Purpose
To compute the matrices of an H-infinity (sub)optimal controller
| AK | BK |
K = |----|----|,
| CK | DK |
from the state feedback matrix F and output injection matrix H as
determined by the SLICOT Library routine SB10QD.
Specification
SUBROUTINE SB10RD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, F, LDF, H, LDH, TU, LDTU, TY,
$ LDTY, X, LDX, Y, LDY, AK, LDAK, BK, LDBK, CK,
$ LDCK, DK, LDDK, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDF, LDH, LDTU, LDTY, LDWORK, LDX, LDY,
$ M, N, NCON, NMEAS, NP
DOUBLE PRECISION GAMMA
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ F( LDF, * ), H( LDH, * ), TU( LDTU, * ),
$ TY( LDTY, * ), X( LDX, * ), Y( LDY, * )
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.
GAMMA (input) DOUBLE PRECISION
The value of gamma. It is assumed that gamma is
sufficiently large so that the controller is admissible.
GAMMA >= 0.
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.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
F (input) DOUBLE PRECISION array, dimension (LDF,N)
The leading M-by-N part of this array must contain the
state feedback matrix F.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1,M).
H (input) DOUBLE PRECISION array, dimension (LDH,NP)
The leading N-by-NP part of this array must contain the
output injection matrix H.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
TU (input) DOUBLE PRECISION array, dimension (LDTU,M2)
The leading M2-by-M2 part of this array must contain the
control transformation matrix TU, as obtained by the
SLICOT Library routine SB10PD.
LDTU INTEGER
The leading dimension of the array TU. LDTU >= max(1,M2).
TY (input) DOUBLE PRECISION array, dimension (LDTY,NP2)
The leading NP2-by-NP2 part of this array must contain the
measurement transformation matrix TY, as obtained by the
SLICOT Library routine SB10PD.
LDTY INTEGER
The leading dimension of the array TY.
LDTY >= max(1,NP2).
X (input) DOUBLE PRECISION array, dimension (LDX,N)
The leading N-by-N part of this array must contain the
matrix X, solution of the X-Riccati equation, as obtained
by the SLICOT Library routine SB10QD.
LDX INTEGER
The leading dimension of the array X. LDX >= max(1,N).
Y (input) DOUBLE PRECISION array, dimension (LDY,N)
The leading N-by-N part of this array must contain the
matrix Y, solution of the Y-Riccati equation, as obtained
by the SLICOT Library routine SB10QD.
LDY INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
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).
Workspace
IWORK INTEGER array, dimension (LIWORK), where
LIWORK = max(2*(max(NP,M)-M2-NP2,M2,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, M2*NP2 + NP2*NP2 + M2*M2 +
max(D1*D1 + max(2*D1, (D1+D2)*NP2),
D2*D2 + max(2*D2, D2*M2), 3*N,
N*(2*NP2 + M2) +
max(2*N*M2, M2*NP2 +
max(M2*M2+3*M2, NP2*(2*NP2+
M2+max(NP2,N))))))
where D1 = NP1 - M2, D2 = M1 - NP2,
NP1 = NP - NP2, M1 = M - M2.
For good performance, LDWORK must generally be larger.
Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
max( 1, Q*(3*Q + 3*N + max(2*N, 4*Q + max(Q, N)))).
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the controller is not admissible (too small value
of gamma);
= 2: if the determinant of Im2 + Tu*D11HAT*Ty*D22 is zero.
Method
The routine implements the Glover's and Doyle's formulas [1],[2].References
[1] Glover, K. and Doyle, J.C.
State-space formulae for all stabilizing controllers that
satisfy an Hinf norm bound and relations to risk sensitivity.
Systems and Control Letters, vol. 11, pp. 167-172, 1988.
[2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the input and output transformations.Further Comments
NoneExample
Program Text
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