Systems Knowledge Question

Please answer without looking at Wikipedia or otherwise finding the 
answer.  Please answer, even if the answer is "no".  I'm trying to get a 
measure of the extent of a bit of knowledge, here:

How many of you know, off the top of your head, that the defining 
characteristic of a linear system is superposition?

That if a system obeys superposition it must be linear?

The difference between a time-varying and a non-linear system?

Why the fact that a system obeys superposition vastly eases the task of 
analyzing its dynamic behavior?

Thanks.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

do you want us to poll here, or to send you email so as not to influence 
others?

On 8/7/11 3:15 PM, Tim wrote:
> Please answer without looking at Wikipedia or otherwise finding the
> answer.

... as if Wikipedia can be depended on to get the right answer...

>  Please answer, even if the answer is "no".  I'm trying to get a
> measure of the extent of a bit of knowledge, here:
>
> How many of you know, off the top of your head, that the defining
> characteristic of a linear system is superposition?
>
> That if a system obeys superposition it must be linear?

i still have an issue of rigor with scalers that are irrational.  if any 
decent form of continuity applies or if only rational scalers apply, 
then it's "yes".

>
> The difference between a time-varying and a non-linear system?
>
> Why the fact that a system obeys superposition vastly eases the task of
> analyzing its dynamic behavior?
>

i was trying to answer that the other day with the exponential functions 
and the eigenfunctions and whatnot.  Jerry thought i was being too 
mathematical.

> Thanks.
>
FWIW

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

Tim wrote:

> Please answer without looking at Wikipedia or otherwise finding the
> answer.

Why? What's wrong with consulting other resources? 
Nobody knows _everything_.

Thanks,
Rich

Boredom + Whoring = Whoredom ?



Tim wrote:

> Please answer without looking at Wikipedia or otherwise finding the 
> answer.  Please answer, even if the answer is "no".  I'm trying to get a 
> measure of the extent of a bit of knowledge, here:
> 
> How many of you know, off the top of your head, that the defining 
> characteristic of a linear system is superposition?
> 
> That if a system obeys superposition it must be linear?
> 
> The difference between a time-varying and a non-linear system?
> 
> Why the fact that a system obeys superposition vastly eases the task of 
> analyzing its dynamic behavior?
> 
> Thanks.
> 

On Sun, 07 Aug 2011 12:40:19 -0700, Rich Grise wrote:

> Tim wrote:
> 
>> Please answer without looking at Wikipedia or otherwise finding the
>> answer.
> 
> Why? What's wrong with consulting other resources? Nobody knows
> _everything_.

Because I want to get a gauge of how much people know off the top of 
their heads, of course.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

this must have something to do with the email i just got from Dilip.

so i have to turn around and say no.  besides not being able to connect 
the irrational constants, there is a problem with these complex-to-real 
operations such Re{}, Im{} that satisfy superposition, but cannot get to 
the scaling property correctly with complex scalers.  likewise, as Dilip 
pointed out in 2009, a system that simply conjugates the input also 
satisfies superposition but does not satisfy scaling.

but superposition can take you to scaling for any real and rational scaler.

sorry to bore you to death, Vlad.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On Sun, 07 Aug 2011 14:15:49 -0500, Tim <tim@seemywebsite.please> wrote:

>Please answer without looking at Wikipedia or otherwise finding the 
>answer.  Please answer, even if the answer is "no".  I'm trying to get a 
>measure of the extent of a bit of knowledge, here:
>
>How many of you know, off the top of your head, that the defining 
>characteristic of a linear system is superposition?
>
>That if a system obeys superposition it must be linear?
>
>The difference between a time-varying and a non-linear system?
>
>Why the fact that a system obeys superposition vastly eases the task of 
>analyzing its dynamic behavior?

Yes, of course... I've read your book.   ;-)

-- 
Rich Webb     Norfolk, VA

"Tim"  wrote in message 
news:NsednU-bJLX4eaPTnZ2dnUVZ_t2dnZ2d@web-ster.com...

Please answer without looking at Wikipedia or otherwise finding the
answer.  Please answer, even if the answer is "no".  I'm trying to get a
measure of the extent of a bit of knowledge, here:

How many of you know, off the top of your head, that the defining
characteristic of a linear system is superposition?

That if a system obeys superposition it must be linear?

The difference between a time-varying and a non-linear system?

Why the fact that a system obeys superposition vastly eases the task of
analyzing its dynamic behavior?

------------

From memory, a learn system is not simply that of superposition. A linear 
function, of which a system is represented mathematically, must poses the 
linearity properties. That if addition, scalar multiplication, homogeneity, 
and one more that I can't recall(since I was forbidden to look at wiki 
you'll have to do it for me ;). Basically,

A linear system essentially has a well defined Fourier transform, causality, 
superposition, etc... But there are superposition itself does not make a 
linear system.

Essentially a if you add two linear systems together you get another linear 
system(that is the idea of superposition). But there are systems that when 
added together give linear systems and non-linear systems.

In any case LTI systems are the most simple types of systems and generally 
are equivalent to solving systems of linear equations(ultimately). Hence it 
is most natural that these would bet he first types of systems studied and 
the most machinery developed for them. Much of mathematics deals with these 
types of systems.

Most non-linear systems are intractable. Not only do they have no way to 
algebraically solve them their numerical solutions are unstable.

Obviously if you can break a system into components, study the individual 
components, and easily "reassemble" the system you have a drastically 
reduced the complexity. This is why "superposition" is important. Again, 
it's more important the concept of linearity and because linear systems 
"add".

Another way of looking at it is if one knows the response to the impulse 
function of a linear system one knows how the function will respond to any 
function since any function can be written as a convolution/integral/sum of 
the impulse functions. Since the system is linear, the operations commute:

let S(.) be the system and f(t) be the input
S(f(t)) = S(int(f(t)*dirac(t))) = int(f(t)*S(dirac(t)))

So if S(dirac(t)) is known or easily obtained(which it almost always is) 
then the "response"(or what we know as the transfer function is very easily 
obtained. For non-linear systems S(.) cannot commute and we can't make such 
implications.

But note there is a similar way to analyze non-linear systems but instead of 
using the dirac function we use white noise. In this case the decomposition 
is much more complex but supposedly it is possible(mathematically). There 
simply is not enough machinery/intelligence to make analyzing such systems 
productive.

One other point that makes non-linear systems so complex is that a very 
slight change in the structure can result in drastically different outcomes. 
This is not true for linear systems. Small perturbations result in small 
perturbations. Of course there are non-linear systems that are approximately 
linear or can be linearly approximated(for example, the simple pendulum).

Most electrical components are approximately linear and therefor when 
combining such components we get an approximately linear system. When a 
non-linear component is introduced we generally approximately it as a 
piecewise linear system(transistors "regions", etc...).

In some sense linearity is all that most people can comprehend.

On 8/08/2011 7:15 a.m., Tim wrote:
> Please answer without looking at Wikipedia or otherwise finding the
> answer.  Please answer, even if the answer is "no".  I'm trying to get a
> measure of the extent of a bit of knowledge, here:
>
> How many of you know, off the top of your head, that the defining
> characteristic of a linear system is superposition?
>
> That if a system obeys superposition it must be linear?
>
> The difference between a time-varying and a non-linear system?
>
> Why the fact that a system obeys superposition vastly eases the task of
> analyzing its dynamic behavior?
>
> Thanks.
>

for an input f1(t)=acos(w1t) then the corresponding output will be 
a'cos(w1t+phi1)

similarly for another input f2(t) =bcos(w2t) the output will be 
b'cos(w2t+phi2).

Then if the system is linear on applying both inputs f1(t)+f2(t) we get 
the overall output that we got for each individual input.

a'cos(w1t+phi1) +b'cos(w2t+phi2)

Time varying, the coefficients of the defining differential equation 
change with time.

if a system input f(t) has an output g(t) then it is time-invariant if

f(t-tau) is the input and g(t-tau) is the output.

On Sun, 07 Aug 2011 17:34:05 -0500, DonMack wrote:

> "Tim"  wrote in message
> news:NsednU-bJLX4eaPTnZ2dnUVZ_t2dnZ2d@web-ster.com...
> 
> Please answer without looking at Wikipedia or otherwise finding the
> answer.  Please answer, even if the answer is "no".  I'm trying to get a
> measure of the extent of a bit of knowledge, here:
> 
> How many of you know, off the top of your head, that the defining
> characteristic of a linear system is superposition?
> 
> That if a system obeys superposition it must be linear?
> 
> The difference between a time-varying and a non-linear system?
> 
> Why the fact that a system obeys superposition vastly eases the task of
> analyzing its dynamic behavior?
> 
> ------------
> 
> From memory, a learn system is not simply that of superposition. A
> linear function, of which a system is represented mathematically, must
> poses the linearity properties. That if addition, scalar multiplication,
> homogeneity, and one more that I can't recall(since I was forbidden to
> look at wiki you'll have to do it for me ;). Basically,
> 
> A linear system essentially has a well defined Fourier transform,
> causality, superposition, etc... But there are superposition itself does
> not make a linear system.
> 
> Essentially a if you add two linear systems together you get another
> linear system(that is the idea of superposition). But there are systems
> that when added together give linear systems and non-linear systems.
> 
> In any case LTI systems are the most simple types of systems and
> generally are equivalent to solving systems of linear
> equations(ultimately). Hence it is most natural that these would bet he
> first types of systems studied and the most machinery developed for
> them. Much of mathematics deals with these types of systems.
> 
> Most non-linear systems are intractable. Not only do they have no way to
> algebraically solve them their numerical solutions are unstable.
> 
> Obviously if you can break a system into components, study the
> individual components, and easily "reassemble" the system you have a
> drastically reduced the complexity. This is why "superposition" is
> important. Again, it's more important the concept of linearity and
> because linear systems "add".
> 
> Another way of looking at it is if one knows the response to the impulse
> function of a linear system one knows how the function will respond to
> any function since any function can be written as a
> convolution/integral/sum of the impulse functions. Since the system is
> linear, the operations commute:
> 
> let S(.) be the system and f(t) be the input S(f(t)) =
> S(int(f(t)*dirac(t))) = int(f(t)*S(dirac(t)))
> 
> So if S(dirac(t)) is known or easily obtained(which it almost always is)
> then the "response"(or what we know as the transfer function is very
> easily obtained. For non-linear systems S(.) cannot commute and we can't
> make such implications.
> 
> But note there is a similar way to analyze non-linear systems but
> instead of using the dirac function we use white noise. In this case the
> decomposition is much more complex but supposedly it is
> possible(mathematically). There simply is not enough
> machinery/intelligence to make analyzing such systems productive.
> 
> One other point that makes non-linear systems so complex is that a very
> slight change in the structure can result in drastically different
> outcomes. This is not true for linear systems. Small perturbations
> result in small perturbations. Of course there are non-linear systems
> that are approximately linear or can be linearly approximated(for
> example, the simple pendulum).
> 
> Most electrical components are approximately linear and therefor when
> combining such components we get an approximately linear system. When a
> non-linear component is introduced we generally approximately it as a
> piecewise linear system(transistors "regions", etc...).
> 
> In some sense linearity is all that most people can comprehend.

Thanks for the answer, Don.

Thanks also for pointing out that superposition does not always imply 
linearity -- IIRC, it does either for any 'reasonable' signal or any 
'reasonable' system, but there is some combination of oddball signals and/
or oddball systems that exemplifies the opposite.

It's one of those things where the engineers in the audience will snort 
and say "get real!" while the mathematicians will nod their heads sagely 
and say "why yes, that's a good point".

I need to track it down, now -- off to Wikipedia*!

(This all came about because a fellow that I know from aeromodeling 
invoked linearity and superposition on a control line stunt web forum.  
It makes perfect sense to him, because he's a systems engineer.  I, on 
the other hand, felt that it was a bit much to expect given that unless 
you design analog electronics or control loops you don't have to remember 
that rule to do your daily job).

* I didn't say _I_ couldn't consult Wikipedia!!

-- 
www.wescottdesign.com