plot the output c(t) using mathlab and show setting time on you graphf

syms t tau;
A = [0 2; -2 -5];
B = [0; 1];
C = [2 1];
x0 = [1; 2];

Phi_t = expm(A*t);

x_h = Phi_t * x0;

u_tau = 1; % Unit step function
x_p = int(Phi_t * B, tau, 0, t);

x_t = x_h + x_p;

y_t = C * x_t;

disp('State-transition matrix Phi(t):');

State-transition matrix Phi(t):

disp(Phi_t);

fplot(Phi_t(1,1))

disp('Homogeneous solution x_h(t):');

Homogeneous solution x_h(t):

disp(x_h);

fplot(x_h)

disp('Particular solution x_p(t):');

Particular solution x_p(t):

disp(x_p);

disp('Full state vector x(t):');

Full state vector x(t):

disp(x_t);

disp('Output y(t):');

Output y(t):

disp(y_t);

Answer 1

[Torsten]

Use "fplot" as done in your code above.

[cf]

A=1

B=-1

inverse laplance of 1/s=1

inverse laplance of 1/(s+5)=e^-5t

c(t)= 1+e^-5t

setting time is 0.7832

[Sam Chak]

Hi @cf

The system you originally provided in your question is linear and the input signal is a unit step function. However, there is discrepancy in the results. Can you rectify the issue?

syms t tau;
A = [0 2; -2 -5];
B = [0; 1];
C = [2 1];
x0 = [1; 2];

Phi_t = expm(A*t);

x_h = Phi_t * x0;

u_tau = 1; % Unit step function
x_p = int(Phi_t * B, tau, 0, t)

x_p = 

x_t = x_h + x_p;

y_t = C * x_t;

% disp('State-transition matrix Phi(t):');
% disp(Phi_t);
% fplot(Phi_t(1,1))
% 
% disp('Homogeneous solution x_h(t):');
% disp(x_h);
% % fplot(x_h)
% 
% disp('Particular solution x_p(t):');
% disp(x_p);

% disp('Full state vector x(t):');
% disp(x_t);

disp('Output y(t):');

Output y(t):

disp(y_t);

figure
fplot(y_t, [0, 6]), hold on

%% parameters
A       = [0, 2; -2, -5];
B       = [0; 1];
C       = [2, 1];
x0      = [1; 2];   % initial values: x1(0) = 1, x2(0) = 2
u_tau   = 1;        % Unit step function

%% state-space representation
function [dxdt, y] = stateSpace(t, x, A, B, C, u_tau)
    dxdt = A*x + B*u_tau;   % state  equation
    y    = C*x;             % output equation, check: y(0) = 2*x1(0) + 1*x2(0) = 4
end

%% call ode45 solver
tspan   = [0, 6];
[t, x]  = ode45(@(t, x) stateSpace(t, x, A, B, C, u_tau), tspan, x0);
[~, y]  = stateSpace(t', x', A, B, C, u_tau);
plot(t, y, '-.', 'linewidth', 1.5, 'color', '#FA477A'), grid on, xlabel('t'), ylabel('y(t)')
legend('Manual Integration', 'Numerical Integration')
title('Output response, y(t)')