Page 278 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

280 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics

T
∞
at the sampling interval h = , X kT (z) = x[kT]= X kT−h (z).Asdis-
n+1 k=0
cussed in [8], G kT−h (z) can be expressed as:
b 0 + b 1z −1 + ... + b n−1z −(n−1)
G kT−h (z) = . (18.12)
1 +¯a 1z −1 +¯a 2z −2 + ... +¯a nz −n
Hence, G kT (z) and G kT−h (z) have the same denominators. Conse-
quently, one may have

G kT−h (z)X kT−h (z) − G kT (z)X kT (z) = 0 (18.13)

which further implies

¯
θ kT−h (z)Y kT (z) − θ kT (z)Y kT−h (z) = 0 (18.14)
¯
T
T
¯
¯ ¯
¯
¯
where θ kT (z) =[b 1 ,b 2 ,...,b n ] and θ kT−h (z) =[b 0 ,b 1 ,...,b n−1 ] .
To facilitate identification, we rewrite Eq. (18.14)as
T
y[kT]= E (h)ϕ (18.15)
where
⎧
⎪E(h) =[−y[kT − h],...,−y[kT − (n − 1)h],y[kT − 2h],...,
⎪
⎨
T
y[kT − (n + 1)h]] .
⎪
⎪ T
ϕ = [b 1 ,b 2 ,...,b n ,b 1 ,b 2 ,...,b n−1 ]
⎩ 1 ¯ ¯ ¯
b 0
Then by assuming b 0 = 0 at each k, the RLS algorithm can be used to
estimate ϕ as
⎧
T
⎪ ˆϕ(k) =ˆϕ(k − 1) + K(k)[y(k) − E (k) ˆϕ(k − 1)]
⎪
⎨
T
K(k) = P(k − 1)E(k)[E (k)P(k − 1)E(k) + 1 ] −1 (18.16)
λ(k)
⎪
P(k) =[I − K(k)E (k)]P(k − 1)
⎪ T
⎩
where λ> 0 is the weight of the algorithm.
Note that b i may contain a scalar factor as a result of normalizing b 0 = 1
([21]). Without loss of generality, we set b 0 = 1, and the regularized param-
eters of numerator can be obtained from (18.16).
18.3.3 Estimation of Preisach Non-linearity
After transfer function G(z) is identified, the unmeasurable variable x(k)
can be computed by using G(z) and y(k), and thus x(k) can be used to