How do i find a difference equation?

I have this first order differential equation given:

i need to find a difference equation using MATLAB. I would appreciate if someone could help me with that. Thanks

[Wayne King]

what is u(t) here? is that the step function?

[LeBron]

its the input

Answer 1

[Azzi Abdelmalek]

Choose your sample time Te

Te=0.1                     % Your sample time
N=1.5
D=[1 -0.5]
Modelc=tf(N,D)             % Your continue transfer function
Modeld=c2d(Modelc,Te)      % Your discrete transfer function
[Nd,Dd]=tfdata(Modeld,'v');

% From Y(z)/U(z)=Nd(z)/Dd(z) you can find the relation between y{n] and u[n]

[LeBron]

I am kinda confused i need to get the answer in this form: y(n+1)=1.05 y(n) + 0.15 u(n) as in previous question this is what i calculated without using Matlab

[Azzi Abdelmalek]

No, when you discretize a continuous system you have to take in account the two blocks ADC and DAC. If H(p) is the transfer function of your continuous system then, if you discretize, using a zoh method: the discrete transfer function G(z) will be

G(z)=(z/(z-1))*Ztransform(inverse_transform(H(p)/p))

[LeBron]

It would be better if you tell me exactly what to type in matlab so i should type everything u have told me in first post plus this last thing G(z)=(z/(z-1))*Ztransform(inverse_transform(H(p)/p))

then i should get my final answer?

[Azzi Abdelmalek]

LeBron, I gave you the code which allows to find your discrete transfer function G(z). Do you know how to find the difference equation from G(z)? Other thing

G(z)=(z/(z-1))*Ztransform(inverse_transform(H(p)/p))

is not a code, it's what the above code is doing

[Roger Stafford]

I would advise you to look at the article

http://en.wikipedia.org/wiki/Runge–Kutta_methods

It gives an explanation of various Runga-Kutta methods of approximating the solution to ordinary differential equations of the kind you have. The discussion of RK4 shows you one method which is a fourth order approximation wherein it is assumed you can sample your u(t) at every h/2 interval with a step size of h in t. The f(t,y) they describe would be

 f(t,y) = 1.5*u(t)+.5*y

in your case, and the corresponding difference equations are given there.

The article also describes other possible kinds of difference equations that one can use with differing amounts of accuracy.

[Azzi Abdelmalek]

He is asking for a difference equation, not a numeric resolution.

[LeBron]

Azzi, No, I dont know how to find the difference equation from G(z). Thanks

[Azzi Abdelmalek]

For example

       1       Y(z)
G(z)=----- = ------
      z + 2    U(z)

Then

Y(z)(z+2)=U(z)

zY(z)+2Y(z)=U(z)

y[n+1]+2y[n]=u[n]

[Roger Stafford]

Azzi, the Runga-Kutta method does give a set of difference equations, in fact a very respectable set indeed.

[LeBron]

Hey Azzi thank you so much for all your help. I really appreciate it. People like you make this forum a great place.