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Disturbances d(i), residuals e(t) = y(t)—y(t), and random noise attributed to inputs, outputs and state variables are also represented by column vectors with appropriate dimensions in a similar manner.

4.3.1 Time Series Models

Time series models have been popular in many fields ranging from modeling stock prices to climate. They could be cast as a regression problem where the regressor variables are the previous values of the same variable and past values of inputs and disturbances. They are also called black box models because they describe the relationship of the present value of the output to external variables but do not provide any knowledge about the physical description of the processes they represent.

A general linear discrete time model for a single variable y(t) can be written as y(t) = rj(t) + w(t) (4.28)

where w(t) is a disturbance term such as measurement noise and r)(t) is the noise-free output r](t) = G(q,6)u(t) (4.29)

with the rational function G(q, 6) and input u(t). The function G(q,6) relates the inputs to noise-free outputs whose values are not known because the measurements of the outputs are corrupted by disturbances such as measurement noise. The parameters of G(q, 6) (such as bi in Eq. 4.30) are represented by the vector 9, and q is called the shift operator (Eq. 4.31). Assume that relevant information for the current value of output y(t) is provided by past values of y{t) for ny previous time instances and past values of u(t) for nu previous instances. The relationship between these variables is v{t) + fMt - 1) + • • • + fnvT]{t - ny)

= biu(t) + b2u(t - 1) + • • • + bnuu(t - (nu - 1)) (4.30)

where i = 1,2, ...,ny and bi, i = 1,2,... ,nu are parameters to be determined from data. Defining the shift operator q as y(t-\) = q~ly{t) (4.31)

Eq. (4.30) can be written using two polynomials in q

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