**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 for the ALPHA-stable part of the system.

SUBROUTINE AB09ND( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, ALPHA, $ A, LDA, B, LDB, C, LDC, D, LDD, NS, HSV, TOL1, $ TOL2, IWORK, DWORK, LDWORK, IWARN, INFO ) C .. Scalar Arguments .. CHARACTER DICO, EQUIL, JOB, ORDSEL INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK, $ M, N, NR, NS, P DOUBLE PRECISION ALPHA, 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. JOB CHARACTER*1 Specifies the model reduction approach to be used as follows: = 'B': use the square-root SPA method; = 'N': use the balancing-free square-root SPA 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. For a system with NU ALPHA-unstable eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N), NR is set as follows: if ORDSEL = 'F', NR is equal to NU+MIN(MAX(0,NR-NU),NMIN), where NR is the desired order on entry, and NMIN is the order of a minimal realization of the ALPHA-stable part of the given system; NMIN is determined as the number of Hankel singular values greater than NS*EPS*HNORM(As,Bs,Cs), where EPS is the machine precision (see LAPACK Library Routine DLAMCH) and HNORM(As,Bs,Cs) is the Hankel norm of the ALPHA-stable part of the given system (computed in HSV(1)); if ORDSEL = 'A', NR is the sum of NU and the number of Hankel singular values greater than MAX(TOL1,NS*EPS*HNORM(As,Bs,Cs)). ALPHA (input) DOUBLE PRECISION Specifies the ALPHA-stability boundary for the eigenvalues of the state dynamics matrix A. For a continuous-time system (DICO = 'C'), ALPHA <= 0 is the boundary value for the real parts of eigenvalues, while for a discrete-time system (DICO = 'D'), 0 <= ALPHA <= 1 represents the boundary value for the moduli of eigenvalues. The ALPHA-stability domain does not include the boundary. 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. The resulting A has a block-diagonal form with two blocks. For a system with NU ALPHA-unstable eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N), the leading NU-by-NU block contains the unreduced part of A corresponding to ALPHA-unstable eigenvalues in an upper real Schur form. The trailing (NR+NS-N)-by-(NR+NS-N) block contains the reduced part of A corresponding to ALPHA-stable eigenvalues. 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). NS (output) INTEGER The dimension of the ALPHA-stable subsystem. HSV (output) DOUBLE PRECISION array, dimension (N) If INFO = 0, the leading NS elements of HSV contain the Hankel singular values of the ALPHA-stable part of the original system ordered decreasingly. HSV(1) is the Hankel norm of the ALPHA-stable subsystem.

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(As,Bs,Cs), where c is a constant in the interval [0.00001,0.001], and HNORM(As,Bs,Cs) is the Hankel-norm of the ALPHA-stable part of the given system (computed in HSV(1)). If TOL1 <= 0 on entry, the used default value is TOL1 = NS*EPS*HNORM(As,Bs,Cs), where NS is the number of ALPHA-stable eigenvalues of A and EPS is the machine precision (see LAPACK Library Routine DLAMCH). This value is appropriate to compute a minimal realization of the ALPHA-stable part. If ORDSEL = 'F', the value of TOL1 is ignored. TOL2 DOUBLE PRECISION The tolerance for determining the order of a minimal realization of the ALPHA-stable part of the given system. The recommended value is TOL2 = NS*EPS*HNORM(As,Bs,Cs). This value is used by default if TOL2 <= 0 on entry. If TOL2 > 0, then TOL2 <= TOL1.

IWORK INTEGER array, dimension (MAX(1,2*N)) On exit, if INFO = 0, IWORK(1) contains the order of the minimal realization of the ALPHA-stable part 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,N*(2*N+MAX(N,M,P)+5) + N*(N+1)/2). For optimum performance LDWORK should be larger.

IWARN INTEGER = 0: no warning; = 1: with ORDSEL = 'F', the selected order NR is greater than NSMIN, the sum of the order of the ALPHA-unstable part and the order of a minimal realization of the ALPHA-stable part of the given system. In this case, the resulting NR is set equal to NSMIN. = 2: with ORDSEL = 'F', the selected order NR is less than the order of the ALPHA-unstable part of the given system. In this case NR is set equal to the order of the ALPHA-unstable part.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: the computation of the ordered real Schur form of A failed; = 2: the separation of the ALPHA-stable/unstable diagonal blocks failed because of very close eigenvalues; = 3: the computation of Hankel singular values failed.

Let be the following 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 AB09ND 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+NS-N) <= INFNORM(G-Gr) <= 2*[HSV(NR+NS-N+1)+...+HSV(NS)], 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 following procedure is used to reduce a given G: 1) Decompose additively G as G = G1 + G2 such that G1 = (As,Bs,Cs,D) has only ALPHA-stable poles and G2 = (Au,Bu,Cu,0) has only ALPHA-unstable poles. 2) Determine G1r, a reduced order approximation of the ALPHA-stable part G1. 3) Assemble the reduced model Gr as Gr = G1r + G2. To reduce the ALPHA-stable part G1, if JOB = 'B', the square-root balancing-based SPA method of [1] is used, and for an ALPHA-stable system, the resulting reduced model is balanced. If JOB = 'N', the balancing-free square-root SPA method of [2] is used to reduce the ALPHA-stable part G1. By setting TOL1 = TOL2, the routine can be used to compute Balance & Truncate approximations as well.

[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.

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**

* AB09ND 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 = 2*NMAX ) INTEGER LDWORK PARAMETER ( LDWORK = NMAX*( 2*NMAX + $ MAX( NMAX, MMAX, PMAX ) + 5 ) + $ ( NMAX*( NMAX + 1 ) )/2 ) * .. Local Scalars .. DOUBLE PRECISION ALPHA, TOL1, TOL2 INTEGER I, INFO, IWARN, J, M, N, NR, NS, P CHARACTER*1 DICO, EQUIL, JOB, 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 AB09ND * .. 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, $ DICO, JOB, 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 AB09ND( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, $ ALPHA, A, LDA, B, LDB, C, LDC, D, LDD, $ NS, 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,NS ) 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 (' AB09ND EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from AB09ND = ',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 ALPHA-stable part are') END

AB09ND EXAMPLE PROGRAM DATA (Continuous system) 7 2 3 0 -.6D0 1.D-1 1.E-14 C N 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

AB09ND EXAMPLE PROGRAM RESULTS The order of reduced model = 5 The Hankel singular values of ALPHA-stable part are 1.9178 0.8621 0.7666 0.0336 0.0246 The reduced state dynamics matrix Ar is -0.5181 -1.1084 0.0000 0.0000 0.0000 8.8157 -0.5181 0.0000 0.0000 0.0000 0.0000 0.0000 0.5847 0.0000 1.9230 0.0000 0.0000 0.0000 -1.6606 0.0000 0.0000 0.0000 -4.3823 0.0000 -3.2922 The reduced input/state matrix Br is -1.2837 1.2837 -0.7522 0.7522 -0.6379 -0.6379 2.0656 -2.0656 -3.9315 -3.9315 The reduced state/output matrix Cr is -0.1380 -0.6445 -0.6416 -0.6293 0.2526 0.6246 0.0196 0.0000 0.4107 0.0000 0.1380 0.6445 -0.6416 0.6293 0.2526 The reduced input/output matrix Dr is 0.0582 -0.0090 0.0015 -0.0015 -0.0090 0.0582