Page 162 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

the scalars d l0, d l1, d r0, d r1,and , p are only used for analysis and not
included in the control implementation. We refer to Chapter 7.

10.2.1 Prescribed Performance Function and Error Transform
To study the transient and steady-state performance of tracking error e(t),
+ +
a positive decreasing smooth function μ(t) : R → R with lim μ(t) =
t→∞
μ ∞ > 0 is chosen as the prescribed performance function (PPF). As orig-
inally proposed by [21,22] and explained in the previous chapters, it is
sufficient to retain prescribed convergence response of tracking error e(t) if
the condition (10.4) holds:

−δμ(t)< e(t)< δμ(t), ∀t > 0 (10.4)
¯
¯
where δ,δ> 0 are positive constants.
Similar to those shown in Chapter 3–Chapter 5, μ(t) is chosen as μ(t) =
(μ 0 − μ ∞ )e −κt + μ ∞ with μ 0 >μ ∞ and κ> 0. And a smooth and strictly
increasing function S(z 1 ) of the transformed error z 1 ∈ R is defined such
that (10.4) can be represented as

e(t) = μ(t)S(z 1 ) (10.5)

with S(z 1 ) being
δe − δe −z 1
¯ z 1
S(z 1 ) = (10.6)
e 1 + e −z 1
z
Since S(z 1 ) is strictly monotonic increasing and μ(t) ≥ μ ∞ > 0, its inverse
function can be deduced as

e(t) 1 λ(t) + δ
−1
z 1 = S = ln (10.7)
μ(t) 2 δ − λ(t)
¯
where λ(t) = e(t)/μ(t).
¯
For any initial condition e(0), if we select appropriate μ(0), δ,and δ such
that −δμ(0)< e(0)< δμ(0), and assume z 1 is controlled to be bounded
¯
¯
(i.e., z 1 ∈ L ∞ ,∀t > 0), then −δ < S(z 1 )< δ holds, and thus the PPF error
condition −δμ(t)< e(t)< δμ(t) is guaranteed.
¯
Lemma 10.1. [21]: System (10.1) is invariant under the error transformation
(10.7), and the stabilization of transformed error z 1 is sufficient to guarantee tracking
control of (10.1) with prescribed error performance (10.4).