INSTITUTE CALIFORNIA OF TECHNOLOGY Control and Dynamical Systems CDS 101/110a Final Exam R. Murray 4 December 2002
The exam consists of four questions, worth a total of 60 points. The point values for each section are shown on the right. The time limit is 3 hours, in one sitting. Budget your time to complete as much of the exam as possible. If time does not permit a complete answer, indicate how you would proceed as explicitly as possible. The exam is open book. You may use any of the optional texts (Friedland, Franklin-Powell and Emami-Naeni, Leonard and Levin, or Kuo), course handouts, lecture and class notes, course problem sets and solutions, and your own handwritten notes. No other books are allowed. You may use a computer or calculator for carrying out numerical computations. MATLAB may be used but is not required. All of your answers must be hand written or hand drawn (do not turn in code or computer plots). If you use MATLAB, please make sure to provide descriptions of how you achieved your solution; do not just include the nal answer. You are not allowed to use the Internet during the exam (except for accessing local computing resources, such as MATLAB/SIMULINK), but you may download or print out copies of presentations, notes, FAQs, or other material posted on the course web site (CDS 101 or 110). You are not allowed to print out contents of other sites for use while taking the exam (although you can take handwritten notes on the sites and use your own notes in the exam). The exam is due by 5 p.m. Friday, December 13, in the box outside 102 Steele. Please write your solutions in a fresh exam book (blue book). We have to grade a large collections of exams in a short time and it makes things much simpler to manage if everyone uses a bluebook. Please note that students in CDS 101 and CDS 110a/ChE 105 are required to answer dierent sets of questions (or portions of questions), depending on which course you are taking. Please be careful to answer only the questions for the course you are taking. No additional credit will be given for solutions provided to questions that are not for the course you are taking. Questions with no indication of the course should be answered by all students.
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Problem 1 (CDS 101 only)
Choose any ve of the feedback systems listed below. For each system you choose, answer the following questions: (a) (CDS 101) Draw a block diagram for the system, with the plant and controller separately [10] identied (you can use words to describe the contents of each block). For the plant, describe a plausible model of the system and give the state, inputs and outputs, and the dynamics of the model. You may give your answer in words, but please be as precise as possible. (b) (CDS 101) Provide a performance specication for the system, using the signals from your [5] block diagram in part (a). Your performance specication should make sense for the problem, but should be in terms of features of the step response and or frequency response of the system. (c) (CDS 101) Describe the main sources of uncertainty for the system and indicate whether or [5] not the feedback system compensates for these uncertainties. You should list at least two sources of uncertainty for each system. Be as explicit as possible. Systems to choose from: 1.1 Thermostat controlled heating of a room. 1.2 Human being balancing on a tightrope. 1.3 Autopilot control of heading and speed in an airplane. 1.4 Lateral (x) control of the Caltech ducted fan 1.5 Congestion control of the Internet using TCP Reno. 1.6 Insect ight during periods of straight motion. 1.7 Segway human transportation system. 1.8 Pupil control system for tracking a moving object with your eyes. 1.9 Population control for a rabbit/fox ecosystem. 1.10 Primary mirror control system for the Keck telescope. Please make sure to identify the number of the system when responding to the three questions above. Your answers will be graded based on your grasp of control concepts and not how well you know the details of the particular system (so it is OK if you make up some plausible description of the system).
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Problem 2 (CDS 101 and CDS 110)
In this problem, you will answer a series questions about each of the systems illustrated in the Bode plots below. You should summarize your answer to this question by creating a matrix with the rows corresponding to each system and the columns corresponding the the answers for questions ad. You should also include the analysis you used to obtain your answers (separate from summary the chart). If you use MATLAB, you should still give a description of the key features that justify your answers.
Bode Diagrams
From: U(1) 20 10 5 0 20 5
Bode Diagrams
From: U(1)
0
Phase (deg); Magnitude (dB)
Phase (deg); Magnitude (dB)
40
10 15 20 0 50 100 To: Y(1) 150 200 250 300 2 10
60
80 0 50 100 To: Y(1) 150 200 250 300 2 10
10
1
10
0
10
1
10
1
10
0
10
1
Frequency (rad/sec)
Frequency (rad/sec)
2.1
Bode Diagrams
From: U(1) 50 20 15 10 0 5 0
2.2
Bode Diagrams
From: U(1)
Phase (deg); Magnitude (dB)
Phase (deg); Magnitude (dB)
50
5 10 15 20 0
100 0 50 100 To: Y(1) 150 200 250 300 350 400 2 10
1 0 1
50 To: Y(1)
100
150
10
10
10
200 2 10
10
1
10
0
10
1
Frequency (rad/sec)
Frequency (rad/sec)
2.3 For each of the Bode plots above, answer the following:
2.4
(a) Assuming that the open loop systems have no right half plane poles, is the closed loop system [5] (assuming unity feedback) asymptotically stable or unstable?
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(b) Which of the following Nyquist plots corresponds to the Bode plots? Assume that all open [5] loop poles have non-positive real part (they might have zero real part).
Nyquist Diagrams
From: U(1) 2
8
Nyquist Diagrams
From: U(1)
1.5
6
1
4
0.5
2
Imaginary Axis
Imaginary Axis
To: Y(1)
0
To: Y(1)
0
0.5
2
1
4
1.5
6
2 2 1.5 1 0.5 0 0.5 1
8 3
2
1
0
1
2
3
4
Real Axis
Real Axis
(N1)
Nyquist Diagrams
From: U(1) 6
25
(N2)
Nyquist Diagrams
From: U(1)
20
4
15
10
2
Imaginary Axis
Imaginary Axis
5 To: Y(1)
To: Y(1)
0
0
5
2
10
15
4
20
6 2
0
2
4
6
8
10
25 10 5 0 5 10 15
Real Axis
Real Axis
(N3)
(N4)
Please note that not all features of the plots above are completely discernible (a limitation of Nyquist plots), so you may need to use multiple features to sort things out. Dont forget to describe how you obtained your answer.
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(c) (CDS 110 only) Which of the following sets of unit step responses are possible for the given [5] frequency response? Note that the scales are dierent on each plot.
Step Response
From: U(1) 3.5 1
Step Response
From: U(1)
0.9 3 0.8
2.5
0.7
0.6
Amplitude
Amplitude
2 To: Y(1)
To: Y(1)
0.5
1.5 0.4
1
0.3
0.2 0.5 0.1
0 0 2 4 6 8 10 12 14 16 18 20
0 0 10 20 30 40 50 60 70 80
Time (sec.)
Time (sec.)
(S1)
Step Response
From: U(1) 250
1.4
(S2)
Step Response
From: U(1)
200
1.2
150
1
100
Amplitude
To: Y(1)
0
To: Y(1) 0.6 0.4 0.2 0 0 20 40 60 80 100 120 140 160 180 200
50
50
100
150
200 0 5 10 15 20 25 30 35 40
Time (sec.)
Amplitude
0.8
Time (sec.)
(S3)
(S4)
(d) (CDS 110 only) What is the maximum and minimum amount of additional positive loop gain [5] for which each system is stable?
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Problem 3 (CDS 101 an...