Question

Consider a mass spring system having a unit mass and with spring constant

If y is the displacement from rest then
y'' + y = u(t)
where u is the applied force used to control the system.
Write this second order equation as a first order system in the from
x' = Ax + Bu.

Answer 1

To convert the second order differential equation y'' + y = u(t) into a first order system for a mass-spring system, we introduce state variables for displacement and velocity, which allows us to express the system in a matrix form x' = Ax + Bu.

Step-by-step explanation:

To write the second order equation y'' + y = u(t) for a mass-spring system as a first order system, we need to introduce state variables to reduce the order of the differential equation. Let's define x1 to be y, the displacement from rest, and let x2 be y', the velocity of the mass. Our state variables then become x = [x1, x2]T. Using these state variables, we can rewrite our second order equation as a set of first order equations:

x1' = x2
x2' = -x1 + u(t)

Thus, the system in matrix form is x' = Ax + Bu, where A = [[0, 1], [-1, 0]] and B = [[0], [1]] which represents the first order system of the mass-spring system.