANFIS model 18: y(k-1) u(k-5) --> trn=0.1533, chk=0.1531
ANFIS model 19: y(k-1) u(k-6) --> trn=0.1952, chk=0.1853
Currently selected inputs: y(k-1) u(k-4)

Selecting input 3 ...
ANFIS model 20: y(k-1) u(k-4) y(k-2) --> trn=0.0808, chk=0.0822
ANFIS model 21: y(k-1) u(k-4) y(k-3) --> trn=0.0806, chk=0.0836
ANFIS model 22: y(k-1) u(k-4) y(k-4) --> trn=0.0817, chk=0.0855
ANFIS model 23: y(k-1) u(k-4) u(k-1) --> trn=0.0886, chk=0.0912
ANFIS model 24: y(k-1) u(k-4) u(k-2) --> trn=0.0835, chk=0.0843
ANFIS model 25: y(k-1) u(k-4) u(k-3) --> trn=0.0609, chk=0.0604
ANFIS model 26: y(k-1) u(k-4) u(k-5) --> trn=0.0848, chk=0.0867
ANFIS model 27: y(k-1) u(k-4) u(k-6) --> trn=0.0890, chk=0.0894
Currently selected inputs: y(k-1) u(k-3) u(k-4)

Elapsed time = 0.279812

For input selection, another more computationally 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);
            [~, t_err, ~, ~, 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
[~, 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]);
h_gca = gca;
h_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','fontsize',10);
ylabel('RMSE','fontsize',10);
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=0.0962
ANFIS model = 2: y(k-1) y(k-2) u(k-2) --> trn=0.0852, chk=0.0862
ANFIS model = 3: y(k-1) y(k-2) u(k-3) --> trn=0.0474, chk=0.0485
ANFIS model = 4: y(k-1) y(k-2) u(k-4) --> trn=0.0808, chk=0.0822
ANFIS model = 5: y(k-1) y(k-2) u(k-5) --> trn=0.1023, chk=0.0991
ANFIS model = 6: y(k-1) y(k-2) u(k-6) --> trn=0.1021, chk=0.0974
ANFIS model = 7: y(k-1) y(k-3) u(k-1) --> trn=0.1231, chk=0.1206
ANFIS model = 8: y(k-1) y(k-3) u(k-2) --> trn=0.1047, chk=0.1085
ANFIS model = 9: y(k-1) y(k-3) u(k-3) --> trn=0.0587, chk=0.0626
ANFIS model = 10: y(k-1) y(k-3) u(k-4) --> trn=0.0806, chk=0.0836
ANFIS model = 11: y(k-1) y(k-3) u(k-5) --> trn=0.1261, chk=0.1311
ANFIS model = 12: y(k-1) y(k-3) u(k-6) --> trn=0.1210, chk=0.1151
ANFIS model = 13: y(k-1) y(k-4) u(k-1) --> trn=0.1420, chk=0.1353
ANFIS model = 14: y(k-1) y(k-4) u(k-2) --> trn=0.1224, chk=0.1229
ANFIS model = 15: y(k-1) y(k-4) u(k-3) --> trn=0.0700, chk=0.0765
ANFIS model = 16: y(k-1) y(k-4) u(k-4) --> trn=0.0817, chk=0.0855
ANFIS model = 17: y(k-1) y(k-4) u(k-5) --> trn=0.1337, chk=0.1405
ANFIS model = 18: y(k-1) y(k-4) u(k-6) --> trn=0.1421, chk=0.1333
ANFIS model = 19: y(k-2) y(k-3) u(k-1) --> trn=0.2393, chk=0.2264
ANFIS model = 20: y(k-2) y(k-3) u(k-2) --> trn=0.2104, chk=0.2077
ANFIS model = 21: y(k-2) y(k-3) u(k-3) --> trn=0.1452, chk=0.1497
ANFIS model = 22: y(k-2) y(k-3) u(k-4) --> trn=0.0958, chk=0.1047
ANFIS model = 23: y(k-2) y(k-3) u(k-5) --> trn=0.2048, chk=0.2135
ANFIS model = 24: y(k-2) y(k-3) u(k-6) --> trn=0.2388, chk=0.2326
ANFIS model = 25: y(k-2) y(k-4) u(k-1) --> trn=0.2756, chk=0.2574
ANFIS model = 26: y(k-2) y(k-4) u(k-2) --> trn=0.2455, chk=0.2400
ANFIS model = 27: y(k-2) y(k-4) u(k-3) --> trn=0.1726, chk=0.1797
ANFIS model = 28: y(k-2) y(k-4) u(k-4) --> trn=0.1074, chk=0.1157
ANFIS model = 29: y(k-2) y(k-4) u(k-5) --> trn=0.2061, chk=0.2133
ANFIS model = 30: y(k-2) y(k-4) u(k-6) --> trn=0.2737, chk=0.2836
ANFIS model = 31: y(k-3) y(k-4) u(k-1) --> trn=0.3842, chk=0.3605
ANFIS model = 32: y(k-3) y(k-4) u(k-2) --> trn=0.3561, chk=0.3358
ANFIS model = 33: y(k-3) y(k-4) u(k-3) --> trn=0.2719, chk=0.2714
ANFIS model = 34: y(k-3) y(k-4) u(k-4) --> trn=0.1763, chk=0.1808
ANFIS model = 35: y(k-3) y(k-4) u(k-5) --> trn=0.2132, chk=0.2240
ANFIS model = 36: y(k-3) y(k-4) u(k-6) --> trn=0.3460, chk=0.3601

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

if ishghandle(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)],'fontsize',10);
disp(['[na nb d] = ' num2str(nn)]);
xlabel('Time Steps','fontsize',10);
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)],'fontsize',10);
xlabel('Time Steps','fontsize',10);
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: 8


Start training ANFIS ...

   1 	 0.0474113 	 0.0485325

Designated epoch number reached --> ANFIS training completed at epoch 1.

[na nb d] = 5  10   2

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)],'fontsize',10);
xlabel('Time Index','fontsize',10);
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)],'fontsize',10);
xlabel('Time Index','fontsize',10);
ylabel('');

Conclusion

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

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