The system identification literature provides a host of different Black-box mod- elling structures including Volterra series [27], non-linear autoregressive moving average with exogenous input (NARMAX) [28, 72], neural networks [29, 75] and fuzzy models [71, 117, 120] to name a few. All of the above structures can be unified by considering them as a basis function expansion of some function that maps the past data to the space of the output [113].
The system identification task is then to find some mapping from the past inputs, uk−1 = [u1 u2 . . . uk−1]T and outputs, yk−1 = [y1 y2 . . . yk−1]T to future outputs
yk = f(uk−1, yk−1) +ek (3.4) where uk and yk are the input and output at sample k. and ek is the error between the predicted output f(uk−1, yk−1)and the observed output yk at sample k.
This thesis is primarily concerned with the identification and analysis of non- linear models using the NARX/NARMAX modelling framework, however an as- pect of the analysis of linear models is discussed in Chapter 7. The discussion is therefore focussed on models of the NARX/NARMAX form.
3.3.1 Linear black-box models
The traditional, and widely used, representation of linear Black-box models is given by [75]
where A(q) =1−a1q−1−. . .−anaq−na B(q) =b1q−1+. . .+bnbq−nb C(q) =1+c1q−1+. . .+cncq−nc (3.6) D(q) =1+d1q−1+. . .+dndq−nd F(q) =1+ f1q−1+. . .+ fnfq−nf
where a, b, c, d and f ∈R are the model parameters that weight the relevant lagged input, output or error term. ekis assumed to be drawn from a zero mean indepen- dent identically distributed (i.i.d.) white noise sequence, q represents the forward shift operator such that q−1y
k = yk−1. The choice of A, B, C, D and F defines a number of different linear model structures of which some of the most common are; autoregressive (AR), autoregressive with exogenous input(ARX) and autore- gressive moving average with exogenous input (ARMAX) which are given by the following equations AR model: B(q) =0, C(q) =D(q) =1 yk =a1yk−1+a2yk−2+. . .+anayk−na+ek (3.7) ARX model:C(q) =D(q) =F(q) =1 yk =a1yk−1+. . .+anayk−na+b1uk−1+. . .+bnbuk−nb (3.8) ARMAX model: D(q) =F(q) =1 yk =a1yk−1+. . .+anayk−na+b1uk−1+. . .+bnbuk−nb +ek+c1ek−1+. . .+cncek−nc (3.9)
Note that the linear models defined by equations (3.7)-(3.9) are all simple basis function expansions in the form of Equation (3.4). The AR and ARX models de- fined above are linear-in-the-parameters allowing the use of simple optimal param- eter estimation techniques such as least squares. In contrast the ARMAX model includes lagged error terms causing it to be non-linear in the parameters requiring a more complicated estimation procedure.
3.3.2 Non-linear black-box models
If the system to be modelled displayed non-linear behaviour, such as is discussed in Section 3.1.2, then in order to accurately model the system dynamics, the model used to represent it must also be non-linear. The NARX model [72] provides a simple non-linear extension to the ARX model given by Equation (3.8). The NARX model is given by
yk = f(xk) +ek (3.10)
where xk = (yk−1, . . . , yk−ny, uk−1, . . . , uk−nu), and nu, ny represent the maximum lags in the input and output respectively. The non-linear function f(·)is decom- posed into a sum of weighted basis functions such that
f(xk) = M
∑
m=1 θmφm(xk) (3.11) =φkθ (3.12) where θ= h θ1, θ2, . . . , θM iT ∈R (3.13) φk = h φ1(xk), φ2(xk), . . . , φM(xk) i (3.14) where θm is the m’th model parameter, φm(xk) the m’th basis function and m is the total number of model terms. Substituting Equation (3.12) into Equation (3.10) and then considering the entire data record, the vector of system outputs is given in matrix form asy=Φθ+e (3.15)
where y = [y1, y2, . . . , yN]T, e = [e1, e2, . . . , eN]T and Φ is known as the regression matrix and is given by
Φ=hφ1T, φ2T, . . . , φTN
iT
. (3.16)
The NARX model basis functions can take various forms including polyno- mial, wavelet, or radial functions [14]. Although f(·) is a non-linear function it still maintains the property of linear-in-the-parameters providing that the basis functions are linearly independent of each other.
The non-linear extension to the ARMAX model is the NARMAX model [72] given by Equation (3.10) where instead
As before with the NARX model, the NARMAX model is also represented by the linear sum of basis function given by Equation (3.12). Like the ARMAX model however, the NARMAX model is not linear-in-the-parameters because the error terms depend on the model output resulting in biased parameter estimates.
In this thesis modelling is performed using models of the NARMAX class. This choice of model class is made based on some of the advantages the NARMAX methodology possesses. One of the key advantages of NARMAX is the wealth of supporting literature available for both identification and analysis purposes. More specifically however, NARMAX models are capable of modelling a wide range of non-linear systems with a compact, parsimonious model description. Perhaps most importantly for this thesis, NARMAX models can be mapped directly into the frequency domain as GFRFs allowing non-linear frequency domain model based analysis, the GFRF concept will be explained in Chapter 4.