**Purpose**

To compute a reduced order model (Ar,Br,Cr,Dr) for an original state-space representation (A,B,C,D) by using either the square-root or the balancing-free square-root Singular Perturbation Approximation (SPA) model reduction method in conjunction with stable coprime factorization techniques.

SUBROUTINE AB09GD( DICO, JOBCF, FACT, JOBMR, EQUIL, ORDSEL, N, M, $ P, NR, ALPHA, A, LDA, B, LDB, C, LDC, D, LDD, $ NQ, HSV, TOL1, TOL2, TOL3, IWORK, DWORK, $ LDWORK, IWARN, INFO ) C .. Scalar Arguments .. CHARACTER DICO, EQUIL, FACT, JOBCF, JOBMR, ORDSEL INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK, M, N, $ NQ, NR, P DOUBLE PRECISION ALPHA, TOL1, TOL2, TOL3 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. JOBCF CHARACTER*1 Specifies whether left or right coprime factorization is to be used as follows: = 'L': use left coprime factorization; = 'R': use right coprime factorization. FACT CHARACTER*1 Specifies the type of coprime factorization to be computed as follows: = 'S': compute a coprime factorization with prescribed stability degree ALPHA; = 'I': compute a coprime factorization with inner denominator. JOBMR CHARACTER*1 Specifies the model reduction approach to be used as follows: = 'B': use the square-root Balance & Truncate method; = 'N': use the balancing-free square-root Balance & Truncate method. 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(NR,NQ,NMIN), where NR is the desired order on entry, NQ is the order of the computed coprime factorization of the given system, and NMIN is the order of a minimal realization of the extended system (see METHOD); NMIN is determined as the number of Hankel singular values greater than NQ*EPS*HNORM(Ge), where EPS is the machine precision (see LAPACK Library Routine DLAMCH) and HNORM(Ge) is the Hankel norm of the extended system (computed in HSV(1)); if ORDSEL = 'A', NR is equal to the number of Hankel singular values greater than MAX(TOL1,NQ*EPS*HNORM(Ge)). ALPHA (input) DOUBLE PRECISION If FACT = 'S', the desired stability degree for the factors of the coprime factorization (see SLICOT Library routines SB08ED/SB08FD). ALPHA < 0 for a continuous-time system (DICO = 'C'), and 0 <= ALPHA < 1 for a discrete-time system (DICO = 'D'). If FACT = 'I', ALPHA is not used. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading N-by-N part of this array must contain the original 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. 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). NQ (output) INTEGER The order of the computed extended system Ge (see METHOD). HSV (output) DOUBLE PRECISION array, dimension (N) If INFO = 0, it contains the NQ Hankel singular values of the extended system Ge ordered decreasingly (see METHOD).

TOL1 DOUBLE PRECISION If ORDSEL = 'A', TOL1 contains the tolerance for determining the order of reduced extended system. For model reduction, the recommended value is TOL1 = c*HNORM(Ge), where c is a constant in the interval [0.00001,0.001], and HNORM(Ge) is the Hankel-norm of the extended system (computed in HSV(1)). The value TOL1 = NQ*EPS*HNORM(Ge) is used by default if TOL1 <= 0 on entry, where EPS is the machine precision (see LAPACK Library Routine DLAMCH). If ORDSEL = 'F', the value of TOL1 is ignored. TOL2 DOUBLE PRECISION The tolerance for determining the order of a minimal realization of the extended system Ge (see METHOD). The recommended value is TOL2 = NQ*EPS*HNORM(Ge). This value is used by default if TOL2 <= 0 on entry. If TOL2 > 0, then TOL2 <= TOL1. TOL3 DOUBLE PRECISION The absolute tolerance level below which the elements of B or C are considered zero (used for controllability or observability tests by the coprime factorization method). If the user sets TOL3 <= 0, then an implicitly computed, default tolerance TOLDEF is used: TOLDEF = N*EPS*NORM(C'), if JOBCF = 'L', or TOLDEF = N*EPS*NORM(B), if JOBCF = 'R', where EPS is the machine precision, and NORM(.) denotes the 1-norm of a matrix.

IWORK INTEGER array, dimension (MAX(1,2*N,PM)) where PM = P, if JOBCF = 'L', PM = M, if JOBCF = 'R'. On exit with INFO = 0, IWORK(1) contains the order of the minimal realization of the system. 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(1,LW1) if JOBCF = 'L' and FACT = 'S', LDWORK >= MAX(1,LW2) if JOBCF = 'L' and FACT = 'I', LDWORK >= MAX(1,LW3) if JOBCF = 'R' and FACT = 'S', LDWORK >= MAX(1,LW4) if JOBCF = 'R' and FACT = 'I', where LW1 = N*(2*MAX(M,P) + P) + MAX(M,P)*(MAX(M,P) + P) + MAX( N*P+MAX(N*(N+5), 5*P, 4*M), LWR ), LW2 = N*(2*MAX(M,P) + P) + MAX(M,P)*(MAX(M,P) + P) + MAX( N*P+MAX(N*(N+5), P*(P+2), 4*P, 4*M), LWR ), LW3 = (N+M)*(M+P) + MAX( 5*M, 4*P, LWR ), LW4 = (N+M)*(M+P) + MAX( M*(M+2), 4*M, 4*P, LWR ), and LWR = 2*N*N + N*(MAX(N,M+P)+5) + N*(N+1)/2. For optimum performance LDWORK should be larger.

IWARN INTEGER = 0: no warning; = 10*K+I: I = 1: with ORDSEL = 'F', the selected order NR is greater than the order of the computed coprime factorization 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; K > 0: K violations of the numerical stability condition occured when computing the coprime factorization using pole assignment (see SLICOT Library routines SB08CD/SB08ED, SB08DD/SB08FD).

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: the reduction of A to a real Schur form failed; = 2: a failure was detected during the ordering of the real Schur form of A, or in the iterative process for reordering the eigenvalues of Z'*(A + H*C)*Z (or Z'*(A + B*F)*Z) along the diagonal; see SLICOT Library routines SB08CD/SB08ED (or SB08DD/SB08FD); = 3: the matrix A has an observable or controllable eigenvalue on the imaginary axis if DICO = 'C' or on the unit circle if DICO = 'D'; = 4: the computation of Hankel singular values failed.

Let be the 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, and let G be the corresponding transfer-function matrix. The subroutine AB09GD determines 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) with the transfer-function matrix Gr, by using the singular perturbation approximation (SPA) method in conjunction with a left coprime factorization (LCF) or a right coprime factorization (RCF) technique: 1. Compute the appropriate stable coprime factorization of G: -1 -1 G = R *Q (LCF) or G = Q*R (RCF). 2. Perform the model reduction algorithm on the extended system ( Q ) Ge = ( Q R ) (LCF) or Ge = ( R ) (RCF) to obtain a reduced extended system with reduced factors ( Qr ) Ger = ( Qr Rr ) (LCF) or Ger = ( Rr ) (RCF). 3. Recover the reduced system from the reduced factors as -1 -1 Gr = Rr *Qr (LCF) or Gr = Qr*Rr (RCF). The approximation error for the extended system satisfies HSV(NR) <= INFNORM(Ge-Ger) <= 2*[HSV(NR+1) + ... + HSV(NQ)], where INFNORM(G) is the infinity-norm of G. If JOBMR = 'B', the balancing-based square-root SPA method of [1] is used for model reduction. If JOBMR = 'N', the balancing-free square-root SPA method of [2] is used for model reduction. By setting TOL1 = TOL2, the routine can be used to compute Balance & Truncate approximations. If FACT = 'S', the stable coprime factorization with prescribed stability degree ALPHA is computed by using the algorithm of [3]. If FACT = 'I', the stable coprime factorization with inner denominator is computed by using the algorithm of [4].

[1] Liu Y. and Anderson B.D.O. Singular Perturbation Approximation of Balanced Systems. Int. J. Control, Vol. 50, pp. 1379-1405, 1989. [2] Varga A. Balancing-free square-root algorithm for computing singular perturbation approximations. Proc. 30-th IEEE CDC, Brighton, Dec. 11-13, 1991, Vol. 2, pp. 1062-1065. [3] Varga A. Coprime factors model reduction method based on square-root balancing-free techniques. System Analysis, Modelling and Simulation, Vol. 11, pp. 303-311, 1993. [4] Varga A. A Schur method for computing coprime factorizations with inner denominators and applications in model reduction. Proc. ACC'93, San Francisco, CA, pp. 2130-2131, 1993.

The implemented methods rely on accuracy enhancing square-root or balancing-free square-root techniques. 3 The algorithms require less than 30N floating point operations.

None

**Program Text**

* AB09GD 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( 2*NMAX, MMAX, PMAX ) ) * The formula below uses that NMAX = MMAX = PMAX. INTEGER LDWORK PARAMETER ( LDWORK = 10*NMAX*NMAX + 5*NMAX ) * .. Local Scalars .. DOUBLE PRECISION ALPHA, TOL1, TOL2, TOL3 INTEGER I, INFO, IWARN, J, M, N, NQ, NR, P CHARACTER*1 DICO, EQUIL, FACT, JOBCF, JOBMR, 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 AB09GD * .. 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, P, NR, ALPHA, TOL1, TOL2, TOL3, $ DICO, JOBCF, FACT, JOBMR, 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,D). CALL AB09GD( DICO, JOBCF, FACT, JOBMR, EQUIL, ORDSEL, $ N, M, P, NR, ALPHA, A, LDA, B, LDB, C, LDC, $ D, LDD, NQ, HSV, TOL1, TOL2, TOL3, 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, NQ ) 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 80 I = 1, P WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1, M ) 80 CONTINUE END IF END IF END IF END IF STOP * 99999 FORMAT (' AB09GD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from AB09GD = ',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 of coprime factors are') END

AB08GD EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 -1.e-1 .1 1.E-10 1.E-10 C L I B S A -0.04165 0.0000 4.9200 0.4920 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 0.0545 0.0000 0.0000 0.0000 0.0000 0.0000 -0.49200 0.004165 0.0000 4.9200 0.0000 0.0000 0.0000 0.0000 0.5210 -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

AB09GD EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The Hankel singular values of coprime factors are 13.6047 9.4106 1.7684 0.7456 0.6891 0.0241 0.0230 The reduced state dynamics matrix Ar is 0.0521 -0.1491 0.0032 -0.0242 0.0181 0.2341 0.2615 0.0009 -0.0171 -0.0362 0.1170 0.0076 -0.5471 2.0904 -0.8098 0.1675 -0.2122 -2.2113 -2.4097 -0.4139 0.0390 -0.5061 0.8787 -0.3166 -3.2955 The reduced input/state matrix Br is 1.0449 0.5863 -0.0490 -1.9210 -1.3930 0.0540 -1.7206 -0.8039 0.6358 -2.0542 The reduced state/output matrix Cr is 0.4331 0.1125 -1.2534 1.6965 -0.6773 1.1171 0.3963 0.6102 0.1213 -0.0841 0.0736 -1.8815 0.0134 0.8457 2.0413 The reduced input/output matrix Dr is 0.0480 0.0003 -0.0017 0.0001 0.0005 0.0460