We consider integral control of a mass-spring-damper system, that is a coupled
system
$$\ddot x(t) + 5\dot x(t) + 4x(t) = u(t),$$
$$\dot u(t) = k(r – x(t))$$
where k is a positive parameter and r is a desired set
point.
Verify that if the initial conditions are zero $($i.e. $x(0) = 0$, $\dot x(0) = 0 $ and
$u(0) = 0$$)$, then,
$$X(s) = \frac{k}{s(s^2+5s+4)+k}\cdot \frac{r}{s}$$
How do I go about reaching this solution?
Best Answer
Apply the Laplace transform to both equations, term by term (since the Laplace transform is a linear operator). Then you will have two equations in terms of $X (s) $ and $U (s) $. Combine the equations to eliminate $U (s) $. Then perform the algebra to isolate $X (s)$ on one side of the equation.