Purpose
To compute the matrices of an H-infinity optimal n-state
controller
| AK | BK |
K = |----|----|,
| CK | DK |
using modified Glover's and Doyle's 1988 formulas, for the system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | D11 D12 | | C | D |
| C2 | D21 D22 |
and for the estimated minimal possible value of gamma with respect
to GTOL, 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, and then to compute the matrices
of the closed-loop system
| AC | BC |
G = |----|----|,
| CC | DC |
if the stabilizing controller exists.
It is assumed that
(A1) (A,B2) is stabilizable and (C2,A) is detectable,
(A2) D12 is full column rank and D21 is full row rank,
(A3) | A-j*omega*I B2 | has full column rank for all omega,
| C1 D12 |
(A4) | A-j*omega*I B1 | has full row rank for all omega.
| C2 D21 |
Specification
SUBROUTINE SB10AD( JOB, N, M, NP, NCON, NMEAS, GAMMA, A, LDA,
$ B, LDB, C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK,
$ LDCK, DK, LDDK, AC, LDAC, BC, LDBC, CC, LDCC,
$ DC, LDDC, RCOND, GTOL, ACTOL, IWORK, LIWORK,
$ DWORK, LDWORK, BWORK, LBWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, JOB, LBWORK, LDA, LDAC, LDAK, LDB, LDBC,
$ LDBK, LDC, LDCC, LDCK, LDD, LDDC, LDDK, LDWORK,
$ LIWORK, M, N, NCON, NMEAS, NP
DOUBLE PRECISION ACTOL, GAMMA, GTOL
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AC( LDAC, * ), AK( LDAK, * ),
$ B( LDB, * ), BC( LDBC, * ), BK( LDBK, * ),
$ C( LDC, * ), CC( LDCC, * ), CK( LDCK, * ),
$ D( LDD, * ), DC( LDDC, * ), DK( LDDK, * ),
$ DWORK( * ), RCOND( 4 )
Arguments
Input/Output Parameters
JOB (input) INTEGER
Indicates the strategy for reducing the GAMMA value, as
follows:
= 1: Use bisection method for decreasing GAMMA from GAMMA
to GAMMAMIN until the closed-loop system leaves
stability.
= 2: Scan from GAMMA to 0 trying to find the minimal GAMMA
for which the closed-loop system retains stability.
= 3: First bisection, then scanning.
= 4: Find suboptimal controller only.
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/output) DOUBLE PRECISION
The initial value of gamma on input. It is assumed that
gamma is sufficiently large so that the controller is
admissible. GAMMA >= 0.
On output it contains the minimal estimated gamma.
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).
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).
AC (output) DOUBLE PRECISION array, dimension (LDAC,2*N)
The leading 2*N-by-2*N part of this array contains the
closed-loop system state matrix AC.
LDAC INTEGER
The leading dimension of the array AC.
LDAC >= max(1,2*N).
BC (output) DOUBLE PRECISION array, dimension (LDBC,M-NCON)
The leading 2*N-by-(M-NCON) part of this array contains
the closed-loop system input matrix BC.
LDBC INTEGER
The leading dimension of the array BC.
LDBC >= max(1,2*N).
CC (output) DOUBLE PRECISION array, dimension (LDCC,2*N)
The leading (NP-NMEAS)-by-2*N part of this array contains
the closed-loop system output matrix CC.
LDCC INTEGER
The leading dimension of the array CC.
LDCC >= max(1,NP-NMEAS).
DC (output) DOUBLE PRECISION array, dimension (LDDC,M-NCON)
The leading (NP-NMEAS)-by-(M-NCON) part of this array
contains the closed-loop system input/output matrix DC.
LDDC INTEGER
The leading dimension of the array DC.
LDDC >= max(1,NP-NMEAS).
RCOND (output) DOUBLE PRECISION array, dimension (4)
For the last successful step:
RCOND(1) contains the reciprocal condition number of the
control transformation matrix;
RCOND(2) contains the reciprocal condition number of the
measurement transformation matrix;
RCOND(3) contains an estimate of the reciprocal condition
number of the X-Riccati equation;
RCOND(4) contains an estimate of the reciprocal condition
number of the Y-Riccati equation.
Tolerances
GTOL DOUBLE PRECISION
Tolerance used for controlling the accuracy of GAMMA
and its distance to the estimated minimal possible
value of GAMMA.
If GTOL <= 0, then a default value equal to sqrt(EPS)
is used, where EPS is the relative machine precision.
ACTOL DOUBLE PRECISION
Upper bound for the poles of the closed-loop system
used for determining if it is stable.
ACTOL <= 0 for stable systems.
Workspace
IWORK INTEGER array, dimension (LIWORK)
LIWORK INTEGER
The dimension of the array IWORK.
LIWORK >= max(2*max(N,M-NCON,NP-NMEAS,NCON,NMEAS),N*N)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
value of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= LW1 + max(1,LW2,LW3,LW4,LW5 + MAX(LW6,LW7)),
where
LW1 = N*M + NP*N + NP*M + M2*M2 + NP2*NP2;
LW2 = max( ( N + NP1 + 1 )*( N + M2 ) +
max( 3*( N + M2 ) + N + NP1, 5*( N + M2 ) ),
( N + NP2 )*( N + M1 + 1 ) +
max( 3*( N + NP2 ) + N + M1, 5*( N + NP2 ) ),
M2 + NP1*NP1 + max( NP1*max( N, M1 ),
3*M2 + NP1, 5*M2 ),
NP2 + M1*M1 + max( max( N, NP1 )*M1,
3*NP2 + M1, 5*NP2 ) );
LW3 = max( ND1*M1 + max( 4*min( ND1, M1 ) + max( ND1,M1 ),
6*min( ND1, M1 ) ),
NP1*ND2 + max( 4*min( NP1, ND2 ) +
max( NP1,ND2 ),
6*min( NP1, ND2 ) ) );
LW4 = 2*M*M + NP*NP + 2*M*N + M*NP + 2*N*NP;
LW5 = 2*N*N + M*N + N*NP;
LW6 = max( M*M + max( 2*M1, 3*N*N +
max( N*M, 10*N*N + 12*N + 5 ) ),
NP*NP + max( 2*NP1, 3*N*N +
max( N*NP, 10*N*N + 12*N + 5 ) ));
LW7 = M2*NP2 + NP2*NP2 + M2*M2 +
max( ND1*ND1 + max( 2*ND1, ( ND1 + ND2 )*NP2 ),
ND2*ND2 + max( 2*ND2, ND2*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 ) ) ) ) );
M1 = M - M2, NP1 = NP - NP2,
ND1 = NP1 - M2, ND2 = M1 - NP2.
For good performance, LDWORK must generally be larger.
BWORK LOGICAL array, dimension (LBWORK)
LBWORK INTEGER
The dimension of the array BWORK. LBWORK >= 2*N.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: if the matrix | A-j*omega*I B2 | had not full
| C1 D12 |
column rank in respect to the tolerance EPS;
= 2: if the matrix | A-j*omega*I B1 | had not full row
| C2 D21 |
rank in respect to the tolerance EPS;
= 3: if the matrix D12 had not full column rank in
respect to the tolerance SQRT(EPS);
= 4: if the matrix D21 had not full row rank in respect
to the tolerance SQRT(EPS);
= 5: if the singular value decomposition (SVD) algorithm
did not converge (when computing the SVD of one of
the matrices |A B2 |, |A B1 |, D12 or D21);
|C1 D12| |C2 D21|
= 6: if the controller is not admissible (too small value
of gamma);
= 7: if the X-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 8: if the Y-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 9: if the determinant of Im2 + Tu*D11HAT*Ty*D22 is
zero [3];
= 10: if there are numerical problems when estimating
singular values of D1111, D1112, D1111', D1121';
= 11: if the matrices Inp2 - D22*DK or Im2 - DK*D22
are singular to working precision;
= 12: if a stabilizing controller cannot be found.
Method
The routine implements the Glover's and Doyle's 1988 formulas [1], [2], modified to improve the efficiency as described in [3]. JOB = 1: It tries with a decreasing value of GAMMA, starting with the given, and with the newly obtained controller estimates of the closed-loop system. If it is stable, (i.e., max(eig(AC)) < ACTOL) the iterations can be continued until the given tolerance between GAMMA and the estimated GAMMAMIN is reached. Otherwise, in the next step GAMMA is increased. The step in the all next iterations is step = step/2. The closed-loop system is obtained by the formulas given in [2]. JOB = 2: The same as for JOB = 1, but with non-varying step till GAMMA = 0, step = max(0.1, GTOL). JOB = 3: Combines the JOB = 1 and JOB = 2 cases for a quicker procedure. JOB = 4: Suboptimal controller for current GAMMA only.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, MA, 1995.
[3] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
Fortran 77 routines for Hinf and H2 design of continuous-time
linear control systems.
Rep. 98-14, Department of Engineering, Leicester University,
Leicester, U.K., 1998.
Numerical Aspects
The accuracy of the result depends on the condition numbers of the input and output transformations and on the condition numbers of the two Riccati equations, as given by the values of RCOND(1), RCOND(2), RCOND(3) and RCOND(4), respectively. This approach by estimating the closed-loop system and checking its poles seems to be reliable.Further Comments
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