Math 448/548 - Example 5.2 - Docking Problem Revisited

This Matlab code illustrates the Eigenvalue Method for studying stability of steady states for DISCRETE-TIME dynamical system described in the docking problem.

Contents

Declare the variables

clear all, close all, clc

syms x1 x2
c = 5; % time to make a control adjustment
w = 10; % delay time before the next observation
k = 0.1; % ratio between acceleration and velocity (control law)

The discrete-time dynamical system is given by delta_x1=f1; delta_x2=f2 where

f1 = -k*w*x1-k*c*x2;
f2 = x1-x2;

or, equivalently, x1=g1(=x1+f1), x2=g2(=x2+f2):

g1 = x1+f1;
g2 = x2+f2;

Stability Analysis via Eigenvalue Method

First, compute the Jacobian matrix

A = [diff(g1,x1), diff(g1,x2);
     diff(g2,x1), diff(g2,x2)];

Eigenvalues are

lambda = eig(A)'
 
lambda =
 
[ -1/2*i*2^(1/2),  1/2*i*2^(1/2)]
 
 

and have modulus

double(abs(lambda(1))), double(abs(lambda(2)))
ans =

    0.7071


ans =

    0.7071

Conclusion: Based on the eigenvalues obtained above, one concludes that the origin (0,0) is a stable (sink) equilibrium. You can also visualize this by plotting a few trajectories of the dynamical system, e.g. using pplane7.m