File: bstap.sci

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function [Q]=bstap(sl)
// Best approximant Q of Sl
//      ||Sl-Q||    =  ||Tsl||
//              inf
// ||Tsl|| norm of hankel operator
//-- sl is assumed antistable
//-- Q : best stable approximation of Sl
//!
//balancing
//-----------------------------------
// Copyright INRIA
slt=gtild(sl);slt=balreal(slt);sl=gtild(slt),
// D such that DB1'+sC1 = 0 , DD' = s**2I
//-------------------------------------------------
  [a,b,c]=sl(2:4),[n,n]=size(a),[m,n]=size(c),
  x=-obs_gram(sl),s=x(1,1),
  [w,r]=rowcomp(x-s*eye()),r=n-r,
  b1=b(1:r,:),c1=c(:,1:r),
  [u1,s1,v1]=svd(-c1'),[u2,s2,v2]=svd(b1),
  v2=v2(:,1:m),dd=s*v1*v2',
//
//--------------------------------
  a22=a(r+1:n,r+1:n),
  b2=b(r+1:n,:),c2=c(:,r+1:n),
  sig=x(r+1:n,r+1:n),
  bb=-inv(s**2*eye()-sig*sig)*(sig*b2+s*c2'*dd),
  aa=-(a22+b2*bb')',cc=c2*sig+dd*b2',
  Q=syslin('c',aa,bb,cc,dd),
endfunction