Links between NLq neural control theory and μ robust control theory

We derive sufficient conditions for I/O stability with finite L ₂-gain of multilayer recurrent neural networks, with parametric uncertainties upon the interconnection matrices. This is done by considering perturbed NL_q systems. NL_qs are discrete time nonlinear dynamical systems in state space form, containing q layers with alternating linear and static nonlinear operators that satisfy a sector condition. Within the present framework, linear fractional transformations with real diagonal uncertainty block are interpreted as perturbed NL_qs for q=1. While in μ theory uncertainties upon nominal linear models are investigated, uncertainties upon nominal nonlinear models can be studied in NL_q theory. It is shown how the state space upper bound test of μ theory related to the case of a real diagonal uncertainty block, follows as a special case from the derived theorems for q=1 (one layer)