%
%  A.A. Rodriguez
%  EEE480: Feedback Systems
%  Copyright (c) Spring 1998
%

%
% Use Routh Table to analyze the following polynomials.
% These poynomials were taken from 
%
% Katsuhiko Ogata
% Modern Control ENgineering
% Third Edition
% Prentice Hall
% 1997
%
% pages 232-238
%

%
% EXAMPLE 1
%
roots( [ 1 2 3 4 5])

% It's roots are
%                0.2878 + 1.4161i
%                0.2878 - 1.4161i
%               -1.2878 + 0.8579i
%               -1.2878 - 0.8579i
%
% Form Routh table to get:
%                              s^4:  1   3   5
%                              s^3:  2   4
%                              s^2:  1   5
%                              s^1: -6
%                              s^0:  5
%
% 2 sign changes in first column idicate presence of 
% 2 right half plane roots.
%


%
% EXAMPLE 2
%
roots([1 2 1 2])

% It's roots are
%                 -2.0000          
%                  0 + 1.0000i
%                  0 - 1.0000i
%
% Routh table will have a row of zeros.
%
% Use the row above the row of zeros to form the so-called auxilliary poynomial.
% Forming the auxilliary polynomial (2s^2 + 2 = 0) will those roots which are
% symmetric with respect to the origin; in this case we have roots on the 
% imaginary axis.
%
% Differentiate the auxilliary polynomial with respect to s  to get 4s and
% replace the row of zeros with the resulting coefficeints; in this case 4.
%
% Complete the Routh table.
% There will be no sign changes in the first column.
%

%
% EXAMPLE 3
%
roots([1 0 -3 2])

% It's roots are
%                  -2.0000          
%                   1.0000 + 0.0000i
%                   1.0000 - 0.0000i
%
% Form the Routh table. You will get a zero in the first column.
% Replace the zero with a small positive quantity epsilon.
% Complete the Routh table.
% You will conclude that there are 2 sign changes in the first column.
%


%
% EXAMPLE 4
%
roots([1 2 24 48 -25 -50])

% It's roots are
%                     0.0000 + 5.0000i
%                     0.0000 - 5.0000i
%                    -2.0000          
%                    -1.0000          
%                     1.0000  
%
% Form the Routh Table
%                          s^5:   1  24  -25
%                          s^4:   2  48  -50
%                          s^3:   0   0
%
% You will get a row of zeros.
% Go to the row above the row of zeros and form
% the auxilliary polynomial: P(s) = 2s^4 + 48s^2 - 50 = 0
% Find its roots.

roots([2 0 48  0 -50])
%
%
% It's roots are:
%                 0 + 5.0000i
%                 0 - 5.0000i
%                 1.0000          
%                -1.0000  
%
% The auxilliary polynomial gives us roots of the original polynomial
% which are symmetric with respect to the origin.
%
% Differentiate the auxilliary polynomial with respect to s to get:
% 8s^3 + 96 s. Use the cofficients of this polynomial to replace the original 
% row (s^3 row) of zeros. 
% Complete the Routh table using this new row.
% This yields the following:
%
%                          s^5:    1     24    -25
%                          s^4:    2     48    -50
%                          s^3:    8     96
%                          s^2:   24    -50
%                          s^1:  112.7
%                          s^0:  -50
%
% The table shows 1 sign change in the first column from which we 
% conclude that there is 1 root in the right half plane.