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subroutine ssxmc(n,m,a,na,b,ncont,indcon,nblk,z,
1 wrka,wrk1,wrk2,iwrk,tol,mode)
c! calling sequence
c subroutine ssxmc(n,m,a,na,b,ncont,indcon,nblk,z,
c 1 wrka,wrk1,wrk2,iwrk,tol,mode)
c
c integer n,m,na,ncont,indcon,nblk(n),iwrk(m),mode
c
c real*8 a(na,n),b(na,m),z(na,n),wrka(n,m)
c real*8 wrk1(m),wrk2(m),tol
c
c arguments in
c
c n integer
c -the order of original state-space representation;
c declared first dimension of nblk,wrka; declared
c second dimension of a (and z, if mode .ne. 0)
c
c m integer
c -the number of system inputs; declared first dimension
c of iwrk,wrk1,wrk2; declared second dimension of b,wrka
c
c a double precision(n,n)
c -the original state dynamics matrix. note that this
c matrix is overwritten here
c
c na integer
c -the declared first dimension of a,b (and z, if
c mode .ne. 0). note that na .ge. n
c
c b double precision(n,m)
c -the original input/state matrix. note that this
c matrix is overwritten here
c
c tol double precision
c -if greater than the machine precision, tol is used
c as zero tolerance in rank determination when trans-
c forming (a,b,c): otherwise (eg tol = 0.0d+0), the
c machine precision is used
c
c mode integer
c -mode = 0 if accumulation of the orthogonal trans-
c formation z is not required, and non-zero if this
c matrix is required
c
c arguments out
c
c a double precision(ncont,ncont)
c -the upper block hessenberg state dynamics matrix of
c a controllable realization for the original system
c
c b double precision(ncont,m)
c -the transformed input/state matrix
c
c ncont integer
c -the order of controllable state-space representation
c
c indcon integer
c -the controllability index of transformed
c system representation
c
c nblk integer(indcon)
c -the dimensions of the diagonal blocks of the trans-
c formed a
c
c z double precision(n,n)
c -the orthogonal similarity transformation which
c reduces the given system to orthogonal canonical
c form. note that, if mode .eq. 0, z is not referenced
c and so can be a scalar dummy variable
c
c!working space
c
c wrka double precision(n,m)
c
c wrk1 double precision(m)
c
c wrk2 double precision(m)
c
c iwrk integer(m)
c
c!purpose
c
c to reduce the linear time-invariant multi-input system
c
c dx/dt = a * x + b * u,
c
c where a and b are (n x n) and (n x m) matrices respectively,
c to orthogonal canonical form using (and optionally accum-
c ulating) orthogonal similarity transformations.
c
c!method
c
c b is first qr-decomposed and the appropriate orthogonal
c similarity transformation applied to a. leaving the first
c rank(b) states unchanged, the resulting lower left block
c of a is now itself qr-decomposed and this new orthogonal
c similarity transformation applied. continuing in this
c manner, a completely controllable state-space pair (acont,
c bcont) is found for the given (a,b), where acont is upper
c block hessenberg with each sub-diagonal block of full row
c rank, and bcont is zero apart from its (independent) first
c rank(b) rows. note finally that the system controllability
c indices are easily calculable from the dimensions of the
c blocks of acont.
c
c!reference
c
c konstantinov, m.m., petkov, p.hr. and christov, n.d.
c "orthogonal invariants and canonical forms for linear
c controllable systems"
c proc. ifac 8th world congress, 1981.
c
c!auxiliary routines
c
c dqrdc (linpack)
c
c!originator
c
c p.hr.petkov, higher institute of mechanical and
c electrical engineering, sofia, bulgaria, april 1981
C Copyright SLICOT
c
c!comments
c
c none
c
c!user-supplied routines
c
c none
c!
c*******************************************************************
c
c
integer nblk(n),iwrk(m)
c
double precision a(na,n),b(na,m),z(na,n),tol
double precision wrka(n,m),wrk1(m),wrk2(m)
c
c local variables:
c
c
double precision abnorm,temp,thrtol
c
c common /smprec/eps
c
c common block smprec is shared with routine ddata which provides
c a value for eps, a machine-dependent parameter which specifies
c the relative precision of drealing-point arithmetic
c
c
c call ddata
c
abnorm = 0.0d+0
ist = 0
ncont = 0
indcon = 0
ni = 0
nb = n
mb = m
c
c use the larger of tol, eps in rank determination
c
c toleps = dble(n * n) * max(tol,eps)
c
if (mode .eq. 0) go to 30
c
c initialize z to identity matrix
c
do 20 i = 1, n
c
do 10 j = 1, n
10 z(i,j) = 0.0d+0
c
z(i,i) = 1.0d+0
20 continue
c
30 do 50 i = 1, n
c
do 40 j = 1, m
wrka(i,j) = b(i,j)
b(i,j) = 0.0d+0
40 continue
c
50 continue
c
60 ist = ist + 1
c
c qr decomposition with column pivoting
c
do 70 j = 1, mb
70 iwrk(j) = 0
c
call dqrdc(wrka,n,nb,mb,wrk1,iwrk,wrk2,1)
c
irnk = 0
mm = min(nb,mb)
if (abs(wrka(1,1)) .gt. abnorm) abnorm = abs(wrka(1,1))
c thresh = toleps * abnorm
c
c rank determination
c
thrtol=tol*abnorm*dble(n*n)
do 100 i = 1,mm
temp=abs(wrka(i,i))
if(temp.gt.thrtol.and.1.0d+0+temp.gt.1.0d+0) irnk = i
100 continue
c
if (irnk .eq. 0) go to 360
nj = ni
ni = ncont
ncont = ncont + irnk
indcon = indcon + 1
nblk(indcon) = irnk
lu = min(irnk,nb-1)
if (lu .eq. 0) go to 200
c
c premultiply appropriate row block of a by qtrans
c
call hhdml(lu,n,n,ni,ni,nb,nb,wrka,n,wrk1,a,na,11,ierr)
c
c postmultiply appropriate column block of a by q
c
call hhdml(lu,n,n,0,ni,n,nb,wrka,n,wrk1,a,na,00,ierr)
c
c if required, accumulate transformations
c
if (mode .ne. 0) call hhdml(lu,n,n,0,ni,n,nb,wrka,n,wrk1,z,na,
1 00,ierr)
c
200 if (irnk .lt. 2) go to 230
c
do 220 i = 2, irnk
im1 = i - 1
c
do 210 j = 1, im1
210 wrka(i,j) = 0.0d+0
c
220 continue
c
c backward permutation of the columns
c
230 do 270 j = 1, mb
if (iwrk(j) .lt. 0) go to 270
k = iwrk(j)
iwrk(j) = -k
240 continue
if (k .eq. j) go to 260
c
do 250 i = 1, irnk
temp = wrka(i,k)
wrka(i,k) = wrka(i,j)
wrka(i,j) = temp
250 continue
c
iwrk(k) = -iwrk(k)
k = -iwrk(k)
go to 240
260 continue
270 continue
c
if (ist .gt. 1) go to 300
c
c form b
c
do 290 i = 1, irnk
c
do 280 j = 1, m
280 b(i,j) = wrka(i,j)
c
290 continue
c
go to 330
c
c form a
c
300 do 320 i = 1, irnk
ia = ni + i
c
do 310 j = 1, mb
ja = nj + j
310 a(ia,ja) = wrka(i,j)
c
320 continue
c
330 if (irnk .eq. nb) go to 360
c
mb = irnk
nb = nb - irnk
c
do 350 i = 1, nb
ia = ncont + i
c
do 340 j = 1, mb
ja = ni + j
wrka(i,j) = a(ia,ja)
a(ia,ja) = 0.0d+0
340 continue
c
350 continue
go to 60
c
360 continue
c
return
end
|
