SUBROUTINE SB01MD( NCONT, N, A, LDA, B, WR, WI, Z, LDZ, G, DWORK,
$ INFO )
C
C PURPOSE
C
C To determine the one-dimensional state feedback matrix G of the
C linear time-invariant single-input system
C
C dX/dt = A * X + B * U,
C
C where A is an NCONT-by-NCONT matrix and B is an NCONT element
C vector such that the closed-loop system
C
C dX/dt = (A - B * G) * X
C
C has desired poles. The system must be preliminarily reduced
C to orthogonal canonical form using the SLICOT Library routine
C AB01MD.
C
C ARGUMENTS
C
C Input/Output Parameters
C
C NCONT (input) INTEGER
C The order of the matrix A as produced by SLICOT Library
C routine AB01MD. NCONT >= 0.
C
C N (input) INTEGER
C The order of the matrix Z. N >= NCONT.
C
C A (input/output) DOUBLE PRECISION array, dimension
C (LDA,NCONT)
C On entry, the leading NCONT-by-NCONT part of this array
C must contain the canonical form of the state dynamics
C matrix A as produced by SLICOT Library routine AB01MD.
C On exit, the leading NCONT-by-NCONT part of this array
C contains the upper quasi-triangular form S of the closed-
C loop system matrix (A - B * G), that is triangular except
C for possible 2-by-2 diagonal blocks.
C (To reconstruct the closed-loop system matrix see
C FURTHER COMMENTS below.)
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,NCONT).
C
C B (input/output) DOUBLE PRECISION array, dimension (NCONT)
C On entry, this array must contain the canonical form of
C the input/state vector B as produced by SLICOT Library
C routine AB01MD.
C On exit, this array contains the transformed vector Z * B
C of the closed-loop system.
C
C WR (input) DOUBLE PRECISION array, dimension (NCONT)
C WI (input) DOUBLE PRECISION array, dimension (NCONT)
C These arrays must contain the real and imaginary parts,
C respectively, of the desired poles of the closed-loop
C system. The poles can be unordered, except that complex
C conjugate pairs of poles must appear consecutively.
C
C Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
C On entry, the leading N-by-N part of this array must
C contain the orthogonal transformation matrix as produced
C by SLICOT Library routine AB01MD, which reduces the system
C to canonical form.
C On exit, the leading NCONT-by-NCONT part of this array
C contains the orthogonal matrix Z which reduces the closed-
C loop system matrix (A - B * G) to upper quasi-triangular
C form.
C
C LDZ INTEGER
C The leading dimension of array Z. LDZ >= MAX(1,N).
C
C G (output) DOUBLE PRECISION array, dimension (NCONT)
C This array contains the one-dimensional state feedback
C matrix G of the original system.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (3*NCONT)
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C METHOD
C
C The method is based on the orthogonal reduction of the closed-loop
C system matrix (A - B * G) to upper quasi-triangular form S whose
C 1-by-1 and 2-by-2 diagonal blocks correspond to the desired poles.
C That is, S = Z'*(A - B * G)*Z, where Z is an orthogonal matrix.
C
C REFERENCES
C
C [1] Petkov, P. Hr.
C A Computational Algorithm for Pole Assignment of Linear
C Single Input Systems.
C Internal Report 81/2, Control Systems Research Group, School
C of Electronic Engineering and Computer Science, Kingston
C Polytechnic, 1981.
C
C NUMERICAL ASPECTS
C 3
C The algorithm requires 0(NCONT ) operations and is backward
C stable.
C
C FURTHER COMMENTS
C
C If required, the closed-loop system matrix (A - B * G) can be
C formed from the matrix product Z * S * Z' (where S and Z are the
C matrices output in arrays A and Z respectively).
C
C CONTRIBUTORS
C
C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, Aug. 1997.
C Supersedes Release 2.0 routine SB01AD by Control Systems Research
C Group, Kingston Polytechnic, United Kingdom, May 1981.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Closed loop spectrum, closed loop systems, eigenvalue assignment,
C orthogonal canonical form, orthogonal transformation, pole
C placement, Schur form.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDZ, N, NCONT
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(*), DWORK(*), G(*), WI(*), WR(*),
$ Z(LDZ,*)
C .. Local Scalars ..
LOGICAL COMPL
INTEGER I, IM1, K, L, LL, LP1, NCONT2, NI, NJ
DOUBLE PRECISION B1, P, Q, R, S, T
C .. External Functions ..
DOUBLE PRECISION DDOT
EXTERNAL DDOT
C .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DLARTG, DLASET, DROT,
$ DSCAL, XERBLA
C .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
C .. Executable Statements ..
C
INFO = 0
C
C Test the input scalar arguments.
C
IF( NCONT.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.NCONT ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, NCONT ) ) THEN
INFO = -4
ELSE IF( LDZ.LT.MAX( 1, N ) ) THEN
INFO = -9
END IF
C
IF ( INFO.NE.0 ) THEN
C
C Error return
C
CALL XERBLA( 'SB01MD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF ( NCONT.EQ.0 .OR. N.EQ.0 )
$ RETURN
C
C Return if the system is not complete controllable.
C
IF ( B(1).EQ.ZERO )
$ RETURN
C
IF ( NCONT.EQ.1 ) THEN
C
C 1-by-1 case.
C
P = A(1,1) - WR(1)
A(1,1) = WR(1)
G(1) = P/B(1)
Z(1,1) = ONE
RETURN
END IF
C
C General case. Save the contents of WI in DWORK.
C
NCONT2 = 2*NCONT
CALL DCOPY( NCONT, WI, 1, DWORK(NCONT2+1), 1 )
C
B1 = B(1)
B(1) = ONE
L = 0
LL = 0
20 CONTINUE
L = L + 1
LL = LL + 1
COMPL = DWORK(NCONT2+L).NE.ZERO
IF ( L.NE.NCONT ) THEN
LP1 = L + 1
IF ( LL.NE.2 ) THEN
IF ( COMPL ) THEN
C
C Compute complex eigenvector.
C
DWORK(NCONT) = ONE
DWORK(NCONT2) = ONE
P = WR(L)
T = DWORK(NCONT2+L)
Q = T*DWORK(NCONT2+LP1)
DWORK(NCONT2+L) = ONE
DWORK(NCONT2+LP1) = Q
C
DO 40 I = NCONT, LP1, -1
IM1 = I - 1
DWORK(IM1) = ( P*DWORK(I) + Q*DWORK(NCONT+I) -
$ DDOT( NCONT-IM1, A(I,I), LDA, DWORK(I), 1 ) )
$ /A(I,IM1)
DWORK(NCONT+IM1) = ( P*DWORK(NCONT+I) + DWORK(I) -
$ DDOT( NCONT-IM1, A(I,I), LDA, DWORK(NCONT+I), 1 ) )
$ /A(I,IM1)
40 CONTINUE
C
ELSE
C
C Compute real eigenvector.
C
DWORK(NCONT) = ONE
P = WR(L)
C
DO 60 I = NCONT, LP1, -1
IM1 = I - 1
DWORK(IM1) = ( P*DWORK(I) -
$ DDOT( NCONT-IM1, A(I,I), LDA, DWORK(I), 1 ) )
$ /A(I,IM1)
60 CONTINUE
C
END IF
END IF
C
C Transform eigenvector.
C
DO 80 K = NCONT - 1, L, -1
IF ( LL.NE.2 ) THEN
R = DWORK(K)
S = DWORK(K+1)
ELSE
R = DWORK(NCONT+K)
S = DWORK(NCONT+K+1)
END IF
CALL DLARTG( R, S, P, Q, T )
DWORK(K) = T
IF ( LL.NE.2 ) THEN
NJ = MAX( K-1, L )
ELSE
DWORK(NCONT+K) = T
NJ = L - 1
END IF
C
C Transform A.
C
CALL DROT( NCONT-NJ+1, A(K,NJ), LDA, A(K+1,NJ), LDA, P, Q )
C
IF ( COMPL .AND. LL.EQ.1 ) THEN
NI = NCONT
ELSE
NI = MIN( K+2, NCONT )
END IF
CALL DROT( NI, A(1,K), 1, A(1,K+1), 1, P, Q )
C
IF ( K.EQ.L ) THEN
C
C Transform B.
C
T = B(K)
B(K) = P*T
B(K+1) = -Q*T
END IF
C
C Accumulate transformations.
C
CALL DROT( NCONT, Z(1,K), 1, Z(1,K+1), 1, P, Q )
C
IF ( COMPL .AND. LL.NE.2 ) THEN
T = DWORK(NCONT+K)
DWORK(NCONT+K) = P*T + Q*DWORK(NCONT+K+1)
DWORK(NCONT+K+1) = P*DWORK(NCONT+K+1) - Q*T
END IF
80 CONTINUE
C
END IF
C
IF ( .NOT.COMPL ) THEN
C
C Find one element of G.
C
K = L
R = B(L)
IF ( L.NE.NCONT ) THEN
IF ( ABS( B(LP1) ).GT.ABS( B(L) ) ) THEN
K = LP1
R = B(LP1)
END IF
END IF
P = A(K,L)
IF ( K.EQ.L ) P = P - WR(L)
P = P/R
C
CALL DAXPY( LP1, -P, B, 1, A(1,L), 1 )
C
G(L) = P/B1
IF ( L.NE.NCONT ) THEN
LL = 0
GO TO 20
END IF
ELSE IF ( LL.EQ.1 ) THEN
GO TO 20
ELSE
C
C Find two elements of G.
C
K = L
R = B(L)
IF ( L.NE.NCONT ) THEN
IF ( ABS( B(LP1)).GT.ABS( B(L) ) ) THEN
K = LP1
R = B(LP1)
END IF
END IF
P = A(K,L-1)
Q = A(K,L)
IF ( K.EQ.L ) THEN
P = P - ( DWORK(NCONT+L)/DWORK(L-1) )*DWORK(NCONT2+L)
Q = Q - WR(L) +
$ ( DWORK(NCONT+L-1)/DWORK(L-1) )*DWORK(NCONT2+L)
END IF
P = P/R
Q = Q/R
C
CALL DAXPY( LP1, -P, B, 1, A(1,L-1), 1 )
CALL DAXPY( LP1, -Q, B, 1, A(1,L), 1 )
C
G(L-1) = P/B1
G(L) = Q/B1
IF ( L.NE.NCONT ) THEN
LL = 0
GO TO 20
END IF
END IF
C
C Transform G.
C
CALL DGEMV( 'No transpose', NCONT, NCONT, ONE, Z, LDZ, G, 1,
$ ZERO, DWORK, 1 )
CALL DCOPY( NCONT, DWORK, 1, G, 1 )
CALL DSCAL( NCONT, B1, B, 1 )
C
C Annihilate A after the first subdiagonal.
C
IF ( NCONT.GT.2 )
$ CALL DLASET( 'Lower', NCONT-2, NCONT-2, ZERO, ZERO, A(3,1),
$ LDA )
C
RETURN
C *** Last line of SB01MD ***
END