SYST 611 Homework 2

Issued:  Friday September 11, 2009

Due: Thursday, September 17, 2009

Notes

1. The automobile obsolescence model is on page 94 of Luenberger, but he uses washing machines instead of automobiles.
2. Luenberger pages 90-112 was covered in class.
3. Z-transforms are discussed in Luenberger pages 255-266.

Three questions, equal weight.

1.       Moving Average.

Suppose a sequence of raw data is denoted as y(k).  A simple four-point moving average produces a sequence y(k) such that each y(k) is the average of data points y(k), y(k−1), y(k−2), y(k−3).  Find a representation for the average of the form:

x(k+1) = Ax(k) + bu(k)

where x(k) is a 3x1 vector, A is a 3x3 matrix, and b is a 3x1 vector.

2.    Obsolescence Model.  

a.    Suppose that the input u(k) of new machines in the example in section 4.1 of Luenberger (page 94) is chosen to exactly equal the number of machines going out of service that year.  Write the corresponding state space model. 

b.    Repeat part (a) using the assumption that, in addition to replacements, there are new purchases amounting to y percent of the total number of machines in service.

3.       Z-transforms.

Using the Z-transform method, find an expression for y(k) given the following difference equation:

Hints.

(1)    At some point you will have a cubic equation (an equation with a cubed term).  The equation will have three roots (values where the equation goes to zero).  If the three roots are r1, r2, and r3, then the original equation factors into (x – r1)(x – r2)(x – r3), where x is the unknown variable. 

(2)    To find the roots of a cubic, you can use any resource at your disposal.