Advanced Control, Spring Semester 2014/2015
Instructor: Dr. Tarek A. Tutunji
Homework # 1
Due Wed April 8, 2015
Problem 1: Transient Response
𝜔2
A general 2nd order transfer function is 𝐺(𝑠) = 𝑠2 +2∁𝜔𝑛 𝑠+𝜔2 .
𝑛
𝑛
Derive the expressions for the rise time, overshoot, and settling time, for 0 < 
Problem 2: Speed Control System
Consider the field-control DC servo motor system with proportional control in the figure below
Assume that the moment of inertia I = 0.75 Kg.m2 and damping constant C = 0.5 Nms/rad and the constants K2=
5Nm/A and H1=0.1 Vs/rad. For simplicity assume that the motor inductance is zero
1) Re-draw the circuit to have a PD controller
2) Draw the block diagram of the system showing all transfer functions
3) Find the controller parameters Kp and KD to obtain maximum overshoot of 10% and settling time of two
second when vi(t) is a unit step function
4) Calculate the steady-value o(t) when vi(t) is 10 Volts
Problem 3: Root locus
𝐾(𝑠+2)
Consider the open-loop transfer function: 𝐺(𝑠) = 𝑠2 +2𝑠+3.
a) Sketch the root locus by hand
b) Verify results using MATLAB
Problem 4: Frequency Analysis
0.5
Consider the transfer function 𝐺(𝑠) = 𝑠(𝑠2 +𝑠+1).
a) Check stability using Nyquist criterion
b) Draw the bode plots by hand and calculate the GM and PM.
c) Verify part (b) the using MATLAB
Problem 5: State-Space Analysis
The state-space equations are:
𝑥̇ = 𝐴𝑥 + 𝐵𝑢
𝑦 = 𝐶𝑥
0 1
1
0 0
0
Given that 𝐴 = [−1 0
0 ] , 𝐵 = [1 0 ] , 𝐶 = [
0
−1 0 −1.5
1 −1
a) Calculate the transition matrix (s)
b) Calculate the output y(t) for unit step input
1 0
]
0 1