Page 277 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Identification and Control of Hammerstein Systems With Hysteresis Non-linearity 279

We further rewrite (18.7) in a parameterized form as
T
y[(k + 1)h]= F [kh]φ (18.8)

where

T
F[kh] = [−y[(k − 1)h],...,−y[(k − n)h],x[(k − 1)h],...,x[(k − n)h]]
.
¯ ¯
¯
φ =[¯a 1 , ¯a 2 ,..., ¯a n ,b 1 ,b 2 ,...,b n ] T
It is noted that the unmeasurable variable x appears in (18.8), so that it
is not possible to identify the coefficients based on (18.8). However, the
input sequence x(k) is non-zero only if t = kT = k(n + 1)h, i.e., if we set
l = k(n + 1),then x[(l − 1)h]= x[(l − 2)h]= ... = x[(l − n)h]= 0is true. In
this case, we can get the following equation:
T
y[kT]= ¯ F [h]φ ¯ (18.9)
where

T
¯ F[h] = [−y[kT − h],−y[kT − 2h],...,−y[kT − nh]]
.
φ ¯ =[¯ a 1 , ¯a 2 ,..., ¯a n ] T
Clearly, Eq. (18.9) contains the output measurements y(k) only, and thus
can be used to identify the unknown parameters φ (e.g., the denominator
¯
coefficients of G(z)).
Then, φ can be estimated by using standard algorithms based on (18.9).
¯
In this chapter, the recursive least squares (RLS) method is used. At each k,
the RLS algorithm is given as

⎧
ˆ
T
ˆ
⎪ ˆ ¯ = φ(k − 1) + K(k)[y(k) − ¯ F (k)φ(k − 1)]
¯
⎪φ(k)
¯
⎨
T
K(k) = P(k − 1) ¯ F(k)[ ¯ F (k)P(k − 1) ¯ F(k) + 1 ] −1 (18.10)
λ(k)
⎪
P(k) =[I − K(k) ¯ F (k)]P(k − 1)
⎪ T
⎩
where λ> 0 is the weight of RLS algorithm.
We now further identify the numerator coefficients of G(z).For this
purpose, we consider the following two sequences
∞
−k
Y kT (z) = y[kT]z = G kT (z)X kT (z)
k=1
(18.11)
∞
−k
Y kT−h (z) = y[kT − h]z = G kT−h (z)X kT−h (z)
k=1