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−% STEADYSTATE returns a possible value for the steady state of an ssm.
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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−%
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−% DESCRIPTION: STEADYSTATE returns a possible value for the steady state
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−%              of the state space of an ssm with given inputs.
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−%
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−% CALL: [pl_out] = steadyState(sys, pl)
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−%
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−% INPUTS:
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−%         - sys, an ssm object
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−%
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−% OUTPUTS:
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−%         _ pl_out contains 'state', the random state position
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−%
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−% <a href="matlab:utils.helper.displayMethodInfo('ssm', 'steadyState')">Parameters Description</a>
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−%
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−% VERSION: $Id: steadyState.m,v 1.11 2011/04/08 08:56:23 hewitson Exp $
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−%
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−%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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−
+
−% TO DO: Check input aos for the timestep, tsdata, and ssm.timestep
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−% options to be defined (NL case)
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−% add check if one input mach no ssm input variable
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−% allow use of other LTPDA functions to generate white noise
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−
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−
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−function varargout = steadyState(varargin)
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−  
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−  %% starting initial checks
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−  
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−  % Check if this is a call for parameters
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−  if utils.helper.isinfocall(varargin{:})
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−    varargout{1} = getInfo(varargin{3});
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−    return
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−  end
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−  
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−  utils.helper.msg(utils.const.msg.MNAME, ['running ', mfilename]);
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−  
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−  % Collect input variable names
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−  in_names = cell(size(varargin));
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−  for ii = 1:nargin,in_names{ii} = inputname(ii);end
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−  
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−  % Collect all SSMs and plists
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−  [sys, ssm_invars, rest] = utils.helper.collect_objects(varargin(:), 'ssm', in_names);
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−  [pl, invars2, rest]  = utils.helper.collect_objects(rest(:), 'plist');
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−  if ~isempty(rest)
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−    pl = combine(pl, plist(rest{:}));
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−  end
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−  pl = combine(pl, getDefaultPlist());
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−  
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−  %% begin function body
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−  
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−  if numel(sys)~=1
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−    error('simulate needs exactly one ssm as an input')
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−  end
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−  if ~sys.isnumerical
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−    error(['error because system ',sys.name,' is not numerical']);
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−  end
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−  timestep  = sys.timestep;
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−  if timestep==0
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−    error('timestep should not be 0 in steadyState!!')
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−  end
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−  if pl.isparam('noise variable names')
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−    error('The noise option used must be split between "covariance" and "cpsd". "noise variable names" does not exist anymore!')
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−  end
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−  sssizes = sys.sssizes;
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−  %% collecting simulation i/o data
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−  
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−  constants_in = find(pl, 'constants');
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−  cov_in = find(pl, 'covariance');
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−  cpsd_in = find(pl, 'CPSD');
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−  noise_in = blkdiag(cov_in, cpsd_in/(timestep*2));
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−  [U1,S1,V1] = svd(noise_in.');
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−  if (sum(S1<0)>0)
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−    error('Covariance matrix is not positive definite')
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−  end
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−  noise_mat = U1*sqrt(S1);
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−  
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−  %% modifying system's ordering
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−  if find(pl, 'reorganize')
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−    sys = reorganize(sys, pl, 'set', 'for simulate', 'internal', 'internal');
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−  end
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−  
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−  %% getting system's i/o sizes
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−  inputSizes = sys.inputsizes;
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−  
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−  Nnoise = inputSizes(2);
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−  Nconstants = inputSizes(3);
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−  
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−  if numel(diag(noise_in))~=Nnoise
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−    error(['There are ' num2str(numel(diag(noise_in))) ' input noise variances and ' num2str(Nnoise) ' corresponding inputs indexed.' ])
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−  elseif numel(constants_in)~=Nconstants
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−    error(['There are ' num2str(numel(constants_in)) ' input constants and ' num2str(Nconstants) ' corresponding inputs indexed.' ])
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−  end
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−  
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−  A = sys.amats{1,1};
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−  Bnoise   = sys.bmats{1,2} * noise_mat;
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−  Bcst     = sys.bmats{1,3} * reshape(constants_in, Nconstants, 1);
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−  
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−  %% counting powers of 2 to use for initilization
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−  nSteps = 500;
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−  tSteady =  find(pl, 'tSteady');
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−  nPow2 = nextpow2(tSteady/(nSteps*timestep));
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−  
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−  %% simulation loop
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−  A_pow2=cell(1,nPow2);
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−  G_pow2=cell(1,nPow2);
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−  
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−  A_pow2{1} = A;
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−  G_pow2{1} = Bcst;
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−  
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−  %% method 1 :  iterate equations with growing time-step for a very long time
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−  E_pow2=cell(1,nPow2);
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−  E_pow2{1} = Bnoise;
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−  for i_pow2 = 2:nPow2
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−    G_pow2{i_pow2} = G_pow2{i_pow2-1} + A_pow2{i_pow2-1}*G_pow2{i_pow2-1};
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−    E_pow2{i_pow2} = E_pow2{i_pow2-1} + A_pow2{i_pow2-1}*E_pow2{i_pow2-1};
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−    A_pow2{i_pow2} = A_pow2{i_pow2-1}^2;
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−  end
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−  lastX = zeros(size(A,1),1);
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−  for i_pow2 = fliplr(1:nPow2)
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−    A = A_pow2{i_pow2};
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−    G = G_pow2{i_pow2};
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−    E = E_pow2{i_pow2};
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−    noise_array = randn(size(E,2), nSteps);
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−    for i_steps = 1:nSteps
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−      lastX  = A*lastX +  G + E*noise_array(:,i_steps) ;
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−    end
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−  end
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−  
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−  %% method 2 : compute the limit state-mean and covariance as i_pow2 tends to infinity
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−  %   P_pow2=cell(1,nPow2);
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−  %   P_pow2{1} = Bnoise*Bnoise.';
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−  %   for i_pow2 = 2:nPow2
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−  %     G_pow2{i_pow2} = G_pow2{i_pow2-1} + A_pow2{i_pow2-1}*G_pow2{i_pow2-1}; % taking step response to 2 longer time;
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−  %     P_pow2{i_pow2} = P_pow2{i_pow2-1} + A_pow2{i_pow2-1}*P_pow2{i_pow2-1}*(A_pow2{i_pow2-1}.');% taking state covariance to 2 longer time;
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−  %     A_pow2{i_pow2} = A_pow2{i_pow2-1}^2;
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−  %   end
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−  %   [U1,S1,V1] = svd(P_pow2{nPow2});
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−  %   lastX = U1*sqrt(S1)*randn(size(A,1),1) + G_pow2{nPow2};
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−  
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−  %% construct output analysis object
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−  plist_out = plist('state', ssm.blockMatRecut(lastX,sssizes,1) );
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−  varargout  = {plist_out};
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−end
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−
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−
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−%--------------------------------------------------------------------------
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−% Get Info Object
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−%--------------------------------------------------------------------------
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−function ii = getInfo(varargin)
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−  
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−  if nargin == 1 && strcmpi(varargin{1}, 'None')
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−    sets = {};
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−    pl   = [];
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−  else
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−    sets = {'Default'};
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−    pl   = getDefaultPlist;
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−  end
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−  % Build info object
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−  ii = minfo(mfilename, 'ssm', 'ltpda', utils.const.categories.op, '$Id: steadyState.m,v 1.11 2011/04/08 08:56:23 hewitson Exp $', sets, pl);
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−end
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−
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−%--------------------------------------------------------------------------
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−% Get Default Plist
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−%--------------------------------------------------------------------------
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−function pl = getDefaultPlist()
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−  pl = plist();
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−  
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−  p = param({'cpsd variable names', 'A cell-array of strings specifying the desired input variable names.'}, {} );
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−  pl.append(p);
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−  
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−  p = param({'cpsd', 'The covariance of this noise between input ports for the <i>time-continuous</i> noise model.'}, []);
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−  pl.append(p);
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−  
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−  p = param({'covariance variable names', 'A cell-array of strings specifying the desired input variable names.'}, {} );
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−  pl.append(p);
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−  
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−  p = param({'covariance', 'The covariance of this noise between input ports for the <i>time-continuous</i> noise model.'}, []);
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−  pl.append(p);
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−  
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−  p = param({'constant variable names', 'A cell-array of strings of the desired input variable names.'}, {});
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−  pl.append(p);
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−  
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−  p = param({'constants', 'Array of DC values for the different corresponding inputs.'}, paramValue.DOUBLE_VALUE(zeros(0,1)));
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−  pl.append(p);
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−  
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−  p = param({'tSteady', 'The settling time used in the calculation, in the same unit as the ssm''s timestep'}, paramValue.DOUBLE_VALUE(10^6) );
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−  pl.append(p);
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−  
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−  p = param({'reorganize', 'When set to 0, this means the ssm does not need be modified to match the requested i/o. Faster but dangerous!'}, paramValue.TRUE_FALSE);
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−  pl.append(p);
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−  
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−end
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−