Page 72 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

APPC of Servo Systems With Continuously Differentiable Friction Model 63

Moreover, according to (4.3), we can further rewrite the friction dynamics
−T F (x 2 ) as
T
−T F (x 2 ) = α φ(x 2 ) (4.16)
T
where α =[α 1 ,α 2 ,α 3 ] are the friction coefficients and φ =[−(tanh(β 1x 2 )−
T
tanh(β 2x 2 )),−tanh(β 3x 2 ),−x 2 ] is a vector.
T T T T T T
Define =[W ,α ] and =[ ,φ ] , then one can represent
error Eq. (4.14)as
T
s ˙ = r + ε + gu (4.17)
In (4.17), the friction dynamics shown in (4.16) are lumped into the
T
NN approximation (4.15) resulting in a more compact form . More-
T
over, we define an unknown scalar θ = as the lumped adaptive param-
eter of HONN (4.15) and friction (4.16), and then a scalar θ (independent
ˆ
of the number of NN nodes) rather than the vectors W and α is updated
online, such that the computational costs can be reduced significantly. This
is different to conventional NN controllers, e.g., [7], [22], and [9].

4.3.2 Control Design and Stability Analysis

The control u can be specified as
2
k 1s θs T ˆ ε s
ˆ
u =− − − (4.18)
r 2η 2 ˆ ε|s|+ σ 1
2
˙ s T
θ = r − σ 2 θ ˆ (4.19)
ˆ
2η 2
˙ ˆ ε = r a |s| − σ 3 ˆε (4.20)

where > 0, a > 0, k 1 > 0, η> 0, and σ 1, σ 2,and σ 3 > 0are design
parameters.
We have the following result:

Theorem 4.1. Consider adaptive control system consisting of plant (4.1)with
the error transform (4.10), control (4.18) and adaptive laws (4.19)–(4.20), then:
i) All signals in the closed-loop system are semi-globally uniformly ultimately
bounded (SGUUB);
ii) The prescribed control performance (4.6)ispreserved.

Proof. i) Select a Lyapunov function as
1 2 g 2 g 2
V = s + θ ˜ + ˜ ε (4.21)
2 2 2 a