Discrete Time Generalized Cellular Neural Networks within Nl Q Theory

Generalized Cellular Neural Networks (GCCNs) were recently introduced by Guzelis & Chua. Instead of one single CNN, a set of CNNs was considered, interconnected in a feedforward, cascade or feedback way. The framework was in continuous time with su cient conditions for global asymptotic and I/O stability and the relation with classical nonlinear control theory such as the Lur'e problem was revealed. In this paper GCNNs are considered in a discrete time setting. The original system description is brought into a so-called NLq system form using state augmentation. NLq's are a general class of nonlinear systems in state space form with a typical feature of having q 'layers' with alternating linear and nonlinear operators that satisfy a sector condition. Within NLq theory su cient conditions for global asymptotic and asymptotic stability are available. The results are closely related to modern control theory (H1 theory and theory). Stability criteria are formulated as Linear Matrix Inequalities (LMIs). Checking stability involves the solution to a convex optimization problem. Furthermore it is shown by examples that existing GCCN con gurations result into q = 1 values. Hence if one considers the q value of the NLq as a measure of complexity of the overall system, GCNNs are still have a low complexity from the analysis point of view. In addition more complex neural network architectures with q > 1 are discussed.