Immersion and invariance:
a new tool for stabilization and adaptive control of nonlinear systems
Dr A. Astolfi, (Imperial College)
A new method to design asymptotically stabilizing and adaptive control
laws for nonlinear systems is
presented. The method relies upon the notions of system immersion and
manifold invariance and does not require the knowledge of a
(control) Lyapunov function. The construction of the stabilizing control
laws resembles the construction used in nonlinear
regulator theory to derive the (invariant) output zeroing manifold and its
friend. The method is well suited in
situations where we know a stabilizing controller of a nominal reduced order
model, which we would like to robustify with respect
to high order dynamics. This is achieved by designing a control law that
immerses the full system dynamics into the reduced order
one. We also show that in this new framework the adaptive control problem
can be formulated from a new perspective that, under
some suitable structural assumptions, allows to modify the classical
certainty equivalent controller and derive parameter update
laws such that stabilization is achieved. It is interesting to note
that our construction does not require a linear
parameterization, furthermore, viewed from a Lyapunov perspective, it
provides a procedure to add cross terms between the
parameter estimates and the plant states.
Finally, it is shown that the proposed approach yields new
stabilizing control laws for systems in feedback and feedforward form
and allows to relax one of the classical standing
assumptions of forwarding,
i.e. the local exponential stability of one subsystem.
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