1-) Students who design a balance robot for the Digital Control Systems course control the robot they have developed with the negative feedback system shown in the figure.

Transfer function of the balance robot ( ) = 1/( ^2*( +2))

a. A proportional controller, indicated by ( ) = ( > 0), was preferred to control the system. Does this robot stay in balance when the unit step function is applied to the system’s input? Prove the reason by drawing the root-place-curve graph.

b. This time, it is desired to control the system using a PD, Proportional + Derivative type controller ( ) = ( + ) ( , > 0). Determine the range that a robot will receive so that this robot can stay in balance against the unit step input.

c. By taking = 1 on the PD controller in (b), the closed loop system Show the positions of the poles on the graph in the s-plane for > 0.

d. The dominant poles of the system using the controller in (c) Calculate the value of the K gain so that it can produce a 52% maximum overhead versus unit input.

e. In order for this system’s steady state error to be a finite value other than 0, determine an appropriate input and calculate the steady state error to be generated by the controller in option (c).

2-) As seen on the figure, the polar curve of the control system has been graphed according to ≅ 0

and ≅ ∞ for K = 1. It is known that the given system is a minimum-phase system. In addition, the

table shows the angular frequency of the system, and the reel and imaginer components of the

system at the point is giv

a. Complete the polar curve of this system and obtain the Nyquist diagram.

b. b. Perform stability analysis using the Nyquist graph.

c. Please determine the order of the system by explaining the reason.

d. Please find the gain margin, the phase margin, the gain pass frequency and phase pass

frequency by using Nyquist diagram and the table.

e. Determine the stability of the system using the calculated gain margin and phase margin.