discrete time histories generation

how can I generate discrete time series? for part C (image below) this is my attempt:

omega  = sqrt(2);
zeta   = 1/sqrt(2);
sympref('AbbreviateOutput', false);

syms x(t) y(t) u
eqn    = diff(x, 2) + 2*zeta*omega*diff(x) + (omega^2)*x == (omega^2)*u;
[V, S] = odeToVectorField(eqn)


A = [0 0; -2 -2];
B = [2; 0];
C = [1 0];
D = 0;
sys = ss(A, B, C, D)

However, for part D, not sure what would be a right approach? Any help will be greatly appreciated. Thanks much.

Answer 1

[arushi]

Hi Cesar, 

 I understand that you want to generate the discrete time series for part D using the results of the variables in part C.

Please find the code below for the solution of part C in which the values A, B, C, D are used as calculated in the part C :

% Define the parameters
omega = sqrt(2);
zeta = 1/sqrt(2);
w = 2*pi/100;
N_TI = 100;
N_T2 = 50;

% Initialize variables
x_TI = zeros(1, N_TI);
x_T2 = zeros(1, N_T2);

% Generate discrete-time histories for TI system
for k = 3:N_TI
    u = sign(cos(2*w*k)); % Input signal
    x_TI(k) = -2*x_TI(k-1) - 2*x_TI(k-2) + 2*u; % Update state variable
end

% Generate discrete-time histories for T2 system
for k = 3:N_T2
    u = sign(cos(2*w*k)); % Input signal
    x_T2(k) = -2*x_T2(k-1) - 2*x_T2(k-2) + 2*u; % Update state variable
end

% Plot the results
k_TI = 0:N_TI-1;
k_T2 = 0:N_T2-1;

figure;
hold on;
plot(k_TI, x_TI, 'b', 'LineWidth', 1.5);
plot(k_T2, x_T2, 'r', 'LineWidth', 1.5);
xlabel('k');
ylabel('x(k)');
legend('TI system', 'T2 system');
title('Discrete-time histories');

Hope this answer helps.