Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$ \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

share|improve this question

closed as unclear what you're asking by Normal Human, graydad, Batominovski, Clayton, Michael Galuza Aug 18 at 4:19

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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.
share|improve this answer

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.

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.