**Purpose**

To compute, for a given open-loop model (A,B,C,D), and for given state feedback gain F and full observer gain G, such that A+B*F and A+G*C are stable, a reduced order controller model (Ac,Bc,Cc) using a coprime factorization based controller reduction approach. For reduction of coprime factors, a stability enforcing frequency-weighted model reduction is performed using either the square-root or the balancing-free square-root versions of the Balance & Truncate (B&T) model reduction method.

SUBROUTINE SB16CD( DICO, JOBD, JOBMR, JOBCF, ORDSEL, N, M, P, NCR, $ A, LDA, B, LDB, C, LDC, D, LDD, F, LDF, G, LDG, $ HSV, TOL, IWORK, DWORK, LDWORK, IWARN, INFO ) C .. Scalar Arguments .. CHARACTER DICO, JOBCF, JOBD, JOBMR, ORDSEL INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, $ LDF, LDG, LDWORK, M, N, NCR, P DOUBLE PRECISION TOL C .. Array Arguments .. INTEGER IWORK(*) DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), $ DWORK(*), F(LDF,*), G(LDG,*), HSV(*)

**Mode Parameters**

DICO CHARACTER*1 Specifies the type of the open-loop system as follows: = 'C': continuous-time system; = 'D': discrete-time system. JOBD CHARACTER*1 Specifies whether or not a non-zero matrix D appears in the given state space model, as follows: = 'D': D is present; = 'Z': D is assumed a zero matrix. JOBMR 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. JOBCF CHARACTER*1 Specifies whether left or right coprime factorization of the controller is to be used as follows: = 'L': use left coprime factorization; = 'R': use right coprime factorization. ORDSEL CHARACTER*1 Specifies the order selection method as follows: = 'F': the resulting controller order NCR is fixed; = 'A': the resulting controller order NCR 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. N also represents the order of the original state-feedback controller. M (input) INTEGER The number of system inputs. M >= 0. P (input) INTEGER The number of system outputs. P >= 0. NCR (input/output) INTEGER On entry with ORDSEL = 'F', NCR is the desired order of the resulting reduced order controller. 0 <= NCR <= N. On exit, if INFO = 0, NCR is the order of the resulting reduced order controller. NCR is set as follows: if ORDSEL = 'F', NCR is equal to MIN(NCR,NCRMIN), where NCR is the desired order on entry, and NCRMIN is the number of 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)); NCR can be further reduced to ensure HSV(NCR) > HSV(NCR+1); if ORDSEL = 'A', NCR is equal to the number of Hankel singular values greater than MAX(TOL,N*EPS*S1). 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 INFO = 0, the leading NCR-by-NCR part of this array contains the state dynamics matrix Ac of the reduced controller. 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 open-loop system input/state matrix B. On exit, this array is overwritten with a NCR-by-M B&T approximation of the matrix B. 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 open-loop system state/output matrix C. On exit, this array is overwritten with a P-by-NCR B&T approximation of the matrix C. LDC INTEGER The leading dimension of array C. LDC >= MAX(1,P). D (input) DOUBLE PRECISION array, dimension (LDD,M) On entry, if JOBD = 'D', the leading P-by-M part of this array must contain the system direct input/output transmission matrix D. The array D is not referenced if JOBD = 'Z'. LDD INTEGER The leading dimension of array D. LDD >= MAX(1,P), if JOBD = 'D'; LDD >= 1, if JOBD = 'Z'. F (input/output) DOUBLE PRECISION array, dimension (LDF,N) On entry, the leading M-by-N part of this array must contain a stabilizing state feedback matrix. On exit, if INFO = 0, the leading M-by-NCR part of this array contains the output/state matrix Cc of the reduced controller. LDF INTEGER The leading dimension of array F. LDF >= MAX(1,M). G (input/output) DOUBLE PRECISION array, dimension (LDG,P) On entry, the leading N-by-P part of this array must contain a stabilizing observer gain matrix. On exit, if INFO = 0, the leading NCR-by-P part of this array contains the input/state matrix Bc of the reduced controller. LDG INTEGER The leading dimension of array G. LDG >= MAX(1,N). HSV (output) DOUBLE PRECISION array, dimension (N) If INFO = 0, HSV contains the N frequency-weighted Hankel singular values ordered decreasingly (see METHOD).

TOL DOUBLE PRECISION If ORDSEL = 'A', TOL contains the tolerance for determining the order of reduced controller. The recommended value is TOL = c*S1, where c is a constant in the interval [0.00001,0.001], and S1 is the largest Hankel singular value (computed in HSV(1)). The value TOL = N*EPS*S1 is used by default if TOL <= 0 on entry, where EPS is the machine precision (see LAPACK Library Routine DLAMCH). If ORDSEL = 'F', the value of TOL is ignored.

IWORK INTEGER array, dimension (LIWORK) LIWORK = 0, if JOBMR = 'B'; LIWORK = N, if JOBMR = 'F'. 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 >= 2*N*N + MAX( 1, 2*N*N + 5*N, N*MAX(M,P), N*(N + MAX(N,MP) + MIN(N,MP) + 6)), where MP = M, if JOBCF = 'L'; MP = P, if JOBCF = 'R'. For optimum performance LDWORK should be larger.

IWARN INTEGER = 0: no warning; = 1: with ORDSEL = 'F', the selected order NCR is greater than the order of a minimal realization of the controller; = 2: with ORDSEL = 'F', the selected order NCR corresponds to repeated singular values, which are neither all included nor all excluded from the reduced controller. In this case, the resulting NCR is set automatically to the largest value such that HSV(NCR) > HSV(NCR+1).

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: eigenvalue computation failure; = 2: the matrix A+G*C is not stable; = 3: the matrix A+B*F is not stable; = 4: the Lyapunov equation for computing the observability Grammian is (nearly) singular; = 5: the Lyapunov equation for computing the controllability Grammian is (nearly) singular; = 6: the computation of Hankel singular values failed.

Let be the 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, and let Go(d) be the open-loop transfer-function matrix -1 Go(d) = C*(d*I-A) *B + D . Let F and G be the state feedback and observer gain matrices, respectively, chosen such that A+BF and A+GC are stable matrices. The controller has a transfer-function matrix K(d) given by -1 K(d) = F*(d*I-A-B*F-G*C-G*D*F) *G . The closed-loop transfer function matrix is given by -1 Gcl(d) = Go(d)(I+K(d)Go(d)) . K(d) can be expressed as a left coprime factorization (LCF) -1 K(d) = M_left(d) *N_left(d), or as a right coprime factorization (RCF) -1 K(d) = N_right(d)*M_right(d) , where M_left(d), N_left(d), N_right(d), and M_right(d) are stable transfer-function matrices. The subroutine SB16CD determines the matrices of a reduced controller d[z(t)] = Ac*z(t) + Bc*y(t) u(t) = Cc*z(t), (2) with the transfer-function matrix Kr, using the following stability enforcing approach proposed in [1]: (1) If JOBCF = 'L', the frequency-weighted approximation problem is solved min||[M_left(d)-M_leftr(d) N_left(d)-N_leftr(d)][-Y(d)]|| , [ X(d)] where -1 G(d) = Y(d)*X(d) is a RCF of the open-loop system transfer-function matrix. The B&T model reduction technique is used in conjunction with the method proposed in [1]. (2) If JOBCF = 'R', the frequency-weighted approximation problem is solved min || [ -U(d) V(d) ] [ N_right(d)-N_rightr(d) ] || , [ M_right(d)-M_rightr(d) ] where -1 G(d) = V(d) *U(d) is a LCF of the open-loop system transfer-function matrix. The B&T model reduction technique is used in conjunction with the method proposed in [1]. If ORDSEL = 'A', the order of the controller is determined by computing the number of Hankel singular values greater than the given tolerance TOL. The Hankel singular values are the square roots of the eigenvalues of the product of two frequency-weighted Grammians P and Q, defined as follows. If JOBCF = 'L', then P is the controllability Grammian of a system of the form (A+BF,B,*,*), and Q is the observability Grammian of a system of the form (A+GC,*,F,*). This choice corresponds to an input frequency-weighted order reduction of left coprime factors [1]. If JOBCF = 'R', then P is the controllability Grammian of a system of the form (A+BF,G,*,*), and Q is the observability Grammian of a system of the form (A+GC,*,C,*). This choice corresponds to an output frequency-weighted order reduction of right coprime factors [1]. For the computation of truncation matrices, the B&T approach is used in conjunction with accuracy enhancing techniques. If JOBMR = 'B', the square-root B&T method of [2,4] is used. If JOBMR = 'F', the balancing-free square-root version of the B&T method [3,4] is used.

[1] Liu, Y., Anderson, B.D.O. and Ly, O.L. Coprime factorization controller reduction with Bezout identity induced frequency weighting. Automatica, vol. 26, pp. 233-249, 1990. [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. [3] 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, 1991. [4] Varga, A. Coprime factors model reduction method based on square-root balancing-free techniques. System Analysis, Modelling and Simulation, Vol. 11, pp. 303-311, 1993.

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**

* SB16CD 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, LDDC, LDF, LDG PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX, $ LDD = PMAX, LDDC = MMAX, LDF = MMAX, LDG = NMAX ) INTEGER LDWORK, LIWORK, MPMAX PARAMETER ( LIWORK = 2*NMAX, MPMAX = MAX( MMAX, PMAX ) ) PARAMETER ( LDWORK = 2*NMAX*NMAX + $ MAX( 2*NMAX*NMAX + 5*NMAX, $ NMAX*( NMAX + MAX( NMAX, MPMAX ) $ + MIN( NMAX, MPMAX ) + 6 ) ) $ ) CHARACTER DICO, JOBCF, JOBD, JOBMR, ORDSEL INTEGER I, INFO, IWARN, J, M, N, NCR, P DOUBLE PRECISION TOL * .. Local Arrays .. DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX), $ D(LDD,MMAX), DWORK(LDWORK), $ F(LDF,NMAX), G(LDG,PMAX), HSV(NMAX) INTEGER IWORK(LIWORK) * .. External Subroutines .. EXTERNAL SB16CD * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. 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, NCR, TOL, $ DICO, JOBD, JOBMR, JOBCF, 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 ) READ ( NIN, FMT = * ) ( ( F(I,J), J = 1,N ), I = 1,M ) READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,P ), I = 1,N ) * Find a reduced ssr for (A,B,C,D). CALL SB16CD( DICO, JOBD, JOBMR, JOBCF, ORDSEL, N, M, P, $ NCR, A, LDA, B, LDB, C, LDC, D, LDD, F, LDF, $ G, LDG, HSV, TOL, IWORK, DWORK, LDWORK, $ IWARN, INFO ) * IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99998 ) INFO ELSE WRITE ( NOUT, FMT = 99997 ) NCR WRITE ( NOUT, FMT = 99987 ) WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,N ) IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99996 ) DO 20 I = 1, NCR WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NCR ) 20 CONTINUE IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99993 ) DO 40 I = 1, NCR WRITE ( NOUT, FMT = 99995 ) ( G(I,J), J = 1,P ) 40 CONTINUE IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99992 ) DO 60 I = 1, M WRITE ( NOUT, FMT = 99995 ) ( F(I,J), J = 1,NCR ) 60 CONTINUE END IF END IF END IF END IF STOP * 99999 FORMAT (' SB16CD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from SB16CD = ',I2) 99997 FORMAT (' The order of reduced controller = ',I2) 99996 FORMAT (/' The reduced controller state dynamics matrix Ac is ') 99995 FORMAT (20(1X,F8.4)) 99993 FORMAT (/' The reduced controller input/state matrix Bc is ') 99992 FORMAT (/' The reduced controller state/output matrix Cc 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 frequency-weighted Hankel singular values are:') END

SB16CD EXAMPLE PROGRAM DATA (Continuous system) 8 1 1 2 0.1E0 C D F R F 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.0150 0.7650 0 0 0 0 0 0 -0.7650 -0.0150 0 0 0 0 0 0 0 0 -0.0280 1.4100 0 0 0 0 0 0 -1.4100 -0.0280 0 0 0 0 0 0 0 0 -0.0400 1.850 0 0 0 0 0 0 -1.8500 -0.040 0.0260 -0.2510 0.0330 -0.8860 -4.0170 0.1450 3.6040 0.2800 -.996 -.105 0.261 .009 -.001 -.043 0.002 -0.026 0.0 4.472135954999638e-002 6.610515358414598e-001 4.698598960657579e-003 3.601363251422058e-001 1.032530880771415e-001 -3.754055214487997e-002 -4.268536964759344e-002 3.287284547842979e-002 4.108939884667451e-001 8.684600000000012e-002 3.852317308197148e-004 -3.619366874815911e-003 -8.803722876359955e-003 8.420521094001852e-003 1.234944428038507e-003 4.263205617645322e-003

SB16CD EXAMPLE PROGRAM RESULTS The order of reduced controller = 2 The frequency-weighted Hankel singular values are: 3.3073 0.7274 0.1124 0.0784 0.0242 0.0182 0.0101 0.0094 The reduced controller state dynamics matrix Ac is -0.4334 0.4884 -0.1950 -0.1093 The reduced controller input/state matrix Bc is -0.4231 -0.1785 The reduced controller state/output matrix Cc is -0.0326 -0.2307