From: Pascal Gahinet <pascal@mathworks.com>
Newsgroups: comp.soft-sys.matlab,comp.software-eng,sci.math,sci.engr.control
Subject: Re: a pencil based approach for solving the Riccati equation
Date: Thu, 26 Mar 1998 13:15:12 -0500

Dear Bogdan

The DARE command in the versions 4.0 and above of the
Control System Toolbox solve discrete-time Riccati equations
using precisely this method.

Included is the help for DARE:

- pascal


 DARE  Solve discrete-time algebraic Riccati equations.

    [X,L,G,RR] = DARE(A,B,Q,R,S,E)  computes the unique symmetric
    stabilizing solution X of the discrete-time algebraic Riccati
    equation
                                       -1
     E'XE = A'XA - (A'XB + S)(B'XB + R)  (A'XB + S)' + Q

    or, equivalently (if R is nonsingular)
                                 -1             -1                 -1
     E'XE = F'XF - F'XB(B'XB + R)  B'XF + Q - SR  S'  with  F:=A-BR  S'.

    When omitted, R,S and E are set to the default values R=I, S=0,
    and E=I.  Additional optional outputs include the gain matrix
                         -1
           G = (B'XB + R)  (B'XA + S'),

    the vector L of closed-loop eigenvalues (i.e., EIG(A-B*G,E)),
    and the Frobenius norm RR of the relative residual matrix.

    See also  CARE.


Bogdan.Marinescu@der.edfgdf.fr wrote:

> Hello World!
>
> I have to solve a Riccati equation associated to a
> discrete-time control problem in which the state matrix
> A is singular. The approach I used is based on the
> computing of the invariant sub-spaces of the associated
> matrix pencil (see, e.g., P. Van Dooren (1981) "Generalized
> eigenvalue approach for solving the Riccati equation",
> SIAM J. on Sci. St. Comp. 2, pp. 121-135 or V. Ionescu
> and M. Weiss (1992) "On computing the stabilising solution
> to the discrete-time Riccati equation", Linear algebra
> and its applications, 174:229-238.
> I will be grateful if somebody could indicate me if there
> is any professional C software to solve this problem.
> Also, I'm not sure if there is any toolbox (Mathworks or not)
> containing an m.(MATLAB) file which solves this problem.
> Please let me know if you have some information about this
> kind of software.
> Thanking you in advance,
>
> Sincerely yours,
>
> Bogdan Marinescu
>
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