13.2 BILINEAR MODEL 
As the simplest example of a bilinear model, we take the first order model 
X141 = AX, + DE p41 + CE Xy (13.7) 
Priestley,~> in studying the bilinear model, assumed that Eq. 13.7 can be expanded into 
the form of Voltera functions, Eq. 11.8 as, 
X= » Suu + y ier Ue Uae 904 (13.8) 
u=o u=ov=o0 
Here, the general transfer functions are 
Ty(@1) = »> BO 
u=o 
oo fo) 
IT3(@1,@2) = S. Sy Luv QED | (13.9) 
u=ov=0 
Assuming Eq. 13.7 is expanded into the form of Eq. 13.8, he derived I"j(), setting 
€,= ein Eq. 13.7, as 
10 
T1(@|1) = ———_. (13.10) 
1(@1) (Ea) 
Similarly setting €, = e'?' + e@2", 
G 
T3(@1,@2) = eterno al {T(@1) +T(@2)} 
Cerri -a} 
1 iw ww 
im =cb e1 e2 
{e wren) —a} e_q e?_a@ 
Particularly along the diagonal w, = w2=a, 
che 
I,@,0) =—>———_. - (13.12) 
2 ) (e* — aye” —a) 
More generally, Priestley showed that 
EMD G2 
TB OS) (13.13) 
(Chie 2) Cea) cn Con) 
From these relations, he found that '}(w1,w2) included all the a, b, and c of 
Eq. 13.7. Thus he says, the bilinear process expressed by Eq. 13.7 is invertible to the 
functional polynomial model of Eq. 13.8. 
This bilinear model can also be approximated he says, by the generalized autore- 
gressive model. The first order bilinear model can be expressed by setting b = 1 in 
Eq. 13.7 (without losing generality), 
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