Example of a block diagram

To clarify the steps, a block diagram is constructed for the system (7.1), i.e.

$\displaystyle m\ddot{x}(t)+b\dot{x}(t)+kx(t)=u(t),$

  1. The highest state derivatives is $ \ddot{x}$ .
  2. Integrating this state once, we obtain $ \dot{x}$ . Integration of $ \dot{x}$ yields $ x$ . This is shown in the figure below. In this figure, $ \ddot{x}$ is represented by ddx and $ \dot{x}$ by dx.
    \includegraphics[scale=.8]{ch7_intx}
  3. Substitution of $ m=1$ , $ b=2$ and $ k=3$ and rewriting (7.1), yields an expression for $ \ddot{x}$ :

    $\displaystyle \ddot{x}=u-2\dot{x}-3x.$ (7.3)

    To transform (7.3) into a block diagram, we use two summation blocks:
    \includegraphics[scale=.8]{ch7_ddx}
    This block is coupled to the blocks obtained in step 2:
    \includegraphics[scale=.8]{ch7_xline}
  4. With the use of gain blocks, we can compute $ -2x$ and $ -3x$ , such that a full block diagram is obtained:
    \includegraphics[scale=.8]{ch7_block}
    Note, that we consider $ x$ to be an output and $ u$ to be an input.


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Esteur 2010-03-22