Question on Wikimization

There is a student, Maria Calle in Spain, who has contacted me regarding a question she has in Control. She posted her question here:
http://www.convexoptimization.com/wikimization/index.php/User:Mtxu
Please dialogue with her there. I already recommended CVX (Grant/Boyd) to Maria as a method of solving the Optimization part of her problem.
thanks, Jon
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Sometimes you have to go to where the information is instead of expecting it to come to you.
Peter Nachtwey
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Maria is a student in Spain who is trying to numerically implement the paper Linear Quadratic Optimal Output Feedback Control For Systems With Poles In A Specified Region, Lisong Yuan, Luke E. K. Achenie, Weisun Jiang http://citeseerx.ist.psu.edu/viewdoc/summary?doi .1.1.42.68
Initilaization of the gain matrix from that paper is described here: Arzelier et al., Pole Assignment of Linear Uncertain Systems in a Sector Via a Lyapunov-Type Approach http://convexoptimization.com/TOOLS/Arzelier.pdf
Maria has taught herself Matlab and CVX http://www.stanford.edu/~boyd/cvx all in the past few days.
She has posted her question on Wikimization.org http://www.convexoptimization.com/wikimization/index.php/Talk:Beginning_with_CVX because it is a wiki that understands LaTeX, HTML, and you can write source code there.
If you are challenged by these papers, please give her a hand.
Jon
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Wikimization.org a crit :

This paper presents a method to determine such a stabilizing controller for STATE feedback control (matrix A+BK). However, initialization in first paper requires an OUTPUT feeback controller (matrix A+BKC). This problem is still open and I don't know of any necessary and sufficient method to get it.
I will however try to give you some ref for sufficiency.
Mat

http://www.convexoptimization.com/wikimization/index.php/Talk:Beginning_with_CVX
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mat a crit :

- C. Crusius and A. Trofino, Sufficient LMI conditions for output feedback control problems, IEEE Trans. on Automat. Control, vol. 44, 1999, pages 1053-1057. - T. Iwasaki and R. Skelton, Parametrization of All Stabilizing Controllers via Quadratic Lyapunov Functions, J. Optimization Theory and Applications, vol. 85, no. 2, 1995, pages 291-307. - D. Peaucelle and D. Arzelier, An Efficient Numerical Solution for H2 Static Output Feedback Synthesis, proceedings of European Control Conference, Porto, Portugal, september 2001, pages 3800-3805. - D. Peaucelle and D. Arzelier, Ellipsoidal Sets for Resilient and Robust Static Output-Feedback, IEEE Trans. on Automat. Control, , 2003.
Mat

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Hi everyone!
I'm Mara, this morning I couldn't write here, I don't know why, but now I can :).
Thanks a lot for all your help.
The teacher gave to me 4 days more.
The exact title of my project is: LQR, selection of weigthing matrix Q and R. I have finded some methods (and I have programed them), but I haven't finded the most important, the method of pole assignment. I must look for where are the poles, and then by the LQR theory with pole assignment I must move them to the LMI region (or to one specific region).
For it I began with the first article, wich is programed yet . But for initializate the gain matrix the authors said that they based their program in another article and in the thesis of one of them ( Is from China and I couldn't find it). This another article is the second one. Until today ( thanks to you) I was thinking that for make output feedback I had just to divide by matrix C (Ho ho). Today I have read a lot of thinks about it and I understand my mistake. But I don't understand why they said in those article was the method for initialize the gain matrix. I've tryed wih anothers methods but the principal program doesn't converge (Also the output feedback described in Optimal control - Lewis ).
I know the second article doesn't work. So Now I don't know how continue whit the proyect. I have a lot of dubts, I can begin with Output Feedback or with another method to select the weigthing matrix Q and R.
What do you advise me?
Thanks a lot for all.
Greetings
Mara
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On Feb 11, 1:51pm, snipped-for-privacy@hotmail.com wrote:

If you know where you want the poles then just place them there. You really must show the transfer function of the system you want to control and let us know what the desired characteristic equation is. LQR can be used to tune SISO systems but I think it is better for MIMO system where there may be conflicting goals that must be optimized. You have not mentioned a cost function and I don't see why a cost function is necessary.
Until you provide information see these
Pole placement is easy. See this ftp://ftp.deltacompsys.com/public/NG/Mathcad%20-%20T0C1-PID%20CTM%20NG.pdf I placed the four poles on the negative real axis to get a critically damped response but you can change the desired characteristic equation to what ever meets your goal. I have found that placing the poles on the negative real axis is safe however. I can also place zeros use a modification of the same technique.
Another method of placing poles is Ackermann's method ftp://ftp.deltacompsys.com/public/NG/Mathcad%20-%20T0C1%20I-PD%20NG%20b.pdf See page 2.
You can place the poles where every you want. No LQR or optimization is necessary.
Peter Nachtwey
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pnachtwey a crit :

Hi Peter,
I think Maria is interested in placing the poles in a specified region (I usually like to consider LMI regions) such that a cost function is minimized.
This can easily be done using two LMIs and I know how to do it (just looking at the Riccati equations or LMIs involved). It would directly result in a constraint on the Q matrix. But she seems to prefer publications with improved methods to do it, and I am not sure I want to give a solution that has not been published yet.
I also think she wants to study the MIMO case (so that she could place the eigenvalues of matrix A+BK).
It is written in http://en.wikipedia.org/wiki/Linear-quadratic_regulator last paragraph of "General description" that control engineers prefer alternative methods like full state feedback (also known as pole placement) to find a controller over the use of the LQR algorithm.
Anyone has ever heard of such a method? I have done pole placement or LQR synthesis but never both at the same time. Is it so obvious?
And by the way, how do you usually fix the R and Q matrices of the cost function? Trial and error or you know a better way to do it?
Have a good day.
Mat

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What cost function? She hasn't provided one! If Maria just wants to place poles then look at what I have done. However, simple pole placement assumes there is an accurate model of the system. If the system changes the poles will drift off from their location on the negative real axis where I put them. I have done quite a bit of testing to see if my solutions will stand up to variations. The LMI pole placement does the same but it sure seems like a lot of work just to place poles.
The best info I found is http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.9447

I have not heard of both at the same time. I don't see why. The goals of each is different. The goal of pole placement is to get a desired response. The goal of LQR is to minimize a cost function.

I have enough experience at this so I can wing it but there are some relationships between the numbers that I think are obvious. ftp://ftp.deltacompsys.com/public/NG/Mathcad%20-%20LQR-Pandiani.pdf You can see that the ratios between the values within the Q and R cost arrays are chosen to get a desired result.
ftp://ftp.deltacompsys.com/public/NG/Mathcad%20-%20LQR%2010pt2.pdf In this example I used the Q array only to get a critically damped response. Not the binomial pattern of the diagonal of the Q array. I use this pattern to get critically damped responses in different applications. If I replaced the 3s with 2s I bet I would get a response similar to a Butterworth filter's response.
Peter Nachtwey
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pnachtwey a crit :

Thanks Peter.
I was actually looking at Chilali's and Gahinet's phd theses this afternoon. The paper you mentioned is better as more concise.
I have also found http://citeseerx.ist.psu.edu/viewdoc/summary?doi .1.1.42.5458 which presents the problem of multiobjective feedback control without (too much) care of robustness issues. I think Maria should have a look at it along with why LQR and H2 syntheses are almost equivalent.

(well, trial and error until experience comes...) but there are some

I understand R and Q are only used to weight relative energies of each control input and output. What about non diagonal matrices? Can this be relevant to meet a particular specification?

I don't understand why using such a pattern produces a critically damped response. I may need some more experience with LQR and digital control.
Thank you.
Mat

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