Page 94 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

5.4 EXPERIMENTAL VALIDATION
5.4.1 Experimental Setup
To validate the proposed control scheme, the turnable servo system in-
troduced in Chapter 4 is used as the test-rig to carry out experiments
(Fig. 4.1). For details of this experimental setup, we refer to Chapter 4
(Section 4.1).
The following four controllers are all implemented in the experiments.
1) ANRISE: This is the control suggested in this chapter, where the RISE
compensation (5.20) and continuous friction model (4.4)areallused.
The controller parameters are given as k 1 = 10, k 2 = 1.5, k s = 0.5, and
β 1 = 5. The embedded friction parameters in (4.4) are chosen as α 1 =
0.02, α 2 = 0.01, α 3 = 0.2. The PPF parameters in (4.5)are μ 0 = 0.15,
μ ∞ = 0.03, κ = 0.4, and δ = 1. The learning parameters of ESN are
given as L = 6, σ = 0.01, = 0.5.
2) ANDSC: This control method was proposed in [22] and described in
Chapter 4.
3) DCRAC: This is an adaptive robust controller with the Coulomb
friction model T f = sgn(x 2 ) asshownin[23]. The adaptive con-
troller is given as u = u a + u s, u a =−ϕ d θ, −ϕ d =[−¨y d ,−¨y d ,−S f (˙y d ),1],
ˆ
1 2
u s = u s1 + u s2, u s1 =−k s1p, u s2 =− h p, k 1 = 400, k s1 = 32, =
4ε
T
diag[25,0,5,1000] , θ(0) =[0.05,0.24,0.1,0] .
T ˆ
4) PID: This is the conventional Proportional-Integral-Derivative (PID)
controller, where the parameter gains are given in Chapter 4.
5.4.2 Experimental Results

For fair comparison in terms of the robustness and generality of these con-
trol methods, all control parameters are determined with a given trajectory
x d = 0.4sin(0.4πt) by using a trial-and-error method to make a tradeoff
between the steady-state performance and transient response. Then these
parameters are fixed and used in all experiments to validate the generality
of different controls under wide operation scenarios.
The aforementioned four controllers are tested for a slowly-varying si-
nusoid reference trajectory, i.e., x d = 0.4sin(0.4πt). The tracking profiles
and the corresponding tracking errors of the four controllers are depicted
in Fig. 5.1 and Fig. 5.2. By comparing the tracking performances of these
controllers, it is clear that the proposed ANRISE gives smaller error than
other controllers, i.e., it achieves better transient and steady-state error. In
particular, although the steady-state tracking performance of ANRISE and