Sketching a zero-input response for an Electrical Engineering problem

Hi everyone, I'm relatively new to MATLAB, so I don't know if anyone can help me. I'm trying to sketch the zero-input response for a system. The system can be described by

(E^2 - 1.56E + 0.81)y[k] = (E + 3)f[k]

with the following conditions: y[-1]=2, and y[-2]=1.

The following code is what I am trying to trouble shoot:

y=[1 2]; 
y=y';
k=-2:16; 
k=k';
for m=1:length(k)-2

Y=1.56*y(m+1)-0.81 *y(m);
Yzi=[Yzi;Y];

end

stem(k,Yzi)

Does anyone know what I am doing wrong?

Answer 1

[Star Strider]

I have no idea how you’re supposed to solve it.

I am not certain what the assignment actually wants you to do. This would be my approach (in terms of powers of 1/z):

a = flip([1  -1.56  0.81]);
b = flip([1  3]);
yi = flip([1 2]);
k=-2:16;
f = filter(b,a,k,yi);

figure(1)
stem(k, f)
grid

But then I’m known to take liberties with problems I don’t understand.

[Orlando Salas]

The question specifically asks this:" Find and sketch the zero-input response for the system described by:

(E^2 - 1.56E + 0.81)y[k] = (E + 3)f[k]

using the initial conditions y[-1] = 2, and y[-2]=1" and I don't know if we can use z-transformations on this problem.... However, you sound knowledgeable on this topic. :D, how about trying to solve a problem like, :

" Using the classical method, find the zero input response for this differential equation: (E^2 + 1)y[k]=(4E-0.7)f[k] with the following conditions y[-3]=1, y[-1] = 0"

[Star Strider]

The term ‘classical method’ leaves much to the imagination. It would help if you could describe it.

My inclination would be to use the z-transform of your equations, then take the inverse to solve the differential equation. It’s been decades since I‘ve dealt with difference equations and their solutions.

[Orlando Salas]

Sorry about that, does the term term "natural-response" help? It's supposed to be similar to the zero-input response. I'm pretty sure that I am looking for yn[k]. My system of equations is supposed to satisfy Q[E](yn[k] + ytheta[k]) = P[E]f[k] when Q[E]yn[k]= 0. I know how to solve it manually, but I am wondering if it is possible to solve via MATLAB....

Edit: I forgot to mention that ytheta[k] represents a forced response rather than a zero-state response.