Page 186 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Dynamic Surface Output Feedback Control of Pure-Feedback Systems 183

where q satisfies q =
n i=0 ((i − 1)γ − (i − 2)), and P is the solution of

T
A P + PA o =−I (11.19)
o
with

⎛ ⎞
−β 1 10 ··· 0
⎜ 01 ⎟
⎜ −β 2 ··· 0 ⎟
⎜ . . . . . ⎟
A o = ⎜ . . . . . . . . . ⎟ . (11.20)
⎜ . ⎟
⎜ ⎟
⎝ −β n 00 ··· 1 ⎠
−β n+1 00 ··· 0
Lemma 11.1 means that there exist positive constants ι and t s such that
for t > t s,wehave ξ − z ≤ ι. As it is pointed out in [14,15], the sepa-
ration principle [16] is trivially fulfilled for finite-time observer, such that
the observer and controller can be developed separately. Hence, in the sta-
bility analysis of the closed-loop systems with this ESO, the observer errors
satisfying ξ − z ≤ ι will be considered.

11.4 CONTROL DESIGN AND STABILITY ANALYSIS
Based on the ESO design, a modified robust adaptive control approach is
developed by incorporating the tracking differentiator (TD) into the dy-
namic surface control (DSC).

11.4.1 Tracking Differentiator
Let signal α r be a function defined on [0,∞) with its n-th derivatives having
a Lipschitz constant L,and then aTD[11]isgiven by

⎧
⎨ ˙ = ϑ 2
ϑ 1
ϑ 2 | ϑ 2 | (11.21)
⎩ ˙ =−rsgn(ϑ 1 − α r + )
ϑ 2
2r
where sgn(·) is the signum function, r represents a positive constant, α r is
the input signal of TD, which are the virtual control signal in each step of
the DSC design.
According to [17]and [18], TD (11.21) is with the time optimal prop-
erty, which can guarantee the finite-time convergence of system states.