**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. The state dynamics matrix A of the original system is an upper quasi-triangular matrix in real Schur canonical form.

SUBROUTINE AB09CX( DICO, 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, 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. 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 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 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 state matrix A is not stable (if DICO = 'C') or not convergent (if DICO = 'D'); = 2: the computation of Hankel singular values failed; = 3: the computation of stable projection failed; = 4: 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 AB09CX 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.

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**Program Text**

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