function [t_k,xtruth,z] = sect_6_1_gen_xz(k_1, k_2, k_3u, mu_q, var_q, R, t_0, t_final, N)
%
% Generates the truth (xtruth) and measurements (z) for the specified nonlinear model.
%
% Inputs: 
%     N = the number of measurements to process (xtruth has then one more sample corresponding to \hat{x](0))
%         between the times t_0 and t_final.
%
% Written by:
% -- 
% John L. Weatherwax                2005-08-14
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

t_k = linspace( t_0, t_final, N+1 );

% Generate the *true* state history
xtruth = zeros(1,N+1);
xtruth(1) = 0; % the initial condtion on the state
for km1 = 1:N
  % integrate the nonlinear system from t_{k-1} to t_k: 
  [t,x_t] = ode45( @(t,x) ( -k_1*x - k_2*(x^4) + k_3u + mu_q ), [t_k(km1), t_k(km1+1)], xtruth(km1) );

  % add process noise to the state: 
  deltaT = t_k(km1+1) - t_k(km1);  
  q      = (var_q/(2*(-k_1)))*(exp(2*(-k_1)*deltaT) - 1); % the process noise covariance
  w      = sqrt(q) * randn(1);
  
  % accumulate the two terms: 
  xtruth(km1+1) = x_t(end) + w;
end

% Generate the measurements (off of the true state):
z = xtruth + sqrt(R)*randn(size(xtruth));