**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.

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,*)

**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.

DWORK DOUBLE PRECISION array, dimension (3*NCONT)

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

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.

[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.

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

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).

**Program Text**

* SB01MD EXAMPLE PROGRAM TEXT * Copyright (c) 2002-2017 NICONET e.V. * * .. 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

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

SB01MD EXAMPLE PROGRAM RESULTS The one-dimensional state feedback matrix G is 1.0000 29.0000 93.0000 -76.0000