Page 40 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Sliding Mode Control of Non-linear Servo Systems With LuGre Friction Model 31

Hence, we can claim the boundedness of the estimated parameters ˆz 0, ˆz 1,
ˆ
ˆ σ 0, ˆσ 1, ζ, T d , ˆ J, and the control errors ˙e, e, ω, δ from (2.23)and (2.26).
ˆ
Based on the definition of ψ(θ), we know that ψ(θ) and ψ(θ) are both
˙
bounded, and thus λ is bounded from (2.23), such that δ is bounded from
(2.26), and the control signal T is bounded from (2.29). Moreover, from
the derivative of (2.27) and the fact that ˙ω is bounded, we have ˙s ∈ L ∞ and
it follows from (2.49)that s ∈ L 2. Consequently, based on Barbalat’s lemma,
we know s converges to zero when time t goes to infinity. Hence, from
(2.23) we can conclude that the tracking error e converges to zero. This
completes the theorem proof.

2.5 SIMULATIONS

To verify the effectiveness of the proposed method, three control methods
are simulated for comparison: (A1) adaptive control [18] but without fric-
tion compensation; (A2) adaptive control with friction compensation [18];
(A3) the proposed method with friction compensation.
In the studied system (2.1) and friction model (2.3)–(2.5), the pa-
2
rameters are set as J = 0.9kgm , σ 0 = 12 Nm, σ 1 = 2.5Nm, σ 2 =
−1
0.2Nm (rad/s) , f c = 0.28 Nm, f s = 0.34 Nm, ω s = 0.01 (rad)/s, and
T d = 0.1sin(4πt). For the proposed sliding mode control, we set k = 50,
λ = 20, υ = 2, α = 200. In the observer and adaptive laws, the parameters
= 0.161, k ζ = 5, k T = 0.01,
are chosen as k 0 = k 1 = 0.01, k σ 0 = 332, k σ 1
= γ ζ = γ J = γ T = 5.
k J = 0.08, and γ σ 0 = γ σ 1
The reference signal to be tracked is θ ref = 2sin(πt), and the tracking tra-
jectories and corresponding tracking errors of three controllers are shown
in Fig. 2.2 and Fig. 2.3. As it can be seen from Fig. 2.2 and Fig. 2.3,the
case A3 with the proposed control can achieve a faster tracking conver-
gence performance than the cases A1 and A2. As shown in Fig. 2.3,the
proposed control can achieve tracking error convergence in 0.3 s and the
steady-state tracking error is about 2 × 10 −4 rad, while adaptive control
with friction compensation (Case A2) will achieve convergence after 10 s
but with small chattering in the steady-state with tracking error around
4 × 10 −4 rad. Compared with the results of Cases A2 and A3, adaptive
control without friction compensation (Case A1) has more sluggish track-
ing response, and its tracking error in the steady-state is about 6×10 −4 rad.
Moreover, comparative parameter identification results of Case A2 and
Case A3 are shown in Figs. 2.4–2.7. We can see that the proposed adaptive
law with offline GSO initialization can achieve better identification perfor-