Question on Wikimization

Sometimes you have to go to where the information is instead of
expecting it to come to you.

Peter Nachtwey

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

Wikimization.org a écrit :

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

mat a écrit :

- 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


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

Hi everyone!

I'm María, 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

María

On Feb 11, 1:51 pm, mt...@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

pnachtwey a écrit :

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

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

pnachtwey a écrit :

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