How to deal with the time response of first order system?

There is a system of differential equations:

x'=ax+by,

y'=cx+dy,

where tau*a'+a=a*. I used to deal with the time response tau in simulink.

Now how to deal with the time response of first order system in matlab by code? Is there a way to solve the problem? Thank you.

Answer 1

[Sam Chak]

@Cola, let's correct your Simulink model first.

The ODE is given by

which can be rearranged into

Integrating both sides and is obtained:

Technically it means that the signal after the integrator block (1/s) is , and that is the output of the system.

To obtain , you need to properly get the integrand, a function that is to be integrated:

So you should perform the subtraction first, and then it multiply with using the Gain block. The signal from the Gain block is fed into the Integrator block (1/s).

If , then the MATLAB code looks something like this:

tau = 1;

% 1st-order Ordinary differential equation

fcn = @(t, x) [(1 - x)/tau];

tspan = [0 10];

init = 0; % initial condition

% Runge-Kutta Dormand-Prince 4/5 solver

[t, x] = ode45(fcn, tspan, init);

plot(t, x)

grid on

xlabel('t')

ylabel('a(t)')

title('Time response of the system')

Result:

[Cola]

@Sam Chak Thank you very much. Your answer is so good and detailed. Thus we can deal with the problem by solving the ODE.