Page 91 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

RISE Based Asymptotic PPC of Servo Systems With Continuously Differentiable Friction Model 83

5.3.3 Stability Analysis
In order to facilitate the stability analysis of the closed-loop control system,
the following Lemma is stated first.
Lemma 5.1. [7] An auxiliary function L(t) is defined as

2
L(t) = r[N B + N d − β 1sgn(z 2 )]+Hsgn(z 2 ) + β 2 z 2 + z 2N B (5.30)
σð 2
where H = is an unknown positive constant, β 1 and β 2 introduced in (5.20)
4
and (5.30) are positive constants, which are selected to fulfill the following condition
1
+ (5.31)
β 1 >ζ N B0 + ζ N d1 (ζ N B1 + ζ N d2 ), β 2 >ζ N B2
k 2
Then the following inequality is satisfied

t

L(σ)dσ ≤ β 1 z 2 (0) − z 2 N B (0) + N d (0) (5.32)
0
Proof. The proof of Lemma 5.1 can be conducted following the proof of
Lemma 1 in [7].

The main results of this chapter can be summarized as follows:
Theorem 5.1. Consider the servo system given by (5.1), the control is given as
(5.19), and the ESN weight is updated via (5.21), then the system is semi-globally
stable. Moreover, the tracking error converges to zero asymptotically, i.e. e 1 (t) → 0
as t →∞, and the transient response of e 1 (t) can be retained within the prescribed
performance bound (5.5).

Proof. An auxiliary function P(t) is defined as

˙ P(t) =−L(t) (5.33)

A positive-definite Lyapunov function is chosen as
1 2 1 2 1 2
V L = z + z + r + Q + P
2
1
2 2 2 (5.34)
1 T
˜
˜
Q = ( )
2
which satisfies the following inequalities
U 1 (y) ≤ V L (y) ≤ U 2 (y) (5.35)