18.031 System functions and the Laplace transform


Posted solutions to pset 2.

Course description

Studies basic continuous control theory as well as representation of functions in the complex frequency domain. Covers generalized functions, unit impulse response, and convolution; and Laplace transform, system (or transfer) function, and the pole diagram. Includes examples from mechanical and electrical engineering.


Haynes Miller hrm@math.mit.edu
Jerry Orloff jorloff@mit.edu


Dates: Tuesday, January 19 -- Friday, January 29, 2016
Classes: Daily, 1:00 -- 3:00, 4-261
Syllabus and calendar
Information and policies
Exam: 1:00, Friday, Jan 29, 2016, 4-261.

Office Hours
Haynes Miller: Thursday Jan. 21 4:30-6 in 2-478; Monday Jan. 25 3-4:30 in 4-261; Thursday Jan. 28 3-4:30 in 4-261.
Jerry Orloff: Thursday Jan 21 3-4:30 in 4-261; Wednesday Jan. 27 3-4:30 in 4-261.

Class stellar page


Class 1: review of LTI systems: constant coefficient ODEs, stability, gain.
Class 2:
(i) frequency response, transfer functions, block diagrams.
(ii) pole diagrams (read sections 1-7)

Classes 3,4: Laplace transform. (read sections 1-14)

Class 5: Step and delta functions.

Class 6: Laplace transform: t-translation rule.

Class 7: Convolution.

Other possibilities
Note on poles and zeros from 2.14

Problem sets

Laplace table
PSet 1
PSet 2


Complex numbers and exponentials
Stability: including the Routh-Hurwitz criteria
Partial fractions decompositions (includes the coverup method)


Answer to last board question from Friday 1/22
Notes on PID controllers


The MIT Mathlets are a set of small applets that dynamically illustrate various mathematical concepts. They can all be found at mathlets.org/mathlets

Here are direct links to some of the applets --MIT mathlets and others-- referenced in 18.031.
Amplitude Phase Second Order I
Amplitude Phase Second Order II
Bode and Nyquist Plots
Amplitude Response: Pole Diagram
Series RLC Circuit
LRC Filters


Here is a description of the Course Mascot, pictured above.
And here is a mathematical account of this kind of feedback mechanism.