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The statement "order of the polynomial B(q) + 1" in the documentation is somewhat ambiguous

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The following statement appears in a number of places in the documentation and refers to the nb parameter of discrete linear systems represented by polynomials, such as ARX, ARMAX, BJ, etc.:
"nb is the order of the polynomial B(q) + 1"
where B(q) is defined as follows for the ARX model for example:
A(q) y(t) = B(q) u(t-nk) + e(t)
However, I find this statement ambiguous. Does it mean "(the order of the polynomial B(q)) + 1", in other words nb = n + 1 where n is the order of B(q)? Or is nb the order of the polynomial (B(q) + 1), i.e. the terms of B(q) + the zeroth-order term 1? (The latter seems unlikely since adding a zeroth-order term should not change the order, if it is defined as the power of the highest term).
To confirm, in the following example, I set na=2, nb=2, nk=2:
>> sys = arx([y u], [2 2 2])
sys =
Discrete-time ARX model: A(z)y(t) = B(z)u(t) + e(t)
A(z) = 1 - 1.1 z^-1 + 0.321 z^-2
B(z) = 1.222 z^-2 + 0.8787 z^-3
Sample time: 1 seconds
Polynomial orders: na=2 nb=2 nk=2
Number of free coefficients: 4
Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.
Estimated using ARX on time domain data.
Fit to estimation data: 72.29% (prediction focus)
FPE: 1.692, MSE: 0.9112
Here, B(z) is a third order polynomial but it includes the delay of nk=2 timesteps (u(t-nk) = z^-2 u(t)) so does that mean B(q) is 1st order in this example, therefore nb = 1 + 1 = 2?
Incidentally, the definition of nb in the documentation for OE models is written in a slightly less ambiguous way:
  • "nb — Order of the B(q) polynomial + 1"
But in the code documentation for arx (help arx) it says the following. Here there is no mention of "+1":
ORDERS = [na nb nk], the orders of A and B polynomials in the arx model.
Paul on 13 Nov 2023
The link in my comment above to the "defintiton of the model" for the arx function shows the structure of the ARX model (and others) where you'll also see the effect of nk on the deterministic input.
I hope you provided feedback on this page: arx.

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