Page 947 - The Mechatronics Handbook

0066_frame_C31.fm Page 10 Wednesday, January 9, 2002 7:49 PM

Figure 31.3(b) shows the magnitude of the MRP attitude error vector s on a logarithmic scale. Again
the large transient errors of the open-loop, adaptation-free control law are visible during the first 20 s
of the maneuver along with the good final convergence characteristics. The ideal LCLD performance is
indicated again through the dotted line. Two versions of the adaptive control law are compared here,
which differ only by whether or not the external disturbance is adaptively estimated too. On this figure
both adaptive laws appear to enforce the desired LCLD very well for the first 40 s of the maneuver. After
this the adaptive law without disturbance learning starts to decay at a slower rate, slower even than the
open-loop (nonadaptive) solution. Including the external disturbance, adaptation clearly improves the
final convergence rate. Note, however, that neither adaptive case starts to deviate from the ideal LCLD
−3
case until the MRP attitude error magnitude has decayed to roughly 10 . This corresponds to having a
principal rotation error of roughly 0.23°. With external disturbance adaptation, the tracking error at
which the LCLD deviations appear is about two orders of magnitude smaller.
The performance of the adaptive control law can be greatly varied by choosing different learning rates.
However, since large initial inertia matrix and external disturbance model errors are present, the adap-
tive learning rates were reduced to avoid radical transient torques. The control torque vector components
u i for various cases are shown in Fig. 31.3(c). The open-loop torques don’t approach the ideal LCLD
torque during the transient part of the maneuver. The torques required by either adaptive case are very
similar. The difference is that the case with external disturbance learning is causing some extra oscillation
of the control about the LCLD case. However, note that with the chosen adaptive learning rates neither
control law exhibits any radical transient torques about the ideal LCLD torque profile. Figure 31.3(d)
illustrates that the adaptive external disturbance estimate F e indeed asymptotically approaches the true
∗
external disturbance F e . By reducing the external disturbance adaptive learning rate g F the transient
e
adaptive estimate errors are kept within a reasonable range.
Figure 31.3(f) shows the absolute performance error in attitude rates. Both cases with adaptation
added show large reductions in attitude rate errors compared to the nonadaptive case.
The purpose of the adaptive control discussed in this example is to enforce the desired LCLD. The previous
figures illustrate that the resulting overall system remains asymptotically stable. Figure 31.3(e) illustrates
the absolute performance error between the actual motion s(t) and the desired linear reference motion
s r (t). This figure demonstrates again the large performance error that results from using the open-loop
control law with the incorrect system model. Adding adaptation improves the transient performance
tracking by up to two orders of magnitude. Without including the external disturbance learning, the
final performance error decay rate flattens out. This error will decay to zero. However, with the given
learning gains, it does so at a slower rate than if no adaptation is taking place. Adding the external
disturbance learning greatly improves the final performance error decay since the system is obtaining an
accurate model of the actual constant disturbance. If the initial model estimates were more accurate,
more aggressive adaptive learning rates could be used, resulting in even better LCLD performance
tracking. This simulation illustrates though that even in the presence of large system uncertainty it is
possible to track the desired LCLD very well.

31.7 Output Feedback Adaptive Control

In contrast to the state-space approaches, the input–output approach treats the plant as a black box that
transforms the applied inputs into the corresponding output space. Stability theory for nonlinear systems
from an input–output viewpoint is important in the context of adaptive output feedback control design.
Solution to the problem of adaptive observer design involving state estimation of systems with unknown
parameters is often the stepping stone towards resolving the output feedback control problem. There
has been fairly recent breakthroughs in this area where the nonlinear adaptive observer design procedure
has been extended to a slightly more general case of systems where the coefficients of the unknown
parameters can depend on the entire state, and not just on the measured part. 15
To a large extent, some powerful results have been made possible by exploiting certain “passivity-
like”conditions coupled with the usual persistent excitation conditions. Crucial to this discussion is the

©2002 CRC Press LLC