Problems:

Problem $1(10*$ pts$)$:

A$.$ You are working at a medical device company, and are helping to design a new needle system that pricks the skin of a patient so that a drop of blood becomes available for a glucose test. The needle tip position $x$ can be modeled as a system with the transfer function:

$H(s)=x\frac{s}{F}(s)=\frac{K{}_n^2}{s^2+2{}_{n}s+{}_n^2}$

Here, $f(t)$ is an input force. It is constrained to be a step function with amplitude $1($e$.$g$.f(t)=u(t).$ Given all the current design parameters, the response that the team is getting is shown below:

Sind Rasoonso

Unfortunately, the problem is that the needle needs to reach a maximum position of $1\mathrm{.}5$ mm $,$ but then after the overshoot, retract to a position of $0\mathrm{.}5$ mm $.$ The peak time should be exactly $0\mathrm{.}2$ seconds. Given that you can control the damping ratio $,$ the natural frequency ${}_n,$ and the gain $K,$ write in a few sentences a proposed strategy for modifying the design to meet the stated objectives. Please make the justification for the strategy clear.