Purpose
To compute the matrices of the H2 optimal controller
| AK | BK |
K = |----|----|,
| CK | DK |
for the normalized discrete-time system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | D11 D12 | | C | D |
| C2 | D21 0 |
where B2 has as column size the number of control inputs (NCON)
and C2 has as row size the number of measurements (NMEAS) being
provided to the controller.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) D12 is full column rank with D12 = | 0 | and D21 is
| I |
full row rank with D21 = | 0 I | as obtained by the
SLICOT Library routine SB10PD,
j*Theta
(A3) | A-e *I B2 | has full column rank for all
| C1 D12 |
0 <= Theta < 2*Pi ,
j*Theta
(A4) | A-e *I B1 | has full row rank for all
| C2 D21 |
0 <= Theta < 2*Pi .
Specification
SUBROUTINE SB10SD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
$ X, LDX, Y, LDY, RCOND, TOL, IWORK, DWORK,
$ LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, LDX, LDY, M, N, NCON, NMEAS, NP
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( * ), X( LDX, * ), Y( LDY, * )
LOGICAL BWORK( * )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the system. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
NCON (input) INTEGER
The number of control inputs (M2). M >= NCON >= 0,
NP-NMEAS >= NCON.
NMEAS (input) INTEGER
The number of measurements (NP2). NP >= NMEAS >= 0,
M-NCON >= NMEAS.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
LDA INTEGER
The leading dimension of the 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
system input matrix B.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system input/output matrix D. Only the leading
(NP-NP2)-by-(M-M2) submatrix D11 is used.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N part of this array contains the
controller state matrix AK.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
The leading N-by-NMEAS part of this array contains the
controller input matrix BK.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading NCON-by-N part of this array contains the
controller output matrix CK.
LDCK INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.
LDDK INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).
X (output) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array contains the matrix
X, solution of the X-Riccati equation.
LDX INTEGER
The leading dimension of the array X. LDX >= max(1,N).
Y (output) DOUBLE PRECISION array, dimension (LDY,N)
The leading N-by-N part of this array contains the matrix
Y, solution of the Y-Riccati equation.
LDY INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
RCOND (output) DOUBLE PRECISION array, dimension (4)
RCOND contains estimates of the reciprocal condition
numbers of the matrices which are to be inverted and the
reciprocal condition numbers of the Riccati equations
which have to be solved during the computation of the
controller. (See the description of the algorithm in [2].)
RCOND(1) contains the reciprocal condition number of the
matrix Im2 + B2'*X2*B2;
RCOND(2) contains the reciprocal condition number of the
matrix Ip2 + C2*Y2*C2';
RCOND(3) contains the reciprocal condition number of the
X-Riccati equation;
RCOND(4) contains the reciprocal condition number of the
Y-Riccati equation.
Tolerances
TOL DOUBLE PRECISION
Tolerance used in determining the nonsingularity of the
matrices which must be inverted. If TOL <= 0, then a
default value equal to sqrt(EPS) is used, where EPS is the
relative machine precision.
Workspace
IWORK INTEGER array, dimension (max(M2,2*N,N*N,NP2))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= max(1, 14*N*N+6*N+max(14*N+23,16*N),
M2*(N+M2+max(3,M1)), NP2*(N+NP2+3)),
where M1 = M - M2.
For good performance, LDWORK must generally be larger.
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the X-Riccati equation was not solved
successfully;
= 2: if the matrix Im2 + B2'*X2*B2 is not positive
definite, or it is numerically singular (with
respect to the tolerance TOL);
= 3: if the Y-Riccati equation was not solved
successfully;
= 4: if the matrix Ip2 + C2*Y2*C2' is not positive
definite, or it is numerically singular (with
respect to the tolerance TOL).
Method
The routine implements the formulas given in [1]. The X- and Y-Riccati equations are solved with condition estimates.References
[1] Zhou, K., Doyle, J.C., and Glover, K.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, NJ, 1996.
[2] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
Fortran 77 routines for Hinf and H2 design of linear
discrete-time control systems.
Report 99-8, Department of Engineering, Leicester University,
April 1999.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the matrices which are to be inverted and on the condition numbers of the matrix Riccati equations which are to be solved in the computation of the controller. (The corresponding reciprocal condition numbers are given in the output array RCOND.)Further Comments
NoneExample
Program Text
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