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subroutine polmc(nm,ng,n,m,a,b,g,wr,wi,z,inc,invr,ierr,jpvt,
x rm1,rm2,rv1,rv2,rv3,rv4)
c
double precision a(nm,n),b(nm,m),g(ng,n),wr(n),wi(n),z(nm,n),
x rm1(m,m),rm2(m,*),rv1(n),rv2(n),rv3(m),rv4(m)
double precision p,q,r,s,t,zz
integer invr(n),jpvt(m)
logical complx
c!purpose
c this subroutine determines the state feedback matrix g of the
c linear time-invariant multi-input system
c
c dx / dt = a * x + b * u,
c
c where a is a nxn and b is a nxm matrix, such that the
c closed-loop system
c
c dx / dt = (a - b * g) * x
c
c has desired poles. the system must be preliminary reduced into
c orthogonal canonical form using the subroutine trmcf.
c!calling sequence
c
c subroutine polmc(nm,ng,n,m,a,b,g,wr,wi,z,inc,invr,ierr,jpvt,
c x rm1,rm2,rv1,rv2,rv3,rv4)
c
c on input-
c
c nm is an integer variable set equal to the row dimension
c of the two-dimensional arrays a, b and z as
c specified in the dimension statements for a, b and z
c in the calling program,
c
c ng is an integer variable set equal to the row dimension
c of the two-dimensional array g as specified in the
c dimension statement for g in the calling program,
c
c n is an integer variable set equal to the order of the
c matrices a and z. n must be not greater than nm,
c
c m is an integer variable set equal to the number of the
c columns of the matrix b. m must be not greater than
c ng,
c
c a is a working precision real two-dimensional variable with
c row dimension nm and column dimension at least n
c containing the block-hessenberg canonical form of the
c matrix a. the elements below the subdiagonal blocks
c must be equal to zero,
c
c b is a working precision real two-dimensional variable with
c row dimension nm and column dimension at least m
c containing the canonical form of the matrix b. the
c elements below the invr(1)-th row must be equal to zero,
c
c wr,wi are working precision real one-dimensional variables
c of dimension at least n containing the real and
c imaginery parts, respectively, of the desired poles,
c the poles can be unordered except that the complex
c conjugate pairs of poles must appfar consecutively.
c note that on output the imaginery parts of the poles
c may be modified,
c
c z is a working precision real two-dimensonal variale with
c row dimension nm and column dimension at least n
c containing the orthogonal transformation matrix produced
c in trmcf which reduces the system into canonical form,
c
c inc is an integer variable set equal to the controllability
c index of the system,
c
c invr is an integer one-dimensional variable of dimension at
c least inc containing the dimensons of the
c controllable subsystems in the canonical form.
c
c on output-
c
c a contains the upper quast-triangular form of the closed-
c loop system matrix a - b * g, that is triangular except
c of possible 2x2 blocks on the diagonal,
c
c b contains the transformed matrix b,
c
c g is a working precision real two-dimensional variable with
c row dimension ng and column dimension at least n
c containing the state feedback matrix g of the original
c system,
c
c z contains the orthogonal matrix which reduces the closed-
c loop system matrix a - b * g to the upper quasi-
c triangular form,
c
c ierr is an integer variable set equal to
c zero for normal return,
c 1 if the system is not completely controllable,
c
c jpvt is an integer temporary one-dimensonal array of
c dimension at least m used in the solution of linear
c equations,
c
c rm1 is a working precision real temporary two-dimensonal
c array of dimension at least mxm used in the solution
c of linear equations,
c
c rm2 is a working precision real temporary two-dimensional
c array od dimension at least mxmax(2,m) used in the
c solution of linear equations,
c
c rv1, are working precision real temporary one-dimensional
c rv2 arrays of dimension at least n used to hold the
c real and imaginery parts, respectively, of the
c eigenvectors during the reduction,
c
c rv3, are working precision real temporary one-dimensional
c rv4 arrays of dimension at least m used in the solution
c of linear equations.
c
c!auxiliary routines
c
c sqrsm
c fortran abs,min,sqrt
c!originator
c p.hr.petkov, higher institute of mechanical and electrical
c engineering, sofia, bulgaria.
c modified by serge Steer INRIA
c Copyright SLICOT
c!
c
ierr = 0
m1 = invr(1)
l = 0
10 l = l + 1
mr = invr(inc)
if (inc .eq. 1) go to 350
lp1 = l + m1
inc1 = inc - 1
mr1 = invr(inc1)
nr = n - mr + 1
nr1 = nr - mr1
complx = wi(l) .ne. 0.0d+0
do 15 i = nr, n
rv1(i) = 0.0d+0
if (complx) rv2(i) = 0.0d+0
15 continue
c
rv1(nr) = 1.0d+0
if (.not. complx) go to 20
if (mr .eq. 1) rv2(nr) = 1.0d+0
if (mr .gt. 1) rv2(nr+1) = 1.0d+0
t = wi(l)
wi(l) = 1.0d+0
wi(l+1) = t * wi(l+1)
c
c compute and transform eigenvector
c
20 do 200 ip = 1, inc
if (ip .eq. inc .and. inc .eq. 2) go to 200
if (ip .eq. inc) go to 120
c
do 40 ii = 1, mr
i = nr + ii - 1
c
do 30 jj = 1, mr1
j = nr1 + jj - 1
rm1(ii,jj) = a(i,j)
30 continue
c
40 continue
c
if (ip .eq. 1) go to 70
c
c scaling
c
s = 0.0d+0
mp1 = mr + 1
np1 = nr + mp1
c
do 50 ii = 1, mp1
i = nr + ii - 1
s = s + abs(rv1(i))
if (complx) s = s + abs(rv2(i))
50 continue
c
do 60 ii = 1, mp1
i = nr + ii - 1
rv1(i) = rv1(i) / s
if (complx) rv2(i) = rv2(i) / s
60 continue
c
if (complx .and. np1 .le. n) rv2(np1) = rv2(np1) / s
70 if (ip .eq. 1) mp1 = 1
np1 = nr + mp1
c
do 100 ii = 1, mr
i = nr + ii - 1
s = wr(l) * rv1(i)
c
do 80 jj = 1, mp1
j = nr + jj - 1
s = s - a(i,j) * rv1(j)
80 continue
c
rm2(ii,1) = s
if (.not. complx) go to 100
rm2(ii,1) = rm2(ii,1) + wi(l+1) * rv2(i)
s = wr(l+1) * rv2(i) + wi(l) * rv1(i)
c
do 90 jj = 1, mp1
c la ligne suivante a ete rajoutee par mes soins
j = nr + jj - 1
s = s - a(i,j) * rv2(j)
90 continue
c
if (np1 .le. n) s = s - a(i,np1) * rv2(np1)
rm2(ii,2) = s
100 continue
c
c solving linear equations for the eigenvector elements
c
nc = 1
if (complx) nc = 2
call dqrsm(rm1,m,mr,mr1,rm2,m,nc,rm2,m,ir,jpvt,
x rv3,rv4)
if (ir .lt. mr) go to 600
c
do 110 ii = 1, mr1
i = nr1 + ii - 1
rv1(i) = rm2(ii,1)
if (complx) rv2(i) = rm2(ii,2)
110 continue
c
if (ip .eq. 1 .and. inc .gt. 2) go to 195
120 nj = nr
if (ip .lt. inc) nj = nr1
ni = nr + mr - 1
inc2 = inc - ip + 2
if (ip .gt. 1) ni = ni + invr(inc2)
if (ip .gt. 2) ni = ni + 1
if (complx .and. ip .gt. 2) ni = min(ni+1,n)
kmr = mr1
if (ip .gt. 1) kmr = mr
c
do 190 kk = 1, kmr
ll = 1
k = nr + mr - kk
if (ip .eq. 1) k = nr - kk
130 p = rv1(k)
if (ll .eq. 2) p = rv2(k)
q = rv1(k+1)
if (ll .eq. 2) q = rv2(k+1)
s = abs(p) + abs(q)
p = p / s
q = q / s
r = sqrt(p*p+q*q)
t = s * r
rv1(k) = t
if (ll .eq. 2) rv2(k) = t
rv1(k+1) = 0.0d+0
if (ll .eq. 2) rv2(k+1) = 0.0d+0
p = p / r
q = q / r
c
c transform a
c
do 140 j = nj, n
zz = a(k,j)
a(k,j) = p * zz + q * a(k+1,j)
a(k+1,j) = p * a(k+1,j) - q * zz
140 continue
c
do 150 i = 1, ni
zz = a(i,k)
a(i,k) = p * zz + q * a(i,k+1)
a(i,k+1) = p * a(i,k+1) - q * zz
150 continue
c
if (k .eq. lp1 .and. ll .eq. 1 .or. k .gt. lp1) go to 170
c
c transform b
c
do 160 j = 1, m
zz = b(k,j)
b(k,j) = p * zz + q * b(k+1,j)
b(k+1,j) = p * b(k+1,j) - q * zz
160 continue
c
c accumulate transformations
c
170 do 180 i = 1, n
zz = z(i,k)
z(i,k) = p * zz + q * z(i,k+1)
z(i,k+1) = p * z(i,k+1) - q * zz
180 continue
c
if (.not. complx .or. ll .eq. 2) go to 190
zz = rv2(k)
rv2(k) = p * zz + q * rv2(k+1)
rv2(k+1) = p * rv2(k+1) - q * zz
if (k + 2 .gt. n) go to 190
k = k + 1
ll = 2
go to 130
190 continue
c
if (ip .eq. inc) go to 200
195 mr = mr1
nr = nr1
if (ip .eq. inc1) go to 200
inc2 = inc - ip - 1
mr1 = invr(inc2)
nr1 = nr1 - mr1
200 continue
c
if (complx) go to 250
c
c find one column of g
c
do 220 ii = 1, m1
i = l + ii
c
do 210 j = 1, m
210 rm1(ii,j) = b(i,j)
c
rm2(ii,1) = a(i,l)
220 continue
c
call dqrsm(rm1,m,m1,m,rm2,m,1,g(1,l),ng,ir,jpvt,rv3,rv4)
if (ir .lt. m1) go to 600
c
do 240 i = 1, lp1
c
do 230 j = 1, m
230 a(i,l) = a(i,l) - b(i,j) * g(j,l)
c
240 continue
c
go to 330
c
c find two columns of g
c
250 l = l + 1
if (lp1 .lt. n) lp1 = lp1 + 1
c
do 270 ii = 1, m1
i = l + ii
if (l + m1 .gt. n) i = i - 1
c
c la ligne suivante a ete rajoutee par mes soins
do 260 j = 1 , m
cxxx if(abs(b(i,j)).le.abs(b(l,j))) i=i-1
260 rm1(ii,j) = b(i,j)
c
p = a(i,l-1)
if (i .eq. l) p = p - (rv2(i) / rv1(i-1)) * wi(i)
rm2(ii,1) = p
q = a(i,l)
if (i .eq. l) q = q - wr(i) + (rv2(i-1) / rv1(i-1)) *wi(i)
rm2(ii,2) = q
270 continue
c
call dqrsm(rm1,m,m1,m,rm2,m,2,rm2,m,ir,jpvt,rv3,rv4)
if (ir .lt. m1) go to 600
c
do 290 i = 1, m
c
do 280 jj = 1, 2
j = l + jj - 2
g(i,j) = rm2(i,jj)
280 continue
c
290 continue
c
do 320 i = 1, lp1
c
do 310 jj = 1, 2
j = l + jj - 2
c
do 300 k = 1, m
300 a(i,j) = a(i,j) - b(i,k)*g(k,j)
c
310 continue
c
320 continue
c
if (l .eq. n) go to 500
330 invr(inc) = invr(inc) - 1
if (invr(inc) .eq. 0) inc = inc - 1
if (complx) invr(inc) = invr(inc) - 1
if (invr(inc) .eq. 0) inc = inc - 1
go to 10
c
c find the rest columns of g
c
350 do 370 ii = 1, mr
i = l + ii - 1
c
do 360 j = 1, m
360 rm1(ii,j) = b(i,j)
c
370 continue
c
do 400 ii = 1, mr
i = l + ii - 1
c
do 380 jj = 1, mr
j = l + jj - 1
if (ii .lt. jj) rm2(ii,jj) = 0.0d+0
if (ii .gt. jj) rm2(ii,jj) = a(i,j)
380 continue
c
400 continue
c
ii = 0
410 ii = ii + 1
i = l + ii - 1
if (wi(i) .ne. 0.0d+0) go to 420
rm2(ii,ii) = a(i,i) - wr(i)
if (ii .eq. mr) go to 430
c la ligne suivante a ete rajoutee par mes soins
goto 410
420 rm2(ii,ii) = a(i,i) - wr(i)
rm2(ii,ii+1) = a(i,i+1) - wi(i)
rm2(ii+1,ii) = a(i+1,i) - wi(i+1)
rm2(ii+1,ii+1) = a(i+1,i+1) - wr(i+1)
ii = ii + 1
if (ii .lt. mr) go to 410
430 call dqrsm(rm1,m,mr,m,rm2,m,m,rm2,m,ir,jpvt,rv3,rv4)
if (ir .lt. mr) go to 600
c
do 450 i = 1, m
c
do 440 jj = 1, mr
j = l + jj - 1
g(i,j) = rm2(i,jj)
440 continue
c
450 continue
c
do 480 i = 1, n
c
do 470 j = l, n
c
do 460 k = 1, m
460 a(i,j) = a(i,j) - b(i,k) * g(k,j)
c
470 continue
c
480 continue
c
c transform g
c
500 do 540 i = 1, m
c
do 520 j = 1, n
s = 0.0d+0
c
do 510 k = 1, n
510 s = s + g(i,k) * z(j,k)
c
rv1(j) = s
520 continue
c
do 530 j = 1, n
530 g(i,j) = rv1(j)
c
540 continue
c
go to 610
c
c set error -- the system is not completely controllable
c
600 ierr = 1
610 return
c
c last card of subroutine polmc
c
end
|
