Assume that we have a discrete transfer function $H(z)$ and a discrete state space model $x(k+1) = Ax(k) + Bu(k)$.
I know how to check the stability, but computing the eigenvalues from the denominator of transfer function $H(z)$ and the state space model, I can just check it's eigenvalues as well.
But what if I only want the answer "Is it stable or not stable?". A boolean answer.
I know that taking the determinant of the time continous state space model matrix $A$ will give me a boolean answer. If the determinant value is negative, then the model is unstable. If the determinant value is positive, then the model is stable.
But in this case, I have only the discrete state space model and the discrete transfer function to use. How can I determine a boolean true/false if those models are stable or not stable?
Reason why I'm asking
I'm using recursive least square to estimate a SISO transfer function and I have an idea. If the estimated SISO transfer function is not stable, input some PWM onto the input $u(k)$ because recursive least square needs some stochastic signals to work, as I have heard/read i the books.
But is that true? I tried to do a simulation if a long constant step response of a second order model with high damping. Here we can see that I don't need "stochastic" output signals to be able to identify the system with recursive least squares. Sure, I know that everything looks perfect in a simulation. But in real world, I would have some noise onto my measurement for sure.
And the result where, with a 10:th order transfer function.
If I try to implement heavy random noise.
We can see that everything went just fine.
Even when I changed the input signal $u(k)$
Source to the library I'm using. I also made a C-version too with the same arguments