Page 13 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

microscopic phenomena: friction, viscosity, elasticity, etc. Friction exists
wherever there is motion or tendency for motion between two physical
components, which could cause steady-state errors, limit cycles, or stick-
slip phenomenon at low speed in the motion control systems. Dead-zone,
a static input-output relationship for a range of input values gives no out-
put, can be encountered in motors, hydraulic valves, and even biomedical
actuation systems. Saturation is always imposed on physical actuators, which
limits the maximum control power for the systems. Hysteresis, a dynamic
characteristic with memory, exists in electro-magnetic and piezoelectric
systems and devices. Although the characteristics of friction, dead-zone,
saturation, and hysteresis are different, they are all non-smooth in nature.
In comparison to other smooth non-linearities, such non-smooth dy-
namics are usually difficult to model since they may vary with time. Hence,
control design of systems with non-smooth non-linearities has always been
an important research topic in the control system field. In particular, the
need for effective control methods to deal with non-smooth dynamics in
practical engineering plants has motivated growing research interests and
activities. Various control design methods based on different techniques
and methodologies have been developed and verified in theory and prac-
tice, which, to some extent, can accommodate these non-linearities.
The early and traditional idea to eliminate the harmful effect of such
non-smooth non-linearities is to implement their inverses inside the con-
troller. However, with this idea, the first concern is that the inverses of such
non-linearities, possibly discontinuous, must exist, and moreover they can
be also linearly parameterized as linear functions of the unknown parame-
ters. This imposes stringent assumptions on the studied systems, and creates
certain challenges in the control implementation. For instance, significant
effort should be made to construct accurate models of such non-smooth
dynamics, which is quite time-consuming and cost-demanding. Further-
more, with this inverse model compensation method, other uncertainties
in the systems (e.g., parameter variations, external disturbances) need fur-
ther considerations.
Another control methodology, adaptive control, has been developed
by combining a parameter estimator with appropriate feedback controls,
which can cope with parameter uncertainties in the model. The basic idea
of adaptive control is that the parameters of the controller and/or plant
can be online adjusted based on the collected system information during
the online operations. In particular, some recent effort has been made to-
ward incorporating function approximation such as neural networks (NNs),