Dr. Liu et An. Michel, STABILITY ANALYSIS OF STATE-SPACE REALIZATIONS FOR 2-DIMENSIONAL FILTERS WITH OVERFLOW NONLINEARITIES, IEEE transactions on circuits and systems. 1, Fundamental theory andapplications, 41(2), 1994, pp. 127-137
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.