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Second-order system as first-order system

If we have a second-order system like this:

\begin{aligned} \ddot{q} = f(q, \dot{q}, u) \qquad \end{aligned}

we can always think of it as first-order system with twice as many variables. Let

$x_1 = q$

$x_2 = \dot{q}$

Then

\begin{aligned} \dot{x_1} = x_2\qquad \end{aligned}

\begin{aligned} \dot{x_2} = f(x_1, x_2, u)\qquad \end{aligned}

If we take vector $[\dot{x_1}, \dot{x_2}]$ and plot it over the domain $(x_1, x_2)$ (so we make vector field), it is called phase portrait of the system.

Note: $x_1 = q =$ x-axis, $x_2 = \dot{q}$ = y-axis.

 

To visualize a vector field like this:

$x_1 = q$

$x_2 = \dot{q} = - sin (x_1)$

you can use Octave:

octave:41> x=-5:0.5:5;
octave:42> [q,q_dot]=meshgrid(x,x);
octave:43> quiver(q,q_dot,q_dot,-sin(q));

 

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