  This paper addresses the problem of optimal robust stabilization of a
discrete-time minimum-phase plant in the framework of robust control theory in
the $\ell_1$ setup and under poor a priori information. Coefficients of the
transfer function of the plant nominal model with stable zeros are unknown and
belong to a known bounded polyhedron in the space of coefficients. The gains of
coprime factor perturbations of the plant and the upper bound of external
disturbance are also unknown. The problem under consideration is to design
adaptive controller that minimizes, with the prescribed accuracy, the
worst-case asymptotic upper bound of the output. Solution of the problem is
based on set-membership estimation of unknown parameters and treating the
control criterion as the identification criterion. A hard nonconvex problem of
on-line computation of optimal estimates is reduced, under additional
nonrestrictive assumption, to a linear-fractional programming via a nonlinear
transformation of estimated parameters. Despite the non-identifiability of the
unknown parameters, the proposed adaptive controller guarantees, with the
prescribed accuracy, the same optimal asymptotic upper bound of the output of
adaptive system as the optimal controller for the plant with known parameters.
In addition to the optimality of adaptive control, the proposed solution
provides on-line verification/validation of current estimates and a priori
assumptions.
