Stability analysis of state-space realizations for two-dimensional filters with overflow nonlinearities

We utilize the second method of Lyapunov to establish sufficient conditions for the global asymptotic stability of the trivial solution of percent nonlinear, shift-invariant 2-D (two-dimensional) systems. We apply this result in the stability analysis of 2-D quarter plane state-space digital filters, which are endowed with a general class of overflow nonlinearities. Utilizing the l_∞ vector norm and the p^th power of the l_p vector norm for 1⩽p<∞ as Lyapunov functions, we show that ||A||_p <1, for some p, 1⩽p⩽∞, constitutes a sufficient condition for the global asymptotic stability of the trivial solution of the 2-D nonlinear digital filters where A denotes the coefficient matrix of the filter operating in its linear range and ||·||_p denotes the matrix norm induced by the l_p vector norm. Using quadratic form Lyapunov functions, we also establish sufficient conditions for the global asymptotic stability of the null solution of the 2-D digital filters. These results are very general, since they involve necessary and sufficient conditions under which positive definite matrices can be used to generate the quadratic Lyapunov functions for the 2-D digital filters with overflow nonlinearities. We generalize the above results to a class of m-D (multidimensional) digital filters with overflow nonlinearities. To demonstrate the applicability of our results, we consider a specific example