A direct adaptive control framework for a class of nonlinear matrix second-order dynamical systems with state-dependent uncertainty is developed. The proposed framework guarantees global asymptotic stability of the closed-loop system states associated with the plant dynamics without requiring any knowledge of the system nonlinearities other than the assumption that they are continuous and lower bounded. Generalizations to the case where the system nonlinearities are unbounded are also considered. In the special case of matrix second-order systems with polynomial nonlinearities with unknown coefficients and unknown order, we provide a universal adaptive controller that guarantees closed-loop stability of the plant states.

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