University of Notre Dame
Aerospace and Mechanical Engineering

ME 654: Geometric Nonlinear Control
Midterm Examination

B. Goodwine Fall, 1999

Issued: October 28, 2003 Due: November 4, 2003

Instructions

You have until the beginning of class on Thursday, October 30 to complete this exam. You may consult your course notes, the course text or any other written referernce. You may not consult with other students or faculty other than the instructor.

1.

The equation of motion for the pendulum is given by

$$m l^2 \ddot \theta + d \dot \theta + m g l \sin \theta = 0.$$

(a)

Use a numerical simulation package to construct the phase portrait for the system when there is zero damping, i.e. d = 0. In particular, verify that there is a continuum of limit cycles as well as saddle points.

(b)

Now add positive damping and again construct the phase portrait. What happens to the continuum of limit cycles and the saddles?

2.

A particle of mass m and position vector r = (x,y,z) moves in a potential field W(x,y,z), so that its equation of motion is

$$m \ddot r = -\nabla W$$

where $\nabla W$ represents the gradient of the function W. By putting $\dot x = u$, $\dot y = v$ and $\dot z = w$, express this in terms of first order derivatives. Suppose that W has a minimum at r = 0. Show that the origin of the system is stable by using the Lyapunov function

$$v = W + \frac{1}{2} m \left( u^2 + v^2 + w^2 \right).$$

What do the level curves of v represent physically? Is the origin asymptotically stable?

Suppose an additional non-conservative force f(u,v,w) is introduced so that

$$m \ddot r = -\nabla W + f.$$

Use the same Lyapunov function to give sufficient conditions for f to maintain stability.

3.

A controls application of Lyapunov theory is called ``Back-stepping.'' An outline of this approach is presented in Section 6.8 in Sastry's text. (If you do not have the book, scanned images from the section are here: page 1, page 2, page 3. Prepare approximately 3 pages of lecture notes that would be appropriate to instruct college mechanical engineering seniors on this material. Use Sastry's text as a skeleton, but add (or subtract) material to make it concise and coherent. You may assume that they have as much knowledge of Lyapunov theory as was covered in class. In particular, your lecture notes must contain:

(a)

the technical mathematical details; and,

(b)

the ``big picture,''

i.

what problem is being solved;

ii.

the types of systems to which it can be applied; and,

iii.

the relationship of this material to the Lyapunov theory covered earlier in the course.

Keep in mind that students only write down what you write on the board, so make sure that everything important you have to tell them is in your notes.

4.

Make up an open-book, one-hour, in-class midterm examination for this course. Submit the examination as well as complete solutions. You are free to exercise as much creativity as possible, but must abide by the following rules:

(a)

The test must have something to do with the material covered in the course. You do not have to cover every single topic in the course, but you must cover a reasonable amount to be fair to all the students.

(b)

The questions must be original. You can look at any reference that you want in search of ideas, but your final result should be entirely the product of your own creative efforts.

(c)

A straight-forward ``plug and chug'' test (like a couple of the questions above) which is internally consistent and error-free will receive a minimal passing grade. Extra points will be earned for questions that do some of the following:

• Require analytical skills. For example, ask the students to derive some formula given in class or in the book for which the derivation was not given in class or the book, or ask them to provide theoretical and/or physical explanations of observed phenomena.
• Require synthetic (creative) skills. For example, give problems with solutions that require putting outside material, e.g., from chemistry, mathematics or other engineering courses, together with material from the course; or, give problems that require using methods presented in the course in new ways; or, give problems that appear to have no relationship to nonlinear control or analysis, but whose solutions require techniques used in this course.
• Require evaluative skill or call for value judgments. Ask the students how they would judge the ``goodness'' of, say, a controller design. (It may work, but is it a good design?) Test the breadth and depth of their thinking.

It thus relativizes discourse not just to form--that familiar perversion of the modernist; nor to authorial intention--that conceit of the romantics; nor to a foundational world beyond discourse--that desperate grasping for a separate reality of the mystic and scientist alike; nor even to history and ideology--those refuges of the hermeneuticist; nor even less to language--that hypostasized abstraction of the linguist; nor, ultimately, even to discourse--that Nietzschean playground of world-lost signifiers of the structuralist and grammatologist, but to all or none of these, for it is anarchic, though not for the sake of anarchy but because it refuses to become a fetishized object among objects--to be dismantled, compared, classified, and neutered in that parody of scientific scrutiny known as criticism.

---Stephen Tyler, Writing Culture,James Clifford and George E. Marcus, editors.

Last updated: October 27, 2003.
B. Goodwine (jgoodwin@nd.edu)