Is there any paper studying the following discrete-time system

x(k)=A_1x(k-1)+A_2x(k-2)....A_nx(k-n)

where x is vector and A_i is matrix.
Thanks:)

x(k)=A_1x(k-1)+A_2x(k-2)....A_nx(k-n)

where x is vector and A_i is matrix.

x(k)=A_1x(k-1)+A_2x(k-2)....A_nx(k-n)

where x is vector and A_i is matrix.

sunnewton wrote:

No quote of the original post? I see you're posting from Google -- please read the link at the end of my tag line.

No quote of the original post? I see you're posting from Google -- please read the link at the end of my tag line.

--

Tim Wescott

Wescott Design Services

Tim Wescott

Wescott Design Services

Click to see the full signature.

sunnewton wrote:

>sunnewton wrote:

>

If A_i is time varying then it must be referenced to absolute time, so this would be better stated in the form

x(k) = A_k x(k-1),

or

x(k) = A_ x(k-1),

the difference depending entirely on your mood.

In that case then for an initial condition x(0) you would have

x(k) = A_ A_ ... A_0 x(0)

I'm 99.44% sure that your system would be unconditionally stable if and only if the product A_ ... A_0 always went to zero as k went to infinity. I'm 99% sure that this condition would be met if each A_k had all eigenvalues less than 1, but I'm sure it's not necessary.

If you really meant what you said, particularly if you mean that A ranges from A_1 to A_n with some fixed n, then the system isn't time varying, you're just describing a big system where

[ x(k) ] [ A_1 A_2 ... A_ A_n ] [ x(k-1) ] [ x(k-1) ] [ I 0 ... 0 0 ] [ x(k-2) ] [ ... ] = [ ... ] [ ... ] [ x(k-n+2) ] [ 0 ... I 0 0 ] [ x(x-n+3) ] [ x(k-n+1) ] [ 0 0 ... I 0 ] [ x(k-n+2) ]

This isn't time varying -- it's just poorly constructed, and probably has a gazzilion redundant states.

>sunnewton wrote:

>

If A_i is time varying then it must be referenced to absolute time, so this would be better stated in the form

x(k) = A_k x(k-1),

or

x(k) = A_ x(k-1),

the difference depending entirely on your mood.

In that case then for an initial condition x(0) you would have

x(k) = A_ A_ ... A_0 x(0)

I'm 99.44% sure that your system would be unconditionally stable if and only if the product A_ ... A_0 always went to zero as k went to infinity. I'm 99% sure that this condition would be met if each A_k had all eigenvalues less than 1, but I'm sure it's not necessary.

If you really meant what you said, particularly if you mean that A ranges from A_1 to A_n with some fixed n, then the system isn't time varying, you're just describing a big system where

[ x(k) ] [ A_1 A_2 ... A_ A_n ] [ x(k-1) ] [ x(k-1) ] [ I 0 ... 0 0 ] [ x(k-2) ] [ ... ] = [ ... ] [ ... ] [ x(k-n+2) ] [ 0 ... I 0 0 ] [ x(x-n+3) ] [ x(k-n+1) ] [ 0 0 ... I 0 ] [ x(k-n+2) ]

This isn't time varying -- it's just poorly constructed, and probably has a gazzilion redundant states.

--

Tim Wescott

Wescott Design Services

Tim Wescott

Wescott Design Services

Click to see the full signature.

x(k)=x(k-1)+A(k)x(k-n)
what condition A(k) shoule satisfy such that the vector x(k) is a
bounded signal ?

Pretty much any controls theory textbook.

A good place to start:

http://en.wikipedia.org/wiki/Control_theory#Non-linear_control_systems (follow the links!)

Or google something like

discrete feedback control matrix primer

If you don't want to learn to use google, here's the answer (or as close to the answer as any responsible person should spoon-feed to you!):

http://www.soe.ucsc.edu/~dunbar/docs/papers/CSM_Dunbar.pdf

Another good thing to feed google might be "stability criterion" and some of those other terms...

Have fun!

A good place to start:

http://en.wikipedia.org/wiki/Control_theory#Non-linear_control_systems (follow the links!)

Or google something like

discrete feedback control matrix primer

If you don't want to learn to use google, here's the answer (or as close to the answer as any responsible person should spoon-feed to you!):

http://www.soe.ucsc.edu/~dunbar/docs/papers/CSM_Dunbar.pdf

Another good thing to feed google might be "stability criterion" and some of those other terms...

Have fun!

snipped-for-privacy@hotmail.com wrote:

What an excellent put-down!

If you'd read the OP's follow-on you would have seen that his A_i are distinct and time varying. Even if you hadn't you'd have to wonder why he wants to specify so many of them, and assume that maybe he had a good reason in spite of the fact that his posts from Google groups does make one question his grasp of USENET etiquette.

If you'd read the paper you would have seen that the only thing it says about time-varying systems is that they aren't covered -- so it's a total miss from what the OP needs to see.

If you looked at the form of his equations you'd see that the system in question is as linear as anything else out there, it's just time varying -- so a wiki on non-linear systems is something of a miss.

So perhaps you should save that excellent put-down for a post to which it really applies, since it's so good.

What an excellent put-down!

If you'd read the OP's follow-on you would have seen that his A_i are distinct and time varying. Even if you hadn't you'd have to wonder why he wants to specify so many of them, and assume that maybe he had a good reason in spite of the fact that his posts from Google groups does make one question his grasp of USENET etiquette.

If you'd read the paper you would have seen that the only thing it says about time-varying systems is that they aren't covered -- so it's a total miss from what the OP needs to see.

If you looked at the form of his equations you'd see that the system in question is as linear as anything else out there, it's just time varying -- so a wiki on non-linear systems is something of a miss.

So perhaps you should save that excellent put-down for a post to which it really applies, since it's so good.

--

Tim Wescott

Wescott Design Services

Tim Wescott

Wescott Design Services

Click to see the full signature.

Sorry, it wasn't intended as a putdown. Just a fact.
Context and phrasing of the question suggests that the OP is looking
for a quick answer to a homework question. 3 self-replies in an
attempt to frame one question suggest a lack of preparation prior to
asking for help.

I did take the time to review the OP's history. It is consistent with the above observations.

Of course the referenced paper doesn't contain the answer. There's are***excellent*** discussions of control theory in many, many university
websites, and even pdfs of entire textbooks on line. The OP needs to
learn how to FIND those, not have them handed to him.

All I intended to supply were starting points to begin searching. The fact that the paper was only peripherally applicable confirms that I did not give away the answer. The referenced paper does contain some excellent search terms and introductory concepts that can help the OP find what he's looking for.

If the student wants to do the work, the knowledge is readily available. It's appropriate to ask "I have no freaking clue what to do with this question, can somebody please get me started?"

However, having somebody spoon-feed the student the answers to homework questions prevents him from learning all the things he should run across on the way to answering the initial question.

I did take the time to review the OP's history. It is consistent with the above observations.

Of course the referenced paper doesn't contain the answer. There's are

All I intended to supply were starting points to begin searching. The fact that the paper was only peripherally applicable confirms that I did not give away the answer. The referenced paper does contain some excellent search terms and introductory concepts that can help the OP find what he's looking for.

If the student wants to do the work, the knowledge is readily available. It's appropriate to ask "I have no freaking clue what to do with this question, can somebody please get me started?"

However, having somebody spoon-feed the student the answers to homework questions prevents him from learning all the things he should run across on the way to answering the initial question.

Now I have formulated my question as following
x(k)=ax(k-1)+b(k)x(k-n)+u(k)-------------------------(1),
where x(k), a, b(k) ,u(k) are all scalar, n is positive integer, and
b(k) is time varying, u(k) is control input. I conjecture that if
|a|+|b(k)|<1
then the system is BIBO and further if u(k)=0, limx(k)=constant.

IF I could get any suggestion from literature, I won't come to the google group...there are little about time varying coefficient and it should be noted that the form of system (1) is quite weird!

This is not homework at all~ it is my own conjecture, I just wannna know whether I am right~

IF I could get any suggestion from literature, I won't come to the google group...there are little about time varying coefficient and it should be noted that the form of system (1) is quite weird!

This is not homework at all~ it is my own conjecture, I just wannna know whether I am right~

IF I could get any suggestion from literature, I won't come to the google group...there are little about time varying coefficient and it should be noted that the form of system (1) is quite weird!

This is not homework at all~ it is my own conjecture, I just wannna know whether I am right~

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