Purpose
To compute the output sequence of a linear time-invariant open-loop system given by its discrete-time state-space model (A,B,C,D), where A is an N-by-N upper or lower Hessenberg matrix. The initial state vector x(1) must be supplied by the user.Specification
SUBROUTINE TF01ND( UPLO, N, M, P, NY, A, LDA, B, LDB, C, LDC, D,
$ LDD, U, LDU, X, Y, LDY, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, LDC, LDD, LDU, LDY, M, N, NY, P
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), U(LDU,*), X(*), Y(LDY,*)
Arguments
Mode Parameters
UPLO CHARACTER*1
Indicates whether the user wishes to use an upper or lower
Hessenberg matrix as follows:
= 'U': Upper Hessenberg matrix;
= 'L': Lower Hessenberg matrix.
Input/Output Parameters
N (input) INTEGER
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.
NY (input) INTEGER
The number of output vectors y(k) to be computed.
NY >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
If UPLO = 'U', the leading N-by-N upper Hessenberg part
of this array must contain the state matrix A of the
system.
If UPLO = 'L', the leading N-by-N lower Hessenberg part
of this array must contain the state matrix A of the
system.
The remainder of the leading N-by-N part is not
referenced.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must contain the
input matrix B of the system.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must contain the
output matrix C of the system.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct link matrix D of the system.
LDD INTEGER
The leading dimension of array D. LDD >= MAX(1,P).
U (input) DOUBLE PRECISION array, dimension (LDU,NY)
The leading M-by-NY part of this array must contain the
input vector sequence u(k), for k = 1,2,...,NY.
Specifically, the k-th column of U must contain u(k).
LDU INTEGER
The leading dimension of array U. LDU >= MAX(1,M).
X (input/output) DOUBLE PRECISION array, dimension (N)
On entry, this array must contain the initial state vector
x(1) which consists of the N initial states of the system.
On exit, this array contains the final state vector
x(NY+1) of the N states of the system at instant NY.
Y (output) DOUBLE PRECISION array, dimension (LDY,NY)
The leading P-by-NY part of this array contains the output
vector sequence y(1),y(2),...,y(NY) such that the k-th
column of Y contains y(k) (the outputs at instant k),
for k = 1,2,...,NY.
LDY INTEGER
The leading dimension of array Y. LDY >= MAX(1,P).
Workspace
DWORK DOUBLE PRECISION array, dimension (N)Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
Given an initial state vector x(1), the output vector sequence
y(1), y(2),..., y(NY) is obtained via the formulae
x(k+1) = A x(k) + B u(k)
y(k) = C x(k) + D u(k),
where each element y(k) is a vector of length P containing the
outputs at instant k and k = 1,2,...,NY.
References
[1] Luenberger, D.G.
Introduction to Dynamic Systems: Theory, Models and
Applications.
John Wiley & Sons, New York, 1979.
Numerical Aspects
The algorithm requires approximately ((N+M)xP + (N/2+M)xN) x NY multiplications and additions.Further Comments
The processing time required by this routine will be approximately half that required by the SLICOT Library routine TF01MD, which treats A as a general matrix.Example
Program Text
* TF01ND EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX, NYMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20, NYMAX = 20 )
INTEGER LDA, LDB, LDC, LDD, LDU, LDY
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDU = MMAX, LDY = PMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX )
* .. Local Scalars ..
CHARACTER*1 UPLO
INTEGER I, INFO, J, K, M, N, NY, P
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), U(LDU,NYMAX),
$ X(NMAX), Y(LDY,NYMAX)
* .. External Subroutines ..
EXTERNAL TF01ND
* .. 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, NY, UPLO
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99994 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), I = 1,N ), J = 1,N )
IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) M
ELSE
READ ( NIN, FMT = * ) ( ( B(I,J), I = 1,N ), J = 1,M )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), I = 1,P ), J = 1,N )
READ ( NIN, FMT = * ) ( ( D(I,J), I = 1,P ), J = 1,M )
READ ( NIN, FMT = * ) ( X(I), I = 1,N )
IF ( NY.LE.0 .OR. NY.GT.NYMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) NY
ELSE
READ ( NIN, FMT = * )
$ ( ( U(I,J), I = 1,M ), J = 1,NY )
* Compute y(1),...,y(NY) of the given system.
CALL TF01ND( UPLO, N, M, P, NY, A, LDA, B, LDB, C,
$ LDC, D, LDD, U, LDU, X, Y, LDY, DWORK,
$ INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NY
DO 20 K = 1, NY
WRITE ( NOUT, FMT = 99996 ) K, Y(1,K)
WRITE ( NOUT, FMT = 99995 ) ( Y(J,K), J = 2,P )
20 CONTINUE
END IF
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TF01ND EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TF01ND = ',I2)
99997 FORMAT (' The output sequence Y(1),...,Y(',I2,') is',/)
99996 FORMAT (' Y(',I2,') : ',F8.4)
99995 FORMAT (9X,F8.4,/)
99994 FORMAT (/' N is out of range.',/' N = ',I5)
99993 FORMAT (/' M is out of range.',/' M = ',I5)
99992 FORMAT (/' P is out of range.',/' P = ',I5)
99991 FORMAT (/' NY is out of range.',/' NY = ',I5)
END
Program Data
TF01ND EXAMPLE PROGRAM DATA 3 2 2 10 U 0.0000 -0.0700 0.0000 1.0000 0.8000 -0.1500 0.0000 0.0000 0.5000 0.0000 2.0000 1.0000 -1.0000 -0.1000 1.0000 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000 0.5000 0.0000 0.5000 1.0000 1.0000 1.0000 -0.6922 -1.4934 0.3081 -2.7726 2.0039 0.2614 -0.9160 -0.6030 1.2556 0.2951 -1.5734 1.5639 -0.9942 1.8957 0.8988 0.4118 -1.4893 -0.9344 1.2506 -0.0701Program Results
TF01ND EXAMPLE PROGRAM RESULTS
The output sequence Y(1),...,Y(10) is
Y( 1) : 0.3078
-0.0928
Y( 2) : -1.5275
1.2611
Y( 3) : -1.3026
3.4002
Y( 4) : -0.3512
-0.7060
Y( 5) : -0.5922
5.4532
Y( 6) : -1.1693
1.1846
Y( 7) : -1.3029
2.2286
Y( 8) : 1.7529
-1.9534
Y( 9) : 0.6793
-4.4965
Y(10) : -0.0349
1.1654