**Purpose**

To find a reduced (controllable, observable, or minimal) state- space representation (Ar,Br,Cr) for any original state-space representation (A,B,C). The matrix Ar is in an upper block Hessenberg staircase form.

SUBROUTINE TB01PX( JOB, EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, $ NR, INFRED, TOL, IWORK, DWORK, LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER EQUIL, JOB INTEGER INFO, LDA, LDB, LDC, LDWORK, M, N, NR, P DOUBLE PRECISION TOL C .. Array Arguments .. INTEGER INFRED(*), IWORK(*) DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*)

**Mode Parameters**

JOB CHARACTER*1 Indicates whether the user wishes to remove the uncontrollable and/or unobservable parts as follows: = 'M': Remove both the uncontrollable and unobservable parts to get a minimal state-space representation; = 'C': Remove the uncontrollable part only to get a controllable state-space representation; = 'O': Remove the unobservable part only to get an observable state-space representation. EQUIL CHARACTER*1 Specifies whether the user wishes to preliminarily balance the triplet (A,B,C) as follows: = 'S': Perform balancing (scaling); = 'N': Do not perform balancing.

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. 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 INFRED(1) >= 0 and/or INFRED(2) >= 0, the leading NR-by-NR part of this array contains the upper block Hessenberg state dynamics matrix Ar of a minimal, controllable, or observable realization for the original system, depending on the value of JOB, JOB = 'M', JOB = 'C', or JOB = 'O', respectively. The block structure of the resulting staircase form is contained in the leading INFRED(4) elements of IWORK. If INFRED(1:2) < 0, then A contains the original matrix. LDA INTEGER The leading dimension of the array A. LDA >= MAX(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB,M), if JOB = 'C', or (LDB,MAX(M,P)), otherwise. On entry, the leading N-by-M part of this array must contain the original input/state matrix B; if JOB = 'M', or JOB = 'O', the remainder of the leading N-by-MAX(M,P) part is used as internal workspace. On exit, if INFRED(1) >= 0 and/or INFRED(2) >= 0, the leading NR-by-M part of this array contains the input/state matrix Br of a minimal, controllable, or observable realization for the original system, depending on the value of JOB, JOB = 'M', JOB = 'C', or JOB = 'O', respectively. If JOB = 'C', only the first IWORK(1) rows of B are nonzero. If INFRED(1:2) < 0, then B contains the original matrix. LDB INTEGER The leading dimension of the 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; if JOB = 'M', or JOB = 'O', the remainder of the leading MAX(M,P)-by-N part is used as internal workspace. On exit, if INFRED(1) >= 0 and/or INFRED(2) >= 0, the leading P-by-NR part of this array contains the state/output matrix Cr of a minimal, controllable, or observable realization for the original system, depending on the value of JOB, JOB = 'M', JOB = 'C', or JOB = 'O', respectively. If JOB = 'M', or JOB = 'O', only the last IWORK(1) columns (in the first NR columns) of C are nonzero. If INFRED(1:2) < 0, then C contains the original matrix. LDC INTEGER The leading dimension of the array C. LDC >= MAX(1,M,P), if N > 0. LDC >= 1, if N = 0. NR (output) INTEGER The order of the reduced state-space representation (Ar,Br,Cr) of a minimal, controllable, or observable realization for the original system, depending on JOB = 'M', JOB = 'C', or JOB = 'O'. INFRED (output) INTEGER array, dimension 4 This array contains information on the performed reduction and on structure of resulting system matrices, as follows: INFRED(k) >= 0 (k = 1 or 2) if Phase k of the reduction (see METHOD) has been performed. In this case, INFRED(k) is the achieved order reduction in Phase k. INFRED(k) < 0 (k = 1 or 2) if Phase k was not performed. This can also appear when Phase k was tried, but did not reduce the order, if enough workspace is provided for saving the system matrices (see LDWORK description). INFRED(3) - the number of nonzero subdiagonals of A. INFRED(4) - the number of blocks in the resulting staircase form at the last performed reduction phase. The block dimensions are contained in the first INFRED(4) elements of IWORK.

TOL DOUBLE PRECISION The tolerance to be used in rank determinations when transforming (A, B, C). If the user sets TOL > 0, then the given value of TOL is used as a lower bound for the reciprocal condition number (see the description of the argument RCOND in the SLICOT routine MB03OD); a (sub)matrix whose estimated condition number is less than 1/TOL is considered to be of full rank. If the user sets TOL <= 0, then an implicitly computed, default tolerance (determined by the SLICOT routine TB01UD) is used instead.

IWORK INTEGER array, dimension (N+MAX(M,P)) On exit, if INFO = 0, the first INFRED(4) elements of IWORK return the orders of the diagonal blocks of A. 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 >= 1, and if N > 0, LDWORK >= N + MAX(N, 3*M, 3*P). For optimum performance LDWORK should be larger. If LDWORK >= MAX(1, N + MAX(N, 3*M, 3*P) + N*(N+M+P) ), then more accurate results are to be expected by accepting only those reductions phases (see METHOD), where effective order reduction occurs. This is achieved by saving the system matrices before each phase and restoring them if no order reduction took place in that phase.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

The order reduction is performed in two phases: Phase 1: If JOB = 'M' or 'C', the pair (A,B) is reduced by orthogonal similarity transformations to the controllability staircase form (see [1]) and a controllable realization (Ac,Bc,Cc) is extracted. Ac results in an upper block Hessenberg form. Phase 2: If JOB = 'M' or 'O', the same algorithm is applied to the dual of the controllable realization (Ac,Bc,Cc), or to the dual of the original system, respectively, to extract an observable realization (Ar,Br,Cr). If JOB = 'M', the resulting realization is also controllable, and thus minimal. Ar results in an upper block Hessenberg form.

[1] Van Dooren, P. The Generalized Eigenstructure Problem in Linear System Theory. (Algorithm 1) IEEE Trans. Auto. Contr., AC-26, pp. 111-129, 1981.

3 The algorithm requires 0(N ) operations and is backward stable.

None

**Program Text**

* TB01PX 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 MAXMP PARAMETER ( MAXMP = MAX( MMAX, PMAX ) ) INTEGER LDA, LDB, LDC PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = MAXMP ) INTEGER LIWORK PARAMETER ( LIWORK = NMAX+MAXMP ) INTEGER LDWORK PARAMETER ( LDWORK = NMAX+MAX( NMAX, 3*MAXMP + $ NMAX*( NMAX+MMAX+PMAX ) ) ) * .. Local Scalars .. DOUBLE PRECISION TOL INTEGER I, INFO, J, M, N, NR, P CHARACTER JOB, EQUIL * .. Local Arrays .. DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX), $ DWORK(LDWORK) INTEGER INFRED(4), IWORK(LIWORK) * .. External Subroutines .. EXTERNAL TB01PX * .. 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, TOL, JOB, EQUIL 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), I = 1,N ), J = 1,M ) 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 ) * Find a minimal ssr for (A,B,C). CALL TB01PX( JOB, EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, $ NR, INFRED, TOL, IWORK, DWORK, LDWORK, INFO) * IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99998 ) INFO ELSE WRITE ( NOUT, FMT = 99997 ) NR WRITE ( NOUT, FMT = 99996 ) DO 20 I = 1, NR WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR ) 20 CONTINUE WRITE ( NOUT, FMT = 99993 ) DO 40 I = 1, NR WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M ) 40 CONTINUE WRITE ( NOUT, FMT = 99992 ) DO 60 I = 1, P WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR ) 60 CONTINUE WRITE ( NOUT, FMT = 99994 ) ( INFRED(I), I = 1,4 ) IF ( INFRED(4).GT.0 ) $ WRITE ( NOUT, FMT = 99991 ) ( IWORK(I), $ I = 1,INFRED(4) ) END IF END IF END IF END IF STOP * 99999 FORMAT (' TB01PX EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from TB01PX = ',I2) 99997 FORMAT (' The order of the minimal realization = ',I2) 99996 FORMAT (/' The transformed state dynamics matrix of a minimal re', $ 'alization is ') 99995 FORMAT (20(1X,F8.4)) 99994 FORMAT (/' Information on the performed reduction is ', 4I5) 99993 FORMAT (/' The transformed input/state matrix of a minimal reali', $ 'zation is ') 99992 FORMAT (/' The transformed state/output matrix of a minimal real', $ 'ization is ') 99991 FORMAT (/' The block dimensions are ',/ 20I5) 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) END

TB01PX EXAMPLE PROGRAM DATA 3 1 2 0.0 M N 1.0 2.0 0.0 4.0 -1.0 0.0 0.0 0.0 1.0 1.0 0.0 1.0 0.0 1.0 -1.0 0.0 0.0 1.0

TB01PX EXAMPLE PROGRAM RESULTS The order of the minimal realization = 3 The transformed state dynamics matrix of a minimal realization is 1.0000 2.0000 0.0000 4.0000 -1.0000 0.0000 0.0000 0.0000 1.0000 The transformed input/state matrix of a minimal realization is 1.0000 0.0000 1.0000 The transformed state/output matrix of a minimal realization is 0.0000 1.0000 -1.0000 0.0000 0.0000 1.0000 Information on the performed reduction is -1 -1 2 0