image thumbnail

emgr - Empirical Gramian Framework

by

Christian (view profile)

 

04 Feb 2013 (Updated )

A framework for computating empirical gramians. Compatible with Octave and Matlab.

emgr(f,g,s,t,w,pr,nf,ut,us,xs,um,xm)
function W = emgr(f,g,s,t,w,pr,nf,ut,us,xs,um,xm)
% emgr - Empirical Gramian Framework ( Version: 3.1 )
% by Christian Himpe 2013-2015 ( http://gramian.de )
% released under BSD 2-Clause License ( opensource.org/licenses/BSD-2-Clause )
%
% SYNTAX:
%    W = emgr(f,g,s,t,w,[pr],[nf],[ut],[us],[xs],[um],[xm]);
%
% SUMMARY:
%    emgr - Empirical Gramian Framemwork,
%    computation of empirical gramians for model reduction,
%    system identification and uncertainty quantification.
%    Enables gramian-based nonlinear model order reduction.
%    Compatible with OCTAVE and MATLAB.
%
% ARGUMENTS:
%   (func handle)  f - system function handle; signature: xdot = f(x,u,p)
%   (func handle)  g - output function handle; signature:    y = g(x,u,p)
%        (vector)  s - system dimensions [inputs,states,outputs]
%        (vector)  t - time discretization [start,step,stop]
%          (char)  w - gramian type:
%            * 'c' : empirical controllability gramian (WC)
%            * 'o' : empirical observability gramian (WO)
%            * 'x' : empirical cross gramian (WX)
%            * 'y' : empirical linear cross gramian (WY)
%            * 's' : empirical sensitivity gramian (WS)
%            * 'i' : empirical identifiability gramian (WI)
%            * 'j' : empirical joint gramian (WJ)
% (matrix,vector,scalar) [pr = 0] - parameters, each column is one set
%        (vector,scalar) [nf = 0] - options, 12 components:
%            + zero(0), mean(1), steady(2), median(3), midrange(4), rms(5)
%            + linear(0), log(1), geom(2), single(3), sparse(4) input scales
%            + linear(0), log(1), geom(2), single(3), sparse(4) state scales
%            + unit(0), reciproce(1), dyadic(2), single(3) input rotations
%            + unit(0), reciproce(1), dyadic(2), single(3) state rotations
%            + single(0), double(1), scaled(2) run
%            + default(0), data-driven gramians(1); only: WC,WO,WX,WY
%            + default(0), robust parameters(1); only: WC,WY
%            + default(0), parameter centering(1); only: WC,WX,WY,WS,WI,WJ
%            + default(0), exclusive options:
%                  * use mean-centered(1); only: WS
%                  * use schur-complement(1); only: WI
%                  * non-symmetric cross gramian(1); only: WX,WJ
%            + default(0), enforce gramian symmetry(1); only: WC,WO,WX,WY
%            + Improved-Runge-Kutta-3(0), custom solver global handle(-1)
%  (matrix,vector,scalar) [ut = 1] - input; default: delta impulse
%         (vector,scalar) [us = 0] - steady-state input
%         (vector,scalar) [xs = 0] - steady-state, initial state x0
%  (matrix,vector,scalar) [um = 1] - input scales
%  (matrix,vector,scalar) [xm = 1] - initial-state scales
%
% RETURNS:
%            (matrix)  W - Gramian Matrix (WC, WO, WX, WY only)
%              (cell)  W - {State-,Parameter-} Gramian (WS, WI, WJ only)
%
% CITATION:
%    C. Himpe (2015). emgr - Empirical Gramian Framework (Version 3.1)
%    [Computer Software]. Available from http://gramian.de
%
% SEE ALSO:
%    gram
%
% KEYWORDS:
%    model reduction, empirical gramian, cross gramian, mor
%
% Further information: <http://gramian.de>
%*

    % Version Info
    if(nargin==1 && strcmp(f,'version')), W = 3.1; return; end;

    % Default Arguments
    if(nargin<6) ||(isempty(pr)), pr = 0.0; end;
    if(nargin<7) ||(isempty(nf)), nf = 0;   end;
    if(nargin<8) ||(isempty(ut)), ut = 1.0; end;
    if(nargin<9) ||(isempty(us)), us = 0.0; end;
    if(nargin<10)||(isempty(xs)), xs = 0.0; end;
    if(nargin<11)||(isempty(um)), um = 1.0; end;
    if(nargin<12)||(isempty(xm)), xm = 1.0; end;

    % System Dimensions
    J = s(1);                 % number of inputs
    N = s(2);                 % number of states
    O = s(3);                 % number of outputs

    M = N; if(numel(s)==4), M = s(4); end; % number of non-constant states

    h = t(2);                 % width of time step
    T = round((t(3)-t(1))/h); % number of time steps

    P = size(pr,1);           % number of parameters
    Q = size(pr,2);           % number of parameter sets

    w = lower(w);             % ensure lower case gramian type

    if(isnumeric(ut) && numel(ut)==1 && ut==Inf) % Chirp Input
        ut = @(t) 0.5*cos(pi./t)+0.5;
    end;

    if(isa(ut,'function_handle')) % Discretize Procedural Input
        uf = ut;
        ut = zeros(J,T);
        for l=1:T
            ut(:,l) = uf(l*h);
        end;
    end;

    % Lazy Arguments
    if(isnumeric(g) && g==1), g = @(x,u,p) x; O = N; end;

    if(numel(nf)<12), nf(12)    = 0;  end;
    if(numel(ut)==1), ut(1:J,1) = (1.0/h)*ut; end;
    if(numel(us)==1), us(1:J,1) = us; end;
    if(numel(xs)==1), xs(1:N,1) = xs; end;
    if(numel(um)==1), um(1:J,1) = um; end;
    if(numel(xm)==1), xm(1:N+(w=='y')*(J-N),1) = xm; end;

    if(size(ut,2)==1), ut(:,2:T) = 0.0; end;
    if(size(us,2)==1), us = repmat(us,[1,T]); end;
    if(size(um,2)==1), um = scales(um,nf(2),nf(4)); end;
    if(size(xm,2)==1), xm = scales(xm,nf(3),nf(5)); end;

%% PARAMETRIC SETUP

    if( (nf(8) && w~='o') || w=='s' || w=='i' || w=='j')

        Q = 1;

        if(nf(8) || w=='s') % Assemble (Controllability) Parameter Scales
            if(size(pr,2)==1)
                pm = scales(pr,nf(2),nf(4));
            else
                pn = size(um,2);
                pm = max(pr,[],2)*(1:pn)./pn - min(pr,[],2)*((1:pn)./pn - 1.0);
                pr = min(pr,[],2);
            end;
        end;

        if(w=='i' || w=='j') % Assemble (Observability) Parameter Scales
            if(size(pr,2)==1)
                pm = scales(pr,nf(3),nf(5));
            else
                pn = size(xm,2);
                pm = max(pr,[],2)*(1:pn)./pn - min(pr,[],2)*((1:pn)./pn - 1.0);
                pr = min(pr,[],2);
            end;
        end;

        if(nf(9)) % Parameter Centering
            pr = mean(pm,2);
            pm = bsxfun(@minus,pm,pr);
        end;

        assert(all(pm(:)),'ERROR! emgr: zero parameter scales!');
    end;

%% STATE-SPACE SETUP

    if(w=='c' || w=='o' || w=='x' || w=='y')

        C = size(um,2); % number of input scales
        D = size(xm,2); % number of state scales

        switch(nf(1)) % residuals

            case 1, % mean state
                res = @(d) mean(d,2);

            case 2, % steady state
                res = @(d) d(:,end);

            case 3, % median state
                res = @(d) median(d,2);

            case 4, % midrange
                res = @(d) 0.5*(min(d,2)+max(d,2));

            case 5, % rms
                res = @(d) sqrt(sum(d.*d,2));

            otherwise, % zero state
                res = @(d) zeros(size(d,1),1);
        end;

        switch(nf(6)) % scaled runs

            case 1, % preconditioned run
                nf(6) = 0;
                WT = emgr(f,g,s,t,w,pr,nf,ut,us,xs,um,xm);
                TX = sqrt(diag(WT));
                TX = TX(1:M);
                tx = 1.0./TX;
                F = f; f = @(x,u,p) TX.*F(tx.*x,u,p);
                G = g; g = @(x,u,p)     G(tx.*x,u,p);

            case 2, % steady scaled run
                TU = us(:,1);
                TX = xs;
                TX = TX(1:M);
                TU(TU==0) = 1.0; tu = 1.0./TU;
                TX(TX==0) = 1.0; tx = 1.0./TX;
                F = f; f = @(x,u,p) TX.*F(tx.*x,tu.*u,p);
                G = g; g = @(x,u,p)     G(tx.*x,tu.*u,p);
        end;

        if(nf(7)) % data-driven
            um = ones(J,size(um,2));
            xm = ones(N,size(xm,2));
        end;

        if(nf(8)) % robust parameter
            J = J + P;
            ut = [ut;ones(P,T)];
            us = [us;repmat(pr,[1,T])];
            um = [um;pm];
            if(w=='y'), xm = [xm;pm]; end;
            F = f; f = @(x,u,p) F(x,u(1:J-P),u(J-P+1:J));
            G = g; g = @(x,u,p) G(x,u(1:J-P),u(J-P+1:J));
        end;

        if(nf(12)) % Set Solver
            global CUSTOM_ODE;
            ode = CUSTOM_ODE;
        else
            ode = @irk3;
        end;

        assert(all(um(:)),'ERROR! emgr: zero input scales!');
        assert(all(xm(:)),'ERROR! emgr: zero state scales!');

        W = zeros(N-(M<N && w=='x')*P,N); % preallocate gramian
    end;

%% GRAMIAN COMPUTATION

    switch(w) % by empirical gramian types

        case 'c', % controllability gramian
            for q=1:Q
                pp = pr(:,q);
                for c=1:C
                    for j=1:J % parfor
                        uu = us + bsxfun(@times,ut,sparse(j,1,um(j,c),J,1));
                        x = ode(f,1,h,T,xs,uu,pp);
                        x = bsxfun(@minus,x,res(x))*(1.0/um(j,c));
                        W = W + (x*x'); % huma
                    end;
                end;
            end;
            W = W * (h/(C*Q));

        case 'o', % observability gramian
            o = zeros(O*T,N);
            for q=1:Q
                pp = pr(:,q);
                for d=1:D
                    for n=1:N % parfor
                        xx = xs + sparse(n,1,xm(n,d),N,1);
                        if(M<N)
                            y = ode(f,g,h,T,xx(1:M),us,xx(M+1:end));
                        else
                            y = ode(f,g,h,T,xx,us,pp);
                        end;
                        y = bsxfun(@minus,y,res(y))*(1.0/xm(n,d));
                        o(:,n) = reshape(y,[O*T,1]);
                    end;
                    W = W + (o'*o); % huma
                end;
            end;
            W = W * (h/(D*Q));

        case 'x', % cross gramian
            assert(J==O || nf(10),'ERROR! emgr: non-square system!');
            o = zeros(O,T,N);
            for q=1:Q
                pp = pr(:,q);
                for d=1:D
                    for n=1:N % parfor
                        xx = xs + sparse(n,1,xm(n,d),N,1);
                        if(M<N)
                            y = ode(f,g,h,T,xx(1:M),us,xx(M+1:end));
                        else
                            y = ode(f,g,h,T,xx,us,pp);
                        end;
                        o(:,:,n) = bsxfun(@minus,y,res(y))*(1.0/xm(n,d));
                    end;
                    o = permute(o,[2,3,1]); % Generalized Transposition
                    for c=1:C
                        for j=1:J % parfor
                            uu = us + bsxfun(@times,ut,sparse(j,1,um(j,c),J,1));
                            if(M<N)
                                x = ode(f,1,h,T,xs(1:M),uu,xs(M+1:end));
                            else
                                x = ode(f,1,h,T,xs,uu,pp);
                            end;
                            x = bsxfun(@minus,x,res(x))*(1.0/um(j,c));
                            if(nf(10)), K = 1:O; else K = j; end;
                            for k=K
                                W = W + (x*o(:,:,k)); % huma
                            end;
                        end;
                    end;
                    o = reshape(o,O,T,N);
                end;
            end;
            W = W * (h/(C*D*Q));

        case 'y', % linear cross gramian
            assert(J==O || nf(8),'ERROR! emgr: non-square system!');
            for q=1:Q
                pp = pr(:,q);
                for c=1:C
                    for j=1:J % parfor
                        uu = us + bsxfun(@times,ut,sparse(j,1,um(j,c),J,1));
                        yy = us + bsxfun(@times,ut,sparse(j,1,xm(j,c),J,1));
                        x = ode(f,1,h,T,xs,uu,pp);
                        y = ode(g,1,h,T,xs,yy,pp);
                        x = bsxfun(@minus,x,res(x))*(1.0/um(j,c));
                        y = bsxfun(@minus,y,res(y))*(1.0/xm(j,c));
                        W = W + (x*y'); % huma
                    end;
                end;
            end;
            W = W * (h/(C*Q));

        case 's', % sensitivity gramian
            W = cell(1,2);
            ps = sparse(P,1);
            nf(8) = 0;
            W{1} = emgr(f,g,[J,N,O],t,'c',ps,nf,ut,us,xs,um,xm);
            W{2} = speye(P);
            F = @(x,u,p) f(x,us(:,1),pr + p*u);
            G = @(x,u,p) g(x,us(:,1),pr + p*u);
            up = ones(1,T);
            ps = speye(P);
            for p=1:P
                V = emgr(F,G,[1,N,O],t,'c',ps(:,p),nf,up,0,xs,pm(p,:),xm);
                W{1} = W{1} + V;      % approximate controllability gramian
                W{2}(p,p) = trace(V); % sensitivity gramian
            end;
            if(nf(10))
                W{2} = W{2} - mean(diag(W{2}));
            end;

        case 'i', % identifiability gramian
            W = cell(1,2);
            ps = sparse(P,1);
            V = emgr(f,g,[J,N+P,O,N],t,'o',ps,nf,ut,us,[xs;pr],um,[xm;pm]);
            W{1} = V(1:N,1:N);         % observability gramian
            W{2} = V(N+1:N+P,N+1:N+P); % approximate identifiability gramian
            if(nf(10))
                W{2} = W{2} - V(N+1:N+P,1:N)*ainv(W{1})*V(1:N,N+1:N+P);
            end;

        case 'j', % joint gramian
            W = cell(1,2);
            ps = sparse(P,1);
            V = emgr(f,g,[J,N+P,O,N],t,'x',ps,nf,ut,us,[xs;pr],um,[xm;pm]);
            W{1} = V(1:N,1:N); % cross gramian
            %W{2} = zeros(P);   % cross-identifiability gramian
            W{2} = -0.5*V(1:N,N+1:N+P)'*ainv(W{1}+W{1}')*V(1:N,N+1:N+P);

        otherwise,
            error('ERROR! emgr: unknown gramian type!');
    end;

    if(nf(11) && (w=='c'||w=='o'||w=='x'||w=='y') ), W = 0.5*(W+W'); end;
end

%% ======== SCALING SELECTOR ========
function s = scales(s,d,e)

    switch(d)

        case 0, % linear
            s = s*[0.25,0.50,0.75,1.0];

        case 1, % logarithmic
            s = s*[0.001,0.01,0.1,1.0];

        case 2, % geometric
            s = s*[0.125,0.25,0.5,1.0];

        case 4, % sparse
            s = s*[0.17,0.5,0.77,0.94];

        otherwise, % single
            %s = s;
    end;

    switch(e)

        case 1, % reciproce
            s = [1.0./s,s];

        case 2, % dyadic
            s = s*s';

        case 3, % single
            %s = s;

        otherwise, % unit
            s = [-s,s];
    end;
end

%% ======== FAST APPROXIMATE INVERSION ========
function x = ainv(m)

    d = 1.0./diag(m);
    n = numel(d);
    x = bsxfun(@times,m,-d);
    x = bsxfun(@times,x,d');
    x(1:n+1:end) = d;
end

%% ======== DEFAULT ODE INTEGRATOR ========
function x = irk3(f,g,h,T,z,u,p)

    if(isnumeric(g) && g==1), g = @(x,u,p) x; end;

    k1 = h*f(z,u(:,1),p); % 2nd Order Midpoint RK2 for starting value
    k2 = h*f(z + 0.5*k1,u(:,1),p);
    z = z + k2;
    x(:,1) = g(z,u(:,1),p);

    x(end,T) = 0; % preallocate trajectory

    k1 = h*f(z,u(:,2),p); % 2nd start value (improves impulse response)
    k2 = h*f(z + 0.5*k1,u(:,2),p);
    z = z + k2;
    x(:,2) = g(z,u(:,2),p);

    for t=3:T % 3rd Order Improved Runge-Kutta IRK3
        l1 = h*f(z,u(:,t),p);
        l2 = h*f(z + 0.5*l1,u(:,t),p);
        z = z + (2.0/3.0)*l1 + (1.0/3.0)*k1 + (5.0/6.0)*(l2 - k2);
        x(:,t) = g(z,u(:,t),p);
        k1 = l1;
        k2 = l2;
    end;
end

Contact us