# What dynamic system could these equations represent?

I have equations of a dynamic system. I need to figure out what this physical system is.

The equations are:

\begin{align} \dot{x}_1&=bx_1+kx_2+x_3\\ \dot{x}_2&=x_1\\ \dot{x}_3&=\alpha (u-x_2)-\beta x_3 \end{align}

All I can figure out is that it is maybe a mass-spring-damper system, plus a feedback control, but I am not quite sure about the terms $x_3$ and $\dot{x}_3$. What do these two terms mean?

You're right, it's a second order plant representing the mass-spring-damper system $P\left(s\right)=1/\left(s^2-bs -k\right)$ (stiffness $k$ and damping $b$ should be thus negative) under the closed-loop control action of the compensator $C\left(s\right)$, whose transfer function is:

$C\left(s\right)=\frac{\alpha}{s+\beta}.$

The negative feedback is closed over the position of the mass, while the set-point is given by $u$.

We can easily derive these results by applying the Laplace transform to the system of equations and then substituting $x_1$:

\begin{align} s^2x_2&=bsx_2+kx_2+x_3\\ sx_3&=\alpha (u-x_2)-\beta x_3 \end{align},

from which we obtain the following transfer functions:

\begin{align} P\left(s\right)&=\frac{x_2\left(s\right)}{x_3\left(s\right)}=\frac{1}{s^2-bs-k}\\ C\left(s\right)&=\frac{x_3\left(s\right)}{u\left(s\right)-x_2\left(s\right)}=\frac{\alpha}{s+\beta} \end{align},

corresponding to the closed-loop system here below:

• Hi, @Ugo thanks for the reply! It makes things clear. But I am not quite sure when doing the Laplace Transform of the equations, how do you deal with the x3 in the first equation and x2 in third equation. Thanks! Commented Feb 24, 2015 at 1:32
• I've extended the answer. Commented Feb 24, 2015 at 9:30