Replacement of Lyapunov Function by Locally Convergent Robust Fixed Point Transformations in Model-based Control A Brief Summary

Lyapunov’s 2nd Method is a popular approachin the model-based control of nonlinear systems
since normally it can guarantee global asymptotic
stability. However, the cost of the application of
Lyapunov function frequently can be inefficient and
complicated parameter tuning process containing unnecessary
number of almost arbitrary control parameters,
and vulnerability of the tuning process against
not modeled, unknown external disturbances. Improved
versions of the original, model-based tuning
are especially sensitive to the external perturbations.
All these disadvantages can simply be avoided by the
application of Robust Fixed Point Transformations at
the cost of giving up the guarantee of global asymptotic
stability. Instead of that simple, stable iterative
control of local basin of attraction can be constructed
on the basis of an approximate system model that
can well compensate the effects of unknown external
disturbances, too. Since the basin of convergence of
the method in principle can be left, setting the three
adaptive parameters of this controller needs preliminary
simulation investigations. These statements are
illustrated and substantiated via simulation results obtained
for the adaptive control of various physical systems.