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C1-test-bench-apa/STEPS/full_APA300ML_out_nodes_3D.txt

@@ -0,0 +1,61 @@
+
+ LIST ALL SELECTED NODES.   DSYS=      0
+
+ *** ANSYS - ENGINEERING ANALYSIS SYSTEM  RELEASE 2020 R2          20.2     ***
+ DISTRIBUTED ANSYS Mechanical Enterprise                       
+
+ 00208316  VERSION=WINDOWS x64   10:10:05  MAR 26, 2021 CP=      2.188
+
+ Unknown                                                                       
+
+
+
+    NODE        X             Y             Z           THXY     THYZ     THZX
+        1   0.0000        0.0000       0.28000E-001     0.00     0.00     0.00
+  1228810   0.0000        0.0000      -0.28000E-001     0.00     0.00     0.00
+  1228811 -0.30000E-001   0.0000        0.0000          0.00     0.00     0.00
+  1228812  0.10000E-001   0.0000        0.0000          0.00     0.00     0.00
+  1228813  0.30000E-001   0.0000        0.0000          0.00     0.00     0.00
+
+ LIST MASTERS ON ALL SELECTED NODES.
+ CURRENT DOF SET= UX   UY   UZ   ROTX ROTY ROTZ
+
+ *** ANSYS - ENGINEERING ANALYSIS SYSTEM  RELEASE 2020 R2          20.2     ***
+ DISTRIBUTED ANSYS Mechanical Enterprise                       
+
+ 00208316  VERSION=WINDOWS x64   10:10:05  MAR 26, 2021 CP=      2.188
+
+ Unknown                                                                       
+
+
+     NODE  LABEL     SUPPORT
+         1  UX  
+         1  UY  
+         1  UZ  
+         1  ROTX
+         1  ROTY
+         1  ROTZ
+   1228810  UX  
+   1228810  UY  
+   1228810  UZ  
+   1228810  ROTX
+   1228810  ROTY
+   1228810  ROTZ
+   1228811  UX  
+   1228811  UY  
+   1228811  UZ  
+   1228811  ROTX
+   1228811  ROTY
+   1228811  ROTZ
+   1228812  UX  
+   1228812  UY  
+   1228812  UZ  
+   1228812  ROTX
+   1228812  ROTY
+   1228812  ROTZ
+   1228813  UX  
+   1228813  UY  
+   1228813  UZ  
+   1228813  ROTX
+   1228813  ROTY
+   1228813  ROTZ

C1-test-bench-apa/STEPS/new_APA300ML_K.CSV

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+ LIST ALL SELECTED NODES.   DSYS=      0
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+ *** ANSYS - ENGINEERING ANALYSIS SYSTEM  RELEASE 2020 R2          20.2     ***
+ DISTRIBUTED ANSYS Mechanical Enterprise                       
+
+ 00208316  VERSION=WINDOWS x64   10:10:05  MAR 26, 2021 CP=      2.188
+
+ Unknown                                                                       
+
+
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+  1228813  0.30000E-001   0.0000        0.0000          0.00     0.00     0.00
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+ LIST MASTERS ON ALL SELECTED NODES.
+ CURRENT DOF SET= UX   UY   UZ   ROTX ROTY ROTZ
+
+ *** ANSYS - ENGINEERING ANALYSIS SYSTEM  RELEASE 2020 R2          20.2     ***
+ DISTRIBUTED ANSYS Mechanical Enterprise                       
+
+ 00208316  VERSION=WINDOWS x64   10:10:05  MAR 26, 2021 CP=      2.188
+
+ Unknown                                                                       
+
+
+     NODE  LABEL     SUPPORT
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+         1  UY  
+         1  UZ  
+         1  ROTX
+         1  ROTY
+         1  ROTZ
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+#794 = VERTEX_POINT ( 'NONE', #699 ) ;
+#795 = FACE_OUTER_BOUND ( 'NONE', #359, .T. ) ;
+#796 = CARTESIAN_POINT ( 'NONE',  ( -3.499999999999996891, 0.000000000000000000, -4.000000000000001776 ) ) ;
+#797 = ORIENTED_EDGE ( 'NONE', *, *, #215, .F. ) ;
+#798 = DIRECTION ( 'NONE',  ( -0.000000000000000000, 0.000000000000000000, -1.000000000000000000 ) ) ;
+#799 = CARTESIAN_POINT ( 'NONE',  ( 1.734723475976807094E-15, 12.99999999999999822, -2.000000000000001776 ) ) ;
+#800 = ORIENTED_EDGE ( 'NONE', *, *, #156, .T. ) ;
+#801 = APPLICATION_PROTOCOL_DEFINITION ( 'draft international standard', 'automotive_design', 1998, #398 ) ;
+#802 = LINE ( 'NONE', #191, #145 ) ;
+#803 = VECTOR ( 'NONE', #574, 1000.000000000000000 ) ;
+#804 = AXIS2_PLACEMENT_3D ( 'NONE', #232, #791, #166 ) ;
+#805 = CARTESIAN_POINT ( 'NONE',  ( 4.336808689942017736E-15, 4.000000000000000000, -4.000000000000001776 ) ) ;
+#806 = VERTEX_POINT ( 'NONE', #648 ) ;
+#807 = EDGE_CURVE ( 'NONE', #498, #681, #742, .T. ) ;
+ENDSEC;
+END-ISO-10303-21;

C1-test-bench-apa/src/extractNodes.m

@@ -0,0 +1,91 @@
+function [int_xyz, int_i, n_xyz, n_i, nodes] = extractNodes(filename)
+% extractNodes -
+%
+% Syntax: [n_xyz, nodes] = extractNodes(filename)
+%
+% Inputs:
+%    - filename - relative or absolute path of the file that contains the Matrix
+%
+% Outputs:
+%    - n_xyz -
+%    - nodes - table containing the node numbers and corresponding dof of the interfaced DoFs
+
+arguments
+    filename
+end
+
+fid = fopen(filename,'rt');
+
+if fid == -1
+    error('Error opening the file');
+end
+
+n_xyz = []; % Contains nodes coordinates
+n_i = []; % Contains nodes indices
+
+n_num = []; % Contains node numbers
+n_dof = {}; % Contains node directions
+
+while 1
+    % Read a line
+    nextline = fgetl(fid);
+
+    % End of the file
+    if ~isstr(nextline), break, end
+
+    % Line just before the list of nodes coordinates
+    if contains(nextline, 'NODE') && ...
+            contains(nextline, 'X') && ...
+            contains(nextline, 'Y') && ...
+            contains(nextline, 'Z')
+
+        while 1
+            nextline = fgetl(fid);
+
+            if nextline < 0, break, end
+
+            c = sscanf(nextline, ' %f');
+
+            if isempty(c), break, end
+
+            n_xyz = [n_xyz; c(2:4)'];
+            n_i = [n_i; c(1)];
+        end
+    end
+
+    if nextline < 0, break, end
+
+    % Line just before the list of node DOF
+  if contains(nextline, 'NODE') && ...
+         contains(nextline, 'LABEL')
+
+        while 1
+            nextline = fgetl(fid);
+
+            if nextline < 0, break, end
+
+            c = sscanf(nextline, ' %d %s');
+
+            if isempty(c), break, end
+
+            n_num = [n_num; c(1)];
+
+            n_dof{length(n_dof)+1} = char(c(2:end)');
+        end
+
+        nodes = table(n_num, string(n_dof'), 'VariableNames', {'node_i', 'node_dof'});
+    end
+
+    if nextline < 0, break, end
+end
+
+fclose(fid);
+
+int_i = unique(nodes.('node_i')); % indices of interface nodes
+
+% Extract XYZ coordinates of only the interface nodes
+if length(n_xyz) > 0 && length(n_i) > 0
+    int_xyz = n_xyz(logical(sum(n_i.*ones(1, length(int_i)) == int_i', 2)), :);
+else
+    int_xyz = n_xyz;
+end

C1-test-bench-apa/src/initializeAPA.m

@@ -0,0 +1,101 @@
+function [actuator] = initializeAPA(args)
+% initializeAPA -
+%
+% Syntax: [actuator] = initializeAPA(args)
+%
+% Inputs:
+%    - args -
+%
+% Outputs:
+%    - actuator -
+
+arguments
+    args.type      char   {mustBeMember(args.type,{'2dof', 'flexible frame', 'flexible'})} = '2dof'
+
+    % Actuator and Sensor constants
+    args.Ga  (1,1) double {mustBeNumeric} = 0
+    args.Gs  (1,1) double {mustBeNumeric} = 0
+
+    % For 2DoF
+    args.k   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*380000
+    args.ke  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*4952605
+    args.ka  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*2476302
+
+    args.c   (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*20
+    args.ce  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*200
+    args.ca  (6,1) double {mustBeNumeric, mustBePositive} = ones(6,1)*100
+
+    args.Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056
+
+    % Force Flexible APA
+    args.xi (1,1) double {mustBeNumeric, mustBePositive} = 0.01
+    args.d_align (3,1) double {mustBeNumeric} = zeros(3,1) % [m]
+    args.d_align_bot (3,1) double {mustBeNumeric} = zeros(3,1) % [m]
+    args.d_align_top (3,1) double {mustBeNumeric} = zeros(3,1) % [m]
+
+    % For Flexible Frame
+    args.ks (1,1) double {mustBeNumeric, mustBePositive} = 235e6
+    args.cs (1,1) double {mustBeNumeric, mustBePositive} = 1e1
+end
+
+actuator = struct();
+
+switch args.type
+  case '2dof'
+    actuator.type = 1;
+  case 'flexible frame'
+    actuator.type = 2;
+  case 'flexible'
+    actuator.type = 3;
+end
+
+if args.Ga == 0
+    switch args.type
+      case '2dof'
+        actuator.Ga = -2.5796;
+      case 'flexible'
+        actuator.Ga = 23.2;
+    end
+else
+    actuator.Ga = args.Ga; % Actuator sensitivity [N/V]
+end
+
+if args.Gs == 0
+    switch args.type
+      case '2dof'
+        actuator.Gs = 466664;
+      case 'flexible'
+        actuator.Gs = -4898341;
+    end
+else
+    actuator.Gs = args.Gs; % Sensor sensitivity [V/m]
+end
+
+actuator.k  = args.k;  % [N/m]
+actuator.ke = args.ke; % [N/m]
+actuator.ka = args.ka; % [N/m]
+
+actuator.c  = args.c;  % [N/(m/s)]
+actuator.ce = args.ce; % [N/(m/s)]
+actuator.ca = args.ca; % [N/(m/s)]
+
+actuator.Leq = args.Leq; % [m]
+
+switch args.type
+  case 'flexible frame'
+    actuator.K = readmatrix('APA300ML_b_mat_K.CSV'); % Stiffness Matrix
+    actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix
+    actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m]
+  case 'flexible'
+    actuator.K = readmatrix('full_APA300ML_K.CSV'); % Stiffness Matrix
+    actuator.M = readmatrix('full_APA300ML_M.CSV'); % Mass Matrix
+    actuator.P = extractNodes('full_APA300ML_out_nodes_3D.txt'); % Node coordiantes [m]
+    actuator.d_align = args.d_align;
+    actuator.d_align_bot = args.d_align_bot;
+    actuator.d_align_top = args.d_align_top;
+end
+
+actuator.xi = args.xi; % Damping ratio
+
+actuator.ks = args.ks; % Stiffness of one stack [N/m]
+actuator.cs = args.cs; % Damping of one stack [N/m]

C1-test-bench-apa/src/vectfit3.m

@@ -0,0 +1,866 @@
+function [SER,poles,rmserr,fit,opts]=vectfit3(f,s,poles,weight,opts);
+%
+%function [SER,poles,rmserr,fit,opts]=vectfit3(f,s,poles,weight,opts)
+%function [SER,poles,rmserr,fit]=vectfit3(f,s,poles,weight,opts)
+%function [SER,poles,rmserr,fit]=vectfit3(f,s,poles,weight)
+% 
+%     ===========================================================
+%     =   Fast Relaxed Vector Fitting                           =
+%     =   Version 1.0                                           =
+%     =   Last revised: 08.08.2008                              = 
+%     =   Written by: Bjorn Gustavsen                           =
+%     =   SINTEF Energy Research, N-7465 Trondheim, NORWAY      =
+%     =   bjorn.gustavsen@sintef.no                             =
+%     =   http://www.energy.sintef.no/Produkt/VECTFIT/index.asp =
+%     =   Note: RESTRICTED to NON-COMMERCIAL use                =
+%     ===========================================================
+%
+% PURPOSE : Approximate f(s) with a state-space model 
+%
+%         f(s)=C*(s*I-A)^(-1)*B +D +s*E
+%                       
+%           where f(s) is a singe element or a vector of elements.  
+%           When f(s) is a vector, all elements become fitted with a common
+%           pole set.
+%
+% INPUT :
+%
+% f(s) : function (vector) to be fitted. 
+%        dimension : (Nc,Ns)  
+%                     Nc : number of elements in vector
+%                     Ns : number of frequency samples 
+% 
+% s : vector of frequency points [rad/sec] 
+%        dimension : (1,Ns)  
+%
+% poles : vector of initial poles [rad/sec]
+%         dimension : (1,N)  
+%
+% weight: the rows in the system matrix are weighted using this array. Can be used 
+%         for achieving higher accuracy at desired frequency samples. 
+%         If no weighting is desired, use unitary weights: weight=ones(1,Ns). 
+%
+%         Two dimensions are allowed:
+%           dimension : (1,Ns) --> Common weighting for all vector elements.   
+%           dimension : (Nc,Ns)--> Individual weighting for vector elements.  
+%
+% opts.relax==1 --> Use relaxed nontriviality constraint 
+% opts.relax==0 --> Use nontriviality constraint of "standard" vector fitting
+%
+% opts.stable=0 --> unstable poles are kept unchanged
+% opts.stable=1 --> unstable poles are made stable by 'flipping' them
+%                   into the left half-plane
+% 
+%
+% opts.asymp=1 --> Fitting with D=0,  E=0 
+% opts.asymp=2 --> Fitting with D~=0, E=0 
+% opts.asymp=3 --> Fitting with D~=0, E~=0 
+%
+% opts.spy1=1 --> Plotting, after pole identification (A)
+%              figure(3): magnitude functions
+%                cyan trace  : (sigma*f)fit              
+%                red trace   : (sigma)fit
+%                green trace : f*(sigma)fit - (sigma*f)fit
+%
+% opts.spy2=1 --> Plotting, after residue identification (C,D,E) 
+%              figure(1): magnitude functions
+%              figure(2): phase angles  
+%
+% opts.logx=1 --> Plotting using logarithmic absissa axis             
+%
+% opts.logy=1 --> Plotting using logarithmic ordinate axis
+%
+% opts.errplot=1   --> Include deviation in magnitude plot
+%
+% opts.phaseplot=1 -->Show plot also for phase angle
+%
+%
+% opts.skip_pole=1 --> The pole identification part is skipped, i.e (C,D,E) 
+%                    are identified using the initial poles (A) as final poles.
+%                 
+% opts.skip_res =1 --> The residue identification part is skipped, i.e. only the 
+%                    poles (A) are identified while C,D,E are returned as zero. 
+%
+%
+% opts.cmplx_ss  =1 -->The returned state-space model has real and complex conjugate 
+%                    parameters. Output variable A is diagonal (and sparse). 
+%              =0 -->The returned state-space model has real parameters only.
+%                    Output variable A is square with 2x2 blocks (and sparse).
+%
+% OUTPUT :
+% 
+%     fit(s) = C*(s*I-(A)^(-1)*B +D +s.*E
+%
+% SER.A(N,N)    : A-matrix (sparse). If cmplx_ss==1: Diagonal and complex. 
+%                                Otherwise, square and real with 2x2 blocks. 
+%                           
+% SER.B(N,1)    : B-matrix. If cmplx_ss=1: Column of 1's. 
+%                       If cmplx_ss=0: contains 0's, 1's and 2's)
+% SER.C(Nc,N)   : C-matrix. If cmplx_ss=1: complex
+%                       If cmplx_ss=0: real-only
+% SERD.D(Nc,1)  : constant term (real). Is non-zero if asymp=2 or 3.
+% SERE.E(Nc,1)  : proportional term (real). Is non-zero if asymp=3.
+%
+% poles(1,N)    : new poles 
+%
+% rmserr(1) : root-mean-square error of approximation for f(s). 
+%                   (0 is returned if skip_res==1)  
+% fit(Nc,Ns): Rational approximation at samples. (0 is returned if
+%             skip_res==1).
+%
+%
+% APPROACH: 
+% The identification is done using the pole relocating method known as Vector Fitting [1], 
+% with relaxed non-triviality constraint for faster convergence and smaller fitting errors [2], 
+% and utilization of matrix structure for fast solution of the pole identifion step [3]. 
+%
+%******************************************************************************** 
+% NOTE: The use of this program is limited to NON-COMMERCIAL usage only.
+% If the program code (or a modified version) is used in a scientific work, 
+% then reference should be made to the following:  
+%
+% [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.   
+%
+% [3] D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter,
+%     "Macromodeling of Multiport Systems Using a Fast Implementation of
+%     the Vector Fitting Method", IEEE Microwave and Wireless Components 
+%     Letters, vol. 18, no. 6, pp. 383-385, June 2008.
+%********************************************************************************
+% This example script is part of the vector fitting package (VFIT3.zip) 
+% Last revised: 08.08.2008. 
+% Created by:   Bjorn Gustavsen.
+%
+% 
+def.relax=1;      %Use vector fitting with relaxed non-triviality constraint
+def.stable=1;     %Enforce stable poles
+def.asymp=2;      %Include only D in fitting (not E)   
+def.skip_pole=0;  %Do NOT skip pole identification
+def.skip_res=0;   %Do NOT skip identification of residues (C,D,E) 
+def.cmplx_ss=1;   %Create complex state space model
+def.spy1=0;       %No plotting for first stage of vector fitting
+def.spy2=1;       %Create magnitude plot for fitting of f(s) 
+def.logx=1;       %Use logarithmic abscissa axis
+def.logy=1;       %Use logarithmic ordinate axis 
+def.errplot=1;    %Include deviation in magnitude plot
+def.phaseplot=0;  %exclude plot of phase angle (in addition to magnitiude)
+def.legend=1;     %Do include legends in plots
+
+
+if nargin<5
+  opts=def;
+else 
+  %Merge default values into opts  
+  A=fieldnames(def);    
+  for m=1:length(A)
+    if ~isfield(opts,A(m))
+      dum=char(A(m)); dum2=getfield(def,dum); opts=setfield(opts,dum,dum2);
+    end
+  end  
+end 
+
+
+
+%Tolerances used by relaxed version of vector fitting
+TOLlow=1e-18; TOLhigh=1e18;
+
+[a,b]=size(poles); 
+if s(1)==0 && a==1
+  if poles(1)==0 && poles(2)~=0 
+    poles(1)=-1;
+  elseif poles(2)==0 && poles(1)~=0 
+    poles(2)=-1;
+  elseif poles(1)==0 && poles(2)==0 
+    poles(1)=-1+i*10; poles(2)=-1-i*10;
+  end
+end  
+
+if (opts.relax~=0) && (opts.relax)~=1
+  disp(['    ERROR in vectfit3.m: ==> Illegal value for opts.relax:  ' num2str(opts.asymp)]),return
+end
+if (opts.asymp~=1) && (opts.asymp)~=2 && (opts.asymp)~=3
+  disp(['    ERROR in vectfit3.m: ==> Illegal value for opts.asymp:  ' num2str(opts.asymp)]),return
+end
+if (opts.stable~=0) && (opts.stable~=1) 
+  disp(['    ERROR in vectfit3.m: ==> Illegal value for opts.stable:  ' num2str(opts.stable)]),return
+end
+if (opts.skip_pole~=0) && (opts.skip_pole)~=1
+  disp(['    ERROR in vectfit3.m: ==> Illegal value for opts.skip_pole:  ' num2str(opts.skip_pole)]),return
+end
+if (opts.skip_res~=0) && (opts.skip_res)~=1
+  disp(['    ERROR in vectfit3.m: ==> Illegal value for opts.skip_res:  ' num2str(opts.skip_res)]),return
+end
+if (opts.cmplx_ss~=0) && (opts.cmplx_ss)~=1
+  disp(['    ERROR in vectfit3.m: ==> Illegal value for opts.cmplx_ss:  ' num2str(opts.cmplx_ss)]),return
+end
+
+rmserr=[];%SERC=[];
+[a,b]=size(s);
+if a<b, s=s.'; end 
+
+% Some sanity checks on dimension of input arrays:
+if length(s)~=length(f(1,:))
+  disp('Error in vectfit3.m!!! ==> Second dimension of f does not match length of s.'); 
+  return;
+end  
+if length(s)~=length(weight(1,:))
+  disp('Error in vectfit3.m!!! ==> Second dimension of weight does not match length of s.');
+  return;
+end
+if length(weight(:,1))~=1
+  if length(weight(:,1))~=length(f(:,1))
+    disp('Error in vectfit3.m!!! ==> First dimension of weight is neither 1 nor matches first dimension of f.');  
+    return;
+  end
+end  
+    
+  %set(0,'DefaultLineLineWidth',0.5) ; set(0,'DefaultLineMarkerSize',4) ;
+  %clear b; clear C; 
+  LAMBD=diag(poles);
+  Ns=length(s); N=length(LAMBD); Nc=length(f(:,1));
+  B=ones(N,1); %I=diag(ones(1,N));                
+  SERA=poles;SERC=zeros(Nc,N);SERD=zeros(Nc,1);SERE=zeros(Nc,1);
+  roetter=poles;
+  fit=zeros(Nc,Ns);
+
+  weight=weight.';
+  if length(weight(1,:))==1
+    common_weight=1;
+  elseif length(weight(1,:))==Nc
+    common_weight=0;
+  else
+    disp('ERROR in vectfit3.m: Invalid size of array weight')
+    return
+  end
+
+  if opts.asymp==1
+    offs=0; 
+  elseif opts.asymp==2
+    offs=1;  
+  else
+    offs=2; 
+  end
+  
+  
+  
+%=========================================================================
+%=========================================================================
+%  POLE IDENTIFICATION:
+%=========================================================================
+%=========================================================================
+ 
+  
+if opts.skip_pole~=1  
+  
+
+
+Escale=zeros(1,Nc+1);
+
+
+%=======================================================
+% Finding out which starting poles are complex :
+%=======================================================
+cindex=zeros(1,N);
+for m=1:N 
+  if imag(LAMBD(m,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
+
+%=======================================================
+% Building system - matrix :
+%=======================================================
+ %I3=diag(ones(1,Nc));I3(:,Nc)=[]; 
+  Dk=zeros(Ns,N);
+  for m=1:N
+    if cindex(m)==0      %real pole
+      Dk(:,m)=1./(s-LAMBD(m,m));
+    elseif cindex(m)==1  %complex pole, 1st part
+      Dk(:,m)  =1./(s-LAMBD(m,m)) + 1./(s-LAMBD(m,m)');
+      Dk(:,m+1)=i./(s-LAMBD(m,m)) - i./(s-LAMBD(m,m)');
+    end
+  end      
+ if opts.asymp==1 || opts.asymp==2    
+   Dk(:,N+1)=1; 
+ elseif opts.asymp==3   
+   Dk(:,N+1)=1; 
+   Dk(:,N+2)=s;
+ end
+    
+%Scaling for last row of LS-problem (pole identification)
+scale=0;
+for m=1:Nc
+  if length(weight(1,:))==1
+    scale=scale+(norm(weight(:,1).*f(m,:).'))^2;
+  else
+    scale=scale+(norm(weight(:,m).*f(m,:).'))^2;
+  end  
+end
+scale=sqrt(scale)/Ns; 
+
+ 
+if opts.relax==1   
+
+  %Escale=zeros(1,Nc*(N+offs)+N+1);
+  %scale=norm(f);%/Ns; %Scaling for sigma in LS problem 
+  AA=zeros(Nc*(N+1),N+1);
+  bb=zeros(Nc*(N+1),1);
+  Escale=zeros(1,length(AA(1,:)));
+  for n=1:Nc
+    A=zeros(Ns,(N+offs) +N+1); %b=zeros(Ns*Nc+1,1);
+
+    if common_weight==1
+      weig=weight;
+    else
+      weig=weight(:,n);
+    end 
+
+
+    for m=1:N+offs %left block
+      A(1:Ns,m)=weig.*Dk(1:Ns,m);
+    end
+    inda=N+offs;
+    for m=1:N+1 %right block
+      A(1:Ns,inda+m)=-weig.*Dk(1:Ns,m).*f(n,1:Ns).';    
+    end
+     
+    A=[real(A);imag(A)];
+    
+   
+    %Integral criterion for sigma:
+    offset=(N+offs);
+    if n==Nc
+      for mm=1:N+1  
+        A(2*Ns+1,offset+mm)=real(scale*sum(Dk(:,mm)));
+      end
+    end  
+    [Q,R]=qr(A,0);
+    ind1=N+offs+1;
+    ind2=N+offs+N+1;
+    R22=R(ind1:ind2,ind1:ind2);
+    AA((n-1)*(N+1)+1:n*(N+1),:)=R22;
+    if n==Nc
+      bb((n-1)*(N+1)+1:n*(N+1),1)=Q(end,N+offs+1:end)'*Ns*scale; 
+    end
+  end %for n=1:Nc 
+
+  for col=1:length(AA(1,:))
+    Escale(col)=1/norm(AA(:,col));  
+    AA(:,col)=Escale(col).*AA(:,col);
+  end  
+  x=AA\bb;
+  %size(x),size(Escale)
+  x=x.*Escale.';
+
+end %if opts.relax==0   
+
+
+%Situation: No relaxation, or produced D of sigma extremely small and large. Solve again, without relaxation 
+  if opts.relax==0 | abs(x(end))<TOLlow | abs(x(end))>TOLhigh
+    AA=zeros(Nc*(N),N);
+    bb=zeros(Nc*(N),1);   
+    if opts.relax==0
+      Dnew=1; 
+    else
+      if x(end)==0
+        Dnew=1;      
+      elseif abs(x(end))<TOLlow
+        Dnew=sign(x(end))*TOLlow
+      elseif abs(x(end))>TOLhigh 
+        Dnew=sign(x(end))*TOLhigh   
+      end    
+    end
+
+   for n=1:Nc
+      A=zeros(Ns,(N+offs) +N); %b=zeros(Ns*Nc+1,1);
+      Escale=zeros(1,N);
+      
+      if common_weight==1
+        weig=weight;
+      else
+        weig=weight(:,n);
+      end 
+
+      for m=1:N+offs %left block
+        A(1:Ns,m)=weig.*Dk(1:Ns,m);
+      end
+      inda=N+offs;
+      for m=1:N %right block
+        A(1:Ns,inda+m)=-weig.*Dk(1:Ns,m).*f(n,1:Ns).';    
+      end
+      b=Dnew*weig.*f(n,1:Ns).';     
+      A=[real(A);imag(A)];
+      b=[real(b);imag(b)];       
+      offset=(N+offs);
+      [Q,R]=qr(A,0);
+      ind1=N+offs+1;
+      ind2=N+offs+N;
+      R22=R(ind1:ind2,ind1:ind2);
+      AA((n-1)*N+1:n*N,:)=R22;
+      bb((n-1)*N+1:n*N,1)=Q(:,ind1:ind2).'*b;
+    end %for n=1:Nc 
+    for col=1:length(AA(1,:))
+      Escale(col)=1./norm(AA(:,col));  
+      AA(:,col)=Escale(col).*AA(:,col);
+    end  
+    
+    %if opts.use_normal==1 
+    %  x=AA.'*AA\(AA.'*bb);  
+    %else     
+      x=AA\bb;
+    %end 
+    x=x.*Escale.';
+    x=[x;Dnew];
+
+  end  %if opts.relax==0 | abs(x(end))<TOLlow | abs(x(end))>TOLhigh
+ 
+
+  %************************************  
+
+C=x(1:end-1);
+D=x(end); %NEW!!
+
+%We now change back to make C complex : 
+% **************
+for m=1:N
+  if cindex(m)==1
+    for n=1:1%Nc+1
+      r1=C(m); r2=C(m+1);
+      C(m)=r1+i*r2; C(m+1)=r1-i*r2;
+    end
+  end
+end
+% **************  
+
+if opts.spy1==1
+  Dk=zeros(Ns,N); 
+  for m=1:N
+    Dk(:,m)=1./(s-LAMBD(m,m));
+  end 
+  RES3(:,1)=D+Dk*C; % (sigma)rat
+  freq=s./(2*pi*i); 
+  if opts.logx==1  
+    if opts.logy==1
+      figure(3),
+      h1=loglog(freq,abs(RES3'),'b'); xlim([freq(1) freq(Ns)]);  %sigma*f
+    else %logy=0
+      figure(3),
+      h1=semilogx(freq,abs(RES3'),'b'); xlim([freq(1) freq(Ns)]);
+    end
+  else %logx=0
+    if opts.logy==1
+      figure(3),
+      h1=semilogy(freq,abs(RES3'),'b'); xlim([freq(1) freq(Ns)]);
+    else %logy=0
+      figure(3),
+      h1=plot(s./(2*pi*i),abs(RES3'),'b'); xlim([freq(1) freq(Ns)]);
+    end
+  end
+  figure(3),xlabel('Frequency [Hz]'); ylabel('Magnitude'); 
+  %figure(3), title('Sigma')
+  if opts.legend==1
+    legend([h1(1)],'sigma');
+  end  
+  drawnow;
+end
+
+%=============================================================================
+% We now calculate the zeros for sigma :
+%=============================================================================
+%oldLAMBD=LAMBD;oldB=B;oldC=C;
+m=0;
+for n=1:N
+  m=m+1;
+  if m<N  
+    if( abs(LAMBD(m,m))>abs(real(LAMBD(m,m))) ) %complex number?
+      LAMBD(m+1,m)=-imag(LAMBD(m,m)); LAMBD(m,m+1)=imag(LAMBD(m,m));
+      LAMBD(m,m)=real(LAMBD(m,m));LAMBD(m+1,m+1)=LAMBD(m,m);
+      B(m,1)=2; B(m+1,1)=0; 
+      koko=C(m); C(m)=real(koko); C(m+1)=imag(koko);
+      m=m+1;
+    end
+  end
+end
+
+ZER=LAMBD-B*C.'/D;
+roetter=eig(ZER).';
+unstables=real(roetter)>0;  
+if opts.stable==1
+  roetter(unstables)=roetter(unstables)-2*real(roetter(unstables)); %Forcing unstable poles to be stable...
+end
+roetter=sort(roetter); N=length(roetter);
+
+
+%=============================================
+%Sorterer polene s.a. de reelle kommer f<EFBFBD>rst:
+for n=1:N
+  for m=n+1:N
+    if imag(roetter(m))==0 && imag(roetter(n))~=0
+      trans=roetter(n); roetter(n)=roetter(m); roetter(m)=trans;
+    end
+  end
+end
+N1=0;
+for m=1:N
+  if imag(roetter(m))==0, N1=m; end
+end
+if N1<N, roetter(N1+1:N)=sort(roetter(N1+1:N)); end   % N1: n.o. real poles
+%N2=N-N1;                                             % N2: n.o. imag.poles
+
+roetter=roetter-2*i*imag(roetter); %10.11.97 !!!
+SERA=roetter.';
+
+
+end %if skip_pole~=1  
+
+%=========================================================================
+%=========================================================================
+%  RESIDUE IDENTIFICATION:
+%=========================================================================
+%=========================================================================
+
+if opts.skip_res~=1
+
+%=============================================================================
+% We now calculate SER for f, using the modified zeros of sigma as new poles :
+%========================================================================================
+
+%clear LAMBD A A1 xA1 xxA1 A2 xA2 xxA2 b xb xxb C RES1 RES2;
+
+LAMBD=roetter; 
+
+%B=ones(N,1); 
+
+% Finding out which poles are complex :
+cindex=zeros(1,N);
+for m=1:N 
+  if imag(LAMBD(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
+
+
+%===============================================================================
+% We now calculate the SER for f (new fitting), using the above calculated 
+% zeros as known poles :
+%=============================================================================== 
+if opts.asymp==1
+  A=zeros(2*Ns,N); BB=zeros(2*Ns,Nc);
+elseif opts.asymp==2
+  A=zeros(2*Ns,N+1); BB=zeros(2*Ns,Nc);
+else
+  A=zeros(2*Ns,N+2); BB=zeros(2*Ns,Nc);
+end
+
+ %I3=diag(ones(1,Nc));I3(:,Nc)=[]; 
+  Dk=zeros(Ns,N); 
+    for m=1:N
+      if cindex(m)==0      %real pole
+        Dk(:,m)=1./(s-LAMBD(m));
+      elseif cindex(m)==1  %complex pole, 1st part
+        Dk(:,m)  =1./(s-LAMBD(m)) + 1./(s-LAMBD(m)');
+        Dk(:,m+1)=i./(s-LAMBD(m)) - i./(s-LAMBD(m)');
+      end
+    end 
+
+
+if common_weight==1
+
+ %I3=diag(ones(1,Nc));I3(:,Nc)=[]; 
+  Dk=zeros(Ns,N); 
+    for m=1:N
+      if cindex(m)==0      %real pole
+        Dk(:,m)=weight./(s-LAMBD(m));
+      elseif cindex(m)==1  %complex pole, 1st part
+        Dk(:,m)  =weight./(s-LAMBD(m)) + weight./(s-LAMBD(m)');
+        Dk(:,m+1)=i.*weight./(s-LAMBD(m)) - i.*weight./(s-LAMBD(m)');
+      end
+    end 
+
+  if opts.asymp==1
+    A(1:Ns,1:N)=Dk; 
+  elseif opts.asymp==2
+    A(1:Ns,1:N)=Dk; A(1:Ns,N+1)=weight;
+  else
+    A(1:Ns,1:N)=Dk; A(1:Ns,N+1)=weight; A(1:Ns,N+2)=weight.*s;
+  end
+  for m=1:Nc
+    BB(1:Ns,m)=weight.*f(m,:).';
+  end
+  A(Ns+1:2*Ns,:)=imag(A(1:Ns,:));
+  A(1:Ns,:)=real(A(1:Ns,:));
+  BB(Ns+1:2*Ns,:)=imag(BB(1:Ns,:));
+  BB(1:Ns,:)=real(BB(1:Ns,:));
+
+  if opts.asymp==2
+    A(1:Ns,N+1)=A(1:Ns,N+1);             
+  elseif opts.asymp==3
+    A(1:Ns,N+1)=A(1:Ns,N+1);             
+    A(Ns+1:2*Ns,N+2)=A(Ns+1:2*Ns,N+2);  
+  end
+
+  %clear Escale;
+  Escale=zeros(1,length(A(1,:)));
+  for col=1:length(A(1,:));
+    Escale(col)=norm(A(:,col),2);
+    A(:,col)=A(:,col)./Escale(col);
+  end
+  X=A\BB;
+  for n=1:Nc
+    X(:,n)=X(:,n)./Escale.';
+  end
+
+  %clear A;
+  X=X.';
+  C=X(:,1:N); 
+  if opts.asymp==2
+    SERD=X(:,N+1);
+  elseif opts.asymp==3
+    SERE=X(:,N+2); 
+    SERD=X(:,N+1);
+  end
+
+else %if common_weight==1
+    
+  SERD=zeros(Nc,1); 
+  SERE=zeros(Nc,1); 
+  C=zeros(Nc,N);
+  for n=1:Nc
+
+    if opts.asymp==1
+      A(1:Ns,1:N)=Dk; 
+    elseif opts.asymp==2
+      A(1:Ns,1:N)=Dk; A(1:Ns,N+1)=1;
+    else
+      A(1:Ns,1:N)=Dk; A(1:Ns,N+1)=1; A(1:Ns,N+2)=s;
+    end
+
+    for m=1:length(A(1,:))
+      A(1:Ns,m)=weight(:,n).*A(1:Ns,m);
+    end
+ 
+    BB=weight(:,n).*f(n,:).';
+    A(Ns+1:2*Ns,:)=imag(A(1:Ns,:));
+    A(1:Ns,:)=real(A(1:Ns,:));
+    BB(Ns+1:2*Ns)=imag(BB(1:Ns));
+    BB(1:Ns)=real(BB(1:Ns));
+
+    if opts.asymp==2
+      A(1:Ns,N+1)=A(1:Ns,N+1);             
+    elseif opts.asymp==3
+      A(1:Ns,N+1)=A(1:Ns,N+1);             
+      A(Ns+1:2*Ns,N+2)=A(Ns+1:2*Ns,N+2);  
+    end
+
+    %clear Escale;
+    Escale=zeros(1,length(A(1,:)));
+    for col=1:length(A(1,:));
+      Escale(col)=norm(A(:,col),2);
+      A(:,col)=A(:,col)./Escale(col);
+    end
+    x=A\BB;
+    x=x./Escale.';
+
+    %clear A;
+    C(n,1:N)=x(1:N).'; 
+
+    if opts.asymp==2
+      SERD(n)=x(N+1);
+    elseif opts.asymp==3
+      SERE(n)=x(N+2); 
+      SERD(n)=x(N+1);
+    end
+
+
+  end %for n=1:Nc
+
+    
+end %if common_weight==1    
+  
+
+%=========================================================================
+
+%We now change back to make C complex. 
+for m=1:N
+  if cindex(m)==1
+    for n=1:Nc
+      r1=C(n,m); r2=C(n,m+1);
+      C(n,m)=r1+i*r2; C(n,m+1)=r1-i*r2;
+   end
+  end
+end
+% **************  
+
+
+B=ones(N,1);
+
+%====================================================
+
+SERA  = LAMBD;
+SERB  = B;
+SERC  = C;
+
+
+
+  Dk=zeros(Ns,N); 
+  for m=1:N
+    Dk(:,m)=1./(s-SERA(m));
+  end 
+  for n=1:Nc
+    fit(n,:)=(Dk*SERC(n,:).').';
+    if opts.asymp==2
+      fit(n,:)=fit(n,:)+SERD(n);
+    elseif opts.asymp==3
+      fit(n,:)=fit(n,:)+SERD(n)+s.'.*SERE(n);
+    end
+  end 
+  
+  fit=fit.'; f=f.';
+  diff=fit-f; rmserr=sqrt(sum(sum(abs(diff.^2))))/sqrt(Nc*Ns);
+
+if opts.spy2==1
+  freq=s./(2*pi*i);
+  if opts.logx==1     
+    if opts.logy==1
+      figure(1),
+      h1=loglog(freq,abs(f),'b'); xlim([freq(1) freq(Ns)]);hold on
+      h2=loglog(freq,abs(fit),'r--'); hold off
+      if opts.errplot== 1
+        hold on,h3=loglog(freq,abs(f-fit),'g');hold off; 
+      end
+    else %logy=0 
+      figure(1),
+      h1=semilogx(freq,abs(f),'b'); xlim([freq(1) freq(Ns)]);hold on
+      h2=semilogx(freq,abs(fit),'r--'); hold off
+      if opts.errplot== 1
+        hold on,h3=semilogx(freq,abs(f-fit),'g');hold off; 
+      end
+    end
+    if opts.phaseplot==1
+      figure(2),
+      h4=semilogx(freq,180*unwrap(angle(f))/pi,'b'); xlim([freq(1) freq(Ns)]);hold on
+      h5=semilogx(freq,180*unwrap(angle(fit))/pi,'r--');hold off
+    end  
+  else %logx=0
+    if opts.logy==1
+      figure(1),
+      h1=semilogy(freq,abs(f),'b'); xlim([freq(1) freq(Ns)]);hold on
+      h2=semilogy(freq,abs(fit),'r--'); hold off
+      if opts.errplot== 1
+        hold on,h3=semilogy(freq,abs(f-fit),'g');hold off; 
+      end
+    else %logy=0 
+      figure(1), 
+      h1=plot(freq,abs(f),'b'); xlim([freq(1) freq(Ns)]);hold on
+      h2=plot(freq,abs(fit),'r--'); hold off
+      if opts.errplot== 1
+        hold on,h3=plot(freq,abs(f-fit),'g');hold off; 
+      end
+    end
+    if opts.phaseplot==1    
+      figure(2),
+      h4=plot(freq,180*unwrap(angle(f))/pi,'b'); xlim([freq(1) freq(Ns)]);hold on
+      h5=plot(freq,180*unwrap(angle(fit))/pi,'r--');hold off
+    end  
+  end %logy=0
+  figure(1),
+  xlabel('Frequency [Hz]'); ylabel('Magnitude');
+  %title('Approximation of f');
+  if opts.legend==1
+    if opts.errplot==1
+      legend([h1(1) h2(1) h3(1)],'Data','FRVF','Deviation');
+    else
+      legend([h1(1) h2(1)],'Data','FRVF');       
+    end  
+  end
+  if opts.phaseplot==1
+    figure(2),
+    xlabel('Frequency [Hz]'); ylabel('Phase angle [deg]');
+    %title('Approximation of f');
+    if opts.legend==1
+      legend([h4(1) h5(1)],'Data','FRVF'); 
+    end
+  end
+  drawnow;
+end
+fit=fit.';
+
+
+end %if skip_res~=1
+
+A=SERA;
+poles=A;
+if opts.skip_res~=1
+  B=SERB; C=SERC; D=SERD; E=SERE;
+else
+  B=ones(N,1); C=zeros(Nc,N); D=zeros(Nc,Nc); E=zeros(Nc,Nc); rmserr=0;  
+end;    
+
+
+%========================================
+% Convert into real state-space model
+%========================================
+if opts.cmplx_ss~=1 
+
+  A=diag(sparse(A));
+  
+  cindex=zeros(1,N);
+  for m=1:N 
+    if imag(A(m,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
+  
+  n=0;
+  for m=1:N
+    n=n+1;
+    if cindex(m)==1
+      a=A(n,n); a1=real(a); a2=imag(a);
+      c=C(:,n); c1=real(c); c2=imag(c);
+      b=B(n,:); b1=2*real(b); b2=-2*imag(b);
+      Ablock=[a1 a2;-a2 a1];
+   
+      A(n:n+1,n:n+1)=Ablock;
+      C(:,n)=c1;
+      C(:,n+1)=c2;
+      B(n,:)=b1;
+      B(n+1,:)=b2;
+    end
+  end
+
+else
+  A=sparse(diag(A));    % A is complex, make it diagonal
+end %if cmplx_ss~=1    
+    
+SER.A=A; SER.B=B; SER.C=C; SER.D=D; SER.E=E;
+
+

C1-test-bench-apa/test_apa_1_basic_meas.m

@@ -0,0 +1,99 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Path for functions, data and scripts
+addpath('./src/'); % Path for scripts
+addpath('./mat/'); % Path for data
+
+%% Colors for the figures
+colors = colororder;
+
+%% Measured height for all the APA at the 8 locations
+apa1 = 1e-6*[0, -0.5 , 3.5  , 3.5  , 42   , 45.5, 52.5 , 46];
+apa2 = 1e-6*[0, -2.5 , -3   , 0    , -1.5 , 1   , -2   , -4];
+apa3 = 1e-6*[0, -1.5 , 15   , 17.5 , 6.5  , 6.5 , 21   , 23];
+apa4 = 1e-6*[0, 6.5  , 14.5 , 9    , 16   , 22  , 29.5 , 21];
+apa5 = 1e-6*[0, -12.5, 16.5 , 28.5 , -43  , -52 , -22.5, -13.5];
+apa6 = 1e-6*[0, -8   , -2   , 5    , -57.5, -62 , -55.5, -52.5];
+apa7 = 1e-6*[0, 9    , -18.5, -30  , 31   , 46.5, 16.5 , 7.5];
+
+apa = {apa1, apa2, apa3, apa4, apa5, apa6, apa7};
+
+%% X-Y positions of the measurements points
+W = 20e-3;   % Width [m]
+L = 61e-3;   % Length [m]
+d = 1e-3;    % Distance from border [m]
+l = 15.5e-3; % [m]
+
+pos = [[-L/2 + d,      W/2 - d];
+       [-L/2 + l - d,  W/2 - d];
+       [-L/2 + l - d, -W/2 + d];
+       [-L/2 + d,     -W/2 + d];
+       [L/2 - l + d,   W/2 - d];
+       [L/2 - d,       W/2 - d];
+       [L/2 - d,      -W/2 + d];
+       [L/2 - l + d,  -W/2 + d]]';
+
+%% Using fminsearch to find the best fitting plane
+apa_d = zeros(1, 7); % Measured flatness of the APA
+for i = 1:7
+    fun = @(x)max(abs(([pos; apa{i}]-[0;0;x(1)])'*([x(2:3);1]/norm([x(2:3);1]))));
+    x0 = [0;0;0];
+    [x, min_d] = fminsearch(fun,x0);
+    apa_d(i) = min_d;
+end
+
+%% Load the measured strokes
+load('meas_apa_stroke.mat', 'apa300ml_2s')
+
+%% Generated voltage across the two piezoelectric stack actuators to estimate the stroke of the APA300ML
+figure;
+plot(apa300ml_2s{1}.t - apa300ml_2s{1}.t(1), 20*apa300ml_2s{1}.V, 'k-')
+xlabel('Time [s]'); ylabel('Voltage [V]')
+ylim([-20, 160])
+
+%% Measured displacement as a function of the applied voltage
+figure;
+hold on;
+for i = 1:7
+    plot(20*apa300ml_2s{i}.V, 1e6*apa300ml_2s{i}.d, 'DisplayName', sprintf('APA %i', i))
+end
+hold off;
+xlabel('Voltage [V]'); ylabel('Displacement [$\mu m$]')
+legend('location', 'southwest', 'FontSize', 8)
+xlim([-20, 150]); ylim([-250, 0]);
+
+%% X-Bending Identification
+% Load Data
+bending_X = load('apa300ml_bending_X_top.mat');
+
+% Spectral Analysis setup
+Ts = bending_X.Track1_X_Resolution; % Sampling Time [s]
+Nfft = floor(1/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+% Compute the transfer function from applied force to measured rotation
+[G_bending_X, f]  = tfestimate(bending_X.Track1, bending_X.Track2, win, Noverlap, Nfft, 1/Ts);
+
+%% Y-Bending identification
+% Load Data
+bending_Y = load('apa300ml_bending_Y_top.mat');
+
+% Compute the transfer function
+[G_bending_Y, ~]  = tfestimate(bending_Y.Track1, bending_Y.Track2, win, Noverlap, Nfft, 1/Ts);
+
+figure;
+hold on;
+plot(f, abs(G_bending_X), 'DisplayName', '$X$ bending');
+plot(f, abs(G_bending_Y), 'DisplayName', '$Y$ bending');
+text(280, 5.5e-2,{'280Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+text(412, 1.5e-2,{'412Hz'},'VerticalAlignment','bottom','HorizontalAlignment','center')
+hold off;
+set(gca, 'Xscale', 'log'); set(gca, 'Yscale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude');
+xlim([100, 1e3]); ylim([5e-5, 2e-1]);
+legend('location', 'northeast', 'FontSize', 8)

C1-test-bench-apa/test_apa_2_dynamics.m

@@ -0,0 +1,440 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Path for functions, data and scripts
+addpath('./src/'); % Path for scripts
+addpath('./mat/'); % Path for data
+
+%% Colors for the figures
+colors = colororder;
+
+%% Load measured data - hysteresis
+apa_hyst = load('frf_data_1_hysteresis.mat', 't', 'u', 'de');
+
+% Initial time set to zero
+apa_hyst.t = apa_hyst.t - apa_hyst.t(1);
+
+ampls = [0.1, 0.2, 0.4, 1, 2, 4]; % Excitation voltage amplitudes
+
+%% Measured displacement as a function of the output voltage
+figure;
+
+hold on;
+for i = [6,5,4,2]
+    i_lim = apa_hyst.t > i*5-1 & apa_hyst.t < i*5;
+    plot(20*apa_hyst.u(i_lim), 1e6*detrend(apa_hyst.de(i_lim), 0), ...
+         'DisplayName', sprintf('$V_a = 65 + %.0f \\sin (\\omega t) \\ [V]$', 20*ampls(i)))
+end
+hold off;
+xlabel('Stack Voltage $V_a$ [V]'); ylabel('Displacement $d_e$ [$\mu$m]');
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+xlim([-20, 150]);
+ylim([-120, 120]);
+
+%% Load data for stiffness measurement
+apa_nums = [1 2 4 5 6 8];
+apa_mass = {};
+for i = 1:length(apa_nums)
+    apa_mass(i) = {load(sprintf('frf_data_%i_add_mass_closed_circuit.mat', apa_nums(i)), 't', 'de')};
+    % The initial displacement is set to zero
+    apa_mass{i}.de = apa_mass{i}.de - mean(apa_mass{i}.de(apa_mass{i}.t<11));
+end
+
+added_mass = 6.4; % Added mass [kg]
+
+%% Plot the deflection at a function of time
+figure;
+hold on;
+plot(apa_mass{2}.t(1:100:end)-apa_mass{2}.t(1), 1e6*apa_mass{2}.de(1:100:end), 'k-');
+plot([0,20], [-0.4, -0.4], 'k--', 'LineWidth', 0.5)
+plot([0,20], [-4.5, -4.5], 'k--', 'LineWidth', 0.5)
+plot([0,20], [-37.4, -37.4], 'k--', 'LineWidth', 0.5)
+% first stroke for stiffness measurements
+anArrow = annotation('doublearrow', 'LineWidth', 0.5);
+anArrow.Parent = gca;
+anArrow.Position = [2, -0.4, 0, -37];
+text(2.5, -20, sprintf('$d_1$'), 'horizontalalignment', 'left');
+
+% second stroke for stiffness measurements
+anArrow = annotation('doublearrow', 'LineWidth', 0.5);
+anArrow.Parent = gca;
+anArrow.Position = [18, -37.4, 0, 32.9];
+text(18.5, -20, sprintf('$d_2$'), 'horizontalalignment', 'left');
+
+% annotation('textarrow',[],y,'String',' Growth ','FontSize',13,'Linewidth',2)
+hold off;
+xlabel('Time [s]'); ylabel('Displacement $d_e$ [$\mu$m]');
+
+%% Load Data
+add_mass_oc = load('frf_data_1_add_mass_open_circuit.mat',   't', 'de');
+add_mass_cc = load('frf_data_1_add_mass_closed_circuit.mat', 't', 'de');
+
+%% Zero displacement at initial time
+add_mass_oc.de = add_mass_oc.de - mean(add_mass_oc.de(add_mass_oc.t<11));
+add_mass_cc.de = add_mass_cc.de - mean(add_mass_cc.de(add_mass_cc.t<11));
+
+%% Estimation of the stiffness in Open Circuit and Closed-Circuit
+apa_k_oc = 9.8 * added_mass / (mean(add_mass_oc.de(add_mass_oc.t > 12 & add_mass_oc.t < 12.5)) - mean(add_mass_oc.de(add_mass_oc.t > 20 & add_mass_oc.t < 20.5)));
+apa_k_sc = 9.8 * added_mass / (mean(add_mass_cc.de(add_mass_cc.t > 12 & add_mass_cc.t < 12.5)) - mean(add_mass_cc.de(add_mass_cc.t > 20 & add_mass_cc.t < 20.5)));
+
+%% Identification using sweep sine (low frequency)
+load('frf_data_sweep.mat');
+load('frf_data_noise_hf.mat');
+
+%% Sampling Frequency
+Ts = 1e-4; % Sampling Time [s]
+Fs = 1/Ts; % Sampling Frequency [Hz]
+
+%% "Hanning" windows used for the spectral analysis:
+Nfft = floor(2/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+%% Separation of frequencies: low freqs using sweep sine, and high freq using noise
+% Only used to have the frequency vector "f"
+[~, f] = tfestimate(apa_sweep{1}.u, apa_sweep{1}.de, win, Noverlap, Nfft, 1/Ts);
+i_lf = f <= 350;
+i_hf = f >  350;
+
+%% FRF estimation of the transfer function from u to de
+enc_frf = zeros(length(f), length(apa_nums));
+for i = 1:length(apa_nums)
+    [frf_lf, ~] = tfestimate(apa_sweep{i}.u,    apa_sweep{i}.de,    win, Noverlap, Nfft, 1/Ts);
+    [frf_hf, ~] = tfestimate(apa_noise_hf{i}.u, apa_noise_hf{i}.de, win, Noverlap, Nfft, 1/Ts);
+    enc_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
+end
+
+%% FRF estimation of the transfer function from u to Vs
+iff_frf = zeros(length(f), length(apa_nums));
+for i = 1:length(apa_nums)
+    [frf_lf, ~] = tfestimate(apa_sweep{i}.u,    apa_sweep{i}.Vs,    win, Noverlap, Nfft, 1/Ts);
+    [frf_hf, ~] = tfestimate(apa_noise_hf{i}.u, apa_noise_hf{i}.Vs, win, Noverlap, Nfft, 1/Ts);
+    iff_frf(:, i) = [frf_lf(i_lf); frf_hf(i_hf)];
+end
+
+%% Save the identified dynamics for further analysis
+save('mat/meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
+
+%% Plot the FRF from u to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, abs(enc_frf(:, i)), ...
+         'DisplayName', sprintf('APA %i', apa_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
+ylim([1e-8, 1e-3]);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, 180/pi*angle(enc_frf(:, i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);
+
+%% Plot the FRF from u to Vs
+figure;
+tiledlayout(2, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, abs(iff_frf(:, i)), ...
+         'DisplayName', sprintf('APA %i', apa_nums(i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, 180/pi*angle(iff_frf(:, i)));
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);
+
+%% Long measurement
+long_noise = load('frf_struts_align_3_noise_long.mat', 't', 'u', 'Vs');
+
+% Long window for fine frequency axis
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(10/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+% Transfer function estimation
+[frf_noise, f] = tfestimate(long_noise.u, long_noise.Vs, win, Noverlap, Nfft, 1/Ts);
+[coh_noise, ~] = mscohere(long_noise.u, long_noise.Vs, win, Noverlap, Nfft, 1/Ts);
+
+%% Bode plot of the FRF from u to de
+figure;
+tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+nexttile();
+hold on;
+plot(f, coh_noise, '.-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Coherence [-]');
+hold off;
+xlim([38, 45]);
+ylim([0, 1]);
+
+%% Bode plot of the FRF from u to de
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(frf_noise), '.-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
+hold off;
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(frf_noise), '.-');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 0]);
+
+linkaxes([ax1,ax2],'x');
+xlim([38, 45]);
+
+%% Load the data
+wi_k = load('frf_data_1_sweep_lf_with_R.mat', 't', 'Vs', 'u'); % With the resistor
+wo_k = load('frf_data_1_sweep_lf.mat',        't', 'Vs', 'u'); % Without the resistor
+
+%% Large Hanning window for good low frequency estimate
+Nfft = floor(50/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+%% Compute the transfer functions from u to Vs
+[frf_wo_k, f] = tfestimate(wo_k.u, wo_k.Vs, win, Noverlap, Nfft, 1/Ts);
+[frf_wi_k, ~] = tfestimate(wi_k.u, wi_k.Vs, win, Noverlap, Nfft, 1/Ts);
+
+%% Model for the high pass filter
+C = 5.1e-6; % Sensor Stack capacitance [F]
+R = 80.6e3; % Parallel Resistor [Ohm]
+
+f0 = 1/(2*pi*R*C); % Crossover frequency of RC HPF [Hz]
+
+G_hpf = 0.6*(s/(2*pi*f0))/(1 + s/(2*pi*f0));
+
+%% Compare the HPF model and the measured FRF
+figure;
+tiledlayout(2, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+plot(f, abs(frf_wo_k), 'DisplayName', 'Without $R$');
+plot(f, abs(frf_wi_k), 'DisplayName', 'With $R$');
+plot(f, abs(squeeze(freqresp(G_hpf, f, 'Hz'))), 'k--', 'DisplayName', 'RC model');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([2e-1, 1e0]);
+yticks([0.2, 0.5, 1]);
+legend('location', 'southeast', 'FontSize', 8);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(frf_wo_k));
+plot(f, 180/pi*angle(frf_wi_k));
+plot(f, 180/pi*angle(squeeze(freqresp(G_hpf, f, 'Hz'))), 'k--', 'DisplayName', 'RC');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:45:360); ylim([-5, 60]);
+yticks([0, 15, 30, 45, 60]);
+
+linkaxes([ax1,ax2],'x');
+xlim([0.2, 8]);
+xticks([0.2, 0.5, 1, 2, 5]);
+
+%% Load identification Data
+data = load("2023-03-17_11-28_iff_plant.mat");
+
+%% Spectral Analysis setup
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(5/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+%% Compute the transfer function from applied force to measured rotation
+[G_iff, f]  = tfestimate(data.id_plant, data.Vs, win, Noverlap, Nfft, 1/Ts);
+
+%% Basic manually tuned model
+w0z = 2*pi*42.7; % Zero frequency
+xiz = 0.004; % Zero damping
+
+w0p = 2*pi*95.2; % Pole frequency
+xip = 0.02; % Pole damping
+
+G_iff_model = exp(-2*s*Ts)*0.64*(1 + 2*xiz/w0z*s + s^2/w0z^2)/(1 + 2*xip/w0p*s + s^2/w0p^2)*(s/(s+2*pi*0.4));
+
+%% Identified IFF plant and manually tuned model of the plant
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(G_iff), 'color', colors(2,:), 'DisplayName', 'Identified plant')
+plot(f, abs(squeeze(freqresp(G_iff_model, f, 'Hz'))), 'k--', 'DisplayName', 'Manual fit')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+
+ax2 = nexttile;
+hold on;
+plot(f, 180/pi*angle(G_iff), 'color', colors(2,:));
+plot(f, 180/pi*angle(squeeze(freqresp(G_iff_model, f, 'Hz'))), 'k--')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:45:360);
+ylim([-90, 180])
+
+linkaxes([ax1,ax2],'x');
+xlim([0.2, 1e3]);
+
+%% Integral Force Feedback Controller
+K_iff = -10*(1/(s + 2*pi*20)) * ... % LPF: provides integral action above 20Hz
+            (s/(s + 2*pi*2))  * ... % HPF: limit low frequency gain
+            (1/(1 + s/2/pi/2e3));   % LPF: more robust to high frequency resonances
+
+%% Load Data
+data = load("2023-03-17_14-10_damped_plants_new.mat");
+
+%% Spectral Analysis setup
+Ts = 1e-4; % Sampling Time [s]
+Nfft = floor(1/Ts);
+win = hanning(Nfft);
+Noverlap = floor(Nfft/2);
+
+%% Get the frequency vector
+[~, f]  = tfestimate(data.data(1).id_plant(1:end), data.data(1).dL(1:end), win, Noverlap, Nfft, 1/Ts);
+
+%% Gains used for analysis are between 1 and 50
+i_kept = [5:10];
+
+%% Identify the damped plant from u' to de for different IFF gains
+G_dL_frf = {zeros(1,length(i_kept))};
+
+for i = 1:length(i_kept)
+    [G_dL, ~]  = tfestimate(data.data(i_kept(i)).id_plant(1:end), data.data(i_kept(i)).dL(1:end), win, Noverlap, Nfft, 1/Ts);
+    G_dL_frf(i) = {G_dL};
+end
+
+%% Fit the data with 2nd order transfer function using vectfit3
+opts = struct();
+
+opts.stable = 1;    % Enforce stable poles
+opts.asymp = 1;     % Force D matrix to be null
+opts.relax = 1;     % Use vector fitting with relaxed non-triviality constraint
+opts.skip_pole = 0; % Do NOT skip pole identification
+opts.skip_res = 0;  % Do NOT skip identification of residues (C,D,E)
+opts.cmplx_ss = 0;  % Create real state space model with block diagonal A
+
+opts.spy1 = 0;      % No plotting for first stage of vector fitting
+opts.spy2 = 0;      % Create magnitude plot for fitting of f(s)
+
+Niter = 100; % Number of iteration.
+N = 2; % Order of approximation
+poles = [-25 - 1i*60, -25 + 1i*60]; % First get for the pole location
+
+G_dL_id  = {zeros(1,length(i_kept))};
+
+% Identification just between two frequencies
+f_keep = (f>5 & f<200);
+
+for i = 1:length(i_kept)
+    %% Estimate resonance frequency and damping
+    for iter = 1:Niter
+        [G_est, poles, ~, frf_est] = vectfit3(G_dL_frf{i}(f_keep).', 1i*2*pi*f(f_keep)', poles, ones(size(f(f_keep)))', opts);
+    end
+    G_dL_id(i) = {ss(G_est.A, G_est.B, G_est.C, G_est.D)};
+end
+
+%% Identified dynamics from u' to de for different IFF gains
+figure;
+tiledlayout(1, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile();
+hold on;
+for i = 1:length(i_kept)
+    plot(f, abs(G_dL_frf{i}), 'color', [colors(i,:), 1], 'DisplayName', sprintf('g = %.0f', data.gains(i_kept(i))))
+    plot(f, abs(squeeze(freqresp(G_dL_id{i}, f, 'Hz'))), '--', 'color', [colors(i,:), 1], 'HandleVisibility', 'off')
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+xlabel('Frequency [Hz]'); ylabel('Amplitude $d_e/u^\prime$ [m/V]');
+xlim([10, 1e3]);
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+
+%% Root Locus of the APA300ML with Integral Force Feedback
+% Comparison between the computed root locus from the plant model and the root locus estimated from the damped plant pole identification
+gains = logspace(-1, 3, 1000);
+
+figure;
+hold on;
+G_iff_poles = pole(pade(G_iff_model));
+i = imag(G_iff_poles) > 100; % Only keep relevant poles
+plot(real(G_iff_poles(i)),  imag(G_iff_poles(i)),  'kx', ...
+    'DisplayName', '$g = 0$');
+G_iff_zeros = tzero(G_iff_model);
+i = imag(G_iff_zeros) > 100; % Only keep relevant zeros
+plot(real(G_iff_zeros(i)), imag(G_iff_zeros(i)), 'ko', ...
+    'HandleVisibility', 'off');
+
+for g = gains
+    clpoles = pole(feedback(pade(G_iff_model), g*K_iff, 1));
+    i = imag(clpoles) > 100; % Only keep relevant poles
+    plot(real(clpoles(i)), imag(clpoles(i)), 'k.', ...
+        'HandleVisibility', 'off');
+end
+
+for i = 1:length(i_kept)
+    plot(real(pole(G_dL_id{i})), imag(pole(G_dL_id{i})), 'x', 'color', [colors(i,:), 1], 'DisplayName', sprintf('g = %1.f', data.gains(i_kept(i))));
+end
+xlabel('Real Part')
+ylabel('Imaginary Part')
+axis equal
+ylim([0, 610]);
+xlim([-300,0]);
+legend('location', 'southwest', 'FontSize', 8);

C1-test-bench-apa/test_apa_3_model_2dof.m

@@ -0,0 +1,166 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Path for functions, data and scripts
+addpath('./src/'); % Path for scripts
+addpath('./mat/'); % Path for data
+
+addpath('./STEPS/'); % Path for Simscape Model
+
+%% Linearization options
+opts = linearizeOptions;
+opts.SampleTime = 0;
+
+%% Open Simscape Model
+mdl = 'test_apa_simscape'; % Name of the Simulink File
+open(mdl); % Open Simscape Model
+
+%% Colors for the figures
+colors = colororder;
+
+%% Input/Output definition of the Model
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/u'],  1, 'openinput');  io_i = io_i + 1; % DAC Voltage
+io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
+io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
+
+%% Frequency vector for analysis
+freqs = 5*logspace(0, 3, 1000);
+
+%% Stiffness values for the 2DoF APA model
+k1 = 0.38e6; % Estimated Shell Stiffness [N/m]
+
+w0 = 2*pi*95; % Resonance frequency [rad/s]
+m = 5.7; % Suspended mass [kg]
+ktot = m*(w0)^2; % Total Axial Stiffness to have to wanted resonance frequency [N/m]
+
+ka = 1.5*(ktot-k1); % Stiffness of the (two) actuator stacks [N/m]
+ke = 2*ka; % Stiffness of the Sensor stack [N/m]
+
+%% Damping values for the 2DoF APA model
+c1 = 5;   % Damping for the Shell [N/(m/s)]
+ca = 50;  % Damping of the actuators stacks [N/(m/s)]
+ce = 2*ca; % Damping of the sensor stack [N/(m/s)]
+
+%% Estimation ot the sensor and actuator sensitivities
+% Initialize the structure with unitary sensor and actuator "sensitivities"
+n_hexapod = struct();
+n_hexapod.actuator = initializeAPA(...
+    'type', '2dof', ...
+    'k',  k1,  ...
+    'ka', ka,  ...
+    'ke', ke,  ...
+    'c',  c1, ...
+    'ca', ca, ...
+    'ce', ce, ...
+    'Ga', 1,   ... % Actuator constant [N/V]
+    'Gs', 1    ... % Sensor constant [V/m]
+);
+
+c_granite = 50; % Do not take into account damping added by the air bearing
+
+% Run the linearization
+G_norm = linearize(mdl, io, 0.0, opts);
+G_norm.InputName  = {'u'};
+G_norm.OutputName = {'Vs', 'de'};
+
+% Load Identification Data to estimate the two sensitivities
+load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
+
+% Estimation ot the Actuator sensitivity
+fa = 10; % Frequency where the two FRF should match [Hz]
+[~, i_f] = min(abs(f - fa));
+ga = -abs(enc_frf(i_f,1))./abs(evalfr(G_norm('de', 'u'), 1i*2*pi*fa));
+
+% Estimation ot the Sensor sensitivity
+fs = 600; % Frequency where the two FRF should match [Hz]
+[~, i_f] = min(abs(f - fs));
+gs = -abs(iff_frf(i_f,1))./abs(evalfr(G_norm('Vs', 'u'), 1i*2*pi*fs))/ga;
+
+%% 2DoF APA300ML with optimized parameters
+n_hexapod = struct();
+n_hexapod.actuator = initializeAPA( ...
+    'type', '2dof', ...
+    'k',  k1,  ...
+    'ka', ka,  ...
+    'ke', ke,  ...
+    'c',  c1, ...
+    'ca', ca, ...
+    'ce', ce, ...
+    'Ga', ga, ...
+    'Gs', gs  ...
+);
+
+%% Identification of the APA300ML with optimized parameters
+G_2dof = exp(-s*Ts)*linearize(mdl, io, 0.0, opts);
+G_2dof.InputName  = {'u'};
+G_2dof.OutputName = {'Vs', 'de'};
+
+%% Comparison of the measured FRF and the optimized 2DoF model of the APA300ML
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
+for i = 1:length(apa_nums)
+    plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(G_2dof('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '2DoF Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:))
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);
+
+%% Comparison of the measured FRF and the optimized 2DoF model of the APA300ML
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(iff_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
+for i = 2:length(apa_nums)
+    plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(G_2dof('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '2DoF Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_2dof('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:))
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);

C1-test-bench-apa/test_apa_4_model_flexible.m

@@ -0,0 +1,132 @@
+%% Clear Workspace and Close figures
+clear; close all; clc;
+
+%% Intialize Laplace variable
+s = zpk('s');
+
+%% Path for functions, data and scripts
+addpath('./src/'); % Path for scripts
+addpath('./mat/'); % Path for data
+
+addpath('./STEPS/'); % Path for Simscape Model
+
+%% Linearization options
+opts = linearizeOptions;
+opts.SampleTime = 0;
+
+%% Open Simscape Model
+mdl = 'test_apa_simscape'; % Name of the Simulink File
+open(mdl); % Open Simscape Model
+
+%% Colors for the figures
+colors = colororder;
+
+%% Input/Output definition of the Model
+clear io; io_i = 1;
+io(io_i) = linio([mdl, '/u'],  1, 'openinput');  io_i = io_i + 1; % DAC Voltage
+io(io_i) = linio([mdl, '/Vs'], 1, 'openoutput'); io_i = io_i + 1; % Sensor Voltage
+io(io_i) = linio([mdl, '/de'], 1, 'openoutput'); io_i = io_i + 1; % Encoder
+
+%% Frequency vector for analysis
+freqs = 5*logspace(0, 3, 1000);
+
+%% Identification of the actuator and sensor "constants"
+% Initialize the APA as a flexible body with unity "constants"
+n_hexapod.actuator = initializeAPA(...
+    'type', 'flexible', ...
+    'ga', 1, ...
+    'gs', 1);
+
+c_granite = 50; % Rought estimation of the damping added by the air bearing
+
+% Identify the dynamics
+G_norm = linearize(mdl, io, 0.0, opts);
+G_norm.InputName  = {'u'};
+G_norm.OutputName = {'Vs', 'de'};
+
+% Load Identification Data to estimate the two gains
+load('meas_apa_frf.mat', 'f', 'Ts', 'enc_frf', 'iff_frf', 'apa_nums');
+
+% Actuator Constant in [N/V]
+ga = -mean(abs(enc_frf(f>10 & f<20)))./dcgain(G_norm('de', 'u'));
+
+% Sensor Constant in [V/m]
+gs = -mean(abs(iff_frf(f>400 & f<500)))./(ga*abs(squeeze(freqresp(G_norm('Vs', 'u'), 1e3, 'Hz'))));
+
+%% Idenfify the dynamics of the Simscape model with correct actuator and sensor "constants"
+% Initialize the APA
+n_hexapod.actuator = initializeAPA(...
+    'type', 'flexible', ...
+    'ga',   23.2, ... % Actuator sensitivity [N/V]
+    'gs',   -4.9e6);    % Sensor sensitivity [V/m]
+
+% Identify with updated constants
+G_flex = exp(-Ts*s)*linearize(mdl, io, 0.0, opts);
+G_flex.InputName  = {'u'};
+G_flex.OutputName = {'Vs', 'de'};
+
+%% Comparison of the measured FRF and the "Flexible" model of the APA300ML
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(enc_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
+for i = 1:length(apa_nums)
+    plot(f, abs(enc_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(G_flex('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '"Flexible" Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $d_e/u$ [m/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-8, 1e-3]);
+legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 1);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, 180/pi*angle(enc_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('de', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:))
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);
+
+%% Comparison of the measured FRF and the "Flexible" model of the APA300ML
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(f, abs(iff_frf(:, 1)), 'color', [0,0,0,0.2], 'DisplayName', 'Identified');
+for i = 2:length(apa_nums)
+    plot(f, abs(iff_frf(:, i)), 'color', [0,0,0,0.2], 'HandleVisibility', 'off');
+end
+plot(freqs, abs(squeeze(freqresp(G_flex('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:), 'DisplayName', '"Flexible" Model')
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude $V_s/u$ [V/V]'); set(gca, 'XTickLabel',[]);
+hold off;
+ylim([1e-2, 1e2]);
+legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
+
+ax2 = nexttile;
+hold on;
+for i = 1:length(apa_nums)
+    plot(f, 180/pi*angle(iff_frf(:, i)), 'color', [0,0,0,0.2]);
+end
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_flex('Vs', 'u'), freqs, 'Hz'))), '--', 'color', colors(2,:))
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360); ylim([-180, 180]);
+
+linkaxes([ax1,ax2],'x');
+xlim([10, 2e3]);