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I'm given the following and I 'm asked to linearize around $x_1=0,x_2=0$$$x_1=x_2^2 \\ x_2=e^{x_1}+u$$ Only thing I know how to do is find the value for $u$ which is $-1$. The only problems I've faced are similar to this one I can't understand linearization in which no one has answered. But still, in this one I have two equations and I don't know how I should go about this.

The solution manual gives($\dot x_{1o}=\dot x_{2o}=0):$ $$x_{2o}=0\\u_o=-e^{x_{1o}}\\\dot x_1=x_2^2=>\dot x_{1o}+Δ\dot x_1=(x_{2o}+Δx_2)^2\\\dot x_2=e^{x_1}+u=>\dot x_{2o}+Δ\dot x_2=e^{x_{1o}+Δx_1}+u_o+Δu_1$$

I can understand how he produced the two last equations but why the dots on $x_1$ and $x_2$?

John Katsantas
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    $x_2^2 = 2x_2 +o(x_2) \text{ and } e^{x_1} = 1+x_1+o(x_1)$ using Taylor expansion around 0. – Wyllich Aug 31 '17 at 13:22
  • So I just replace the nonlinear terms? The solution of the manual seems a bit more complicated. I'll upload – John Katsantas Aug 31 '17 at 13:34
  • @Wyllich I added some things. Why is he changing the equations(adding derivatives to the left part)? And the the one with $x_2^2$, shouldn't the right side be zero? – John Katsantas Aug 31 '17 at 13:48
  • Well, the author took the liberty to stop the Taylor expansion of $x_2^2$ at its first (thus constant) term $0$ (as you are working around $x_2=0$. What I wrote was to the 2nd term.

    Concerning the last 2 lines you wrote, the author sets $x \underset{x\to x_0}= x_0 + \Delta x$

     – Wyllich Aug 31 '17 at 13:50
  • I didn't explain correctly. Why don't we do: $\dot x_1+ Δx_1=x_{2o}^2+2x_{2o}Δx_2=0$ I find the right side different from his. There is clearly something I've misunderstood. – John Katsantas Aug 31 '17 at 13:58
  • Your start with 1 linearisation and study the implications. It is less flexible to work with 2 linearisations. – Wyllich Aug 31 '17 at 14:01
  • I'm having a control systems course and this is usually the first question (giving 0,5 points out of 10)of a big problem . It's either this case or the one in my other question so I just want to know how to face these two cases without going in too much detail about linearization. I'm also having trouble finding examples online. What is wrong in the calculation I did in the comment? Also, since we expand around $x_{2o}=0$ why do we have a $Δx_2$ – John Katsantas Aug 31 '17 at 14:06
  • Alright. I'll try to give some physical interpretation. Perhaps the first equation $ x_1 = x_2^2$ translates how an internal parameter $x_1$ responds to an external stimulus $x_2$ which will eventually have influence over $x_1$. If the question is to study the system, as $x_2$ induces the behaviour, it makes sense to study the sequential influence of $x_2$ over $x_1$. – Wyllich Aug 31 '17 at 14:12
  • Let us continue this discussion in chat. – Wyllich Aug 31 '17 at 14:16

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