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Problem No.1
For the characteristic polynomial shown, Using Routh Hurwitz’s stability test to determine the range of k for stable response (show work)
s^4+3s^3+3s^2+6s+(4+k)
Problem No.2
Given the loop transfer function:
G(s)H(s)=(K(s+2))/((s+1)(s+4)(s+6)(s+8))
Determine the following: For parts 1-5, assume K =1.
The loop poles and zeros
The location of the intersection of the asymptotes
The angles of the asymptotes
The number of branches of the root locus
Draw, clearly, the root locus – use arrows to show the movement of the root loci. Do not spend time calculating the exact location of a break away point, simply reasonably estimate it.
What is the absolute value of the gain, K, at the poles s = -6?
The absolute value of the gain, K, at the zero s = -2?
(there is an extra graph sheet, if needed)
Problem No.3
For the given root locus: Make sure your answers are clear and organized.
Helpful hint: The values on the inside perimeter of the graph indicate values of zeta.
1- Determine the needed series gain to establish a damping ratio of 0.68 in the closed loop system. Must show work
2- (Thinking part) With the gain found in (1): Determine the new locations of the poles P1 and P2, the roughly estimated location of the pole P3, and the location of the
zero (Z1) for:
a) the open loop system? { }
b) the closed loop system? { }
3- With the gain found in (1), what are the natural and damping frequencies, ω_n and ω_d in the closed loop system?
4- Determine the Open Loop transfer function, G(s)
6- Determine the range of the series gain K for a stable response.
7- Determine the value of positive, greater than zero, gain K that will result in a marginally stable system, if any.
Problem No. 4
For the modified ship steering system shown,
1- Determine the value of K so that the low frequency gain is 10dB. Then use this value of K for the rest of the problem.
2- Using Matlab, draw (CLEARLY) the magnitude and phase Bode plots and determine:
a- the gain crossover frequency
b- the phase crossover frequency
c- the gain margin
d- the phase margin
3- Based on the results of (2), would the closed loop system be stable or unstable ?
4- Based on the above, answer A OR B (but not both)
There is only one correct answer
A- if you decided that the system is stable, then what is the maximum allowable gain in ratio form that can be added to the system before it becomes unstable ?
B- if you decided that the system is unstable, then list the one or two things that specifically must be done to force the system into a stable behavior.
Problem No.5
In a feedback control system,
a) if the steady state error is nonzero, what type of controller is needed ? (choose one)
(i) Proportional Derivate, (ii) Proportional Integral
b) if the rise time is very slow, what controller is mostly preferred? (choose one)
(i) PD, (ii) PI
c) In the frequency response analysis of the open loop transfer function of a closed loop feedback control system, the following is found:
PM = -20 degrees, GM = 12dB
is the closed loop system stable or unstable ?
Problem No.6
Given the fourth order attitude control system shown, answer the following test questions.
The following must be done in order to receive appropriate credit:
All answers must be clearly and orderly stated and professionally presented.
Anytime a Matlab code or a Simulink is used, you must submit the commented code and /or the Simulink diagrams along with the results.
All plots must have appropriate labels and titles.
Uncompensated System: Let Gc(s) =1 for now (May use Matlab whenever possible)
Determine the range of K so that the steady state error due to a unit ramp input is ≤0.02 (see hints -1 at end of this exam) Must show work.
For the rest of the exam, assume the minimum value of K =0.6
b. For K = 0.6, determine the open loop poles and zeros.
c. Obtain the unit step response of the closed loop system with K set to 0.6 Submit the response and the Matlab code (See hints -2)
d. From the step response in (c) , determine the following
rise time
settling time
% overshoot
steady state value.
e. Challenge (bonus): Plot Error due to a unit ramp input and verify on the plot the steady state error to be 0.1 when K is set to 0.6. Submit the response and the Matlab or Simulink code.
f. Obtain the frequency response of the open loop system and show on the plot the gain and phase margin (note: K is set to 0.6 ) Submit the plot and the Matlab code (see hints -2 at end of this exam)
Compensated System USING MATLAB (Not Simulink) (let K =0.6)
(See hints -3)
Proportional Derivative Control
With Gc(s) = Kd S + Kp
Let Kp = 0.8 and let Kd be a vector of values ranging from 0.02 and the value of 0.2with steps of 0.02 kd=[0.02 0.04 0.06 0.08 0.1 0.2];
Using Matlab, plot the unit step response of the compensated closed loop system for each case of Kd (only submit the response for Kd =0.08) This response must show the rise time, settling time, and % overshoot.
For which value of Kd did the step response look most favorable in meeting requirements?
Kd = ?
Obtain and submit the frequency response of the compensated open loop system Gp*Gc (see hint -3). Indicate the gain and phase margins on the plot
Submit the Matlab code used.
List your findings in the table below:
parameter uncompensated system compensated system (PD)
rise time
settling time
% overshoot
steady state value
GM
PM
Problem No. 7 -- This refers to Problem No.6
The following must be done in order to receive appropriate credit:
All answers must be clearly and orderly stated and professionally presented.
Anytime a Matlab code or a Simulink is used, you must submit the commented code and /or the Simulink diagrams along with the results.
All plots must have appropriate labels and titles.
For the same problem as No. 6, implement a Simulink PID controller design.
Submit the block diagram,
Submit screen shot images of all of the PID tuning steps
Submit a screen shot of the PID controller parameters (Kp. Ki, and Kd).
Submit the frequency and step response of the best possible compensated system
Show the transient and frequency domain performance results ( rise time, settling time, steady state value, % overshoot, Gain and phase margins) in a table as was done above (in problem 6)
Hint: In Simulink,
I have the number 6, it should have been 0.6 for the gain. (should have been 100000*0.6)
-create a new model
In Continuous, select the pole =zero function to create the transfer function
Once created, stretch the block to see the function
After adding the series PID, here is the block diagram.

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