## SB10ED

### H2 optimal state controller for a discrete-time system

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

Purpose

      To compute the matrices of the H2 optimal n-state controller

                            | AK | BK |
                        K = |----|----|
                            | CK | DK |

      for the discrete-time system

                    | A  | B1  B2  |   | A | B |
                P = |----|---------| = |---|---| ,
                    | C1 |  0  D12 |   | C | D |
                    | C2 | D21 D22 |

      where B2 has as column size the number of control inputs (NCON)
      and C2 has as row size the number of measurements (NMEAS) being
      provided to the controller.

      It is assumed that

      (A1) (A,B2) is stabilizable and (C2,A) is detectable,

      (A2) D12 is full column rank and D21 is full row rank,

                j*Theta
      (A3) | A-e       *I  B2  | has full column rank for all
           |    C1         D12 |

           0 <= Theta < 2*Pi ,

                j*Theta
      (A4) | A-e       *I  B1  |  has full row rank for all
           |    C2         D21 |

           0 <= Theta < 2*Pi .

Specification

          SUBROUTINE SB10ED( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
         $                   D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
         $                   RCOND, TOL, IWORK, DWORK, LDWORK, BWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
         $                   LDDK, LDWORK, M, N, NCON, NMEAS, NP
          DOUBLE PRECISION   TOL
    C     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
         $                   BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
         $                   D( LDD, * ), DK( LDDK, * ), DWORK( * ),
         $                   RCOND( * )
          LOGICAL            BWORK( * )

Arguments

Input/Output Parameters

      N       (input) INTEGER
              The order of the system.  N >= 0.

      M       (input) INTEGER
              The column size of the matrix B.  M >= 0.

      NP      (input) INTEGER
              The row size of the matrix C.  NP >= 0.

      NCON    (input) INTEGER
              The number of control inputs (M2).  M >= NCON >= 0,
              NP-NMEAS >= NCON.

      NMEAS   (input) INTEGER
              The number of measurements (NP2).  NP >= NMEAS >= 0,
              M-NCON >= NMEAS.

      A       (input/worksp.) DOUBLE PRECISION array, dimension (LDA,N)
              The leading N-by-N part of this array must contain the
              system state matrix A.
              This array is modified internally, but it is restored on
              exit.

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

      B       (input) DOUBLE PRECISION array, dimension (LDB,M)
              The leading N-by-M part of this array must contain the
              system input matrix B.

      LDB     INTEGER
              The leading dimension of the array B.  LDB >= max(1,N).

      C       (input) DOUBLE PRECISION array, dimension (LDC,N)
              The leading NP-by-N part of this array must contain the
              system output matrix C.

      LDC     INTEGER
              The leading dimension of the array C.  LDC >= max(1,NP).

      D       (input) DOUBLE PRECISION array, dimension (LDD,M)
              The leading NP-by-M part of this array must contain the
              system input/output matrix D.

      LDD     INTEGER
              The leading dimension of the array D.  LDD >= max(1,NP).

      AK      (output) DOUBLE PRECISION array, dimension (LDAK,N)
              The leading N-by-N part of this array contains the
              controller state matrix AK.

      LDAK    INTEGER
              The leading dimension of the array AK.  LDAK >= max(1,N).

      BK      (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
              The leading N-by-NMEAS part of this array contains the
              controller input matrix BK.

      LDBK    INTEGER
              The leading dimension of the array BK.  LDBK >= max(1,N).

      CK      (output) DOUBLE PRECISION array, dimension (LDCK,N)
              The leading NCON-by-N part of this array contains the
              controller output matrix CK.

      LDCK    INTEGER
              The leading dimension of the array CK.
              LDCK >= max(1,NCON).

      DK      (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
              The leading NCON-by-NMEAS part of this array contains the
              controller input/output matrix DK.

      LDDK    INTEGER
              The leading dimension of the array DK.
              LDDK >= max(1,NCON).

      RCOND   (output) DOUBLE PRECISION array, dimension (7)
              RCOND contains estimates the reciprocal condition
              numbers of the matrices which are to be inverted and the
              reciprocal condition numbers of the Riccati equations
              which have to be solved during the computation of the
              controller. (See the description of the algorithm in [2].)
              RCOND(1) contains the reciprocal condition number of the
                       control transformation matrix TU;
              RCOND(2) contains the reciprocal condition number of the
                       measurement transformation matrix TY;
              RCOND(3) contains the reciprocal condition number of the
                       matrix Im2 + B2'*X2*B2;
              RCOND(4) contains the reciprocal condition number of the
                       matrix Ip2 + C2*Y2*C2';
              RCOND(5) contains the reciprocal condition number of the
                       X-Riccati equation;
              RCOND(6) contains the reciprocal condition number of the
                       Y-Riccati equation;
              RCOND(7) contains the reciprocal condition number of the
                       matrix Im2 + DKHAT*D22 .

Tolerances

      TOL     DOUBLE PRECISION
              Tolerance used for controlling the accuracy of the
              transformations applied for diagonalizing D12 and D21,
              and for checking the nonsingularity of the matrices to be
              inverted. If TOL <= 0, then a default value equal to
              sqrt(EPS) is used, where EPS is the relative machine
              precision.

Workspace

      IWORK   INTEGER array, dimension (max(2*M2,2*N,N*N,NP2))

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) contains the optimal
              LDWORK.

      LDWORK  INTEGER
              The dimension of the array DWORK.
              LDWORK >= N*M + NP*(N+M) + M2*M2 + NP2*NP2 +
                        max(1,LW1,LW2,LW3,LW4,LW5,LW6), where
              LW1 = (N+NP1+1)*(N+M2) + max(3*(N+M2)+N+NP1,5*(N+M2)),
              LW2 = (N+NP2)*(N+M1+1) + max(3*(N+NP2)+N+M1,5*(N+NP2)),
              LW3 = M2 + NP1*NP1 + max(NP1*max(N,M1),3*M2+NP1,5*M2),
              LW4 = NP2 + M1*M1 + max(max(N,NP1)*M1,3*NP2+M1,5*NP2),
              LW5 = 2*N*N+max(1,14*N*N+6*N+max(14*N+23,16*N),M2*(N+M2+
                              max(3,M1)),NP2*(N+NP2+3)),
              LW6 = max(N*M2,N*NP2,M2*NP2,M2*M2+4*M2),
              with M1 = M - M2 and NP1 = NP - NP2.
              For good performance, LDWORK must generally be larger.
              Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
              2*Q*(3*Q+2*N)+max(1,(N+Q)*(N+Q+6),Q*(Q+max(N,Q,5)+1),
                      2*N*N+max(1,14*N*N+6*N+max(14*N+23,16*N),
                                Q*(N+Q+max(Q,3)))).

      BWORK   LOGICAL array, dimension (2*N)

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
                                       j*Theta
              = 1:  if the matrix | A-e       *I  B2  | had not full
                                  |      C1       D12 |
                    column rank in respect to the tolerance EPS;
                                       j*Theta
              = 2:  if the matrix | A-e       *I  B1  |  had not full
                                  |      C2       D21 |
                    row rank in respect to the tolerance EPS;
              = 3:  if the matrix D12 had not full column rank in
                    respect to the tolerance TOL;
              = 4:  if the matrix D21 had not full row rank in respect
                    to the tolerance TOL;
              = 5:  if the singular value decomposition (SVD) algorithm
                    did not converge (when computing the SVD of one of
                    the matrices |A-I  B2 |, |A-I  B1 |, D12 or D21).
                                 |C1   D12|  |C2   D21|
              = 6:  if the X-Riccati equation was not solved
                    successfully;
              = 7:  if the matrix Im2 + B2'*X2*B2 is not positive
                    definite, or it is numerically singular (with
                    respect to the tolerance TOL);
              = 8:  if the Y-Riccati equation was not solved
                    successfully;
              = 9:  if the matrix Ip2 + C2*Y2*C2' is not positive
                    definite, or it is numerically singular (with
                    respect to the tolerance TOL);
              =10:  if the matrix Im2 + DKHAT*D22 is singular, or its
                    estimated condition number is larger than or equal
                    to 1/TOL.

Method

      The routine implements the formulas given in [1].

References

      [1] Zhou, K., Doyle, J.C., and Glover, K.
          Robust and Optimal Control.
          Prentice-Hall, Upper Saddle River, NJ, 1996.

      [2] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
          Fortran 77 routines for Hinf and H2 design of linear
          discrete-time control systems.
          Report 99-8, Department of Engineering, Leicester University,
          April 1999.

Numerical Aspects

      The accuracy of the result depends on the condition numbers of the
      matrices which are to be inverted and on the condition numbers of
      the matrix Riccati equations which are to be solved in the
      computation of the controller. (The corresponding reciprocal
      condition numbers are given in the output array RCOND.)

Further Comments

      None

Example

Program Text

    *     SB10ED EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 10, MMAX = 10, PMAX = 10 )
          INTEGER          LDA, LDB, LDC, LDD, LDAK, LDBK, LDCK, LDDK
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
         $                   LDAK = NMAX, LDBK = NMAX, LDCK = PMAX,
         $                   LDDK = PMAX )
          INTEGER          LIWORK
          PARAMETER        ( LIWORK = MAX( 2*MMAX, PMAX, 2*NMAX,
         $                                 NMAX*NMAX ) )
          INTEGER          MPMX
          PARAMETER        ( MPMX = MAX( MMAX, PMAX ) )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 2*MPMX*( 3*MPMX + 2*NMAX ) +
         $                   MAX( ( NMAX + MPMX )*( NMAX + MPMX + 6 ),
         $                   MPMX*( MPMX + MAX( NMAX, MPMX, 5 ) + 1 ),
         $                   2*NMAX*NMAX + MAX( 14*NMAX*NMAX + 6*NMAX +
         $                   MAX( 14*NMAX + 23, 16*NMAX ),
         $                   MPMX*( NMAX + MPMX + MAX( MPMX, 3 ) ) ) ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION TOL
          INTEGER          I, INFO, J, M, N, NCON, NMEAS, NP
    *     .. Local Arrays ..
          LOGICAL          BWORK(2*NMAX)
          INTEGER          IWORK(LIWORK)
          DOUBLE PRECISION A(LDA,NMAX), AK(LDA,NMAX), B(LDB,MMAX),
         $                 BK(LDBK,MMAX), C(LDC,NMAX), CK(LDCK,NMAX),
         $                 D(LDD,MMAX), DK(LDDK,MMAX), DWORK(LDWORK),
         $                 RCOND( 8 )
    *     .. External Subroutines ..
          EXTERNAL         SB10ED
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, M, NP, NCON, NMEAS
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99990 ) N
          ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
             WRITE ( NOUT, FMT = 99989 ) M
          ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
             WRITE ( NOUT, FMT = 99988 ) NP
          ELSE IF ( NCON.LT.0 .OR. NCON.GT.MMAX ) THEN
             WRITE ( NOUT, FMT = 99987 ) NCON
          ELSE IF ( NMEAS.LT.0 .OR. NMEAS.GT.PMAX ) THEN
             WRITE ( NOUT, FMT = 99986 ) NMEAS
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
             READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
             READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
             READ ( NIN, FMT = * ) TOL
             CALL SB10ED( N, M, NP, NCON, NMEAS, A, LDA, B, LDB,
         $                C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
         $                DK, LDDK, RCOND, TOL, IWORK, DWORK, LDWORK,
         $                BWORK, INFO )
             IF ( INFO.EQ.0 ) THEN
                WRITE ( NOUT, FMT = 99997 )
                DO 10 I = 1, N
                   WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,N )
       10       CONTINUE
                WRITE ( NOUT, FMT = 99996 )
                DO 20 I = 1, N
                   WRITE ( NOUT, FMT = 99992 ) ( BK(I,J), J = 1,NMEAS )
       20       CONTINUE
                WRITE ( NOUT, FMT = 99995 )
                DO 30 I = 1, NCON
                   WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,N )
       30       CONTINUE
                WRITE ( NOUT, FMT = 99994 )
                DO 40 I = 1, NCON
                   WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NMEAS )
       40       CONTINUE
                WRITE( NOUT, FMT = 99993 )
                WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 7 )
             ELSE
                WRITE( NOUT, FMT = 99998 ) INFO
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' SB10ED EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (/' INFO on exit from SB10ED =',I2)
    99997 FORMAT (' The controller state matrix AK is'/)
    99996 FORMAT (/' The controller input matrix BK is'/)
    99995 FORMAT (/' The controller output matrix CK is'/)
    99994 FORMAT (/' The controller matrix DK is'/)
    99993 FORMAT (/' The estimated condition numbers are'/)
    99992 FORMAT (10(1X,F8.4))
    99991 FORMAT ( 5(1X,D12.5))
    99990 FORMAT (/' N is out of range.',/' N = ',I5)
    99989 FORMAT (/' M is out of range.',/' M = ',I5)
    99988 FORMAT (/' N is out of range.',/' N = ',I5)
    99987 FORMAT (/' NCON is out of range.',/' NCON = ',I5)
    99986 FORMAT (/' NMEAS is out of range.',/' NMEAS = ',I5)
          END

Program Data

     SB10ED EXAMPLE PROGRAM DATA
       6     5     5     2     2
      -0.7  0.0  0.3  0.0 -0.5 -0.1
      -0.6  0.2 -0.4 -0.3  0.0  0.0
      -0.5  0.7 -0.1  0.0  0.0 -0.8
      -0.7  0.0  0.0 -0.5 -1.0  0.0
       0.0  0.3  0.6 -0.9  0.1 -0.4
       0.5 -0.8  0.0  0.0  0.2 -0.9
      -1.0 -2.0 -2.0  1.0  0.0
       1.0  0.0  1.0 -2.0  1.0
      -3.0 -4.0  0.0  2.0 -2.0
       1.0 -2.0  1.0  0.0 -1.0
       0.0  1.0 -2.0  0.0  3.0
       1.0  0.0  3.0 -1.0 -2.0
       1.0 -1.0  2.0 -2.0  0.0 -3.0
      -3.0  0.0  1.0 -1.0  1.0  0.0
       0.0  2.0  0.0 -4.0  0.0 -2.0
       1.0 -3.0  0.0  0.0  3.0  1.0
       0.0  1.0 -2.0  1.0  0.0 -2.0
       1.0 -1.0 -2.0  0.0  0.0
       0.0  1.0  0.0  1.0  0.0
       2.0 -1.0 -3.0  0.0  1.0
       0.0  1.0  0.0  1.0 -1.0
       0.0  0.0  1.0  2.0  1.0
       0.00000001

Program Results

     SB10ED EXAMPLE PROGRAM RESULTS

     The controller state matrix AK is

      -0.0551  -2.1891  -0.6607  -0.2532   0.6674  -1.0044
      -1.0379   2.3804   0.5031   0.3960  -0.6605   1.2673
      -0.0876  -2.1320  -0.4701  -1.1461   1.2927  -1.5116
      -0.1358  -2.1237  -0.9560  -0.7144   0.6673  -0.7957
       0.4900   0.0895   0.2634  -0.2354   0.1623  -0.2663
       0.1672  -0.4163   0.2871  -0.1983   0.4944  -0.6967

     The controller input matrix BK is

      -0.5985  -0.5464
       0.5285   0.6087
      -0.7600  -0.4472
      -0.7288  -0.6090
       0.0532   0.0658
      -0.0663   0.0059

     The controller output matrix CK is

       0.2500  -1.0200  -0.3371  -0.2733   0.2747  -0.4444
       0.0654   0.2095   0.0632   0.2089  -0.1895   0.1834

     The controller matrix DK is

      -0.2181  -0.2070
       0.1094   0.1159

     The estimated condition numbers are

      0.10000D+01  0.10000D+01  0.25207D+00  0.83985D-01  0.48628D-02
      0.55015D-03  0.49886D+00

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