Page 190 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Dynamic Surface Output Feedback Control of Pure-Feedback Systems 187

Step n: Define the last error as

(11.37)
e n = ξ n − α n−1
whose derivative is

= a(¯x n ) + b(¯x n )v −¨α n−1 ≤ F(¯x n ) + b 0v − ϑ n,2 + ι n . (11.38)
˙ e n
Then, let α n−1 go throughaTD givenby
⎧
⎨ ˙ = ϑ n,2
ϑ n,1
ϑ n,2 | ϑ n,2 | (11.39)
⎩ ˙ =−r nsgn(ϑ n,1 − α n−1 + ).
ϑ n,2
2r n
The Lyapunov function in the n-th step is set as
e 2 n
V n = (11.40)
2
From (11.39)and (11.37), the derivative ˙ V n can be calculated as

˙ V n = e n (F + b 0v −¨α n−1 ) ≤ e n (F − ϑ n,2 + ι n ) + e nb 0v (11.41)

Finally, the actual control signal is chosen to be

ι
v = 1 (−e n−1 − c ne n − ξ n+1 + ϑ n,2 − n tanh( 0.2785ˆ n e n )) (11.42)
ι
b 0 ω
ι
˙ 0.2785ˆ n e n
ι
ˆ ι n = θ n [e n tanh( ) − σ n ˆ n ] (11.43)
ω
where θ n, σ n ,c n are positive constants.
Substituting (11.42)into(11.41)results in
ι
2
≤−c ne + ι ne n − ne n tanh( 0.2785ˆ n e n ) + e n (F − ξ n+1 ) (11.44)
ι
˙ V n
n ω
11.4.3 Stability Analysis
This section will prove the stability and tracking performance of the pro-
posed control system. It is proved that all signals of the overall closed-loop
system are uniformly ultimately bounded (UUB) and the tracking errors
converge to an arbitrarily small residue set.
Theorem 11.1. Consider the closed-loop system consisting of the plant (11.1),
unknown dead-zone non-linearities (11.2), the non-linear ESO (11.15), the TDs
(11.24), (11.30), (11.39), the virtual control (11.28), (11.35), and the actual
control (11.42). Then,