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<HTML>
<HEAD><TITLE>SB10VD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="SB10VD">SB10VD</A></H2>
<H3>
State feedback and output injection matrices for an H2 optimal state controller (continuous-time)
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To compute the state feedback and the output injection
matrices for an H2 optimal n-state controller for the system
| A | B1 B2 | | A | B |
P = |----|---------| = |---|---|
| C1 | 0 D12 | | C | D |
| C2 | D21 D22 |
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 SB10UD. Matrix D is not used
explicitly.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE SB10VD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ F, LDF, H, LDH, X, LDX, Y, LDY, XYCOND, IWORK,
$ DWORK, LDWORK, BWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDC, LDF, LDH, LDWORK, LDX,
$ LDY, M, N, NCON, NMEAS, NP
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), F( LDF, * ), H( LDH, * ),
$ X( LDX, * ), XYCOND( 2 ), Y( LDY, * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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).
F (output) DOUBLE PRECISION array, dimension (LDF,N)
The leading NCON-by-N part of this array contains the
state feedback matrix F.
LDF INTEGER
The leading dimension of the array F. LDF >= max(1,NCON).
H (output) DOUBLE PRECISION array, dimension (LDH,NMEAS)
The leading N-by-NMEAS 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.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (max(2*N,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 >= 13*N*N + 12*N + 5.
For good performance, LDWORK must generally be larger.
BWORK LOGICAL array, dimension (2*N)
</PRE>
<B>Error Indicator</B>
<PRE>
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 Y-Riccati equation was not solved
successfully.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The routine implements the formulas given in [1], [2]. The X-
and Y-Riccati equations are solved with condition and accuracy
estimates [3].
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Zhou, K., Doyle, J.C., and Glover, K.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, NJ, 1996.
[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., 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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
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.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
The Riccati equations are solved by the Schur approach
implementing condition and accuracy estimates.
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
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