Definition of the system state variables, output and input-output equations

The state variables for this system are the elongation of the spring caused by the input displacement of the road (z), and the vertical velocity of the mass (v or z'). The state variables equations are:

z' = v
v' = z'' = [By'(t) + Ky(t) - Bz' - Kz] / M

The output is the mass vertical displacement (z). The output equation is:

z = [By'(t) + Ky(t) - Mz'' - Bz'] / K

Equations in matrix form.

[ z'] = [ 0 1 ] [ z ] + [ 0 0] [ y ]
[ z''] [ -K/M -B/M] [ v ] [ K B] [ y']

The input-output equation is:

Mz'' +Bz' +Kz = By'(t) + Ky(t).

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