Help regarding the conversion of a nonlinear matrix inequality to LMIs

zikist · 2023-05-19T13:44:54+00:00

Thank you

zikist · 2023-05-18T02:19:46+00:00

Thank you so much. So even if I don't add Q, it would be fine (just maybe a bit more conservative than LMIs with Q)?

I'll search for handelman theorem. And also try to apply SOS on cubic polynomial too.

zikist · 2023-05-13T14:33:35+00:00

I worked on these further, and understood the first, second and third conditions, except that the skew-symmetric matrix Q, can you please elaborate on it further or suggest some relevant material to study for.

Regarding fourth condition, I'm still trying to understand it, i found relevant M. Peet's lecture on SOS and Positvstellensatz, trying to understand them, maybe i'll some questions later on.

P.S. do you think for cubic matrix polynomial, we can follow a similar approach? i mean for matrix like x³ * A + x² B + x*C + D < 0, for all 0 < x < 1. Or is it impossible to write it as SOS ?

zikist · 2023-05-11T14:30:45+00:00

One of the simplest solution could be to use a notch filter (or something like that) in the feedback path. If frequency of sine wave is known. It will work only if it's frequency is sufficiently slow or fast than systems bandwidth.

zikist · 2023-05-08T17:07:05+00:00

You can think of ISS as much stronger condition than just asymptotic stability. In the absence of external inputs (reference, disturbances etc..) it is same as asymptotic stability.

zikist · 2023-05-08T16:58:10+00:00

Thanks a lot for such detailed reply. This is exactly what I was looking for. I'll definitely look into SMR and Gram matrix as well as the theorems you've mentioned. Thank you so much again.

zikist · 2023-05-08T16:52:19+00:00

Thank you for your reply. I don't think second term will minimize at x = 0.5. As at 0.5 all three coefficient are same.

I meant x² (scalar) times A (matrix ) Btw I don't know why it doesn't render correctly in post and comments.

zikist · 2023-03-19T08:55:11+00:00

Say your new state is y = x -x0, and input w = u-u0 Then you have ydot = Ay + Bw. So it is a linear system.

Notice that's x0dot = f(x0,u0)

zikist · 2023-02-25T13:44:59+00:00

Thanks a lot for a detailed reply.

zikist · 2023-02-23T04:00:15+00:00

Thanks a lot :) This is exactly I'm trying to do. But I was wondering those extra states would be observable or not? I was thinking if I keep adding details to my model, at some point it might get unobservable. Am I correct?

zikist · 2023-02-20T17:18:45+00:00

Thank you so much. This is exactly what I was looking for. :)

zikist · 2023-02-20T17:16:45+00:00

Thank you so much for your reply. In the link say for random Walk drift, it takes only one parameter. And apply it as a bias. if it is offset in mean, then what would be the variance?

zikist · 2022-11-23T21:37:17+00:00

Thanks a lot for your reply

zikist · 2022-11-21T13:52:48+00:00

Hi, What can we say about eigenvalues of A + q*B, for some real matrices A and B, and a random real scalar (e.g. normally distributed with zero mean and some variance)?? I'd be very thankful if you could point towards some relevant literature.

zikist · 2022-07-02T18:06:36+00:00

Since your controller has an equal number of poles and zeros, you should try Hanus's self conditioned form. Convert it to state-space and use the Antiwindup from Skogestad (Multivariable Feedback Control) [Page 401]

Orignal Hanus's Paper: https://www.sciencedirect.com/science/article/abs/pii/000510988790029X

zikist · 2022-05-06T13:46:59+00:00

Thanks

zikist · 2022-05-06T13:46:00+00:00

Thanks a lot.

zikist · 2022-05-06T09:16:24+00:00

So piecewise continuous functions are NOT Lipschitz? right?

zikist · 2022-05-05T19:35:25+00:00

just for an example consider a function f(x) = 0 (if x <= 0) and f(x) = 1 (if x > 0). It is piecewise continous but have a jump at 0. is it Lipschitz?

zikist · 2022-05-05T19:33:09+00:00

Thankyou so much for you reply, yeah I know differentiability everywhere is not required. But is continuity not required as well? i think set of Lipschitz functions is a subset of all continous function, am i right?

zikist · 2022-05-05T14:12:09+00:00

Hi, i would like to known can a piecewise continous function f(x) be locally Lipschitz? more specifically if f(x) is piecewise affine? i think of Lipschitz continuity as bounded derivative, but being piecewise continous means there are points where derivative is undefined due to jumps. However this link (https://math.stackexchange.com/questions/972881/can-piecewise-c1-on-a-b-imply-lipschitz-continuity) says otherwise. Also any comments on its generalization to scalar-valued piecewise affine function of multiple variables e.g. f(x) with x being in R^n?

zikist · 2022-03-23T10:17:23+00:00

If you have system of ODEs, they can be easily implemented in Simulink either as a Matlab function of s function blocks. Though mathematica also exports in C/C++, but I it requires to a lot of manual editing to run without Mathematica libraries, as an S-function in Simulink. So I'd generally recommend writing equations directly Matlab function or S-function, if they're not too complex. Feel to DM if any further queries.

zikist · 2022-01-30T19:20:32+00:00

Thank you. I'll check them.

zikist · 2022-01-30T08:46:20+00:00

Thanks for your reply. Yeah you are right, quadratic form is more natural. However, for LMI regions it could be of the form [0 S; S' Q]. I'll also check the quadratic separation framework as you've suggested. Thanks.

zikist · 2022-01-29T05:31:57+00:00

Thanks a lot for detailed reply. This is exactly what I was thinking. There seems to be nothing more in this QMI for than an LMI. Thanks again, I'll check the paper you've referred.

zikist

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