Purpose
To compute the state feedback and the output injection
matrices for an H-infinity (sub)optimal n-state controller,
using 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 a given value of gamma, 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
subroutine SB10PD,
(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 SB10QD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, F, LDF, H, LDH, X, LDX, Y, LDY,
$ XYCOND, IWORK, DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDD, LDF, LDH, LDWORK,
$ LDX, LDY, M, N, NCON, NMEAS, NP
DOUBLE PRECISION GAMMA
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), F( LDF, * ),
$ H( LDH, * ), X( LDX, * ), XYCOND( 2 ),
$ 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.
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 (output) DOUBLE PRECISION array, dimension (LDF,N)
The leading M-by-N part of this array contains the state
feedback matrix F.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1,M).
H (output) DOUBLE PRECISION array, dimension (LDH,NP)
The leading N-by-NP part of this array contains the output
injection matrix H.
LDH INTEGER
The leading dimension of the array H. LDH >= max(1,N).
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).
XYCOND (output) DOUBLE PRECISION array, dimension (2)
XYCOND(1) contains an estimate of the reciprocal condition
number of the X-Riccati equation;
XYCOND(2) contains an estimate of the reciprocal condition
number of the Y-Riccati equation.
Workspace
IWORK INTEGER array, dimension max(2*max(N,M-NCON,NP-NMEAS),N*N)
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,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))),
where M1 = M - M2 and NP1 = NP - NP2.
For good performance, LDWORK must generally be larger.
Denoting Q = MAX(M1,M2,NP1,NP2), an upper bound is
max(1,4*Q*Q+max(2*Q,3*N*N + max(2*N*Q,10*N*N+12*N+5))).
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 controller is not admissible (too small value
of gamma);
= 2: if the X-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties);
= 3: if the Y-Riccati equation was not solved
successfully (the controller is not admissible or
there are numerical difficulties).
Method
The routine implements the Glover's and Doyle's formulas [1],[2] modified as described in [3]. The X- and Y-Riccati equations are solved with condition and accuracy estimates [4].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.
[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.
[4] Petkov, P.Hr., Konstantinov, M.M., and Mehrmann, V.
DGRSVX and DMSRIC: Fortan 77 subroutines for solving
continuous-time matrix algebraic Riccati equations with
condition and accuracy estimates.
Preprint SFB393/98-16, Fak. f. Mathematik, Tech. Univ.
Chemnitz, May 1998.
Numerical Aspects
The precision of the solution of the matrix Riccati equations can be controlled by the values of the condition numbers XYCOND(1) and XYCOND(2) of these equations.Further Comments
The Riccati equations are solved by the Schur approach implementing condition and accuracy estimates.Example
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