Project - Position control of mass spring damper system

pulse generator is used as the x1 reference

transfer function is obtained as

F= ma

F = md2x/dt2+c.dx/dt+kx

laplace equation

F(S) = ms2x(s)+csx(s)+kx(s)

X(s)/F(s) = 1/ms2+cs+k

transfer function = 1/(ms2+cs+k)

poles is obtained using the pole placement method

roots[m c k]

will give the poles as we know that its an over damped system the poles are negative and real

jacobian matrices can be found by partial differenciation of x1_dot = x2 & x2_dot = f(t)/m-c/mx2-k/mx1

A = [ 0 1; -k/m -c/m]

B = [0;1/m]

K can be find by using the place command [A, B, P]

we got proportional gain as 0.7434

derivative gain as 0.0324

when we put integrain gain Ki = 1000

we are obtaining the path as x ref which we have given as input with overshoot and a time delay

please find the attached for the same