H∞ State Observer Design

subarto kumar ghosh

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Apr 20, 2021, 1:27:02 PM4/20/21

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Dear Prof. Löfberg,

I am trying to solve the H-infinity state observer related problems. But, I am getting unknown error when the second inequality is set to less than equal 0 (not hard inequality). However, when the second inequality is setting to less than equal (negative and any value), the results are showing that successfully solved but I am not getting the results the same as the results obtained by MATLAB LMI toolbox. For your clarification, I have posted the problems herewith

The abovementioned problem is solved using the MATLAB toolbox. But I have solved the same problem with YALMIP tool box which is attached as m-file.

Actually I am not understanding where I am making error in this code? Please help me.

Thanks in advance.

exam90_33.m

Johan Löfberg

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Apr 20, 2021, 1:37:48 PM4/20/21

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Mosek is telling you Problem status : ILL_POSED

Solution status : PRIMAL_ILLPOSED_CER

My interpretation (which matches with the claimed "solution") is that the infimum is gamma = 0, and this is an unattainable solution achieved by allowing P and W tend to infinity (typical Hinf behaviour, the Hinf optimal controller is numerically bad)

Hence, you have to solve some sensible sub-optimal solution such as

optimize([F,object == 1e-6])

or

optimize(F,object + penalty*(norm(P)+norm(W)))

or anything that stops the solver from trying to return the numerical nonsense solution (gama=0, P=W=inf)

subarto kumar ghosh

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Apr 20, 2021, 2:04:08 PM4/20/21

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Dear Sir,

Thank you very much for your answer. I have used the objective function ==1e-6 which provides the successful solution but I am not getting the same results as the result obtained by the MATLAB LMI TOOLBOX which is mentioned as above.

Is any sort of gain value or results acceptable when the solution shows info: 'Successfully solved (MOSEK)'? Please help me.

Thanks in advance.

Johan Löfberg

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Apr 20, 2021, 2:14:51 PM4/20/21

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the solution you show has driven gama down to close to 10^-9,and at that points do you trust the numerics of the solver, and numerical computations in general? The solution shown is data I would be very concerned about in general, and if I reviewed a paper showing stuff like this I would ask the authors to describe why they get weird numerics like this and if they are aware of the implications, and if they have solved an ill-posed problem. And if you set an equality on gama, any P/W that satisfies the LMIs for that gama are possible solutions

subarto kumar ghosh

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Apr 20, 2021, 2:28:02 PM4/20/21

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Thank you sir. I got your answer.

Wishing you good health and time.

Johan Löfberg

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Apr 20, 2021, 2:40:45 PM4/20/21

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A way to compute a super-optimal solution with gama = 0 is

optimize([P>=0, [P W]*[B2;D2] == 0, A'*P+C1'*W'+P*A+W*C1 <= 0])

P0 = value(P);

W0 = value(W);

L=inv(P0)*W0;

% Arbitrary rescaling makes M1 feasible in the limit

assign(P,P0*1e8);

assign(W,W0*1e8);

assign(gama,1e-8)

max(eig(value(M1)))

in theory...as the solve never will be perfect, and if the equality constrant has residuals, it will destroy things (which is sort of back to the original issue that the setup is ill-posed, as long as P W is in that null-space, gama can be driven to 0), but it is numerically extremely sensitive

subarto kumar ghosh

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Apr 20, 2021, 3:26:09 PM4/20/21

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Thank you so much sir for your answer and explanation.

wlight C

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Jan 28, 2022, 8:53:08 PM1/28/22

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hi,

I am trying to solve the H-infinity output feedback related problem, yet I don't get a reasonable solution. I understand you also solve related H-infinity problem. Could you please give me
some advice to debugging the H-infinity related problem at your convenience?

Thank you very much!

Johan Löfberg

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Jan 29, 2022, 3:32:18 AM1/29/22

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you would have to define what "not reasonable" means

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