Creating a block diagram

Most dynamical systems can be represented in a block diagram. Thereto, a differential equation is splitted in simple blocks. Arrows connecting blocks are signals. The most used blocks are given in the following table:
Integrator


\includegraphics[scale=0.5]{ch7_integrator}

$ y(t)=\int_0^t x(s) ds$
Sum


\includegraphics[scale=0.5]{ch7_sum}

$ z=x+y$
Gain


\includegraphics[scale=0.5]{ch7_gain1}
or
\includegraphics[scale=0.5]{ch7_gain2}
$ y=2 x$

With these blocks, most differential equations can be described. This is done in four steps:
  1. The highest state derivatives are identified;
  2. These states are integrated once or more, to obtain all states;
  3. The highest states are computed with a summation;
  4. The signals entering the summation are computed by means of gains, multiplying known signals or are inputs. In block diagrams, inputs are represented by arrows pointing towards the blocks and outputs as arrows pointing out of the blocks.

In modeling differential equations, one should prefer the use of integrator blocks instead of differentiation blocks.

Remark: This constructive approach can also be used for more complex systems, for example systems in the form:

$\displaystyle \begin{pmatrix}\dot{x}\\ \dot{y}\end{pmatrix}=\begin{pmatrix}f(x,y,t)\\ g(x,y,t)\end{pmatrix}$ (7.2)



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Esteur 2010-03-22