Hello Lukas
I am not sure I understand this scaling thing. Does it means simply to
scale the states by some constant so that the CARE will be well
conditioned? I will try that. If I am wrong I will appreciate if you can
provide me some Ref.
I started to read a little bit on the methods for the solution of CARE.
It seems that when R is nearly singular the "The Inverse-Free
Generalized Schur Method" is preferable than the "Generalized Schur
Method" [1]. He also states that this is the method used in MATLAB for
the case when R is nearly singular.
That might be another possible approach for the solution of this
problem.
Sincerely
Ido
[1] Biswa Nath Datta (2004) ``Numerical Methods for Linear Control
Systems - Design and Analysis". Elsevier
-----Original Message-----
From: Lukas Reichlin <lukas.reichlin.li...@gmail.com>
To: Ido Halperin <idoh...@gmail.com>
Cc: octave-dev@lists.sourceforge.net
Subject: Re: [OctDev] problem in solving CARE (continuos algebraic
Riccati's equation) for large model
Date: Mon, 18 Jun 2012 14:28:12 +0200
On 17.06.2012, at 23:17, Ido Halperin wrote:
> Hello Lukas
>
> I appreciate your quick reply and I a apologize for the delay in my
> response. It took me some time to compile octave. Usually I makes use of
> debian binaries.
>
> a. Relating the results you attached, they seems to me erroneous. I
> compared the stable eigenvalues of the Hamiltonian:
>
> S = Bum*(R\Bum');
> H = [Am -S; -Qm -Am'];
> lambda1 = eig(H)
>
> and the eigenvalues of the closed loop system, according to the 'Km' you
> attached:
> lambda2 = eig(Am - Bum*Km);
>
> and they differ.
>
>
>
> b. I downloaded octave 3.6.2 source and lapack 3.4.1 from netlib and
> compiled them. It didn't helped. It gives the following:
>
>
> octave:33> [Km riccPm] = lqr(Am,Bum,Qm,R);
> error: are: 4: after reordering, roundoff changed values of some complex
> eigenvalues so that leading eigenvalues in the (generalized) Schur form
> no longer satisfy the stability condition; this could also be caused due
> to scaling
> error: called from:
> error: /home/ido/octave/control-2.3.51/care.m at line 162, column 15
> error: /home/ido/octave/control-2.3.51/lqr.m at line 95, column 16
> octave:33> [Km riccPm] = lqr(Am,Bum,Qm,R*3);
> error: are: 5: the computed dimension of the solution does not equal N
> error: called from:
> error: /home/ido/octave/control-2.3.51/care.m at line 162, column 15
> error: /home/ido/octave/control-2.3.51/lqr.m at line 95, column 16
> octave:33>
>
>
> Ido
Hi Ido
Have you tried to scale your equations in order to enhance numerics? SLICOT has
+/- 3 solvers for AREs: SB02MD, SB02OD and SB02RD:
Riccati Equations
SB02MD
Solution of algebraic Riccati equations (Schur vectors method)
SB02MT
Conversion of problems with coupling terms to standard problems
SB02ND
Optimal state feedback matrix for an optimal control problem
SB02OD
Solution of algebraic Riccati equations (generalized Schur method)
SB02PD
Solution of continuous algebraic Riccati equations (matrix sign
function method) with condition and forward error bound estimates
SB02QD
Condition and forward error for continuous Riccati equation solution
SB02RD
Solution of algebraic Riccati equations (refined Schur vectors method)
with condition and forward error bound estimates
SB02SD
Condition and forward error for discrete Riccati equation solution
I chose SB02OD back in 2010 because it was the easiest one to implement.
Unfortunately, SB02OD has no automatic scaling (the other 2 have it) and one of
my profs told me that the algorithm is not the most reliable one. Maybe we
should try SB02RD in conjunction with SB02MT, and I could ask the SLICOT
authors for their advice.
Lukas
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