For the mass-spring-damper system shown in Figure A1, derive a linear time invariant mathematical model expressing the displacement x as a function of the mass M, spring stiffness K and damping parameter C, when an external force F(t) is applied to the mass. List the assumptions you have made.

Put the mass-spring-damper model into the standard second order form shown below. Find the steady state gain, damping ratio, natural frequency and forcing function. $\frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = Ku(t)$

Sketch the response of the mass-spring-damper system for the case that $C = 0$ and explain the significance of this result. On the same plot, sketch the response of a critically damped system. For the case that $M = K = 1$, determine the value of C that yields a critically damped system.

The following differential equation represents the closed-loop dynamics of a proportional-velocity (PV) control system: $\frac{d^2x}{dt^2} + \left(0.3 + k_v\right)\frac{dx}{dt} + 0.5 + k_p = 0.4r(t)$ where $x$ is the output and $r(t)$ is the set point, while $k_v$ and $k_p$ are control gains. Design a controller (i.e. determine the values of the control gains) that yields closed-loop damping $\zeta = 1$ and $\omega_n = 1.2$ rads/s. Find the steady state gain of the closed-loop system, and state whether or not this control system achieves steady state tracking of the set point.

With the aid of sketches and/or any of the equations and analysis considered above, discuss the strengths and weaknesses of PV control in comparison to proportional-integral (PI) control.