Page 153 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

ANDSC of Strict-Feedback Systems With Non-linear Dead-Zone 149

Figure 9.1 ANDSC implementation for non-linear system with dead-zone.

∗2 ∗2 2 2
n−1 σ i g i0 θ i 1 σ ai ε i ∗ k e M i
i = 1,··· ,n−1and ϑ = + + + 0.2785ω i ε + +
i=1 4 2g i0 2g i0 i 4
1 σ n g n0 θ n ∗2 σ an ¯ε n ∗2 ∗
n
2g n0 + 4 + 2g n0 + 0.2875¯ε ω n.
If the control parameters are set to fulfill (9.49), γ and ϑ are all positive.
In addition, the last term in (9.51)

n
c i1 2 z i m i e à im 2
i=1 2 1 − 2tanh ( ) j=1 1−¯τ i ϕ (¯x i )
ij
ω i
is bounded since the functions ϕ ij (¯x i ) are bounded on the compact sets C i
2
and the fact −1 ≤ 1 − 2tanh (z i /ω i ) ≤ 1 holds. Then according to Lya-
punov Theorem, uniformly ultimately bounded (UUB) stability of the
system (9.51) can be guaranteed for small enough ϑ, c i1,and ϕ ij (¯x i ) or
large enough γ . Consequently, it is readily concluded that z i , ˜ , ˜ε i, e i are
θ i
bounded, which further implies that θ i, ˆε i, x i,and s i are all bounded owing
ˆ
∗ ∗
to the boundedness of θ , ε ,and y d , ˙y d . Moreover, the control signals α i
i i
(i.e., |z i | ≥ 0.8814ω i ),
and v are also bounded. For the specific case z i /∈ z i
the last term of (9.51)isnegativeaccordingtoLemma 9.2, then as pointed
out in Step 1, the convergence rate can be improved. Furthermore, from the
formulation of γ i and ϑ i, the size of the errors can be adjusted appropriately
small by small enough ϑ i and large enough γ i.

9.3.3 Practical Implementation
The design of the proposed adaptive neural dynamic surface control
(ANDSC) is based on a recursive procedure, and its implementation can
be conducted in a systematic manner. Fig. 9.1 depicts the schematic dia-
gram of the developed control. The implementation for an n-order system
is presented step-by-step as follows: