Generating the four Kharitonov polynomials and displaying corresponding Hurwitz stability matrix

by Nasser M. Abbasi
Nov 27, 2014

Introduction

Software written in Mathematica to generate the four Kharitonov’s  polynomials from the interval polynomial specification and construct the four Hurwitz stability matrices to test for stability of each polynomial. Examples from chapter 5, “New tools for robustness of linear systems” by Professor B. Ross Barmish are used for illustration.

Example 5.5.2

This function takes interval polynomal and generates the 4 Kharitonov’s polynomials

This function takes the result and generate the Hurwitz matrix and root locations. The polynomial is stable when all leading minors are positive.

Hurwitz Matrix root locations Real part of roots
 9 6 -108 -196 -196
 -1.71185 -1.00624 -1.00624 0.362161 0.362161
 10 10 -110 -196 -392
 -1.5452 -0.854858 -0.854858 0.627459 0.627459
 9 -9 -270 -729 -729
 -2.43433 -1.25737 -1.25737 0.474539 0.474539
 10 25 45 -89 -178
 -1.3877 -0.672514 -0.672514 0.616365 0.616365

Example 5.6.2

Hurwitz Matrix root locations Real part of roots
 0.75 2.125 2.65625
 -2.38347 -0.108264 -0.108264
 1.25 3.125 0.78125
 -10.5718 -0.21411 -0.21411
 0.75 0.5 0.625
 -2.13812 -0.0309384 -0.0309384
 1.25 4 1
 -12.6098 -0.195114 -0.195114

Example 5.10.1

Hurwitz Matrix root locations Real part of roots
 1.95 3.225 7.9575 9.39937 6.41034 6.41034
 -3.2334 -0.299508 -0.116271 -0.116271 -0.0922772 -0.0922772
 2.05 2.775 4.0575 2.49938 0.404656 0.404656
 -3.3032 -0.338496 -0.1981 -0.1981 -0.00605111 -0.00605111
 1.95 2.425 3.5075 3.11438 2.34509 2.34509
 -3.20234 -0.356709 -0.173359 -0.173359 -0.0221182 -0.0221182
 2.05 3.575 8.4075 7.81438 2.68991 2.68991
 -3.33369 -0.281521 -0.17061 -0.17061 -0.0467823 -0.0467823

page 71 example, in conlcusion

Hurwitz Matrix root locations Real part of roots
 2.5 16.3945 76.3375 76.3375
 -2.70998 -1.89535 -0.216088 -0.216088
 9.5 42.3173 119.115 119.115
 -4.33442 -0.269038 -0.269038 -0.0750017
 2.5 8.56797 36.9111 36.9111
 -3.9738 -0.568926 -0.247385 -0.247385
 9.5 66.7352 239.922 239.922
 -3.69136 -0.611201 -0.611201 -0.033736