btabal_cf.m

function [sysr,hsv] = btabal_cf(fact,sys,tol,ord,alpha)
%BTABAL_CF Balance and truncate approximation of coprime factors
%          with balancing.
%          [SYSR,HSV] = BTABAL_CF(FACT,SYS,TOL,ORD,ALPHA)  calculates for 
%          the transfer function
%                                       -1
%               G(lambda) =  C(lambdaI-A) B + D
%
%          of an original system SYS = (A,B,C,D) an approximate  
%          transfer function
%                                          -1
%               Gr(lambda) =  Cr(lambdaI-Ar) Br + Dr
%
%          of a reduced order system SYSR = (Ar,Br,Cr,Dr) using the
%          Balance & Truncate approximation method with balancing 
%          on a Left/Right Coprime Factorization (LCF/RCF) of G.
%          The allowed factorizations are:
%             FACT = 'rcfid' - RCF with inner denominator
%             FACT = 'lcfid' - LCF with inner denominator
%             FACT = 'rcfs'  - RCF with prescribed stability degree ALPHA
%             FACT = 'lcfs'  - LCF with prescribed stability degree ALPHA
%             FACT = ' '     - no coprime factorization is used
%
%          TOL is a tolerance vector [TOL1, TOL2, TOL3], where TOL1 is the  
%          tolerance for model reduction and TOL3 is the tolerance 
%          for controlability/observability tests for computing right/left
%          coprime factorizations. TOL2 is not used.
%
%          ORD specifies the desired order of the reduced system SYSR.
%
%          ALPHA is the stability degree for the eigenvalues of A when
%          using the RCF or LCF with prescribed stability degree. 
%          For a continuous-time system ALPHA <= 0 is the maximum value
%          for the real parts of eigenvalues (default: -0.05), while for
%          a discrete-time system, 1 >= ALPHA >= 0 represents the maximum
%          value for the moduli of eigenvalues (default: 0.95).
%
%          HSV contains the decreasingly ordered Hankel singular values of
%          the extended system (see Method with 'type btabal_cf').
%
%          The order NR of the reduced system SYSR is determined as follows:
%          let NMIN be the order of a minimal realization of the extended
%          system (see Method with 'type btabal_cf'). Then
%          (1) if TOL1 > 0 and  ORD < 0, then NR = min(NRS,NMIN), where 
%              NRS is the number of Hankel singular values greater than TOL1;
%          (2) if ORD >= 0, then NR = MIN(ORD,NMIN).
%

%          Method:
%          The following approach is used in conjunction with either 
%          a stable Left Coprime Factorization (LCF) or a stable 
%          Right Coprime Factorization (RCF) of G:
%        
%          1. Compute the appropriate stable coprime factorization of G:
%                     -1                   -1
%                G = R  *Q (LCF) or G = Q*R   (RCF).
%
%          2. Perform the Balance and Truncate model reduction algorithm,
%             with balancing, on the extended system
%                                           ( Q )
%                Ge = ( Q R ) (LCF) or Ge = ( R )  (RCF)
%
%             to obtain a reduced extended system with reduced factors
%                                               ( Qr )
%                Ger = ( Qr Rr ) (LCF) or Ger = ( Rr )  (RCF).
%
%          3. Recover the reduced system from the reduced factors as
%                       -1                       -1
%                Gr = Rr  *Qr (LCF) or Gr = Qr*Rr   (RCF).
%
%          RELEASE 2.0 of SLICOT Model and Controller Reduction Toolbox.
%          Based on SLICOT RELEASE 5.7, Copyright (c) 2002-2020 NICONET e.V.
%
%          Interface M-function to the SLICOT-based MEX-function SYSRED.
%          A. Varga 07-30-1998. Revised 02-16-1999.
%          Revised, V. Sima 12-01-2002.
%

if ~isa(sys,'lti')
   error('The input system SYS must be an LTI object')
end

ni = nargin;
discr = double(sys.ts > 0);

if ni < 5
   alpha = -0.05;
   if discr
      alpha = 1 + alpha;
   end
end
if ni < 4 
   ord = -1;
end
if ni < 3 
   tol = 0; 
end

k = find(strcmp({'rcfid','lcfid','rcfs','lcfs'},fact)==1);
if isempty(k), k = 0; end

[a,b,c,d]=ssdata(sys);

[ar,br,cr,dr,hsv]=sysred(10*k+1,a,b,c,d,tol,discr,ord,alpha);

sysr = ss(ar,br,cr,dr,sys);


% end btabal_cf