Page 70 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

APPC of Servo Systems With Continuously Differentiable Friction Model 61

transient and steady-state performance can be designed a priori by tuning
δ,
the parameters δ, ¯ κ, μ 0,and μ ∞.
To solve the control problem with prescribed performance (4.6), an
output error transform will be introduced by transforming condition (4.6)
into an equivalent “unconstrained” one [13]. Hence, we define a smooth,
strictly increasing function S(z 1 ) of the transformed error z 1 ∈ R, such that:
i) −δ < S(z 1 )< δ, ∀z 1 ∈ L ∞;
¯
¯
ii) lim S(z 1 ) = δ, and lim S(z 1 ) =−δ;
z 1 →+∞ z 1 →−
From the properties of S(z 1 ), condition (4.6) can be represented as
e(t) = μ(t)S(z 1 ) (4.7)

Since S(z 1 ) is strictly monotonic increasing and the fact μ(t) ≥ μ ∞ > 0
holds, the inverse function of S(z 1 ) exists and can be given by

e(t)
−1
z 1 = S (4.8)
μ(t)
¯
Note that the PPF (4.5), S(z 1 ) and the associated parameters δ, δ, κ, μ 0 ,μ ∞
are all a priori designed. For any initial condition e(0), if parameters μ 0,
δ,and δ are selected such that −δμ(0)< e(0)< δμ(0) and z 1 can be con-
¯
¯
¯
trolled to be bounded (i.e., z 1 ∈ L ∞ ,∀t > 0), then −δ < S(z 1 )< δ holds.
Then the condition −δμ(t)< e(t)< δμ(t) is guaranteed. Consequently, the
¯
tracking control problem of system (4.2) with error constraint (4.6)isnow
transformed to stabilization of the transformed system (4.8).
Lemma 4.1. [13]: The control of system (4.2) is invariant under the error
transform (4.8) with function S(z 1 ) fulfilling the properties i) and ii). Thus the
stabilization of transformed error dynamics z 1 in (4.8) is sufficient to guarantee the
tracking control of system (4.2) with prescribed error performance (4.6).
Proof. Theproofcanbederived basedon(4.6)–(4.8).

Here, we introduce a unified error function S(z 1 ) with properties i) and
ii) as [21], which is given by
¯ z 1
δe − δe −z 1
S(z 1 ) = (4.9)
z
e 1 + e −z 1
Then from (4.8), the transformed error z 1 is derived as
e(t)
1 λ(t) + δ
z 1 = S −1 = ln (4.10)
μ(t) 2 δ − λ(t)
¯
where λ(t) = e(t)/μ(t).