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$ \newcommand{\e}{\boldsymbol \eta} \newcommand{\h}{\boldsymbol h} \newcommand{\T}{T^{\mathsf{ref}}} \newcommand{\g}{\boldsymbol g} \newcommand{\e}{\boldsymbol \eta} \newcommand{\dt}{\partial_{t}} \newcommand{\d}{\delta} \newcommand{\t}{(t)} $ I am trying to understand the following article: Feedback characteristics of nonlinear dynamical systems by A. Lahellec, S. Hallegatte, J.-Y. Grandpeix, P. Dumas and S. Blanco.

In the beginning, it was quite easy to follow:

Consider a state-space model driven by some external forcing $\h\t$:
$$ \partial_t\e(t)= \g\big(\e(t),\h(t)\big), \quad \e\left(t=0\right)=\e_0\label{1}\tag{1} $$ where $\e$ is the state space vector. Once a reference trajectory $\T$ is followed, perturbations to $\eqref{1}$ are solutions to the tangent linear system:
$$ \partial_t\delta\e(t)=A\left(t\right)\delta\e(t)+B(t)\delta\h(t), \quad t\in\left[0,T\right]\label{2}\tag{2} $$ In $\eqref{2}$ $\,\delta\e$ is the deviation of the perturbed system from $\T$ caused by perturbation $\delta\h(t)$ of the reference forcing function. $A(t)=\partial_x\,\g(t) $ and $B(t)=\partial_h\,\g(t)$ are the Jacobian matrices.

To create a feedback loop in the system:

A scalar variable $\varphi$ is added to the system, with $\varphi = f(\eta) $ i.e. a function of the state variable.
The primitive model $\eqref{1}$ is modified to be sensitive to $\varphi$ in such a way that original and modified models are mathematically equivalent:
$$ \partial_t\e(t)=\g'\big(\e(t),\varphi(t),\h(t)\big) =\g\big(\e(t),\h(t)\big) $$ A scalar perturbation $u(t)$ is applied to $\varphi$ only: $\varphi (t) = f(\eta) + u(t)$. The system will respond to that change within the feedback loop: perturbation $u$ on $\varphi$ $\rightarrow$ selected mechanisms $\rightarrow$ rest of the system $\rightarrow$ response $\delta\varphi $.

The opening of the loop (figure on the left) means preventing the system from responding on $\varphi$.

(From here on i don't understand:)

Following the preceding procedure, the original tangent system $\eqref{2}$ is modified into the following two equations:
$$ \begin{cases} \dt\d\e(t) = A^\flat(t)\delta\e(t)+\left\lvert b(t)\right\rangle\d\varphi(t) \\ \d\varphi(t)=\left\langle c(t)\right\rvert\delta\e(t)+u(t) \end{cases} \label{3}\tag{3} $$ The column and row matrices $\,\left\lvert b(t)\right\rangle = \partial_{\varphi}\g'\,$ and $\, \left\langle c(t)\right\rvert = \partial_\eta f\,$ formalize the feedback loop. For $\eqref{3}$ to be equivalent to $\eqref{2}$ the equality $ \,A(t)=A^\flat(t)+\left\lvert b\right\rangle\left\langle c\right\rvert (t) \, $ must hold.

System $\eqref{3}$ takes the form of $\eqref{2}$ when $\delta\varphi$ has been eliminated.


My questions are:

  1. How is $\eqref{2}$ modified into $\eqref{3}$ ? I am not familiar with these notations and i don't get why the term $\,A^\flat\,$ appears and $\,B(t)\, $ disappears. O_O
  2. Consequently, explain why $(4)$ is the condition for $\eqref{2}$ and $\eqref{3}$ to be equivalent.
  3. It seems like $\eqref{3}$ is the equations for closed loop feedback. To open the loop we set $\,\left\langle c \right\rvert\,$ to $0$ and disallow the system perturbation to influence $\varphi$. Is that correct ?

[1] A. Lahellec, S. Hallegatte, J.-Y. Grandpeix, P. Dumas, and S. Blanco, 2008, Feedback characteristics of nonlinear dynamics systems, Europhysics Letters 81 (2008), 60001, doi:10.1209/0295-5075/81/60001

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There seems to be a picture missing. –  Jonas Meyer Mar 19 at 20:59
    
Geralt: Imageshack pics apparently expire. Kindly relocate the image (if at all possible) and reload it (to SE's imgur account as it's done nowadays), or otherwise fix the post. ALL: I just removed the link to the pic as Wayback machine didn't seem to have that particular pic. –  Jyrki Lahtonen Aug 20 at 7:50

2 Answers 2

up vote 0 down vote accepted
  1. It was just the chain rule with confusing notation:
    $d\mathbf{g'} = \frac{\partial \mathbf{g'}}{\partial\mathbf{x}}d\mathbf{x} + \frac{\partial \mathbf{g'}}{\partial\varphi}d\varphi$
    where $\mathbf{x}$ and $\varphi $ are function of t. $\mathbf{g'}(\mathbf{x},\varphi) = \partial_t\mathbf{x} $ and $d\mathbf{g'} = \partial_t\delta \mathbf{x} $. Assume no change in external forcing $\delta\mathbf{h}(\mathbf{t})=0 $
    $ \frac{\partial \mathbf{g'}}{\partial\mathbf{x}} = A^b $ which is just a weird way to denote the Jacobian matrix $\partial_\mathbf{x}\mathbf{g'} $
    $ \frac{\partial \mathbf{g'}}{\partial\varphi} = \left | b \right \rangle $ and $\frac{\partial f}{\partial\mathbf{x}} = \left\langle c \right | $

  2. Assume that the system is autonomous (no external forcing ) it follows immediately from 1.

  3. Yes.
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Thank you for your interest in this article. Let me mention a supplementary document that you may have missed: here

It shows how the linear feedback approach can give insights on even chaotic systems.

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thank you! Chaos was my last semester's subject :D –  Geralt of Rivia Jun 12 '12 at 3:49

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