2.004  Dynamics and Control II

Here are the rules for the final exam:

1) You may bring up to three double-sided pages of notes, hand-written or typed, and as dense as you'd like them. One suggestion would be to bring the two pages that you used in Quizes 1 & 2, if you still have them, plus an additional page with the newest material (e.g. frequency response/Bode plots.) Or you may opt for three brand new pages of your choice.

2) You may not bring any other aid material (e.g. textbooks, lecture notes, or your own beyond the three-page limit.) or electronic aids. Computers, calculators, personal digital assistants (PDAs) and cell phones are explicitly prohibited. As in the quizes, the rules regarding electronics are essentially the same as in a plane during take-off or landing: you may not use anything that has an on-off switch! Or if you do bring such a device then the switch should be off during the exam. In particular, please be considerate of your classmates by turning your cell phones off; it would be extremely rude to the rest of the class if a phone rings while everyone else is trying to concentrate.

Regarding calculators, the arithmetic required to solve the problems will be minimal so it can be done by hand. If you somehow find yourself doing complicated numerical calculations, go back and retrace your steps - you must have gone off track somewhere.

We'll follow the regular office hours schedule: Wednesday Dec. 12, 4-5pm; Thursday Dec. 13, 3:30-5pm. Location: 3-434 as usual.

Following up on Nate's question in class today, the clean story is as follows: First of all, we need to be careful whether we refer to open-loop or closed-loop quantities such as Bode plots, gain & phase margins, etc. The rule that we learnt says that both open-loop gain margin and open-loop phase margin must be positive to guarantee stability of the closed-loop system and it applies only for open-loop stable systems, i.e. systems whose open-loop transfer function has poles on the left-hand side only.

What happens if the open-loop system is unstable (i.e. there is at least one open-loop pole on the left hand side, as in the case of the inverted pendulum from PS 10?) The rule must then be modified in a way that is not readily accessible to us at the moment, since we have not covered the fundamentals of gain/phase margins, namely the so-called Nyquist diagram and Nyquist criterion. (An excellent discussion of these topics is in the Nise textbook, chapters 10.3-6; these chapters are not included in the material that you must study, but if you are curious about the topic, we encourage you to read them or contact us to arrange a private tutorial.) For the specific case of the inverted pendulum, it turns out that the closed-loop pendulum with PD compensation is stable if the gain margin is negative and the phase margin is positive - if that sounds weird and confusing, it is because the inverted pendulum has one open-loop pole on the right hand side.

In the problem set, question 1(m), we actually overlooked this subtlety but you need not worry about it: the gain/phase margin problems that we consider solvable given the material that we covered in the lectures will always involve systems with open-loop poles on the left hand side (i.e. open-loop stable systems). If anything, the weirdness in the homework problem served to alert you that there is more to the topic of controls than we have covered in this class, so you may want to consider signing up for one of the more advanced elective subjects, e.g. 2.12 or 2.14!

 

This semester, subject evaluations will be entered online only, at the following website:

http://web.mit.edu/surveys/subjecteval/

There will be no paper evaluations. The evaluation website will remain open until 5pm on December 14th. Please make sure to login and enter your evaluations before the deadline.