**Purpose**

To compute a reduced order model (Ar,Br,Cr,Dr) for a stable original state-space representation (A,B,C,D) by using the optimal Hankel-norm approximation method in conjunction with square-root balancing.

SUBROUTINE AB09CD( DICO, EQUIL, ORDSEL, N, M, P, NR, A, LDA, B, $ LDB, C, LDC, D, LDD, HSV, TOL1, TOL2, IWORK, $ DWORK, LDWORK, IWARN, INFO ) C .. Scalar Arguments .. CHARACTER DICO, EQUIL, ORDSEL INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK, $ M, N, NR, P DOUBLE PRECISION TOL1, TOL2 C .. Array Arguments .. INTEGER IWORK(*) DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), $ DWORK(*), HSV(*)

**Mode Parameters**

DICO CHARACTER*1 Specifies the type of the original system as follows: = 'C': continuous-time system; = 'D': discrete-time system. EQUIL CHARACTER*1 Specifies whether the user wishes to preliminarily equilibrate the triplet (A,B,C) as follows: = 'S': perform equilibration (scaling); = 'N': do not perform equilibration. ORDSEL CHARACTER*1 Specifies the order selection method as follows: = 'F': the resulting order NR is fixed; = 'A': the resulting order NR is automatically determined on basis of the given tolerance TOL1.

N (input) INTEGER The order of the original state-space representation, i.e. the order of the matrix A. N >= 0. M (input) INTEGER The number of system inputs. M >= 0. P (input) INTEGER The number of system outputs. P >= 0. NR (input/output) INTEGER On entry with ORDSEL = 'F', NR is the desired order of the resulting reduced order system. 0 <= NR <= N. On exit, if INFO = 0, NR is the order of the resulting reduced order model. NR is set as follows: if ORDSEL = 'F', NR is equal to MIN(MAX(0,NR-KR+1),NMIN), where KR is the multiplicity of the Hankel singular value HSV(NR+1), NR is the desired order on entry, and NMIN is the order of a minimal realization of the given system; NMIN is determined as the number of Hankel singular values greater than N*EPS*HNORM(A,B,C), where EPS is the machine precision (see LAPACK Library Routine DLAMCH) and HNORM(A,B,C) is the Hankel norm of the system (computed in HSV(1)); if ORDSEL = 'A', NR is equal to the number of Hankel singular values greater than MAX(TOL1,N*EPS*HNORM(A,B,C)). A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading N-by-N part of this array must contain the state dynamics matrix A. On exit, if INFO = 0, the leading NR-by-NR part of this array contains the state dynamics matrix Ar of the reduced order system in a real Schur form. LDA INTEGER The leading dimension of array A. LDA >= MAX(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB,M) On entry, the leading N-by-M part of this array must contain the original input/state matrix B. On exit, if INFO = 0, the leading NR-by-M part of this array contains the input/state matrix Br of the reduced order system. LDB INTEGER The leading dimension of array B. LDB >= MAX(1,N). C (input/output) DOUBLE PRECISION array, dimension (LDC,N) On entry, the leading P-by-N part of this array must contain the original state/output matrix C. On exit, if INFO = 0, the leading P-by-NR part of this array contains the state/output matrix Cr of the reduced order system. LDC INTEGER The leading dimension of array C. LDC >= MAX(1,P). D (input/output) DOUBLE PRECISION array, dimension (LDD,M) On entry, the leading P-by-M part of this array must contain the original input/output matrix D. On exit, if INFO = 0, the leading P-by-M part of this array contains the input/output matrix Dr of the reduced order system. LDD INTEGER The leading dimension of array D. LDD >= MAX(1,P). HSV (output) DOUBLE PRECISION array, dimension (N) If INFO = 0, it contains the Hankel singular values of the original system ordered decreasingly. HSV(1) is the Hankel norm of the system.

TOL1 DOUBLE PRECISION If ORDSEL = 'A', TOL1 contains the tolerance for determining the order of reduced system. For model reduction, the recommended value is TOL1 = c*HNORM(A,B,C), where c is a constant in the interval [0.00001,0.001], and HNORM(A,B,C) is the Hankel-norm of the given system (computed in HSV(1)). For computing a minimal realization, the recommended value is TOL1 = N*EPS*HNORM(A,B,C), where EPS is the machine precision (see LAPACK Library Routine DLAMCH). This value is used by default if TOL1 <= 0 on entry. If ORDSEL = 'F', the value of TOL1 is ignored. TOL2 DOUBLE PRECISION The tolerance for determining the order of a minimal realization of the given system. The recommended value is TOL2 = N*EPS*HNORM(A,B,C). This value is used by default if TOL2 <= 0 on entry. If TOL2 > 0, then TOL2 <= TOL1.

IWORK INTEGER array, dimension (LIWORK) LIWORK = MAX(1,M), if DICO = 'C'; LIWORK = MAX(1,N,M), if DICO = 'D'. On exit, if INFO = 0, IWORK(1) contains NMIN, the order of the computed minimal realization. DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal value of LDWORK. LDWORK INTEGER The length of the array DWORK. LDWORK >= MAX( LDW1, LDW2 ), where LDW1 = N*(2*N+MAX(N,M,P)+5) + N*(N+1)/2, LDW2 = N*(M+P+2) + 2*M*P + MIN(N,M) + MAX( 3*M+1, MIN(N,M)+P ). For optimum performance LDWORK should be larger.

IWARN INTEGER = 0: no warning; = 1: with ORDSEL = 'F', the selected order NR is greater than the order of a minimal realization of the given system. In this case, the resulting NR is set automatically to a value corresponding to the order of a minimal realization of the system.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: the reduction of A to the real Schur form failed; = 2: the state matrix A is not stable (if DICO = 'C') or not convergent (if DICO = 'D'); = 3: the computation of Hankel singular values failed; = 4: the computation of stable projection failed; = 5: the order of computed stable projection differs from the order of Hankel-norm approximation.

Let be the stable linear system d[x(t)] = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) (1) where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1) for a discrete-time system. The subroutine AB09CD determines for the given system (1), the matrices of a reduced order system d[z(t)] = Ar*z(t) + Br*u(t) yr(t) = Cr*z(t) + Dr*u(t) (2) such that HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)], where G and Gr are transfer-function matrices of the systems (A,B,C,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the infinity-norm of G. The optimal Hankel-norm approximation method of [1], based on the square-root balancing projection formulas of [2], is employed.

[1] Glover, K. All optimal Hankel norm approximation of linear multivariable systems and their L-infinity error bounds. Int. J. Control, Vol. 36, pp. 1145-1193, 1984. [2] Tombs M.S. and Postlethwaite I. Truncated balanced realization of stable, non-minimal state-space systems. Int. J. Control, Vol. 46, pp. 1319-1330, 1987.

The implemented methods rely on an accuracy enhancing square-root technique. 3 The algorithms require less than 30N floating point operations.

None

**Program Text**

* AB09CD 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 = 20, MMAX = 20, PMAX = 20 ) INTEGER LDA, LDB, LDC, LDD PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, $ LDD = PMAX ) INTEGER LIWORK PARAMETER ( LIWORK = MAX( NMAX, MMAX ) ) INTEGER LDWORK PARAMETER ( LDWORK = MAX( NMAX*( 2*NMAX + $ MAX( NMAX, MMAX, PMAX ) + 5 ) + $ ( NMAX*( NMAX + 1 ) )/2, $ NMAX*( MMAX + PMAX + 2 ) + $ 2*MMAX*PMAX + $ MIN( NMAX, MMAX ) + MAX( 3*MMAX + 1, $ MIN( NMAX, MMAX ) + PMAX ) ) ) * .. Local Scalars .. DOUBLE PRECISION TOL1, TOL2 INTEGER I, INFO, IWARN, J, M, N, NR, P CHARACTER*1 DICO, EQUIL, ORDSEL * .. Local Arrays .. DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX), $ D(LDD,MMAX), DWORK(LDWORK), HSV(NMAX) INTEGER IWORK(LIWORK) * .. External Subroutines .. EXTERNAL AB09CD * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. 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, P, NR, TOL1, TOL2, DICO, EQUIL, ORDSEL IF ( N.LT.0 .OR. N.GT.NMAX ) THEN WRITE ( NOUT, FMT = 99990 ) N ELSE READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N ) IF ( M.LT.0 .OR. M.GT.MMAX ) THEN WRITE ( NOUT, FMT = 99989 ) M ELSE READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1, N ) IF ( P.LT.0 .OR. P.GT.PMAX ) THEN WRITE ( NOUT, FMT = 99988 ) P ELSE READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P ) READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P ) * Find a reduced ssr for (A,B,C). CALL AB09CD( DICO, EQUIL, ORDSEL, N, M, P, NR, $ A, LDA, B, LDB, C, LDC, D, LDD, HSV, TOL1, $ TOL2, IWORK, DWORK, LDWORK, IWARN, INFO ) * IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99998 ) INFO ELSE WRITE ( NOUT, FMT = 99997 ) NR WRITE ( NOUT, FMT = 99987 ) WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,N ) IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 ) DO 20 I = 1, NR WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR ) 20 CONTINUE IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 ) DO 40 I = 1, NR WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M ) 40 CONTINUE IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 ) DO 60 I = 1, P WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR ) 60 CONTINUE WRITE ( NOUT, FMT = 99991 ) DO 70 I = 1, P WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M ) 70 CONTINUE END IF END IF END IF END IF STOP * 99999 FORMAT (' AB09CD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from AB09CD = ',I2) 99997 FORMAT (' The order of reduced model = ',I2) 99996 FORMAT (/' The reduced state dynamics matrix Ar is ') 99995 FORMAT (20(1X,F8.4)) 99993 FORMAT (/' The reduced input/state matrix Br is ') 99992 FORMAT (/' The reduced state/output matrix Cr is ') 99991 FORMAT (/' The reduced input/output matrix Dr is ') 99990 FORMAT (/' N is out of range.',/' N = ',I5) 99989 FORMAT (/' M is out of range.',/' M = ',I5) 99988 FORMAT (/' P is out of range.',/' P = ',I5) 99987 FORMAT (/' The Hankel singular values are') END

AB09CD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 1.E-1 1.E-14 C N A -0.04165 0.0000 4.9200 -4.9200 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 0.0000 0.0000 0.5450 0.0000 0.0000 0.0000 -0.5450 0.0000 0.0000 0.0000 0.0000 0.0000 4.9200 -0.04165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 -5.2100 -12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3.3300 -3.3300 0.0000 0.0000 12.500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 12.500 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

AB09CD EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The Hankel singular values are 2.5139 2.0846 1.9178 0.7666 0.5473 0.0253 0.0246 The reduced state dynamics matrix Ar is -0.5038 -5.3070 -3.2250 0.0000 0.0000 1.8355 -0.5038 -2.6289 0.0000 0.0000 0.0000 0.0000 -1.5171 0.0000 0.0000 0.0000 0.0000 0.0000 -1.2925 -9.0718 0.0000 0.0000 0.0000 0.5047 -1.2925 The reduced input/state matrix Br is -1.5343 1.5343 -0.3614 0.3614 -1.1096 1.1096 -4.5325 -4.5325 -0.7396 -0.7396 The reduced state/output matrix Cr is 1.8971 -0.3055 -2.1124 0.4421 -2.1023 -0.0394 1.1112 -0.3119 0.0000 0.0000 -1.8971 0.3055 2.1124 0.4421 -2.1023 The reduced input/output matrix Dr is 0.0126 -0.0126 0.0005 -0.0005 -0.0126 0.0126