Doubts about the definition of a linear system

[zoltrix]

hello

I have doubts about linear system
suppose that x is the input and y the output
if x1 then y1
if x2 then y2
the system is linear if
if (x1+x2) then (y1+y2)
now, take a mass m
you can write
f = m*a
however suppose that m changes with position
m = m(x)
above equations should be still valid
if(f1+f2) then (a1+a2)
is it a linear system ?

 

[sophiecentaur]

is it a linear system ?

I'm tempted to say "who cares what you call it?" because the results are the results, however you classify them.

Your x1, x2 etc, definition works when it describes a straight line but the SUVAT problems that you will have done(?) don't show a straight line relationship between x and t . In a 'linear' spring, the stored energy is not a linear function of extension, although the force is linear with extension.

In a system with varying mass (typically a rocket), the changes in energy are definitely not linear with time so you could say it's not a linear system. In the case of a rocket, m could well be a linear function of time (not straightforward position).

So where does this take you? Probably best not to worry with what to call things.

 

[zoltrix]

where does this take me to ?

in many accademic books you can find the above definition of linear system but it is a partial and consequently deceiving definition
if you stick to it, a system with a variable mass or variable moment of inertia, quite common in automation, should be a linear system but it is not
you must also add :

the transfer function must not depend on the state of the system

in a linear spring the potential energy is not a linear function of the extension
but take the k factor :
dF = k * dx
with dx = x-x0
k must be a constant ,it must not depend on x
this is a key difference between a linear and non linear spring which should be emphasized

I dont think it is a trivial stuff

 

[sophiecentaur]

I dont think it is a trivial stuff

Not trivial but it can be sidestepped by avoiding using the classification of a system as linear. I have never seen the term being used as an essential part of a calculation (afair). If a transfer function shows a linear relationship over a range of input variables then that's perhaps nice to know. There are many perfectly behaved systems within limits but they can turn chaotic outside those limits.

Do you feel compelled to use the term in something you are planning to publish? Perhaps you could find some weasel words (caveats) to ease your conscience.

 

[zoltrix]

an exhaustive definition of linear system is important , for example, in automation and control
if you try to control a non linear system using a theory which is valid only for a linear one then you get wrong results

 

[The Fez]

F=ma is not a valid equation if m isn't constant. 2nd law of motion states "The rate of change of momentum of a body is directly proportional to the applied force..." In other words, F=d(mv)/dt. only if m is constant can you pull it out of the derivative, so F=d(mv)/dt = m(dv/dt) = ma.

 

[DaveE]

The definition is clear. Your model either is or isn't linear. How that relates to your real world problem may be a different story.

https://en.wikipedia.org/wiki/Linear_system

 

[zoltrix]

"the superposition principle, or equivalently both the additivity and homogeneity properties"
are not enough to define a linear system
you must also add :
" without restrictions (that is, for all inputs, all scaling constants and all time.)"
the latter sentence is often omitted or given for granted

 

[sophiecentaur]

you must also add :
" without restrictions (that is, for all inputs, all scaling constants and all time.)"

This is the danger of the tail wagging the dog? If you don't specify an operational range then could any practical system be treated as linear?
This is surely just a philosophical worry. Nothing wrong with that but in practice, the tight requirement would exclude every real world system from being described as linear. Science has more sense than that because limits would (should) always be applied to the applicability of any theory.

At the expense of using another idiom we should not throw out the baby with the bathwater.

 

[zoltrix]

no real system is linear but this has nothing to do with the definition of linear system
in science you describe a model taking into account only some parameters
the math model might be linear
if so , the definition of linearity can and must be exhaustive

 

[sophiecentaur]

the definition of linearity can and must be exhaustive

as long as the range of interest is clearly stated. Let's face it, we derive the differential calculus by assuming linearity for small variations. Then, to make things pukkah, we include the requirement for functions to be continuous and differentiable - within a given range.