Due Thursday, March 23

Here is a zip file named *kharitonov.zip*
that contains 4 MATLAB .m files. Download the zip file, unzip it, and store the
4 files in your MATLAB working directory. If you have trouble downloading the
zip file or unzipping it, send me an email, and I will send the .m files to you
by email.

In MATLAB, run the script file *xmpl_421_kharit.m*. This will generate in
the MATLAB workspace a frequency vector *w* and 4 arrays that contain the
intervals of uncertainty, {*Q*1, *Q*2, *Q*3, *Q*4} for 4
families of interval polynomials. The number of rows in each *Q* is the
number of coefficients in the polynomial, *n*+1. The ordering of the rows
is least significant coefficient, *a*_{0}, at the top of the array.
The first column is the lower bound, and the second column is the upper bound.

Once you have run the script file *xmpl_421_kharit.m*, the function that
you will use for this homework assignment is *kharitonov.m*. The syntax for
using this function is:

[value, K] = kharitonov(Q, w);

where *Q* is the intervals of uncertainty for the
coefficients, *w* is the frequency where the polynomials will be evaluated,
*value* is the value set, and *K* is an array containing the
Kharitonov polynomials. Each row of the array corresponds to one Kharitonov
polynomial. The other two functions (*kharpoly.m* and *plotkhar.m*)
are called by *kharitonov.m*, so you do not have to worry about them.

For each of the 4 *Q*
arrays (one at a time), generate the value set and Kharitonov polynomials. A
plot of the value sets is made automatically for each *Q*. In addition to
looking at the entire value set plot, you will need to zoom in on the origin to
determine the characteristics of the plot in sufficient detail to draw
conclusions about robust stability. For each of the polynomial families:

- Determine whether or not the family is robustly stable by examination of the plot of the value set. Describe how you are able to reach that conclusion. Provide plots of the value sets, including several zoomed-in plots.
- Determine the roots of the 4 Kharitonov polynomials, and compare the locations of the roots with your conclusion about robust stability from the value set plot.
- Compare the locations of the roots of the 4 Kharitonov polynomials with the plot of the value set, and see if there is any relationship between how close the roots are to the imaginary axis and how close the value sets come to the origin of the complex plane.

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*Lastest revision on
Friday, May 26, 2006 9:41 PM
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