Page 79 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Table 4.2 Tracking performance IAE for y = Asin(0.4πt)
d
Amplitude (rad) A = 0.4 A = 0.6 A = 0.8 A = 1 A = 1.2
PID 0.1998 0.2247 0.3275 0.3325 0.6785
ANC 0.1944 0.1842 0.2184 0.1721 0.3697
ANDSC 0.1347 0.1849 0.2502 0.3249 0.6788
APPC 0.0664 0.0980 0.1024 0.1165 0.2299

Figure 4.5 Output tracking profiles of different controllers.

2) Case 2: Sinusoidal Waves with Varying Amplitudes
Since the influence of friction non-linearities are more notable at low
speed regions, to show the compensation for friction, we select a sinusoidal
signal y d = Asin(2πt/5) with a fixed period T = 5 s and varying amplitude
A = 0.4–1.2 rad as the reference. Comparative performance is summarized
in Table 4.2. It is also found that the proposed APPC performs better than
other control schemes due to the introduction of the PPF design and the
friction compensation via the continuously differentiable friction model
(4.3). Moreover, in the low speed regimes (e.g., A = 0.4, 0.6), ANDSC
performs slightly better than ANC, while in the middle/high speed regime
(e.g., A = 0.8–1.2) its performance is deteriorated. Among all case studies,
PID control gives larger error, which exactly illustrates how the addition of
the adaptive element allows for the compensation of time-varying dynamics
to improve the overall control performance.
As an example, Fig. 4.5 and Fig. 4.6 show the tracking profiles and
the corresponding tracking errors for y d = 0.4sin(0.4πt) with different
controllers. One can find from Fig. 4.5 that the proposed APPC can com-
pensate for the dynamics of friction effectively, i.e., it gives smaller tracking