STABILITY ANALYSIS OF STATE-SPACE REALIZATIONS FOR 2-DIMENSIONAL FILTERS WITH OVERFLOW NONLINEARITIES

We utilize the second method of Lyapunov to establish sufficient condi tions for the global asymptotic stability of the trivial solution of p ercent nonlinear, shift-invariant 2-D (two-dimensional) systems. We ap ply this result in the stability analysis of 2-D quarter plane state-s pace digital filters, which are endowed with a general class of overfl ow nonlinearities. Utilizing the l(infinity) vector norm and the p(th) power of the l(p) vector norm for 1 less-than-or-equal-to p < infinit y as Lyapunov functions, we show that parallel-to A parallel-to p, < 1 , for some p, 1 less-than-or-equal-to p less-than-or-equal-to infinity , constitutes a sufficient condition for the global asymptotic stabili ty 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 parallel-to . parallel-to 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 stabili ty of the null solution of the 2-D digital filters. These results are very general, since they involve necessary and sufficient conditions u nder which positive definite matrices can be used to generate the quad ratic Lyapunov functions for the 2-D digital filters with overflow non linearities. We generalize the above results to a class of m-D (multid imensional) digital filters with overflow nonlinearities. To demonstra te the applicability of our results, we consider a specific example.