y(t) + a1y(t − 1) + . . . + ana y(t − na)

= b1u(t − 1) + . . . + bnb u(t − nb) + e(t) (1)

with y(t) the  output  signal,  and  u(t)  the  input  signal of the model, and a1, a2, . . . , ana b1, b2, . . . , bnb unknown parameters. The use of these kinds of models in estimation and identification problems is essentially based on the argument that a least  squares  identification  criterion  is an optimization problem that is analytically    solvable.

Since the white noise term e(t) here enters as a direct error in the difference equation, the model is often called an equation error model. The adjustable parameters in this case are:

y1. (sim)

8

y1. (sim)

8

6 6

4 4

2 2

0 0

−2 −2

−4 −4

−6 −6

100 200 300 400 500 600 700 800

Time (sec)

100 200 300 400 500 600 700 800

Time (sec)

Fig. 2. ARX modeled data (- - -) v/s actual data  (—)

θ = [a1 . . . ana   b1 . . . bnb ]

Fig. 3. ARMAX modeled data (- - -) v/s actual data (—)

B(q) C(q)

G(q, θ) = ; H(q, θ) =

If we introduce

A(q)

A(q)

A(q) = 1 + a1q−1 + . . . + a B(q) = 1 + b1q−1 + . . . + b

We see that the model corresponds to

B(q)

q−na q−nb

1

The predictor for the ARMAX model can be obtained as

yˆ(t|θ)  = B(q)u(t) + [1 − A(q)]y(t) (5)

+[C(q) − 1]ε(t, θ)

where

G(q, θ) =

A(q)

; H(q, θ) =

A(q)

ε(t, θ) = y(t) − yˆ(t|θ)

Computing the predictor for the system above we get

yˆ(t|θ) = B(q)u(t) + [1 − A(q)]y(t) (2)

Now we introduce the vector

ϕ(t) = [−y(t − 1) . . . − y(t − na)

u(t − 1) . . . u(t − nb)]

Then we can write the above equation in the following form

yˆ(t|θ) = θT .ϕ(t) = ϕT (t).θ (3) The predictor is a scalar product between a known data

In this case our regression vector would  be

ϕ(t) = [−y(t − 1) . . . − y(t − na)

u(t − 1) . . . u(t − nb)

ε(t − 1, θ) . . . ε(t − nc, θ)]

Fig. 3 displays the ARMAX modeled data versus  the actual  data.

3.3 Box-Jenkins model