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+++ b/m-toolbox/classes/+utils/@math/ctfit.m	Wed Nov 23 19:22:13 2011 +0100
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+% CTFIT fits a continuous model to a frequency response.
+% 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% DESCRIPTION:
+%
+%     Fits a continuous model to a frequency response using relaxed
+%     s-domain vector fitting algorithm [1, 2]. Model function is expanded
+%     in partial fractions:
+%
+%              r1              rN
+%     f(s) = ------- + ... + ------- + d
+%            s - p1          s - pN
+%
+% CALL:
+%
+%     [res,poles,dterm,mresp,rdl] = ctfit(y,f,poles,weight,fitin)
+%
+% INPUTS:
+%
+%     - y: Is a vector wuth the frequency response data.
+%     - f: Is the frequency vector in Hz.
+%     - poles: are a set of starting poles.
+%     - weight: are a set of weights used in the fitting procedure.
+%     - fitin: is a struct containing fitting options and parameters. fitin
+%     fields are:
+%       - fitin.stable = 0; fit without forcing poles to be stable.
+%       - fitin.stable = 1; force poles to be stable flipping unstable
+%       poles in the left side of the complex plane. s -> s - 2*conj(s).
+%       - fitin.dterm = 0; fit with d = 0.
+%       - fitin.dterm = 1; fit with d different from 0.
+%       - fitin.polt = 0; fit without plotting results.
+%       - fitin.plot = 1; plot fit results.
+%
+% OUTPUT:
+%
+%     - res: vector or residues.
+%     - poles: vector of poles.
+%     - dterm: direct term d.
+%     - mresp: frequency response of the fitted model
+%     - rdl: residuals y - mresp
+%
+% REFERENCES:
+%
+%     [1] B. Gustavsen and A. Semlyen, "Rational approximation of frequency
+%         domain responses by Vector Fitting", IEEE Trans. Power Delivery
+%         vol. 14, no. 3, pp. 1052-1061, July 1999.
+%     [2] B. Gustavsen, "Improving the Pole Relocating Properties of Vector
+%         Fitting", IEEE Trans. Power Delivery vol. 21, no. 3, pp.
+%         1587-1592, July 2006.
+%
+% NOTE:
+%
+%     This function cannot handle more than one frequency response per time
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% VERSION: $Id: ctfit.m,v 1.2 2008/10/24 06:19:23 hewitson Exp $
+%
+% HISTORY:     12-09-2008 L Ferraioli
+%                 Creation
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+function [res,poles,dterm,mresp,rdl] = ctfit(y,f,poles,weight,fitin)
+  
+  %% Collecting inputs
+  
+  % Default input struct
+  defaultparams = struct('stable',0, 'dterm',0, 'plot',0);
+  
+  names = {'stable','dterm','plot'};
+  
+  % collecting input and default params
+  if ~isempty(fitin)
+    for jj=1:length(names)
+      if isfield(fitin, names(jj))
+        defaultparams.(names{1,jj}) = fitin.(names{1,jj});
+      end
+    end
+  end
+  
+  stab = defaultparams.stable; % Enforce pole stability is is 1
+  dt = defaultparams.dterm; % 1 to fit with direct term
+  plotting = defaultparams.plot; % set to 1 if plotting is required
+  
+  %% Inputs in column vectors
+  
+  [a,b] = size(y);
+  if a < b % shifting to column
+    y = y.';
+  end
+  
+  [a,b] = size(f);
+  if a < b % shifting to column
+    f = f.';
+  end
+  
+  [a,b] = size(poles);
+  if a < b % shifting to column
+    poles = poles.';
+  end
+  
+  clear w
+  w = weight;
+  [a,b] = size(w);
+  if a < b % shifting to column
+    w = w.';
+  end
+  
+  N = length(poles); % Model order
+  
+  if dt
+    dl = 1; % Fit with direct term
+  else
+    dl = 0; % Fit without direct term
+  end
+  
+  % definition of s
+  s = 1i.*2.*pi.*f;
+  
+  Nz = length(s);
+  
+  %% Marking complex and real poles
+  
+  % cindex = 1; pole is complex, next conjugate pole is marked with cindex
+  % = 2. cindex = 0; pole is real
+  cindex=zeros(N,1);
+  for m=1:N
+    if imag(poles(m))~=0
+      if m==1
+        cindex(m)=1;
+      else
+        if cindex(m-1)==0 || cindex(m-1)==2
+          cindex(m)=1; cindex(m+1)=2;
+        else
+          cindex(m)=2;
+        end
+      end
+    end
+  end
+  
+  %% Initializing the augmented problem matrices
+  
+  
+  
+  % Matrix initialinzation
+  BA = zeros(Nz+1,1);
+  AA = zeros(Nz+1,2*N+dl+1);
+  Ak=zeros(Nz,N+1);
+  
+  % Defining Ak
+  for jj = 1:N
+    if cindex(jj) == 1 % conjugate complex couple of poles
+      Ak(:,jj) = (1./(s-poles(jj)))+(1./(s-poles(jj+1)));
+      Ak(:,jj+1) = (1i./(s-poles(jj)))-(1i./(s-poles(jj+1)));
+    elseif cindex(jj) == 0 % real pole
+      Ak(:,jj) = 1./(s-poles(jj));
+    end
+  end
+  
+  
+  Ak(1:Nz,N+1) = ones(Nz,1);
+  
+  
+  for m=1:N+dl % left columns
+    AA(1:Nz,m)=w.*Ak(1:Nz,m);
+  end
+  for m=1:N+1 %Rightmost blocks
+    AA(1:Nz,N+dt+m)=-w.*(Ak(1:Nz,m).*y);
+  end
+  
+  % Scaling factor
+  clear sc
+  sc = norm(w.*y)/Nz;
+  
+  % setting the last row of AA and BA for the relaxion condition
+  for qq = 1:N+1
+    AA(Nz+1,N+dl+qq) = real(sc*sum(Ak(:,qq)));
+  end
+  
+  AA = [real(AA);imag(AA)];
+  
+  %   AAstr1 = AA; % storing
+  
+  % Last element of the solution vector
+  BA(Nz+1) = Nz*sc;
+  
+  % solving for real and imaginary part of the solution vector
+  nBA = [real(BA);imag(BA)];
+  
+  % Normalization factor
+  nf = zeros(2*N+dl+1,1);
+  for pp = 1:2*N+dl+1
+    nf(pp,1) = norm(AA(:,pp),2); % Euclidean norm
+    AA(:,pp) = AA(:,pp)./nf(pp,1); % Normalization
+  end
+  
+  
+  %% Solving augmented problem
+  
+  % XA = pinv(AA)*nBA;
+  % XA = inv((AA.')*AA)*(AA.')*nBA;
+  
+  % XA = AA.'*AA\AA.'*nBA;
+  
+  XA = AA\nBA;
+  
+  XA = XA./nf; % renormalization
+  
+  %% Finding zeros of sigma
+  
+  lsr = XA(N+dl+1:2*N+dl,1); % collect the least square results
+  
+  Ds = XA(end); % direct term of sigma
+  
+  % Real poles have real residues, complex poles have comples residues
+  rs = zeros(N,1);
+  for tt = 1:N
+    if cindex(tt) == 1 % conjugate complex couple of poles
+      rs(tt,1) = lsr(tt)+1i*lsr(tt+1);
+      rs(tt+1,1) = lsr(tt)-1i*lsr(tt+1);
+    elseif cindex(tt) == 0 % real pole
+      rs(tt,1) = lsr(tt);
+    end
+  end
+  
+  % [snum, sden] = residue(rs,poles,Ds);
+  %
+  % % ceking for numerical calculation errors
+  % for jj = 1:length(snum)
+  %   if ~isequal(imag(snum(jj)),0)
+  %     snum(jj)=real(snum(jj));
+  %   end
+  % end
+  %
+  % % Zeros of sigma are poles of F
+  % szeros = roots(snum);
+  
+  DPOLES = diag(poles);
+  B = ones(N,1);
+  C = rs.';
+  for kk = 1:N
+    if cindex(kk) == 1
+      DPOLES(kk,kk)=real(DPOLES(kk,kk));
+      DPOLES(kk,kk+1)=imag(DPOLES(kk,kk));
+      DPOLES(kk+1,kk)=-1*imag(DPOLES(kk,kk));
+      DPOLES(kk+1,kk+1)=real(DPOLES(kk,kk));
+      B(kk,1) = 2;
+      B(kk+1,1) = 0;
+      C(1,kk) = real(C(1,kk));
+      C(1,kk+1) = imag(C(1,kk));
+    end
+  end
+  
+  H = DPOLES-B*C/Ds;
+  szeros = eig(H);
+  
+  %% Ruling out unstable poles
+  
+  % This option force the poles of f to stay inside the left side of the
+  % complex plane
+  
+  if stab
+    unst = real(szeros)>0;
+    szeros(unst) = szeros(unst)-2*real(szeros(unst)); % Mirroring respect to the complex axes
+  end
+  N = length(szeros);
+  
+  %% Separating complex poles from real poles and ordering
+  
+  rnpoles = [];
+  inpoles = [];
+  for tt = 1:N
+    if imag(szeros(tt)) == 0
+      % collecting real poles
+      rnpoles = [rnpoles; szeros(tt)];
+    else
+      % collecting complex poles
+      inpoles = [inpoles; szeros(tt)];
+    end
+  end
+  
+  % Sorting complex poles in order to have them in the expected order a+jb
+  % and a-jb a>0 b>0
+  inpoles = sort(inpoles);
+  npoles = [rnpoles;inpoles];
+  npoles = npoles - 2.*1i.*imag(npoles);
+  
+  %% Marking complex and real poles
+  
+  cindex=zeros(N,1);
+  for m=1:N
+    if imag(npoles(m))~=0
+      if m==1
+        cindex(m)=1;
+      else
+        if cindex(m-1)==0 || cindex(m-1)==2
+          cindex(m)=1; cindex(m+1)=2;
+        else
+          cindex(m)=2;
+        end
+      end
+    end
+  end
+  
+  %% Initializing direct problem
+  
+  % Matrix initialinzation
+  B = w.*y;
+  AD = zeros(Nz,N+dl);
+  Ak=zeros(Nz,N+dl);
+  
+  % Defining Ak
+  for jj = 1:N
+    if cindex(jj) == 1 % conjugate complex couple of poles
+      Ak(:,jj) = (1./(s-npoles(jj)))+(1./(s-npoles(jj+1)));
+      Ak(:,jj+1) = (1i./(s-npoles(jj)))-(1i./(s-npoles(jj+1)));
+    elseif cindex(jj) == 0 % real pole
+      Ak(:,jj) = 1./(s-npoles(jj));
+    end
+  end
+  
+  if dt
+    Ak(1:Nz,N+dl) = ones(Nz,1); % considering the direct term
+  end
+  
+  % Defining AD
+  for m=1:N+dl
+    AD(1:Nz,m)=w.*Ak(1:Nz,m);
+  end
+  
+  
+  AD = [real(AD);imag(AD)];
+  nB = [real(B);imag(B)];
+  
+  % Normalization factor
+  nf = zeros(N+dl,1);
+  for pp = 1:N+dl
+    nf(pp,1) = norm(AD(:,pp),2); % Euclidean norm
+    AD(:,pp) = AD(:,pp)./nf(pp,1); % Normalization
+  end
+  
+  %% Solving direct problem
+  
+  % XD = inv((AD.')*AD)*(AD.')*nB;
+  % XD = AD.'*AD\AD.'*nB;
+  % XD = pinv(AD)*nB;
+  XD = AD\nB;
+  
+  XD = XD./nf; % Renormalization
+  
+  %% Final residues and poles of f
+  
+  if dt
+    lsr = XD(1:end-1); % Fitting with direct term
+  else
+    lsr = XD(1:end); % Fitting without direct term
+  end
+  
+  res = zeros(N,1);
+  % Real poles have real residues, complex poles have comples residues
+  for tt = 1:N
+    if cindex(tt) == 1 % conjugate complex couple of poles
+      res(tt) = lsr(tt)+1i*lsr(tt+1);
+      res(tt+1) = lsr(tt)-1i*lsr(tt+1);
+    elseif cindex(tt) == 0 % real pole
+      res(tt) = lsr(tt);
+    end
+  end
+  
+  clear poles
+  poles = npoles;
+  
+  if dt
+    dterm = XD(end);
+  else
+    dterm = 0;
+  end
+  
+  %% freq resp of the fit model
+  
+  % f = pfparams.freq;
+  r = res;
+  p = poles;
+  d = dterm;
+  
+  Nf = length(f);
+  N = length(p);
+  
+  rsp = zeros(Nf,1);
+  indx = (0:length(d)-1).';
+  for ii = 1:Nf
+    for jj = 1:N
+      rsptemp = r(jj)/(1i*2*pi*f(ii)-p(jj));
+      rsp(ii) = rsp(ii) + rsptemp;
+    end
+    % Direct terms response
+    rsp(ii) = rsp(ii) + sum(((1i*2*pi*f(ii))*ones(length(d),1).^indx).*d);
+  end
+  
+  mresp = rsp;
+  
+  % Residual
+  rdl = y - mresp;
+  
+  %% Plotting response
+  
+  if plotting
+    figure(1)
+    subplot(2,1,1);
+    loglog(f,abs(y),'k')
+    hold on
+    loglog(f,abs(mresp),'r')
+    loglog(f,abs(rdl),'b')
+    xlabel('Frequency [Hz]')
+    ylabel('Amplitude')
+    legend('Original', 'CTFIT','Residual')
+    hold off
+    
+    subplot(2,1,2);
+    semilogx(f,angle(y),'k')
+    hold on
+    semilogx(f,angle(mresp),'r')
+    xlabel('Frequency [Hz]')
+    ylabel('Phase [Rad]')
+    legend('Original', 'CTFIT')
+  end
+  hold off
+end
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