Page 35 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Table 2.4 The identification result of dynamic parameters
2
Parameters σ 0 (Nm/rad) σ 1 (Nm s/rad) J (kg/m )
True value 12 2.5 0.9
Estimate value 12.0972 2.4682 0.8971

ˆ i
i
where e i = T − T represents the dynamic identification error for the
d_m m m
i-th velocity signal of the m-th glowworm.
Then, following similar procedures of that for static parameter identifi-
cations, we can perform the identification of dynamic parameters by using
GSO. The identification results are shown in Table 2.4.

2.4 CONTROLLER DESIGN AND STABILITY ANALYSIS

In this section, an adaptive non-linear sliding mode controller is proposed
to achieve output tracking, where an online adaptive law with finite-time
convergence is incorporated into sliding mode control.

2.4.1 Adaptive Non-linear Sliding Mode Control Design

Define the position tracking error as

(2.21)
e = θ − θ ref
where e is the tracking error, θ ref is the reference position signal.
The dynamic system (2.1) can be converted into the following form

θ ˙ θ 0 0 0
= A + T − T f − T d (2.22)
˙ ω ω 1 J 1 J 1 J

01
a 11 a 12
with A = = .
a 21 a 22 00
The non-linear sliding surface is designed as
s =˙e + (N − ψ(θ)P)e =˙e + λe (2.23)

where λ = N − ψ(θ)P,and ψ(θ) is a negative non-linear differentiable
function given by

2
ψ(θ) =−υ exp(−α(θ − θ ref ) ) (2.24)