**Purpose**

To compute a reduced order model (Ar,Br,Cr) for a stable original state-space representation (A,B,C) by using either the square-root or the balancing-free square-root Balance & Truncate model reduction method. The state dynamics matrix A of the original system is an upper quasi-triangular matrix in real Schur canonical form. The matrices of the reduced order system are computed using the truncation formulas: Ar = TI * A * T , Br = TI * B , Cr = C * T .

SUBROUTINE AB09AX( DICO, JOB, ORDSEL, N, M, P, NR, A, LDA, B, LDB, $ C, LDC, HSV, T, LDT, TI, LDTI, TOL, IWORK, $ DWORK, LDWORK, IWARN, INFO ) C .. Scalar Arguments .. CHARACTER DICO, JOB, ORDSEL INTEGER INFO, IWARN, LDA, LDB, LDC, LDT, LDTI, LDWORK, $ M, N, NR, P DOUBLE PRECISION TOL C .. Array Arguments .. INTEGER IWORK(*) DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), 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 Balance & Truncate method; = 'N': use the balancing-free square-root Balance & Truncate method. 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 TOL.

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,NMIN), where 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(TOL,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 in a real Schur canonical 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). 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. T (output) DOUBLE PRECISION array, dimension (LDT,N) If INFO = 0 and NR > 0, the leading N-by-NR part of this array contains the right truncation matrix T. LDT INTEGER The leading dimension of array T. LDT >= MAX(1,N). TI (output) DOUBLE PRECISION array, dimension (LDTI,N) If INFO = 0 and NR > 0, the leading NR-by-N part of this array contains the left truncation matrix TI. LDTI INTEGER The leading dimension of array TI. LDTI >= MAX(1,N).

TOL DOUBLE PRECISION If ORDSEL = 'A', TOL contains the tolerance for determining the order of reduced system. For model reduction, the recommended value is TOL = 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 TOL = N*EPS*HNORM(A,B,C), where EPS is the machine precision (see LAPACK Library Routine DLAMCH). This value is used by default if TOL <= 0 on entry. If ORDSEL = 'F', the value of TOL is ignored.

IWORK INTEGER array, dimension (LIWORK) LIWORK = 0, if JOB = 'B', or LIWORK = N, if JOB = 'N'. 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*(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 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 state matrix A is not stable (if DICO = 'C') or not convergent (if DICO = 'D'); = 2: the computation of Hankel singular values failed.

Let be the stable linear system d[x(t)] = Ax(t) + Bu(t) y(t) = Cx(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 AB09AX determines for the given system (1), the matrices of a reduced NR order system d[z(t)] = Ar*z(t) + Br*u(t) yr(t) = Cr*z(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) and (Ar,Br,Cr), respectively, and INFNORM(G) is the infinity-norm of G. If JOB = 'B', the square-root Balance & Truncate method of [1] is used and, for DICO = 'C', the resulting model is balanced. By setting TOL <= 0, the routine can be used to compute balanced minimal state-space realizations of stable systems. If JOB = 'N', the balancing-free square-root version of the Balance & Truncate method [2] is used. By setting TOL <= 0, the routine can be used to compute minimal state-space realizations of stable systems.

[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 Moudui, P. Borne, S. G. Tzafestas (Eds.), Vol. 2, pp. 42-46.

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.

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**Program Text**

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