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Professor Edison Tse
Problem Set
#1
(not to be turned in)
1.
(a) Write the following series (the Fibonacci series, where each element is the sum of the previous two)
in the form of a linear, two-state invariant dynamic system.
1, 1, 2, 3, 5, 8, 13, 21, 34.
...
(b) What are the eigenvalues of the 2X2 state matrix of the system?
(c) Consider now the dynamic system with a single state variable z(k) that is the ratio of two consecutive
elements of the series above: z(k) = a(k+1)/a(k), where a(k) is the kth element of the series. What value
does z(k) tend to as k tends to infinity? You can do this by solving for the limiting behavior of a non-
linear dynamic system in z (how?), or you can just do repeated calculations (with MS Excel, for example).
How does your answer compare to your answer in (b)?
2.
Consider the dynamic system described by the following equations:
x(k+1) = y(k) - .5*x(k) + 5
y(k+1) = .6*y(k) +
λ
*x(k) + 5
(a)
Assume x(0) = 5 and y(0) = 0. Model the system in a spreadsheet, and plot its state in x-y space as k
increments from 0 to 15 for each of 3 values of
λ
: -1, 0, 1. I.e., plot (x(0),y(0)), (x(1),y(1)), .
..,
(x(15),y(15)). What kind of behavior does the system exhibit in each case?