a$)$ The plant in isalation comprisenta only the ball, betam arnd pivot

and has the transfer function $P(s)=\frac{w31}{s^{11}}.$

Determine the polen of the plant.

ii$)$ Comment on the stability of the plant based on the pole

values.

iii$)$ Qualitativaly juatify your arnawer basaed on the phyaica of

the plant.

A position control system with a PO controller which is capable of

moving the ball to a given position along the beam $($a desired

position$)$ is shown in Figure $3\mathrm{.}2.$

Figure $3\mathrm{.}2.$ Bal and beam postion control system with a PD controller.

armpatude of $15cm$ for $N_p=7$ and $K_s=20$ is depicted in Figure

Fipure $3\mathrm{.}3.$ Step reaperise of the panibion control myatem.

c$)$ The poles of the closed loop system are:

$p_1=-99\mathrm{.}367,p_{2\mathrm{.}3}=-0\mathrm{.}3163+-j0\mathrm{.}2189,$

mark the pole positions on the complex plane. Use this

information to discuss the stability of the syalam. Explain

qualitatively the relations between the pole positions and the step

resporise in Figure $3\mathrm{.}3.$

d$)$ A different physical system with simple real poles is connected in

a closed$-$loop configuration without a controller. When subjected

to a reference step input, the output exhibits a steady$-$state error

as would be expected. To eliminate the steady$-$state error, a PID

controller is added. Which term of the PID controller would yos

adlust? Would is be increased or decreased? Justify your answer.