**Purpose**

To compute a reduced order model (Ar,Br,Cr,Dr) for an original state-space representation (A,B,C,D) by using the square-root or balancing-free square-root Balance & Truncate (B&T) or Singular Perturbation Approximation (SPA) model reduction methods. The computation of truncation matrices TI and T is based on the Cholesky factor S of a controllability Grammian P = S*S' and the Cholesky factor R of an observability Grammian Q = R'*R, where S and R are given upper triangular matrices. For the B&T approach, the matrices of the reduced order system are computed using the truncation formulas: Ar = TI * A * T , Br = TI * B , Cr = C * T . (1) For the SPA approach, the matrices of a minimal realization (Am,Bm,Cm) are computed using the truncation formulas: Am = TI * A * T , Bm = TI * B , Cm = C * T . (2) Am, Bm, Cm and D serve further for computing the SPA of the given system.

SUBROUTINE AB09IX( DICO, JOB, FACT, ORDSEL, N, M, P, NR, $ SCALEC, SCALEO, A, LDA, B, LDB, C, LDC, D, LDD, $ TI, LDTI, T, LDT, NMINR, HSV, TOL1, TOL2, $ IWORK, DWORK, LDWORK, IWARN, INFO ) C .. Scalar Arguments .. CHARACTER DICO, FACT, JOB, ORDSEL INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDT, LDTI, $ LDWORK, M, N, NMINR, NR, P DOUBLE PRECISION SCALEC, SCALEO, TOL1, TOL2 C .. Array Arguments .. INTEGER IWORK(*) DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), $ DWORK(*), HSV(*), T(LDT,*), TI(LDTI,*)

**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 B&T method; = 'F': use the balancing-free square-root B&T method; = 'S': use the square-root SPA method; = 'P': use the balancing-free square-root SPA method. FACT CHARACTER*1 Specifies whether or not, on entry, the matrix A is in a real Schur form, as follows: = 'S': A is in a real Schur form; = 'N': A is a general dense square matrix. 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,NMINR), where NR is the desired order on entry and NMINR is the number of the Hankel singular values greater than N*EPS*S1, where EPS is the machine precision (see LAPACK Library Routine DLAMCH) and S1 is the largest Hankel singular value (computed in HSV(1)); NR can be further reduced to ensure HSV(NR) > HSV(NR+1); if ORDSEL = 'A', NR is equal to the number of Hankel singular values greater than MAX(TOL1,N*EPS*S1). SCALEC (input) DOUBLE PRECISION Scaling factor for the Cholesky factor S of the controllability Grammian, i.e., S/SCALEC is used to compute the Hankel singular values. SCALEC > 0. SCALEO (input) DOUBLE PRECISION Scaling factor for the Cholesky factor R of the observability Grammian, i.e., R/SCALEO is used to compute the Hankel singular values. SCALEO > 0. 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. If FACT = 'S', A is in a real Schur form. 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, if JOB = 'S' or JOB = 'P', the leading P-by-M part of this array must contain the original input/output matrix D. On exit, if INFO = 0 and JOB = 'S' or JOB = 'P', the leading P-by-M part of this array contains the input/output matrix Dr of the reduced order system. If JOB = 'B' or JOB = 'F', this array is not referenced. LDD INTEGER The leading dimension of array D. LDD >= 1, if JOB = 'B' or JOB = 'F'; LDD >= MAX(1,P), if JOB = 'S' or JOB = 'P'. TI (input/output) DOUBLE PRECISION array, dimension (LDTI,N) On entry, the leading N-by-N upper triangular part of this array must contain the Cholesky factor S of a controllability Grammian P = S*S'. On exit, if INFO = 0, and NR > 0, the leading NMINR-by-N part of this array contains the left truncation matrix TI in (1), for the B&T approach, or in (2), for the SPA approach. LDTI INTEGER The leading dimension of array TI. LDTI >= MAX(1,N). T (input/output) DOUBLE PRECISION array, dimension (LDT,N) On entry, the leading N-by-N upper triangular part of this array must contain the Cholesky factor R of an observability Grammian Q = R'*R. On exit, if INFO = 0, and NR > 0, the leading N-by-NMINR part of this array contains the right truncation matrix T in (1), for the B&T approach, or in (2), for the SPA approach. LDT INTEGER The leading dimension of array T. LDT >= MAX(1,N). NMINR (output) INTEGER The number of Hankel singular values greater than MAX(TOL2,N*EPS*S1). Note: If S and R are the Cholesky factors of the controllability and observability Grammians of the original system (A,B,C,D), respectively, then NMINR is the order of a minimal realization of the original system. HSV (output) DOUBLE PRECISION array, dimension (N) If INFO = 0, it contains the Hankel singular values, ordered decreasingly. The Hankel singular values are singular values of the product R*S.

TOL1 DOUBLE PRECISION If ORDSEL = 'A', TOL1 contains the tolerance for determining the order of the reduced system. For model reduction, the recommended value lies in the interval [0.00001,0.001]. If TOL1 <= 0 on entry, the used default value is TOL1 = N*EPS*S1, where EPS is the machine precision (see LAPACK Library Routine DLAMCH) and S1 is the largest Hankel singular value (computed in HSV(1)). If ORDSEL = 'F', the value of TOL1 is ignored. TOL2 DOUBLE PRECISION The tolerance for determining the order of a minimal realization of the system. The recommended value is TOL2 = N*EPS*S1. This value is used by default if TOL2 <= 0 on entry. If TOL2 > 0, and ORDSEL = 'A', then TOL2 <= TOL1.

IWORK INTEGER array, dimension (LIWORK), where LIWORK = 0, if JOB = 'B'; LIWORK = N, if JOB = 'F'; LIWORK = 2*N, if JOB = 'S' or 'P'. 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, 2*N*N + 5*N, N*MAX(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 NMINR, the order of a minimal realization of the given system; in this case, the resulting NR is set automatically to NMINR; = 2: with ORDSEL = 'F', the selected order NR corresponds to repeated singular values, which are neither all included nor all excluded from the reduced model; in this case, the resulting NR is set automatically to the largest value such that HSV(NR) > HSV(NR+1).

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: the computation of Hankel singular values failed.

Let be the stable linear system d[x(t)] = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t), (3) where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1) for a discrete-time system. The subroutine AB09IX determines for the given system (3), the matrices of a reduced NR order system d[z(t)] = Ar*z(t) + Br*u(t) yr(t) = Cr*z(t) + Dr*u(t), (4) by using the square-root or balancing-free square-root Balance & Truncate (B&T) or Singular Perturbation Approximation (SPA) model reduction methods. The projection matrices TI and T are determined using the Cholesky factors S and R of a controllability Grammian P and an observability Grammian Q. The Hankel singular values HSV(1), ...., HSV(N) are computed as singular values of the product R*S. If JOB = 'B', the square-root Balance & Truncate technique of [1] is used. If JOB = 'F', the balancing-free square-root version of the Balance & Truncate technique [2] is used. If JOB = 'S', the square-root version of the Singular Perturbation Approximation method [3,4] is used. If JOB = 'P', the balancing-free square-root version of the Singular Perturbation Approximation method [3,4] is used.

[1] 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. [2] Varga A. Efficient minimal realization procedure based on balancing. Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991, A. El Moudni, P. Borne, S. G. Tzafestas (Eds.), Vol. 2, pp. 42-46. [3] Liu Y. and Anderson B.D.O. Singular Perturbation Approximation of balanced systems. Int. J. Control, Vol. 50, pp. 1379-1405, 1989. [4] Varga A. Balancing-free square-root algorithm for computing singular perturbation approximations. Proc. 30-th CDC, Brighton, Dec. 11-13, 1991, Vol. 2, pp. 1062-1065.

The implemented method relies on accuracy enhancing square-root or balancing-free square-root methods.

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**Program Text**

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