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I have been struggling with an ostensibly simple problem, that is how to apply perturbation analysis principles on a system of linear differential equations with linear perturbation of the following form:

$$ \dot{x}=\frac{1}{\epsilon} a_0 x + a_1 z + bu\\ \dot{z} = c_0 x + c_1 z $$

where $x,z,u\in\Re$ and $0<\epsilon<<1$. How can I decompose my system and write the solution $x(t),z(t)$ as an asymptotic expansion in $\epsilon$ ?

Update 1: Let us consider the case where $t=\mathcal{O}(\epsilon)$. Then, put $\tau=\frac{t}{\epsilon}=\mathcal{O}(1)$ and the above system becomes:

$$ \frac{d x(\tau)}{d\tau}= a_0 x(\tau) + \epsilon a_1 z(\tau) + \epsilon bu\\ \frac{d z(\tau)}{d\tau} = \epsilon c_0 x + \epsilon c_1 z $$

and let us now set $x(\tau)=x(\frac{t}{\epsilon})=x_0(\tau)+\epsilon x_1(\tau)+\mathcal{O}(\epsilon^2)$ and similarly $z(\tau)=z_0(\tau)+\epsilon z_1(\tau)+\mathcal{O}(\epsilon^2)$. Then we have the following inner system:

$$ \frac{d x_0}{d\tau}=a_0 x_0(\tau)\\ \frac{d x_1}{d\tau}=a_0 x_1(\tau) + a_1 z_0(\tau) + bu(\tau)\\ \frac{d z_0}{d\tau}=0\\ \frac{d z_1}{d\tau}=c_0 x_0(\tau) + c_1 z_0(\tau) $$

and then we may apply matching to both the outer (see @Jon's answer below) and the inner solution.

Question : Can it be considered expedient to consider an asymptotic expansion of the input variable $u$ like $u(t)=\frac{1}{\epsilon}\sum_{i\geq 0}\epsilon^i u_i(t)$?

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1 Answer 1

I would do the following. Just put $\lambda=\frac{1}{\epsilon}$ and now $\lambda\gg 1$. Now you will get

$$ \lambda\dot{x}= a_0 x + \lambda a_1 z + \lambda bu\\ \dot{z} = c_0 x + c_1 z. $$

Then put



that gives the first few equations for the perturbation series

$$ \dot{x_0}= a_1 z_0 + bu\\ \dot{z}_0 = c_0 x_0 + c_1 z_0. $$

$$ \dot{x_1}= a_0 x_0 + a_1 z_1\\ \dot{z}_1 = c_0 x_1 + c_1 z_1. $$

and so on.

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Thanks for the answer, Jon. What about terms like $\lambda x_{-1}(t)$? Or maybe determining outer solution (for $t=\mathcal{O}(1)$), inner solutions (for $t=\mathcal{O}(\epsilon)$) and then matching these two? How do these apply to control systems where you have the input $u$? Should one consider an expansion in $u$ as well? – Pantelis Sopasakis May 7 '12 at 11:57
@PantelisSopasakis: There is no term $x_{-1}$. Just start your expansion as shown. Yes, it would be interesting to match $\epsilon$ and $\frac{1}{\epsilon}$ expansions as you should be able to recover a complete analytical solution. I assumed that $u$ is some kind of forcing term having no dynamics. – Jon May 7 '12 at 13:40
I found a wonderful text by P.Kokotovic in Lecture Notes in Control and Information Sciences, "Singular Perturbations and Asymptotic Analysis in Control Systems," Springer-Verlag Berlin 1987. You don't need terms like $x_{-1}$ if your fast subsystem is as. stable. Then, multiple time scales ($t=\mathcal{O}(1)$ and $t=\mathcal{O}(\epsilon)$) are necessary for describing your system's overall dynamics. If $u$ follows some dynamics, then you formulate the whole thing as a simple perturbed system and you solve it. What I still don't get is why $u(t)=\mathcal{O}(\epsilon)$ is a valid assumption? – Pantelis Sopasakis May 7 '12 at 14:00
You can find the book here if you have access. – Pantelis Sopasakis May 7 '12 at 14:01
@PantelisSopasakis: The assumption $u(t)=O(\epsilon)$ just holds if the external forcing is a small perturbation to your system. But this is just arbitrary and you can do as you like. On the other hand, you can add some dynamics to it and this should be in some way connected to the other variables of the system otherwise it remains an arbitrary forcing. – Jon May 8 '12 at 9:51

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