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eigen value-equilibrium points.m
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eigen value-equilibrium points.m
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1;
## Function : find_equilibrium_points()
## ----------------------------------------------------
## Input: x1_dot, x2_dot (Both are symbolic functions defined in terms of x1
## and x2)
## Output: eqbm_points ( Solution in form of a matrix in which each entry is a
## solution)
## Purpose: This function takes the symbolic functions x1_dot and x2_dot and
## converts them into equations where x1_dot = 0 and x2_dot = 0. Then
## it proceeds to solve the equation (by calling the solve() function
## in octave) and returns all possible values of x1 and x2 in form of a
## matrix.
function eqbm_points = find_equilibrium_points(x1_dot, x2_dot)
x1_dot == 0;
x2_dot == 0;
################## ADD YOUR CODE HERE ######################
eqbm_points =solve(x1_dot,x2_dot);
############################################################
endfunction
## Function : find_jacobian_matrices()
## ----------------------------------------------------
## Input: eqbm_points, x1_dot, x2_dot (eqbm_points is matrix generated in
## find_equilibrium_points() function.
## Output: jacobian_matrices
## Purpose: This function takes the eqbm_points and x1_dot and x2_dot as input.
## It computes the jacobian matrix for x1_dot and x2_dot (It should be
## (2x2 symbolic array). Then it substitutes the calculated values of
## x1 and x2 for each of the equilibrium_points (stored in eqbm_points
## variable). The jacobian_matrices variable is a cell array in which
## each element is a 2x2 J matrix calculated for each of the corresponding
## equilibrium_points.
## Hint: It might be helpful to look up jacobian() and subs() function in octave.
function jacobian_matrices = find_jacobian_matrices(eqbm_points, x1_dot, x2_dot)
syms x1 x2
solutions = {};
jacobian_matrices = {};
for k = 1:length(eqbm_points)
x_1 = eqbm_points{k}.x1;
x_2 = eqbm_points{k}.x2;
soln = [x_1 x_2];
solutions{k} = soln;
endfor
################## ADD YOUR CODE HERE ######################
j1=jacobian(x1_dot,[x1 x2]);
j2=jacobian(x2_dot,[x1 x2]);
for i=1:length(eqbm_points)
f1=double(subs(j1,[x1 x2],solutions{i}));
f2=double(subs(j2,[x1 x2],solutions{i}));
final=[f1;f2];
jacobian_matrices{i}=final;
endfor
############################################################
endfunction
## Function : check_eigen_values()
## ----------------------------------------------------
## Input: eqbm_points, jacobian_matrices
## Output: eigen_values , stability
## Purpose: This function takes the eqbm_points and jacobian_matrices as input.
## For each jacobian matrix stored in jacobian_matrices, the eigen values
## of matrix are calculated and stored in the cell array eigen_values.
## Subsequently the eigen values are checked in this function. If for any
## jacobian matrix the eigen values have positive real part, the system is
## unstable at the corresponding equilibrium point. If all eigen values
## have negative real part, the system is stable at the corresponding
## equilibrium point.
## Hint: It might be helpful to look up eig() function in octave
function [eigen_values stability] = check_eigen_values(eqbm_pts, jacobian_matrices)
stability = {};
eigen_values = {};
for k = 1:length(jacobian_matrices)
matrix = jacobian_matrices{k};
flag = 1;
################## ADD YOUR CODE HERE ######################
eigen_values{k}=eig(matrix);
for i=1:length(eigen_values{k})
if real(eigen_values{k}(i))>0,
flag=0;
endif
endfor
############################################################
if flag == 1
fprintf("The system is stable for equilibrium point (%d, %d) \n",double(eqbm_pts{k}.x1),double(eqbm_pts{k}.x2));
stability{k} = "Stable";
else
fprintf("The system is unstable for equilibrium point (%d, %d) \n",double(eqbm_pts{k}.x1),double(eqbm_pts{k}.x2));
stability{k} = "Unstable";
endif
endfor
endfunction
## Function : main_function()
## ----------------------------------------------------
## Input: x1_dot, x2_dot
## Output: equilibrium_points, jacobian_matrices, eigen_values , stability
## Purpose: It takes x1_dot and x2_dot as argument. First the equilibrium points
## are calculated. For each equilibrium point, the jacobian matrix is
## calculated. Then the stability for each equilibrium point is determined
## by computing the eigen_values and checking the real parts of eigen
## values.
## This equation returns equilibrium_points, jacobians, eigen_values,
## stability. Each of these is a cell array.
## YOU ARE NOT ALLOWED TO MAKE ANY CHANGES TO THIS FUNCTION
function [equilibrium_points jacobians eigen_values stability] = main_function(x1_dot, x2_dot)
pkg load symbolic # Load the octave symbolic library
syms x1 x2 # Define symbolic variables x1 and x1
equilibrium_points = find_equilibrium_points(x1_dot, x2_dot);
jacobians = find_jacobian_matrices(equilibrium_points, x1_dot, x2_dot);
[eigen_values stability] = check_eigen_values(equilibrium_points, jacobians);
endfunction