Now we consider motor shaft position$/$angle $(t)$ as our output. We study position control $($not speed control$)$ with a closed loop PID controller $(K_p+K_{d}s+\frac{k_t}{s})$ and unity feedback. We consider a reference step input $r_1(t)=u_S(t)$ and reference ramp input $r_2(t)=0\mathrm{.}25.t*u_s(t).$

For DC motor position control, we use the transfer function $F(s)$ given as

$F(s)=\frac{k}{s+1}*\frac{1}{s}$

where the $\frac{1}{s}$ term acts to integrate the velocity and thus give motor shaft angle $(t).$

Consider proportional control and set $K_d=0=k_l.$

a$)$ System type.

i$.$ Assuming stability, what is the system type? $($Why$?)$

ii$.$ Based on system type, what can you say about steady$-$state error to step $r_1(t).$ Does this agree with your results from the Final Value Theorem in LAB $02?$

iii. Based on system type, what can you say about steady$-$state error to ramp $r_2(t).$ Does this agree with your results from the Final Value Theorem?

iv$.$ How does the information provided by the Final Value Theorem prediction of steady state error differ from that provided by system type?

b$)$ Stability & Performance.

i$.$ Using the Routh Array, determine values of the gain $K_p$ that stabilize the system.

ii$.$ For a stable system, based on your Final Value Theorem results from LAB $02$ PRELAB, how can you reduce the expected steady state error in response to step $r_1(t).$

iii. For a stable system, based on your Final Value Theorem results from LAB $02$ PRELAB, how can you reduce the expected steady state error in response to ramp $r_2(t).$

Consider proportional$-$integral $($PI$)$ control and set $K_d=0$ with both $K_p$ and $K_l$ nonzero. We have added an integrator to the forward path!

c$)$ System type

i$.$ Assuming stability, what is the system type? $($Why$?)$

ii$.$ Based on system type, what can you say about steady$-$state error to step $r_1(t).$ Does this agree with your results from the Final Value Theorem in LAB $02?$

iii. Based on system type, what can you say about steady$-$state error to ramp $r_2(t).$ Does this agree with your results from the Final Value Theorem in LAB $02?$

d$)$ Stability & Performance.

i$.$ Set $K_p=5$ and, using the Routh Array, determine values of the gain $K_l$ that stabilize the system.

ii$.$ Set $K_l=5$ and, using the Routh Array, determine values of the gain $K_p$ that stabilize the system.

iii. For $K=23,=0\mathrm{.}154,K_p=5,$ vary the value of $K_l=1,5,15,25.$ Draw a map of the pole locations for each. Predict the behavior of the system based on the pole locations. Complete the following table

$\backslash$table$[[K_l,$Pole Locations,Behavior? $(t0$