SB10KD

Positive feedback controller for a discrete-time system

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

Purpose

  To compute the matrices of the positive feedback controller

           | Ak | Bk |
       K = |----|----|
           | Ck | Dk |

  for the shaped plant

           | A | B |
       G = |---|---|
           | C | 0 |

  in the Discrete-Time Loop Shaping Design Procedure.

Specification
      SUBROUTINE SB10KD( N, M, NP, A, LDA, B, LDB, C, LDC, FACTOR,
     $                   AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK, RCOND,
     $                   IWORK, DWORK, LDWORK, BWORK, INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDDK,
     $                   LDWORK, M, N, NP
      DOUBLE PRECISION   FACTOR
C     .. Array Arguments ..
      INTEGER            IWORK( * )
      LOGICAL            BWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
     $                   BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
     $                   DK( LDDK, * ), DWORK( * ), RCOND( 4 )

Arguments

Input/Output Parameters

  N       (input) INTEGER
          The order of the plant.  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.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The leading N-by-N part of this array must contain the
          system state matrix A of the shaped plant.

  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 of the shaped plant.

  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 of the shaped plant.

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

  FACTOR  (input) DOUBLE PRECISION
          = 1  implies that an optimal controller is required;
          > 1  implies that a suboptimal controller is required
               achieving a performance FACTOR less than optimal.
          FACTOR >= 1.

  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,NP)
          The leading N-by-NP 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 M-by-N part of this array contains the
          controller output matrix Ck.

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

  DK      (output) DOUBLE PRECISION array, dimension (LDDK,NP)
          The leading M-by-NP part of this array contains the
          controller matrix Dk.

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

  RCOND   (output) DOUBLE PRECISION array, dimension (4)
          RCOND(1) contains an estimate of the reciprocal condition
                   number of the linear system of equations from
                   which the solution of the P-Riccati equation is
                   obtained;
          RCOND(2) contains an estimate of the reciprocal condition
                   number of the linear system of equations from
                   which the solution of the Q-Riccati equation is
                   obtained;
          RCOND(3) contains an estimate of the reciprocal condition
                   number of the linear system of equations from
                   which the solution of the X-Riccati equation is
                   obtained;
          RCOND(4) contains an estimate of the reciprocal condition
                   number of the matrix Rx + Bx'*X*Bx (see the
                   comments in the code).

Workspace
  IWORK   INTEGER array, dimension (2*max(N,NP+M))

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

  LDWORK  INTEGER
          The dimension of the array DWORK.
          LDWORK >= 15*N*N + 6*N +
                    max( 14*N+23, 16*N, 2*N+NP+M, 3*(NP+M) ) +
                    max( N*N, 11*N*NP + 2*M*M + 8*NP*NP + 8*M*N +
                              4*M*NP + NP ).
          For good performance, LDWORK must generally be larger.

  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;
          = 1:  the P-Riccati equation is not solved successfully;
          = 2:  the Q-Riccati equation is not solved successfully;
          = 3:  the X-Riccati equation is not solved successfully;
          = 4:  the iteration to compute eigenvalues failed to
                converge;
          = 5:  the matrix Rx + Bx'*X*Bx is singular;
          = 6:  the closed-loop system is unstable.

Method
  The routine implements the method presented in [1].

References
  [1] McFarlane, D. and Glover, K.
      A loop shaping design procedure using H_infinity synthesis.
      IEEE Trans. Automat. Control, vol. AC-37, no. 6, pp. 759-769,
      1992.

Numerical Aspects
  The accuracy of the results depends on the conditioning of the
  two Riccati equations solved in the controller design. For
  better conditioning it is advised to take FACTOR > 1.

Further Comments
  None
Example

Program Text

*     SB10KD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          NMAX, MMAX, PMAX
      PARAMETER        ( NMAX = 10, MMAX = 10, PMAX = 10 )
      INTEGER          LDA, LDAK, LDB, LDBK, LDC, LDCK, LDDK
      PARAMETER        ( LDA  = NMAX, LDAK = NMAX, LDB  = NMAX,
     $                   LDBK = NMAX, LDC  = PMAX, LDCK = MMAX,
     $                   LDDK = MMAX )
      INTEGER          LIWORK
      PARAMETER        ( LIWORK = 2*MAX( NMAX, MMAX + PMAX ) )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK = 15*NMAX*NMAX + 6*NMAX +
     $                            MAX( 14*NMAX + 23, 16*NMAX,
     $                                 2*NMAX+PMAX+MMAX,
     $                                 3*(PMAX+MMAX) ) +
     $                            MAX( NMAX*NMAX,
     $                                 11*NMAX*PMAX + 2*MMAX*MMAX +
     $                                 8*PMAX*PMAX + 8*MMAX*NMAX +
     $                                 4*MMAX*PMAX + PMAX ) )
*     .. Local Scalars ..
      DOUBLE PRECISION FACTOR
      INTEGER          I, INFO, J, M, N, NP
*     .. Local Arrays ..
      LOGICAL          BWORK(2*NMAX)
      INTEGER          IWORK(LIWORK)
      DOUBLE PRECISION A(LDA,NMAX), AK(LDA,NMAX), B(LDB,MMAX),
     $                 BK(LDBK,PMAX), C(LDC,NMAX), CK(LDCK,NMAX),
     $                 DK(LDDK,PMAX), DWORK(LDWORK), RCOND(4)
*     .. External Subroutines ..
      EXTERNAL         SB10KD
*     .. 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
      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
         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 = * ) FACTOR
         CALL SB10KD( N, M, NP, A, LDA, B, LDB, C, LDC, FACTOR, AK,
     $                LDAK, BK, LDBK, CK, LDCK, DK, LDDK, RCOND,
     $                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,NP )
   20       CONTINUE
            WRITE ( NOUT, FMT = 99995 )
            DO 30 I = 1, M
               WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,N )
   30       CONTINUE
            WRITE ( NOUT, FMT = 99994 )
            DO 40 I = 1, M
               WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NP )
   40       CONTINUE
            WRITE( NOUT, FMT = 99993 )
            WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 4 )
         ELSE
            WRITE( NOUT, FMT = 99998 ) INFO
         END IF
      END IF
      STOP
*
99999 FORMAT (' SB10KD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10KD =',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 (/' NP is out of range.',/' NP = ',I5)
      END
Program Data
 SB10KD EXAMPLE PROGRAM DATA
   6     2     2   
   0.2  0.0  0.3  0.0 -0.3 -0.1
  -0.3  0.2 -0.4 -0.3  0.0  0.0
  -0.1  0.1 -0.1  0.0  0.0 -0.3
   0.1  0.0  0.0 -0.1 -0.1  0.0
   0.0  0.3  0.6  0.2  0.1 -0.4
   0.2 -0.4  0.0  0.0  0.2 -0.2
  -1.0 -2.0
   1.0  3.0 
  -3.0 -4.0 
   1.0 -2.0 
   0.0  1.0
   1.0  5.0  
   1.0 -1.0  2.0 -2.0  0.0 -3.0
  -3.0  0.0  1.0 -1.0  1.0 -1.0
   1.1
Program Results
 SB10KD EXAMPLE PROGRAM RESULTS


 The controller state matrix AK is

   0.0337   0.0222   0.0858   0.1264  -0.1872   0.1547
   0.4457   0.0668  -0.2255  -0.3204  -0.4548  -0.0691
  -0.2419  -0.2506  -0.0982  -0.1321  -0.0130  -0.0838
  -0.4402   0.3654  -0.0335  -0.2444   0.6366  -0.6469
  -0.3623   0.3854   0.4162   0.4502   0.0065   0.1261
  -0.0121  -0.4377   0.0604   0.2265  -0.3389   0.4542

 The controller input matrix BK is

   0.0931  -0.0269
  -0.0872   0.1599
   0.0956  -0.1469
  -0.1728   0.0129
   0.2022  -0.1154
   0.2419  -0.1737

 The controller output matrix CK is

  -0.3677   0.2188   0.0403  -0.0854   0.3564  -0.3535
   0.1624  -0.0708   0.0058   0.0606  -0.2163   0.1802

 The controller matrix DK is

  -0.0857  -0.0246
   0.0460   0.0074

 The estimated condition numbers are

  0.11269D-01  0.17596D-01  0.18225D+00  0.75968D-03

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