www.gusucode.com > ident_featured 案例代码 matlab源码程序 > ident_featured/idnlgreydemo4.m
%% Narendra-Li Benchmark System: Nonlinear Grey Box Modeling of a Discrete-Time System
% This example shows how to identify the parameters of a complex yet
% artificial nonlinear discrete-time system with one input and one output.
% The system was originally proposed and discussed by Narendra and Li in
% the article
%
% K. S. Narendra and S.-M. Li. "Neural networks in control systems". In
% Mathematical Perspectives on Neural Networks (P. Smolensky,
% M. C. Mozer, and D. E. Rumelhard, eds.), Chapter 11, pages 347-394,
% Lawrence Erlbaum Associates, Hillsdale, NJ, USA, 1996.
%
% and has been considered in numerous discrete-time identification
% examples.
% Copyright 2005-2016 The MathWorks, Inc.
%% Discrete-Time Narendra-Li Benchmark System
% The discrete-time equations of the Narendra-Li system are:
%
% x1(t+Ts) = (x1(t)/(1+x1(t)^2)+p(1))*sin(x2(t))
% x2(t+Ts) = x2(t)*cos(x2(t)) + x1(t)*exp(-(x1(t)^2+x2(t)^2)/p(2))
% + u(t)^3/(1+u(t)^2+p(3)*cos(x1(t)+x2(t))
%
% y(t) = x1(t)/(1+p(4)*sin(x2(t))+p(5)*sin(x1(t)))
%
% where x1(t) and x2(t) are the states, u(t) the input signal, y(t) the
% output signal and p a parameter vector with 5 elements.
%% IDNLGREY Discrete-Time Narendra-Li Model
% From an IDNLGREY model file point of view, there is no difference between
% a continuous- and a discrete-time system. The Narendra-Li model file
% must therefore return the state update vector dx and the output y, and
% should take t (time), x (state vector), u (input), p (parameter(s)) and
% varargin as input arguments:
%
% function [dx, y] = narendrali_m(t, x, u, p, varargin)
% %NARENDRALI_M A discrete-time Narendra-Li benchmark system.
%
% % Output equation.
% y = x(1)/(1+p(4)*sin(x(2)))+x(2)/(1+p(5)*sin(x(1)));
%
% % State equations.
% dx = [(x(1)/(1+x(1)^2)+p(1))*sin(x(2)); ... % State 1.
% x(2)*cos(x(2))+x(1)*exp(-(x(1)^2+x(2)^2)/p(2)) ... % State 2.
% + u(1)^3/(1+u(1)^2+p(3)*cos(x(1)+x(2))) ...
% ];
%
% Notice that we have here chosen to condense all 5 parameters into one
% parameter vector, where the i:th element is referenced in the usual
% MATLAB way, i.e., through p(i).
%%
% With this model file as a basis, we next create an IDNLGREY object
% reflecting the modeling situation. Worth stressing here is that the
% discreteness of the model is specified through a positive value of Ts (=
% 1 second). Notice also that Parameters only holds one element, a 5-by-1
% parameter vector, and that SETPAR are used to specify that all the
% components of this vector are strictly positive.
FileName = 'narendrali_m'; % File describing the model structure.
Order = [1 1 2]; % Model orders [ny nu nx].
Parameters = {[1.05; 7.00; 0.52; 0.52; 0.48]}; % True initial parameters (a vector).
InitialStates = zeros(2, 1); % Initial initial states.
Ts = 1; % Time-discrete system with Ts = 1s.
nlgr = idnlgrey(FileName, Order, Parameters, InitialStates, Ts, 'Name', ...
'Discrete-time Narendra-Li benchmark system', ...
'TimeUnit', 's');
nlgr = setpar(nlgr, 'Minimum', {eps(0)*ones(5, 1)});
%%
% A summary of the entered Narendra-Li model structure is obtained through
% the PRESENT command:
present(nlgr);
%% Input-Output Data
% Two input-output data records with 300 samples each, one for estimation
% and one for validation purposes, are available. The input vector used for
% both these records is the same, and was chosen as a sum of two sinusoids:
%
% u(t) = sin(2*pi*t/10) + sin(2*pi*t/25)
%
% for t = 0, 1, ..., 299 seconds. We create two different IDDATA objects to
% hold the two data records, ze for estimation and zv for validation
% purposes.
load(fullfile(matlabroot, 'toolbox', 'ident', 'iddemos', 'data', 'narendralidata'));
ze = iddata(y1, u, Ts, 'Tstart', 0, 'Name', 'Narendra-Li estimation data', ...
'ExperimentName', 'ze');
zv = iddata(y2, u, Ts, 'Tstart', 0, 'Name', 'Narendra-Li validation data', ...
'ExperimentName', 'zv');
%%
% The input-output data that will be used for estimation are shown in a
% plot window.
figure('Name', ze.Name);
plot(ze);
%%
% *Figure 1:* Input-output data from a Narendra-Li benchmark system.
%% Performance of the Initial Narendra-Li Model
% Let us now use COMPARE to investigate the performance of the initial
% Narendra-Li model. The simulated and measured outputs for |ze| and |zv|
% are generated, and fit to |ze| is shown by default. Use the plot context
% menu to switch the data experiment to |zv|. The fit is not that bad for
% either dataset (around 74 %). Here we should point out that COMPARE by
% default estimates the initial state vector, in this case one initial
% state vector for |ze| and another one for |zv|.
compare(merge(ze, zv), nlgr);
%%
% *Figure 2:* Comparison between true outputs and the simulated outputs of
% the initial Narendra-Li model.
%%4
% By looking at the prediction errors obtained via PE, we realize that the
% initial Narendra-Li model shows some systematic and periodic differences
% as compared to the true outputs.
pe(merge(ze, zv), nlgr);
%%
% *Figure 3:* Prediction errors obtained with the initial Narendra-Li
% model. Use context menu to switch between experiments.
%% Parameter Estimation
% In order to reduce the systematic discrepancies we estimate all 5
% parameters of the Narendra-Li model structure using NLGREYEST. The
% estimation is carried out based on the estimation data set ze only.
opt = nlgreyestOptions('Display', 'on');
nlgr = nlgreyest(ze, nlgr, opt);
%% Performance of the Estimated Narendra-Li Model
% COMPARE is once again employed to investigate the performance of the
% estimated model, but this time we use the model's internally stored
% initial state vector ([0; 0] for both experiments) without any initial
% state vector estimation.
clf
opt = compareOptions('InitialCondition','m');
compare(merge(ze, zv), nlgr, opt);
%%
% *Figure 4:* Comparison between true outputs and the simulated outputs of
% the estimated Narendra-Li model.
%%
% The improvement after estimation is significant, which perhaps is best
% illustrated through a plot of the prediction errors:
figure;
pe(merge(ze, zv), nlgr);
%%
% *Figure 5:* Prediction errors obtained with the estimated Narendra-Li
% model.
%%
% The modeling power of the estimated Narendra-Li model is also confirmed
% through the correlation analysis provided through RESID:
figure('Name', [nlgr.Name ': residuals of estimated model']);
resid(zv, nlgr);
%%
% *Figure 6:* Residuals obtained with the estimated Narendra-Li model.
%%
% In fact, the obtained model parameters are also quite close to the ones
% that were used to generate the true model outputs:
disp(' True Estimated parameter vector');
ptrue = [1; 8; 0.5; 0.5; 0.5];
fprintf(' %1.4f %1.4f\n', [ptrue'; getpvec(nlgr)']);
%%
% Some additional model quality results (loss function, Akaike's FPE, and
% the estimated standard deviations of the model parameters) are next
% provided via the PRESENT command:
present(nlgr);
%%
% As for continuous-time input-output systems, it is also possible to use
% STEP for discrete-time input-output systems. Let us do this for two
% different step levels, 1 and 2:
figure('Name', [nlgr.Name ': step responses of estimated model']);
t = (-5:50)';
step(nlgr, 'b', t, stepDataOptions('StepAmplitude',1));
line(t, step(nlgr, t, stepDataOptions('StepAmplitude',2)), 'color', 'g');
grid on;
legend('0 -> 1', '0 -> 2', 'Location', 'NorthWest');
%%
% *Figure 7:* Step responses obtained with the estimated Narendra-Li model.
%%
% We conclude this example by comparing the performance of the
% estimated IDNLGREY Narendra-Li model with some basic linear models. The
% fit obtained for the latter models are considerably lower than what is
% returned for the estimated IDNLGREY model.
nk = delayest(ze);
arx22 = arx(ze, [2 2 nk]); % Second order linear ARX model.
arx33 = arx(ze, [3 3 nk]); % Third order linear ARX model.
arx44 = arx(ze, [4 4 nk]); % Third order linear ARX model.
oe22 = oe(ze, [2 2 nk]); % Second order linear OE model.
oe33 = oe(ze, [3 3 nk]); % Third order linear OE model.
oe44 = oe(ze, [4 4 nk]); % Fourth order linear OE model.
clf
fig = gcf;
Pos = fig.Position; fig.Position = [Pos(1:2)*0.7, Pos(3)*1.3, Pos(4)*1.3];
compare(zv, nlgr, 'b', arx22, 'm-', arx33, 'm:', arx44, 'm--', ...
oe22, 'g-', oe33, 'g:', oe44, 'g--');
L = findobj(gcf,'type','legend');
L.Location = 'best';
%%
% *Figure 8:* Comparison between true output and the simulated outputs of
% a number of estimated Narendra-Li models.
%% Conclusions
% In this example we have used a fictive discrete-time Narendra-Li
% benchmark system to illustrate the basics for performing discrete-time
% IDNLGREY modeling.