Follow along with the video below to see how to install our site as a web app on your home screen.
Note: This feature may not be available in some browsers.
When you do a Taylor expansion in more than one dimension, you need the cross terms too. That's just how the theory works out.
MrAl said:Yes it's cool i guess, but it takes a lot longer to post that's what i dont like about it.
For Q2, there can be cases where there are no differential equations. This is true of linear systems also. However, things are more interesting when there are differential equations and for nonlinear state-space systems there are of course differential equations, but the nonlinear state space form generally has first order differential equations of the form below, where n is the number of states, m is the number of outputs and k is the number of inputs.
State Equations
dx1/dt=f1(x1,x2 ..., u1, u2 ... uk, t)
dx2/dt=f2(x1,x2 ..., u1, u2 ... uk, t)
dx3/dt=f3(x1,x2 ..., u1, u2 ... uk, t)
.
.
.
dxn/dt=fn(x1,x2 ..., u1, u2 ... uk, t)
Output Equations
y1=g1(x1,x2 ..., u1, u2 ... uk, t)
y2=g2(x1,x2 ..., u1, u2 ... uk, t)
y3=g3(x1,x2 ..., u1, u2 ... uk, t)
.
.
.
ym=gm(x1,x2 ..., u1, u2 ... uk, t)
You see here that f1,f2 ... fn and g1, g2 .... gn are nonlinear functions that must be linearized to allow a matrix formulation in linear form. Those matrices then become Jacobian matrices, since they are made of partial derivatives.
All the state equations are of first order. The derivative terms such as as x1(dot), x2(dot), etc. appear on the left and on the right side of equality sign we have an expression which is completely algebraic and involve no derivative terms. In case I'm utterly wrong then please correct me.
You said, "You see here that f1,f2 ... fn and g1, g2 .... gn are nonlinear functions that must be linearized to allow a matrix formulation in linear form". Sorry but I can't get how they are non-linear. For example, f1 is going to be non-linear if it's of the form, say, f1=(x1)(x2)+x3 (type #1). But if it's of the form f1=x1+x2+x3 (type #2) then I don't think it's non- linear because degree of each individual variable is one. As you are saying that f1 is non-linear then what makes you say that f1 is of form type #1 and not of type #2? Perhaps, a non-linear system always give rise to f1, f2, etc. of the form type #1. For instance, specific functions using eqs. 3-17 and 3-18, f1=x2, f2=(-k/m)x1 - (b/m)x2 + (1/m)u, are linear because they represent a system which is linear. Please guide me. Thank you.
OK, this is just a wording issue. Of course you can have special cases where some of the functions f1, f2... or g1, g2 ... etc. are linear. In such cases, you don't need to linearize the equations because they are already linear. However, the process of linearization does not care if they are linear or nonlinear. It works in both cases. It's just that if you linearize something that is linear, you get the same thing back again, which then allows you to prove the equation is linear.
In the special case where all functions are linear, then you already have a linear state space system and there is no need to linearize the equations. However, your original question was specifically about nonlinear systems, so I didn't think to clarify this point.
Thank you.
So, I gather that for a non-linear system the functions f1, f2... or g1, g2 ... etc. could be of both types, i.e. f=(x1)(x2)+x3 (type #1) and f=x1+x2+x3 (type #2). But still linearization could be performed on all f's and g's because only those f's and g's will be affected which are not linear.
Regards
PG
Yes, that's correct. Type #1 would be any linear formula and Type #2 would be any "linearizable" nonlinear function.
I can't stress enough that this subject/material will not be fully understood and will be quickly forgotten if you don't work out practical examples.
I thought I just confirm it. As I wrote it in my post Type #1 was f=(x1)(x2)+x3, and Type #2 was f=x1+x2+x3. So, I will say Type #2 is any linear formula and so on. So, don't you have them in reverse order? Please let me know.
That's one of the benefits of working on real problems. Undergraduate study gives the tools, but real work (grad work can be real work too, contrary to popular opinion) is the real venue for learning.Your point is well taken. Actually I was thinking this too that I'm getting into too much theory without having any concrete example
I believe this way, for you or anyone else, it's really hard to guide especially when the person seeking help and guidance is not physically present in front of you.
However, this assumes the signals and the OPAMPs are referenced to ground. But, if the OPAM is being referenced to a voltage reference, then the circuit is not linear any more.