Page 189 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

By substituting (11.28)into(11.27), one obtains

2
ι
ι
˙ V 1 ≤−c 1e + e 1e 2 − 1e 1 tanh( 0.2785ˆ 1 e 1 ). (11.29)
1 ω
Step i (2 ≤ i ≤ n − 1): To solve the well-known “explosion of complex-
ity” problem in the traditional backstepping design, we let α i−1 go through
aTDgiven by
⎧
⎨ ˙ = ϑ i,2
ϑ i,1
ϑ i,2 | ϑ i,2 | (11.30)
⎩ ˙ =−r 2sgn(ϑ i,1 − α i−1 + ).
ϑ i,2
2r 2
Then, the i-th error surface is defined to be

e i = ξ i − α i−1 . (11.31)

Differentiating e i along (11.15)yields

(11.32)
˙ e i = ξ i+1 −¨α i−1 ≤ ξ i+1 − ϑ i,2 + ι i
where ι i = ι + ι ϑ,2 represents the filter error bound of the i-th state of ESO
and the i-th TD, ϑ i,2 is the second state of the i-th employed TD as depicted
in Fig. 11.1.
Consider a Lyapunov function as

1 2
V i = e . (11.33)
2 i
From (11.31)and (11.32), it can be concluded that

˙ V i ≤ e i (e i+1 + α i − ϑ i,2 + ι i ). (11.34)
Then, the virtual control signal α i is designed as

ι
ι
α i =−e i−1 − c ie i + ϑ i,2 − i tanh( 0.2785ˆ i e i )
ω (11.35)
ι
˙ 0.2785ˆ i e i
ι
ˆ ι i = θ i [e i tanh( ) − σ i ˆ i ],
ω
where θ i ,σ i ,c i are positive constants, and ˆι i represents the estimation of the
filter error bound ι i.
Substituting (11.35)into(11.34)yields
ι
0.2785ˆ ie i
2
ι
˙ V i ≤−c ie + e ie i+1 + e i ι i − ie i tanh( ). (11.36)
i
ω