%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% For training Takagi-Sugeno fuzzy systems using recursive least squares%% By: Kevin Passino% Version: 3/12/99%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clearfigure(1)clf             % Clear the figure that will be used% First set the number of steps for the simulationNrls=301; % One more than the number of iterations you wanttime=0:Nrls-1; % For use in plotting (notice that time starts at zero in the plots but                         % and the index 1 corresponds to a time of zerolambda=1; % Use standard (nonweighted) least squares% Define the parameters for the phi function of the fuzzy systemR=20;c(1,:)=-5.4:0.6:6;sigma(1,:)=0.1*ones(1,R);% Next, pick the initial conditions (the index 1 corresponds to k=0)theta(:,1)=0*ones(2*R,1);% As another option you can run the batch least squares program and generate the% best guess at theta then use it to initialize the RLS in the following manner:%load variables theta40%theta(:,1)=theta40;  % If you use this then you will get good approximation accuracy                               % with very little adjustment from RLS% Add some noise to the initialization so that it is not as good%for i=1:2*R%	theta(i,1)=theta(i,1)+0.25*theta(i,1);%endalpha=100;P(:,:,1)=alpha*eye(2*R);% Note that we do not need to initialize K% Next, define the error that results from the initial choice of parameters:x(1)=6*(-1+2*rand);  % Input data uniformly distributed on (-6,6)z(1)=0.15*(rand-0.5)*2;  % Define the auxiliary variableG(1)=exp(-50*(x(1)-1)^2)-0.5*exp(-100*(x(1)-1.2)^2)+atan(2*x(1))+2.15+...0.2*exp(-10*(x(1)+1)^2)-0.25*exp(-20*(x(1)+1.5)^2)+0.1*exp(-10*(x(1)+2)^2)-0.2*exp(-10*(x(1)+3)^2);if x(1) >= 0	G(1)=G(1)+0.1*(x(1)-2)^2-0.4;endy(1)=G(1)+z(1); % Adds in the influence of the auxiliary variable% Next, compute the estimator outputfor j=1:R	mu(j,1)=exp(-0.5*((x(1)-c(1,j))/sigma(1,j))^2);end	denominator(1)=sum(mu(:,1)); 	for j=1:R	xi(j,1)=mu(j,1)/denominator(1);endphi(:,1)=[xi(:,1)', x(1)*xi(:,1)']';% The estimator output is:yhat(1)=theta(:,1)'*phi(:,1); epsilon(1,1)=y(1)-yhat(1); % Define the estimation error % Next, start the estimator	for k=2:Nrls		% First, generate data from the "unknown" function	x(k)=6*(-1+2*rand);  % Input data uniformly distributed on (-6,6)	z(k)=0.15*(rand-0.5)*2;  % Define the auxiliary variable	G(k)=exp(-50*(x(k)-1)^2)-0.5*exp(-100*(x(k)-1.2)^2)+atan(2*x(k))+2.15+...	0.2*exp(-10*(x(k)+1)^2)-0.25*exp(-20*(x(k)+1.5)^2)+0.1*exp(-10*(x(k)+2)^2)-0.2*exp(-10*(x(k)+3)^2);	if x(k) >= 0		G(k)=G(k)+0.1*(x(k)-2)^2-0.4;	end	y(k)=G(k)+z(k); % Adds in the influence of the auxiliary variable	% Compute the phi vector	for j=1:R		mu(j,k)=exp(-0.5*((x(k)-c(1,j))/sigma(1,j))^2);	end		denominator(k)=sum(mu(:,k)); 		for j=1:R		xi(j,k)=mu(j,k)/denominator(k);	end	phi(:,k)=[xi(:,k)', x(k)*xi(:,k)']';		% Next, compute the RLS update			K(:,k)=P(:,:,k-1)*phi(:,k)/(lambda+phi(:,k)'*P(:,:,k-1)*phi(:,k));	theta(:,k)=theta(:,k-1)+K(:,k)*(y(k)-phi(:,k)'*theta(:,k-1));	P(:,:,k)=(1/lambda)*(eye(size(P(:,:,k-1)))-K(:,k)*phi(:,k)')*P(:,:,k-1);		yhat(k)=theta(:,k)'*phi(:,k);  % The current guess of the estimator	epsilon(k,1)=y(k)-yhat(k);  % Compute the estimation error  (for plotting if you want)		if k <=11  % For the first 10 iterations plot the approximator mapping	% Next, compute the estimator mapping and plot it on the dataxt=-6:0.05:6;for i=1:length(xt),	for j=1:R		mut(j,i)=exp(-0.5*((xt(i)-c(1,j))/sigma(1,j))^2);	end		denominatort(i)=sum(mut(:,i)); 		for j=1:R		xit(j,i)=mut(j,i)/denominatort(i);	end	phit(:,i)=[xit(:,i)', xt(i)*xit(:,i)']';	Fts(i)=theta(:,k)'*phit(:,i);end% Plot the estimator mapping after having k-1 pieces of training datafigure(1)subplot(5,2,k-1)plot(x,y,'ko',xt,Fts,'k')xlabel('x')T=num2str(k-1);T=strcat('y, k=',T);ylabel(T)gridaxis([-6 6 0 4.8])hold onend  % Ends the test to see if 10 data pairs have been usedend% Next, plot the output of the last approximator obtained and the unknown function% First, compute the approximator for i=1:length(xt),	for j=1:R		mut(j,i)=exp(-0.5*((xt(i)-c(1,j))/sigma(1,j))^2);	end		denominatort(i)=sum(mut(:,i)); 		for j=1:R		xit(j,i)=mut(j,i)/denominatort(i);	end	phit(:,i)=[xit(:,i)', xt(i)*xit(:,i)']';	Fts(i)=theta(:,k)'*phit(:,i);end% Next, plotfigure(2)plot(x,y,'ko',xt,Fts,'k')xlabel('x')T=num2str(k-1);T=strcat('y, k=',T);ylabel(T)title('Takagi-Sugeno fuzzy system trained with RLS')gridaxis([-6 6 0 4.8])%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% End of program%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%