Why is this differential equation non-linear?

[PainterGuy]

Hi,

Could you please have a look on the attachment?

Question 1:
Why is this differential equation non-linear? Is it $u=\overset{\cdot }{m}$ which makes it non-linear?

I think one can consider $x_{3}$ , k, and g to be constants. If it is really $u=\overset{\cdot }{m}$ which makes it non-linear then I don't think it's possible to make it linear. Could you please correct me?

Question 2:
What is there in "These expressions" which shows that the system is linear? Is it "0 x u"?

 

[Mayhem]

Because it contains a non-linear term regardless of how you write it.

 

[PainterGuy]

Because it contains a non-linear term regardless of how you write it.

Thank you!

Is that non-linear term $u=\overset{\cdot }{m}$?

What is there in that equation system which shows that the system is not linear?

Thanks a lot for your time and help!

 

[Mayhem]

Thank you!

Is that non-linear term $u=\overset{\cdot }{m}$?

What is there in that equation system which shows that the system is not linear?

Thanks a lot for your time and help!

Since $u = \dot{m}$ and $x_3 = m$ it means that $\frac{k}{x_3}u = \frac{k}{m}\dot{m}$. Usually, products of the same function (like $y^n$, where $n$ is some number) and products of functions/dervative of functions (like $y'y''$ or $yy'''$ and similar) is a quick way to qualitatively check if a differential equation is non-linear. I might be wrong though. My only real training in differential equations consist of a mandatory UG course on analysis.

 

[Office_Shredder]

In the matrix equation you have for $x_1,x_2,x_3$, the equation would be linear if the matrix was full of constants. But it has an $x_3$ term in it.

 

[DaveE]

You said you thought that x₃ = m could be considered a constant.
Then why would they include x₃ as a state variable, and why would they discuss it's derivative?
"The thrust generated is assumed to be proportional to dm/dt" If m was constant that would be zero.

Also, if a system is non-linear, you can't just make it linear. That's what non-linear means. It is common practice to approximate non-linear systems with linear models though. Is that what you're asking about?

 

[PainterGuy]

You said you thought that x₃ = m could be considered a constant.
Then why would they include x₃ as a state variable, and why would they discuss it's derivative?
"The thrust generated is assumed to be proportional to dm/dt" If m was constant that would be zero.

Thank you!

I don't remember much of the control theory but let's try it. I can still figure out the basics.

IMHO, I don't think that the problem is worded properly.

A rocket is propelled forward by a thrust force equal in magnitude, but opposite in direction, to the time-rate of momentum change of the exhaust gas accelerated from the combustion chamber through the rocket engine nozzle. This is the exhaust velocity with respect to the rocket, times the time-rate at which the mass is expelled, or in mathematical terms: ${T}={v}\frac{dm}{dt}$

where T is the thrust generated (force), [itex]\frac{dm}{dt}[\itex] is the rate of change of mass with respect to time (mass flow rate of exhaust), and v is the velocity of the exhaust gases measured relative to the rocket.

Source: https://en.wikipedia.org/wiki/Thrust#Examples

I don't think $\overset{\cdot }{m}$ and $m$ are the same things as implied in the question statement; "m" is mass and m-dot is the amount or mass of exhaust gases released per unit time. Please let me know if I've it right. I understand, as the fuel is burned, the rocket is supposed to 'lose' overall mass.

Also, if a system is non-linear, you can't just make it linear. That's what non-linear means. It is common practice to approximate non-linear systems with linear models though. Is that what you're asking about?

Yes, that's what I'm asking about. I was more interested in the differential equation but looking at the differential equation from practical example would be more productive and helpful. So, how do we proceed? Thanks.

 

[DaveE]

I don't think that the problem is worded properly.

Most HW problems are a bit confusing. I think it takes a lot of knowledge and effort to write these well. Unfortunately, many profs don't put in that effort.

I don't think m⋅ and m are the same things as implied in the question statement; "m" is mass and m-dot is the amount or mass of exhaust gases released per unit time.

Yes, they aren't the same. Like distance and speed, for example. But no, I don't think they said they were the same. Maybe you should reread that sentence. It has the form "x" is distance and dx/dt is the amount of distance covered per unit time. Personally, I don't think those are confusing or equivalent.

Yes, that's what I'm asking about. I was more interested in the differential equation but looking at the differential equation from practical example would be more productive and helpful. So, how do we proceed?

That state space equation is a differential equation, or at least a set of equations. You could multiply it all out and reduce the number of equations with substitutions. However, everyone that does this stuff thinks it's easier in the matrix form, a series of first order equations as opposed to one higher order DE.

This subject is much too complicated to cover in a forum like this. Non-linear dynamics is usually a graduate level subject. The problem is that there are lots of types of non-linear systems. You can linearize them with a bunch of first derivatives, or perturbation theory, but then the question immediately arises, "is your model any good?" You may find that you can't get one good model that works over the range of interest.

I did study this stuff decades ago, but don't remember enough to teach it. You can look on YouTube for some Cornell lectures from Steven Strogatz, which are excellent, but they aren't a cookbook approach. It looks like there are a bunch of useful links from a google search for "linearization in modern control theory". Also Wikipedia has some good stuff.

 

[mpresic3]

I remember this problem given in a textbook, but I cannot remember the textbook. Where was this problem taken from, please.

 

[PainterGuy]

I remember this problem given in a textbook, but I cannot remember the textbook. Where was this problem taken from, please.

I was helping someone with the differential equation part. After your post I asked about the book. It's linear system theory by chen.

 

[mpresic3]

Thanks

 

[PainterGuy]

Yes, they aren't the same. Like distance and speed, for example. But no, I don't think they said they were the same. Maybe you should reread that sentence. It has the form "x" is distance and dx/dt is the amount of distance covered per unit time. Personally, I don't think those are confusing or equivalent.

They said, "The thrust generated is assumed to be proportional to $\overset{\cdot }{m}$, where m is the mass of module."

To me, the statement is saying that that thrust is proportional to the derivative of mass 'm' where 'm' is the mass of module. But, IMHO, the derivative is being taken of mass of gases being emitted at any given time and not of the total mass of module. You see my confusion.

On the other hand, the gas tank is part the module so as the mass of gas tank decreases so does the total mass of module. Perhaps, they are looking at it, in ideal terms, where mass of module is considered separate from the mass of gas tank, and as the gas tank loses mass the module mass still remains constant.

I'd really appreciate if you can comment on it. Thanks in advance!

 

[DaveE]

the derivative is being taken of mass of gases being emitted at any given time and not of the total mass of module.

I'm not sure I understand what you meant, but this sounds a bit like a second derivative.

the gas tank is part the module so as the mass of gas tank decreases so does the total mass of module. Perhaps, they are looking at it, in ideal terms, where mass of module is considered separate from the mass of gas tank, and as the gas tank loses mass the module mass still remains constant.

Suppose you rocket has a mass described as m ≡ M_rocket + m_fuel ≡ M_r + m_f. In this model M_rocket is a constant value, and m_fuel is being burned. Then I think you can see that their model (the rate that fuel is being used is proportional to thrust) makes sense. So Thrust ∝ dm_f/dt.

So now look at dm/dt = d(M_r + m_f)/dt = dM_r/dt + dm_f/dt = dm_f/dt, because the derivative of a constant is zero. Thrust isn't about how much mass there is, it's about how much it is changing.