An electromechanical system has the following equations of motion x + k1x¨ + k2x

An electromechanical system has the following equations of motion
x + k1x¨ + k2x˙ + 3.5 sin(x) = k3i………… (1)
where i indicates the control input and x is the control variable. Using the Routh stability criteria, determine the range for
k1, k2, and k3 to yield a stable system. First you need to linearize the system around equilibrium point of x
∗ = 0. Then
set k1 = 3, k2 = 2.7, and k3 = 1 to design a PID controller yielding a critically damped system with the natural frequency
of ωn = 0.5
rad/s
while applying the step input.