Page 86 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

where μ 0 >μ ∞ and κ> 0 are positive constants, and it fulfills lim μ(t) =
t→∞
μ ∞ > 0.
Then, the transient response of the tracking error e 1 (t) can be guaran-
teed by
−δμ(t)< e 1 (t)<δμ(t) ∀t > 0 (5.5)

where δ is a positive constant. (Note that compared with Chapter 3 and
Chapter 4,weset δ = δ i .) Then −δμ(0) and δμ(0) represent the lower
i
bound and upper bound of the undershoot and maximum overshoot, re-
spectively; κ introduces the convergence rate and μ ∞ denotes the allowable
steady-state error. Thus, the transient response of the control error e 1 (t) can
be prescribed by tuning δ, κ, μ 0,and μ ∞.
Then we can select the following error transform function S(·) fulfilling
the properties given in Chapter 4
δe − δe −z 1
z 1
S(z 1 ) = (5.6)
z
e 1 + e −z 1
where z 1 is the transformed error of e 1 obtained by

e 1 (t)
−1
z 1 = S (5.7)
μ(t)
From the properties of S(z 1 ), condition (5.5) equals to
e 1 (t) = μ(t)S(z 1 ) (5.8)

Then, from (5.6)and (5.6), the transformed error z 1 can be written as
e 1 (t) 1 λ(t) + δ

−1
z 1 = S = ln (5.9)
μ(t) 2 δ − λ(t)
where λ(t) = e 1 (t)/μ(t).
Then as shown in Lemma 4.1, the tracking control of original system
with constraint (5.5) is equivalent to the stabilization of the transformed
error system (5.9).

5.3 RISE BASED ADAPTIVE CONTROL DESIGN AND ANALYSIS
After obtaining the transformed error system (5.9), the problem to be ad-
dressed is to design an appropriate controller such that z 1 is bounded.
Essentially different to available PPF based control strategies, e.g., [13,14],