Path: news.mathworks.com!not-for-mail
From: "Michael Zhang" <cy_xiaoxiao@126.com>
Newsgroups: comp.soft-sys.matlab
Subject: Re: LMI term dimensions incompatible
Date: Thu, 8 May 2008 14:28:04 +0000 (UTC)
Organization: HeNan Institue of Finance &#38; Economics
Lines: 408
Message-ID: <fvv2lk$lua$1@fred.mathworks.com>
References: <fvtuvp$aiv$1@fred.mathworks.com> <fvueq0$mf3$1@fred.mathworks.com> <fvunbr$1tb$1@fred.mathworks.com> <fvur63$oog$1@fred.mathworks.com> <fvus3i$kci$1@fred.mathworks.com> <fvuskh$mq9$1@fred.mathworks.com>
Reply-To: "Michael Zhang" <cy_xiaoxiao@126.com>
NNTP-Posting-Host: webapp-05-blr.mathworks.com
Content-Type: text/plain; charset="ISO-8859-1"
Content-Transfer-Encoding: 8bit
X-Trace: fred.mathworks.com 1210256884 22474 172.30.248.35 (8 May 2008 14:28:04 GMT)
X-Complaints-To: news@mathworks.com
NNTP-Posting-Date: Thu, 8 May 2008 14:28:04 +0000 (UTC)
X-Newsreader: MATLAB Central Newsreader 1379620
Xref: news.mathworks.com comp.soft-sys.matlab:467404



"Johan L?fberg" <loefberg@control.ee.ethz.ch> wrote in 
message <fvuskh$mq9$1@fred.mathworks.com>...
> "Johan L?fberg" <loefberg@control.ee.ethz.ch> wrote in
> message <fvus3i$kci$1@fred.mathworks.com>...
> > "Johan L?fberg" <loefberg@control.ee.ethz.ch> wrote in
> > message <fvur63$oog$1@fred.mathworks.com>...
> > > "Michael Zhang" <cy_xiaoxiao@126.com> wrote in message
> > > <fvunbr$1tb$1@fred.mathworks.com>...
> > > > "Johan L?fberg" <loefberg@control.ee.ethz.ch> wrote 
in 
> > > > message <fvueq0$mf3$1@fred.mathworks.com>...
> > > > > Well, I won't directly answer your question, but 
when I 
> > > > see
> > > > > this, I can see that it must be annoying. My 
advice
> to you
> > > > > is to use more modern tools to setup and solve 
the SDP.
> > > > > 
> > > > > Using YALMIP and the solver SeDuMi
> > > > > http://control.ee.ethz.ch/~joloef/wiki/pmwiki.php
> > > > > http://control.ee.ethz.ch/~joloef/wiki/pmwiki.php?
> > > > n=Solvers.SEDUMI
> > > > > 
> > > > > you would code your problem as below. 
> > > > > 
> > > > > I think this approach would save you a lot of 
agony
> during
> > > > > the modelling.
> > > > > 
> > > > > /johan
> > > > > 
> > > > > 
> > > > > Q = sdpvar(n,n);
> > > > > S1 = sdpvar(n,n);
> > > > > S2 = sdpvar(n,n);
> > > > > 
> > > > > gammasquared = sdpvar(1,1);
> > > > > 
> > > > > A = randn(n,n);
> > > > > Ad = randn(n,n);
> > > > > Bd = randn(n,n);
> > > > > B1 = randn(n,n);
> > > > > B2 = randn(n,n);
> > > > > M = randn(n,n);
> > > > > Cd = randn(n,n);
> > > > > C = randn(n,n);
> > > > > D11 = randn(n,n);
> > > > > D12 = randn(n,n);
> > > > > Dd = randn(n,n);
> > > > > 
> > > > > 
> > > > > % We could do something like this, but then we 
would
> have 
> > > > to
> > > > > find out the
> > > > > % dimensions of the zero blocks, and we are too 
lazy to 
> > > > do that 
> > > > > %B = [ -Q+Bd*S2*Bd'+Ad*S1*Ad' A*Q+B2*M         
> > > > B1          
> > > > >        Bd*S2*Dd'+Ad*S1*Cd'  0       0;
> > > > > %  Q*A'+M'*B2'                -Q            
> > > > 0              
> > > > >     Q*C'+M'*D12'         M'      Q;
> > > > > %  B1'                         0          -
> > > > gammasquared*I  
> > > > >          D11'                0       0;
> > > > > %  Dd*S2*Bd'+Cd*S1*Ad'    C*Q+D12*M         
> > > > D11            
> > > > >   -I+Cd*S1*Cd'+Dd*S2*Dd' 0       0;
> > > > > %  0                      M                 
> > > > 0              
> > > > >      0                  -S2      0;
> > > > > %  0                      Q                 
> > > > 0              
> > > > >      0                   0      -S1 ];
> > > > > 
> > > > > % Instead, we use the blkvar shortcut. It 
automatically
> > > > > fills in the blanks and symmetrizes for you
> > > > > B = blkvar;
> > > > > B(1,1) =  -Q+Bd*S2*Bd'+Ad*S1*Ad';
> > > > > B(1,2) =  A*Q+B2*M;
> > > > > B(1,3) = B1;
> > > > > B(1,4) = Bd*S2*Dd'+Ad*S1*Cd';
> > > > > B(2,2) = -Q;
> > > > > B(2,4) = Q*C'+M'*D12';
> > > > > B(2,5) = M';
> > > > > B(2,6) = Q;
> > > > > B(3,3) = -gammasquared*eye(size(B1,2));
> > > > > B(3,4) = D11';
> > > > > B(4,4) = -eye(size(Cd,1))+Cd*S1*Cd'+Dd*S2*Dd';
> > > > > B(5,5) = -S2;
> > > > > B(6,6) = -S1;
> > > > > 
> > > > > % set up all constraints
> > > > > Constraints = [B < 0, Q>0, S1>0, S2>0];
> > > > > Objective = gammasquared;
> > > > > % Minimize gamma^2
> > > > > solvesdp(Constraints,Objective)
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > 
> > > > > "Michael Zhang" <cy_xiaoxiao@126.com> wrote in 
message
> > > > > <fvtuvp$aiv$1@fred.mathworks.com>...
> > > > > > Is anyone can help me? I can figure out what 
happened 
> > > > to my 
> > > > > > programm.
> > > > > > 
> > > > > > I need to solve a inequality:
> > > > > > 
> > > > > > | -Q+Bd*S2*Bd'+Ad*S1*Ad' A*Q+B2*M   B1
> > > > > > |      Q*A'+M'*B2'         -Q       0
> > > > > > |          B1'              0       -gammar^2*I
> > > > > > |   Dd*S2*Bd'+Cd*S1*Ad'  C*Q+D12*M  D11
> > > > > > |          0                M       0
> > > > > > |          0                Q       0
> > > > > > 
> > > > > >     Bd*S2*Dd'+Ad*S1*Cd'     0       0  |
> > > > > >          Q*C'+M'*D12'       M'      Q  |
> > > > > >              D11'           0       0  |  <0
> > > > > >   -I+Cd*S1*Cd'+Dd*S2*Dd'    0       0  |
> > > > > >              0             -S2      0  |
> > > > > >              0              0      -S1 | 
> > > > > >   
> > > > > > Matrices:Q, S1, S2 are positive-define.
> > > > > > 
> > > > > > Below is programm:
> > > > > > 
===================================================
> > > > > > A = [2 1; 0 1];
> > > > > > Ad = [0.2 0.1; 0 0.1];
> > > > > > B1 = [0.1 0.1]';
> > > > > > B2 = [1 1]';
> > > > > > Bd = [0.1 0.1]';
> > > > > > 
> > > > > > C = [1, 1];
> > > > > > Cd = [0.1, 0.1];
> > > > > > 
> > > > > > D11 = 0.1;
> > > > > > D12 = 1;
> > > > > > Dd = 0.1;
> > > > > > 
> > > > > > gammar = 1;
> > > > > > 
> > > > > > % Initial a LMI system
> > > > > > setlmis([]);
> > > > > > 
> > > > > > % Define Variables
> > > > > > 
> > > > > > % Q is a symmetric matrix, has a block size of 
2 and 
> > > > this 
> > > > > > block is symmetric 
> > > > > > Q = lmivar(1, [2 1]);
> > > > > > 
> > > > > > % S1 a symmeric matrix, size 2 by 2
> > > > > > S1 = lmivar(1, [2 1]);
> > > > > > 
> > > > > > 
> > > > > > % S2 is 1 by 1 matrix
> > > > > > S2 = lmivar(1, [1 0]);
> > > > > > 
> > > > > > 
> > > > > > % Type of 2, size 1 by 2 
> > > > > > M = lmivar(2, [1 2]);
> > > > > > 
> > > > > > 
> > > > > > %I use upper triangular to represent the LMI
> > > > > > 
> > > > > > % pos in (1, 1)
> > > > > > lmiterm([1 1 1 Q], -1, 1);
> > > > > > lmiterm([1 1 1 S2], Bd, Bd');
> > > > > > lmiterm([1 1 1 S1], Ad, Ad');
> > > > > > 
> > > > > > % pos (1, 2)
> > > > > > lmiterm([1 1 2 Q], A, 1);
> > > > > > lmiterm([1 1 2 M], B2, 1);
> > > > > > 
> > > > > > % pos(1, 3)
> > > > > > lmiterm([1 1 3 0], B1);
> > > > > > 
> > > > > > % pos(1, 4)
> > > > > > lmiterm([1 1 4 S2], Bd, Bd');
> > > > > > lmiterm([1 1 4 S1], Ad, Cd');
> > > > > > 
> > > > > > 
> > > > > > % pos(2, 2)
> > > > > > lmiterm([1 2 2 Q], -1, 1);
> > > > > > 
> > > > > > % pos(2, 4)
> > > > > > lmiterm([1 2 4 Q], 1, C');
> > > > > > 
> > > > > > lmiterm([1 2 4 -M], 1, D12');
> > > > > > 
> > > > > > % pos(2, 5)
> > > > > > lmiterm([1 2 5 -M], 1, 1);
> > > > > > 
> > > > > > % pos(2, 6)
> > > > > > lmiterm([1 2 6 Q], 1, 1);
> > > > > > 
> > > > > > % pos(3, 3)
> > > > > > lmiterm([1 3 3 0], -(gammar^2));
> > > > > > 
> > > > > > % pos(3, 4)
> > > > > > lmiterm([1 3 4 0], D11');
> > > > > > 
> > > > > > % pos(4, 4)
> > > > > > lmiterm([1 4 4 0], -1);
> > > > > > lmiterm([1 4 4 S1], Cd, Cd');
> > > > > > lmiterm([1 4 4 S2], Dd, Dd');
> > > > > > 
> > > > > > % pos(5, 5)
> > > > > > lmiterm([1 5 5 S2], -1, 1);
> > > > > > 
> > > > > > 
> > > > > > % pos(6, 6)
> > > > > > lmiterm([1 6 6 S1], -1, 1);
> > > > > > 
> > > > > > lmis = getlmis;
> > > > > > 
> > > > > > [tmin, feas] = feasp(lmis)
> > > > > > 
> > > > > > --------------------------------
> > > > > > When I run it, I get error message:
> > > > > > 
> > > > > > ??? Error using ==> lmiterm at 296
> > > > > > lhs of LMI #1, block (4,1): term dimensions 
> > > > incompatible 
> > > > > > with
> > > > > > other terms in same row
> > > > > > 
> > > > > > Error in ==> paperor at 56
> > > > > > lmiterm([1 1 4 S1], Ad, Cd');
> > > > > > 
> > > > > > I have checked many times and can't figure out 
why it 
> > > > is 
> > > > > > wrong, can anybody point it to me?
> > > > > > 
> > > > > 
> > > > 
> > > > Dear Johan:
> > > > 
> > > > Thank you very much. But I still have some doubts 
about 
> > > > this.
> > > > 
> > > > You recommend me to use YALMIP, but I think the
> program you 
> > > > write for me is not apply to me.
> > > > 
> > > > 1.The original problem is for a discrete time linear
> delay 
> > > > system which I have't write it here, If there 
exists 
> > > > positive-define matrices Q, S1, S2 and a matrix M
> satisfied 
> > > > that LMI, then we call it H-inf quadratically stable
> with H-
> > > > inf norm bound gammar. So I mean gammar cannot be a
> random 
> > > > variable. So you write the program that object is 
to 
> > > > minimize gamma^2, I think it is not suitable.
> > > > 
> > > > 2.In my problem, matrices like A, Ad, Dd, they are 
all 
> > > > specified, you use randn(n,n) to generate them. I 
replace 
> > > > them with specified ones.
> > > > 
> > > 
> > > 
> > > I though gammar was a decision variable that you 
wanted to
> > > optimize. If it is given, well then you just use that
> > > instead and remove the objective from the call to 
solvesdp
> > > and solve the feasibility problem instead
> > > 
> > > You can of course use your own data, I only defined 
random
> > > matrices to get an example that was working.
> > > 
> > > The problem can be solved with YALMIP and SeDuMi 
without a
> > > doubt, it is a trivial SDP problem.
> > 
> > 
> > ..and I noticed you optimized over M too, so you should 
use
> > 
> > M = sdpvar(somesize,someothersize,'full')
> > 
> > 
> 
> To summarize
> 
> A = [2 1; 0 1];
> Ad = [0.2 0.1; 0 0.1];
> B1 = [0.1 0.1]';
> B2 = [1 1]';
> Bd = [0.1 0.1]';
> 
> C = [1, 1];
> Cd = [0.1, 0.1];
> 
> D11 = 0.1;
> D12 = 1;
> Dd = 0.1;
> 
> gammar = 1;
> 
> Q = sdpvar(2,2);
> S1 = sdpvar(2,2);
> S2 = sdpvar(1);
> % Full by definition, but just to rmember it if 
> % we change dimensions later
> M = sdpvar(1,2,'full');
> 
> B = blkvar;
> B(1,1) = -Q+Bd*S2*Bd'+Ad*S1*Ad';
> B(1,2) = A*Q+B2*M;
> B(1,3) = B1;
> B(1,4) = Bd*S2*Dd'+Ad*S1*Cd';
> B(2,2) = -Q;
> B(2,4) = Q*C'+M'*D12';
> B(2,5) = M';
> B(2,6) = Q;
> B(3,3) = -gammar^2*eye(size(B1,2));
> B(3,4) = D11';
> B(4,4) = -eye(size(Cd,1))+Cd*S1*Cd'+Dd*S2*Dd';
> B(5,5) = -S2;
> B(6,6) = -S1;
> 
> % set up all constraints
> Constraints = [B < 0, Q>0, S1>0, S2>0];
> solvesdp(Constraints)
>     yalmiptime: 0.3430
>     solvertime: 1.3440
>           info: 'No problems detected (SeDuMi-1.1)'
>        problem: 0
>         dimacs: [1.4589e-015 0 0 0 7.3848e-016 3.9007e-
015]
> 
> 
> >> double(Q)
> 
> ans =
> 
>     0.2425   -0.2091
>    -0.2091    1.0081
> 
> >> double(S1)
> 
> ans =
> 
>     1.3613   -0.4978
>    -0.4978    2.3019
> 
> >> double(S2)
> 
> ans =
> 
>     2.0517
> 
> >> double(M)
> 
> ans =
> 
>    -0.1367   -0.6933
> 
> 
> 
This LMI example is what I read from a paper, but the 
author's result is different form yours. And he gave 
proofs, I think the author only use matlab and we use other 
toolboxes. I doubt if there really exists result difference 
because of toolboxes?

The paper's address is http://www.sciencedirect.com/science?
_ob=ArticleURL&_udi=B6V21-3WWV8W6-B&_user=10&_coverDate=08%
2F31%
2F1999&_alid=736573613&_rdoc=1&_fmt=high&_orig=search&_cdi=5
689&_sort=d&_docanchor=&view=c&_ct=1&_acct=C000050221&_versi
on=1&_urlVersion=0&_userid=10&md5=8ecef13fb66b50ddc4a3990637
4c61b7