Given the following ARMAX model
for q−1 the backward shift (delay) operator such that
(a) Find the pulse transfer representation of this process (C = D = 0). Convert it to the following equivalent pole-zero and normal state-space forms. Is the system controllable? Is it observable? Show your calculations.
(b) Find the pole-zero or ARX representation of this process (C = 1, D = 0). Convert it to the equivalent state-space form.
(c) Find the pole-zero or ARMAX representation of this process (D = 0). Convert it to the equivalent state-space form.
(d) Find the all-zero or FIR representation of this process (A = 1, C =D = 0). Convert it to the equivalent state-space form.
(e) Find the all-pole or IIR representation of this process(B = 0, C = 0, D = 0). Convert it to the equivalent state-space form.
(f) Find the all-zero or MA representation of this process(A = 1, B =0, D = 0)). Convert it to the equivalent state-space form.
(g) Using the full model above with A, B, C, D polynomials, is it possible to find and equivalent Gauss-Markov representation? If so, find it and convert to the equivalent state-space form. (Hint: Consider the C/D polynomials to be a coloring filter with input
ε
(t) and output e(t).)