Page 191 - IJOCTA-15-4

Proportional integral derivative plus control for nonlinear discrete-time state-dependent parameter. . .

2

1.8
1.5
1.6 1
0.5
1.4
0 0.05 0.1 0.15
1.2
1
0.8
0.6
0.4
0.2
0
0 1 2 3 4 5 6 7 8 9 10

Figure 3. 2-DOF Robotic manipulator 2 2
1.5
1
Table 1. Proposed control parameters. 0.5
1.5
0 0.2 0.4
link 1 link 2
lengthl i (m) 1 1 1
massm i (kg) 0.5 0.5
2
inertiaI i (kg.m ) 5 5
0.5

4.2. MRAC-NNTDE application on 0 0 1 2 3 4 5 6 7 8 9 10
2-DOF robot manipulator

The following appropriate values are chosen for Figure 4. Position tracking
the proposed control (24) and the dynamics model
(4) in order to guarantee high-performance track-

ing using our suggested model-free control ap-
proach: r 1 (t) = r 2 (t) = step, D(t) = [sin(t) + 0.1
T
0.5 ˙q 1 + sin(1.5q 1 ), sin(t) + 1.3 ˙q 2 + 1.8sin(2q 2 )] ,
0
= diag(−10, −10, −10, −10), =
A p B p
T -0.1
0 0 1 0 5 1 0 0 0
, K = 10 × ,
0 0 0 1 0 1 0 0 -0.2
¯
M = diag(0.0001, 0.0001), d = 0.001, x p1 (0) =
-0.3
x p2 (0) = −0.1. The initial conditions and adap-
tation law parameters (19,28) are also chosen -0.4

1 0
as φ 1 = 0.01 × diag(1, 1, 1, 1), φ 2 = , -0.5
0 1
0 1 2 3 4 5 6 7 8 9 10
ˆ
ˆ
φ 3 = 5, ϑ(0) = ˆυ(0) = 1 and W(0) = 0. The
initial values of joints are given as q 1 (0) = 1.5
0.4
and q 2 (0) = 2.1. These parameter selections are
0.35
crucial for achieving the desired tracking perfor-
0.3
mance in the model-free control technique.
0.25
MRAC-NNTDE is evaluated on the uncertain
0.2
dynamics of a 2-DOF robotic manipulator under 0.15
disturbances to demonstrate the effectiveness of 0.1
the suggested approach. As a result, their com-
0.05
parison of the simulation results is shown in Fig- 0
ures 4-7, respectively, for position tracking, track- -0.05
ing error, control inputs, and TDE estimation -0.1
0 1 2 3 4 5 6 7 8 9 10
of unknown dynamics. Moreover, the adaptive
gains to compensate for the unknown dynamics
Figure 5. Tracking error
are given in Figures 8-10.
733