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A natural development of the output error model is to further the properties of the output error. This can be done by assuming that the true process is
vector ϕ(t) and a parameter vector θ Such a model is called
B(q)
y(t) =
C(q)
a linear regression in statistics and the vector ϕ(t) is called regression vector. Fig. 2 displays the ARX modeled data versus the actual data.
where
u(t) + e(t) (6)
F (q) D(q)
3.2 ARMAX modeling
F (q) = 1 + f1q−1
+ . . . + fnf q
−nf
D(q) = 1 + d1q−1 + . . . + d
q−nd
The basic problem with the ARX model is the lack of adequate freedom in describing the properties of the disturbance term. We could add flexibility to that by describing the equation error as a moving average of white noise. This gives the model:
y(t) + a1y(t − 1) + . . . + ana (t − na)
= b1u(t − 1) + . . . + bnb u(t − nb) + e(t) (4)
e(t) + c1e(t − 1) + . . . + cnc e(t − nc) It can be rewritten as
A(q)y(t) = B(q)u(t) + C(q)e(t)
where
In a sense, this is the most natural finite-dimensional parameterization and the transfer functions G and H are independently parameterized as rational functions [9]. In this case the parameter vector is given by
θ = [b1 . . . bnb f1 . . . fnf c1 . . . cnc d1 . . . dnd ]
Fig. 4 displays the BJ modeled data versus the actual data.
3.4 State Space model
In state-space form, the relationship between the input, noise and output signals is written as a system of first order differential or difference equations using an auxiliary state vector x(t). For most physical systems it is easier to construct models with physical insight in continuous
and
C(q) = 1 + c1q−1 + . . . + c
q−n
time than in discrete time, simply because most laws of physics are expressed in continuous time. This means that the modeling normally leads to a representation
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. 4. BJ modeled data (- - -) v/s actual data (—)
x˙ (t) = F (θ)x(t) + tt(θ)u(t) (7)
Here F and tt are matrices of appropriate dimensions (n× n and n× m, respectively for an n-dimensional system
Fig. 5. State-Space modeled data (- - -) v/s actual data (—)
. 1.0324 −0.1613 .