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.

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Tim Wescott
Control system and signal processing consulting
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Tim
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... as if Wikipedia can be depended on to get the right answer...

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".

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.

FWIW

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r b-j                  rbj@audioimagination.com

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Reply to
robert bristow-johnson

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

Thanks, Rich

Reply to
Rich Grise

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.

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r b-j                  rbj@audioimagination.com

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Reply to
robert bristow-johnson

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

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Rich Webb     Norfolk, VA
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Rich Webb

Me, too ;-) ...Jim Thompson

[On the Road, in New York]
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| James E.Thompson, CTO                            |    mens     |
| Analog Innovations, Inc.                         |     et      |
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Reply to
Jim Thompson

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.

Reply to
DonMack

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.

Reply to
fitlike

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!!
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Reply to
Tim Wescott

...

Tim, didn't Dilip Sarwate set you straight? he emailed me right after you posted (and i replied to) this thread.

in 2009 Dilip pointed out simply that complex conjugation satisfies superposition, but not scaling (where i've been thinking that scaling comes from superposition). and i realized that the same also applies to the Re{} and Im{} operators.

for real signals and scalers, i think that superposition will lead you to scaling (so superposition is all that is needed), but i don't know exactly how to take it to the irrational scalers.

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r b-j                  rbj@audioimagination.com

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Reply to
robert bristow-johnson

Scalers:

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Scalars:

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Reply to
John S

it's good to know. sometimes it's dependence instead of dependance (or is it defendent vs. defendant?) phasers vs. phasors (one is star trek, the other is reactive circuits).

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r b-j                  rbj@audioimagination.com

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Reply to
robert bristow-johnson

I thought it was going to be this:

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There _are_ rational scalers: in some states when you do the computation for log scaling off of state lands you have to stick a '3' into the equation where one would normally stick pi*. So in those states, scaling _is_ rational!

  • This has to be part of the source of the "such-and-such a state legislates that pi = 3" hoopla. The truth is that the stuff is going out to bid, and everyone knows how its calculated, so you could stick _any_ number** in there and the prices would still be the same.
** I vote for 42, in memory of Douglas Adams.
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Reply to
Tim Wescott

I'll bet those phasers have reactive circuits in them, though.

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Reply to
Tim Wescott

This is because "linearity" is not well-defined. Speaking as a mathematician, there is "real-linearity" as in: superposition under addition of terms and multiplication with real scalars, and "complex-linearity" which also implies linearity under multiplication of complex scalars. Of course complex conjugation is real-linear, but complex-antilinear (this is the point!), and hence, all linear combinations of conjugation and non-conjugation, as Re() and Im() are hence also real-linear, but not complex-linear.

You need continuity of the operation in question. Without that, of course, not. With that, it is trivial: Every real number can be sufficiently closely approximated by a rational number, and by continuity, then you have linearity in the limit as well.

Greetings, Thomas

Reply to
Thomas Richter

You have to be a bit careful about your universe of basis functions, though--the usual definition requires scale invariance as well as superposition--you have to be able to multiply your signal by any real number and have the response come out multiplied by the same factor.

You also need time-invariance if you want the usual linear systems theory to hold.

And there are generalizations, e.g. an analogue multiplier, which is a bilinear system.

Cheers

Phil Hobbs

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Dr Philip C D Hobbs
Principal Consultant
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Phil Hobbs

Yep, I know that. Engineers think sometimes 'linear' just means it relates to the 'linear IC' databook, though. Alas, I got my first degree in mathematics... communication with engineers is sometimes difficult.

Yep. And the natural question is, 'what are the eigenvectors?'.

I'm unclear on this (I think of time-varying as having a connection with boundary conditions, like Matthieu functions - is that what is intended?). The other connection is with parametric time-variation, i.e. the linear elements aren't constants. I'd characterize a warmup transient in a light bulb as parametric time-variation, but wouldn't call that a time-varying system. The lighbulb warmup amounts to nonlinear behavior (so you can detect RF with a lightbulb).

Oh, sure. The use of Green's functions is a complete answer to superposition situations... but sometimes that doesn't qualify as 'vastly eases' IMHO.

Reply to
whit3rd

so Tim, can you take a clear position on this? must superposition imply linearity?

i've brought this up a couple of times and haven't seen you take a stand on it.

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r b-j                  rbj@audioimagination.com

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Reply to
robert bristow-johnson

That's because I was madly Wiki-ing.

As near as I can tell, it's got to be superposition _and_ scaling (with complex numbers) as you said. I haven't sat down and actually done the math, although the fact that the complex conjugate operation obeys superposition certainly points to a problem in my original assertion.

I was led astray because in real analysis you can start with superposition and deduce scaling -- apparently that's not the case with complex analysis.

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Reply to
Tim Wescott

hey, 4 days ago, i was at the very same place and Dilip emailed me the obvious counter example. i thought the timing was uncanny (8/7/11 3:33 PM EDT) and thought for sure he was also in conversation with you about it. later found that this assumption (by me) was also wrong.

our posts literally just "crossed in the mail".

we're on the same page. in the real and continuous world, superposition is equivalent to linearity.

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r b-j                  rbj@audioimagination.com

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Reply to
robert bristow-johnson

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