% Fall'06 - Spring'07 16.01-16.04 MATLAB Tutorials: Linear Systems
% Instructor: Violeta Ivanova, violeta@mit.edu
% Faculty: Mark Drela, Stephen Hall, Wes Harris, Ian Waitz

% Example 1: Linear Systems: Eigenvalues and Eigenvectors
% This exercise is based on S. Hall's Signals and Systems lectures.

% C1. State space linear equations: x' = Ax
% If  x = (v; i), create the matrix A for the following system:
% dv/dt = -1/2*v - 2*i
% di/dt =  1/2*v - 3*i

A = [-1/2   -2;     1/2     -3]

% C2. Define vector x0 for initial conditions: v(0) = 2 and i(0) = 1

x0 = [2; 1] 

% C3.  Compute eigenvalues and eigenvectors of the matrix A
% Hint: use function EIG with option 'nobalance' for non-symmetric A

[eigvecs, eigvals] = eig(A, 'nobalance')

% Compute a column vector lambda including the two eigenvalues:
lambda = diag(eigvals)

% Compute two column eigvectors eigvec1 and eigvec2:
eigvec1 = eigvecs(:, 1)
eigvec2 = eigvecs(:, 2)
      
% C4. Compute v(t) and i(t) for 0 < t < 5s
% Hint: the solution has exponential form - see handout for formula

% Compute the coefficients' vector a from the initial conditions x0
% Hint: Use operator \ to solve equation x0 = eigenvectors * a
a = eigvecs \ x0

t = [0 : 0.1 : 5];
v = a(1) * eigvec1(1) * exp(lambda(1)*t) + a(2) * eigvec2(1) * exp(lambda(2)*t);
i = a(1) * eigvec1(2) * exp(lambda(1)*t) + a(2) * eigvec2(2) * exp(lambda(2)*t);

% C5. Plot v(t) and i(t) vs. time t

subplot (211);
plot(t, v);
xlabel('Time');
ylabel('Voltage');
subplot(212);
plot(t, i);
xlabel('Time');
ylabel('Current');