So I've been doing a couple of these type of questions. However i am stuck on this one!
Consider the ARMA model $$y(t + 2) + y(t + 1) −0.75y(t) = u(t + 1) + 4u(t)$$
Is it stable; is it invertible?
Convert this ARMA model to a state space model.
Find the general solution of this ARMA model in the case $$u(t) = (−2)t$$ . Find the particular solution corresponding to the initial conditions $$\cases{y(0) = −1 \\ y(1) = 1}$$
So the bit im stuck on is when finding if it is stable on not its a fraction in the quadratic because is gives you $0(B)=1+B-0.75B^2$.
How do you solve this?
This is my answer for the first section I used the quadratic formula to solve $1+B-0.75B^2$. This gave me $-2/3$ and 2, because one of the modulus is smaller than 1 it isn't stable.
Then to find if it is invertible I used the second half of the ARMA model and got the 1-4B=0, thus $B=1/4$, showing it isnt invertible.
To convert it into a arma model is an easy part, however finding the general solution was not. Ive had a go and got $y(t)=A(1/2)^t+B(-3/2)^t+8/5(-2)^t$.
When using the inital conditions i got $y(t)=3/20(1/2)^t-11/4(-3/2)^t+8/5(-2)^t$ which really doesn't sound right! any help would be very appreciated