Page 262 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Identification and Inverse Model Based Control of Uncertain Systems With Backlash 263

17.3.3 Parameter Estimation
In this chapter, we will use the least squares (LS) method to identify the
unknown lumped parameter vector θ in (17.17).
Hence, the estimated parameter θ can be updated by
ˆ

T Q N−1 ϕ N
θ
ˆ
ˆ = θ N−1 + y(N) − ϕ ˆ (17.18)
θ N N N−1 T
ϕ Q N−1 ϕ N + 1
N
where N is the number of the collected data, y(N) is the output of the
N−1 N−1 ) .
k = N-th instant, and Q N−1 = (ϕ T ϕ −1

ˆ
ˆ
Based on the estimated parameter vector θ N = − β 1 (N), ˆγ 11 (N),··· ,
T

ˆ γ 1s (N),−β 2 (N),··· , ˆγ ns (N), ˆα 1 (N),··· , ˆα n (N) , we will further deter-
ˆ
mine the unknown parameters α i ,β i of system (17.12) and parameters
p i ,l,d 1 ,d 2 of backlash (17.13).
First, based on the estimated θ N , we can calculate the coefficients of
ˆ
ˆ α i (N),β i (N) based on (17.5) and the following estimated polynomial
ˆ
−1
A(z ) =1 +ˆα 1 (N)z −1 + ··· + ˆα n (N)z −n
ˆ
(17.19)
−n
−1
ˆ B(z ) =β(N)z −1 + ··· + β n (N)z .
ˆ
ˆ
Hence, the estimation of the coefficients a i in (17.3)can be obtained
based on (17.5) by using the estimated α i ,β i.
Moreover, we will calculate the estimated p i ,l,d 1 ,d 2 based on the de-
N 2
ˆ

rived θ. It is noted that β = 0, thus we can multiply β i on both sides
i=1
of γ ij = β ip i and calculate the sum from 1 to n, such that
n n n
2 2

β i γ ij = β (17.20)
β p j = p j
i i
i=1 i=1 i=1
Hence, the estimation of p j can be obtained by
n ˆ
i=1 β i (N) ˆγ ij (N)
ˆ p j (N) = , j = 1,··· ,s (17.21)
n ˆ 2
i=1 (β i (N))
Moreover, from (17.13), it can be found that the unknown w l is ap-
proximated by the linear combination of p j σ i (0,u(t − i) − l ,l j − l ) obtained
j j
in each subinterval. Hence, p j is the slope of the local linear function σ j .
According to this fact, the estimation of backlash slope l can be obtained
by using the estimated coefficients ˆp j as