Project onto SSM

Contents

function varargout = proj2SSM(z0,projMethod,varargin)
        

PROJ2SSM

This function returns the projection of point z0 on an autonomous SSM. The information of SSM is given in the varargin arguments and we support both linear and nonlinear projection methods.

For projMethod=nonlinear we solve an optimization problem argmin_q $||z_0-W(q)||$ , where the map W gives the expansion of SSM.

For projMethod=linear we find q from z0=Vq, which is solved using the orthonormality of V and W(left eigenvectors) with respect to B matrix

switch projMethod
    case 'nonlinear'
        W_0     = varargin{1}; % expansion of SSM
        autData = varargin{2}; % data for mapping between real and complex coordinates
        pV      = varargin{3}; % initial guess of optimization

        fprintf('Call fminunc to find the projection of point on SSM\n');

        options = optimoptions('fminunc','Display','notify');
        options = optimoptions(options,'StepTolerance',1e-10);
        options = optimoptions(options, 'MaxFunctionEvaluations', 1e5);

        [z,fval,exitflag] = fminunc(@(u) squaDist2pointSSM(z0,u,W_0,autData), pV, options);

        % convert real representation to complex one
        x_real = z(autData.realx,:);
        x_comp = z(autData.compx(1:2:end-1),:)+1i*z(autData.compx(2:2:end),:);
        x  = zeros(autData.dim, 1);
        x(autData.realx,:) = x_real;
        x(autData.compx(1:2:end-1),:) = x_comp;
        x(autData.compx(2:2:end),:)   = conj(x_comp);

        varargout{1} = x;        % projected point on SSM (coordinates in modal domain)
        varargout{2} = fval;     % squared distance to projected point
        varargout{3} = exitflag; % flag for convergence of optimization routine

    case 'linear'
        Wm = varargin{1}; % left eigenvectors correspond to master spectral subspace
        B  = varargin{2};
        q  = Wm'*B*z0;
        varargout{1} = q;

    otherwise
        error('Please select projection method to SSM from: {linear, nonlinear}');
end
        

end
        

Published with MATLAB® R2023a