## SB01MD

### State feedback matrix of a linear time-invariant single-input system

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To determine the one-dimensional state feedback matrix G of the
      linear time-invariant single-input system

            dX/dt = A * X + B * U,

      where A is an NCONT-by-NCONT matrix and B is an NCONT element
      vector such that the closed-loop system

            dX/dt = (A - B * G) * X

      has desired poles. The system must be preliminarily reduced
      to orthogonal canonical form using the SLICOT Library routine
      AB01MD.

Specification

          SUBROUTINE SB01MD( NCONT, N, A, LDA, B, WR, WI, Z, LDZ, G, DWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          INTEGER           INFO, LDA, LDZ, N, NCONT
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(*), DWORK(*), G(*), WI(*), WR(*),
         $                  Z(LDZ,*)

Arguments

Input/Output Parameters

      NCONT   (input) INTEGER
              The order of the matrix A as produced by SLICOT Library
              routine AB01MD.  NCONT >= 0.

      N       (input) INTEGER
              The order of the matrix Z.  N >= NCONT.

      A       (input/output) DOUBLE PRECISION array, dimension
              (LDA,NCONT)
              On entry, the leading NCONT-by-NCONT part of this array
              must contain the canonical form of the state dynamics
              matrix A as produced by SLICOT Library routine AB01MD.
              On exit, the leading NCONT-by-NCONT part of this array
              contains the upper quasi-triangular form S of the closed-
              loop system matrix (A - B * G), that is triangular except
              for possible 2-by-2 diagonal blocks.
              (To reconstruct the closed-loop system matrix see
              FURTHER COMMENTS below.)

      LDA     INTEGER
              The leading dimension of array A.  LDA >= MAX(1,NCONT).

      B       (input/output) DOUBLE PRECISION array, dimension (NCONT)
              On entry, this array must contain the canonical form of
              the input/state vector B as produced by SLICOT Library
              routine AB01MD.
              On exit, this array contains the transformed vector Z * B
              of the closed-loop system.

      WR      (input) DOUBLE PRECISION array, dimension (NCONT)
      WI      (input) DOUBLE PRECISION array, dimension (NCONT)
              These arrays must contain the real and imaginary parts,
              respectively, of the desired poles of the closed-loop
              system. The poles can be unordered, except that complex
              conjugate pairs of poles must appear consecutively.

      Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
              On entry, the leading N-by-N part of this array must
              contain the orthogonal transformation matrix as produced
              by SLICOT Library routine AB01MD, which reduces the system
              to canonical form.
              On exit, the leading NCONT-by-NCONT part of this array
              contains the orthogonal matrix Z which reduces the closed-
              loop system matrix (A - B * G) to upper quasi-triangular
              form.

      LDZ     INTEGER
              The leading dimension of array Z.  LDZ >= MAX(1,N).

      G       (output) DOUBLE PRECISION array, dimension (NCONT)
              This array contains the one-dimensional state feedback
              matrix G of the original system.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (3*NCONT)

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      The method is based on the orthogonal reduction of the closed-loop
      system matrix (A - B * G) to upper quasi-triangular form S whose
      1-by-1 and 2-by-2 diagonal blocks correspond to the desired poles.
      That is, S = Z'*(A - B * G)*Z, where Z is an orthogonal matrix.

References

      [1] Petkov, P. Hr.
          A Computational Algorithm for Pole Assignment of Linear
          Single Input Systems.
          Internal Report 81/2, Control Systems Research Group, School
          of Electronic Engineering and Computer Science, Kingston
          Polytechnic, 1981.

Numerical Aspects

                                    3
      The algorithm requires 0(NCONT ) operations and is backward
      stable.

Further Comments

      If required, the closed-loop system matrix (A - B * G) can be
      formed from the matrix product Z * S * Z' (where S and Z are the
      matrices output in arrays A and Z respectively).

Example

Program Text

    *     SB01MD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 20 )
          INTEGER          LDA, LDZ
          PARAMETER        ( LDA = NMAX, LDZ = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 3*NMAX )
    *     .. Local Scalars ..
          DOUBLE PRECISION TOL
          INTEGER          I, INFO1, INFO2, J, N, NCONT
          CHARACTER*1      JOBZ
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(NMAX), DWORK(LDWORK), G(NMAX),
         $                 WI(NMAX), WR(NMAX), Z(LDZ,NMAX)
    *     .. External Subroutines ..
          EXTERNAL         AB01MD, SB01MD
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, TOL, JOBZ
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99995 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             READ ( NIN, FMT = * ) ( B(I), I = 1,N )
             READ ( NIN, FMT = * ) ( WR(I), I = 1,N )
             READ ( NIN, FMT = * ) ( WI(I), I = 1,N )
    *        First reduce the given system to canonical form.
             CALL AB01MD( JOBZ, N, A, LDA, B, NCONT, Z, LDZ, DWORK, TOL,
         $                DWORK(N+1), LDWORK-N, INFO1 )
    *
             IF ( INFO1.EQ.0 ) THEN
    *           Find the one-dimensional state feedback matrix G.
                CALL SB01MD( NCONT, N, A, LDA, B, WR, WI, Z, LDZ, G, DWORK,
         $                   INFO2 )
    *
                IF ( INFO2.NE.0 ) THEN
                   WRITE ( NOUT, FMT = 99997 ) INFO2
                ELSE
                   WRITE ( NOUT, FMT = 99996 ) ( G(I), I = 1,NCONT )
                END IF
             ELSE
                WRITE ( NOUT, FMT = 99998 ) INFO1
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' SB01MD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB01MD =',I2)
    99997 FORMAT (' INFO on exit from SB01MD =',I2)
    99996 FORMAT (' The one-dimensional state feedback matrix G is',
         $       /20(1X,F8.4))
    99995 FORMAT (/' N is out of range.',/' N = ',I5)
          END

Program Data

     SB01MD EXAMPLE PROGRAM DATA
       4     0.0     I
      -1.0  0.0  2.0 -3.0
       1.0 -4.0  3.0 -1.0
       0.0  2.0  4.0 -5.0
       0.0  0.0 -1.0 -2.0
       1.0  0.0  0.0  0.0
      -1.0 -1.0 -1.0 -1.0
       0.0  0.0  0.0  0.0

Program Results

     SB01MD EXAMPLE PROGRAM RESULTS

     The one-dimensional state feedback matrix G is
       1.0000  29.0000  93.0000 -76.0000

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