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Stabilization and performance synthesis for systems with repeated scalar nonlinearities

The class of nonlinear systems described by a discrete-time state-equation containing a repeated scalar nonlinearity as in recurrent neural networks is considered. Given a plant of this form, sufficient conditions are derived for: 1) the parametrization of all controllers of the same form such that the closed loop is stable in the sense of a diagonally dominant Lyapunov function; and (2) the synthesis of a controller of the same form so that the induced norm of the closed loop is under a prescribed level. Several of these conditions can be written into linear matrix inequalities.