Under the conditions of (20), (17) becomes AV(n) 5 -[p'z(n) + -/+[dn)lk*(n)l'.
(21)Following the arguments of Szego [l], we can show that 4V(n) = 0 if and only if z(n + 1) = s(n) = 0 (Le., at. t.he equilibrium point) and AV(n) < 0 for s(n) # 0. Also, since the right-hand side of (21) is independent of 8, by an argument similar to that of Szego [l], we see that V is posit.ive definite, even when p is negative.Thus t.he system (9) and, hence, the system (1) are asymptotically stable if t.he conditions of (20) are satisfied for some CY 2 0 along with (15). Substitu6ing for 4, c, and r in (20j, we get (22b) 2, P e + 6X + 2 t > o , 9 2 0 , 6 > 0 , 0 5 x 5 1 ;3)One way of eliminating condit.ion 2 is to allow K -+ 0 and L -+ m such t.hat. K L = I; (a constant) so t.hat condition 2 is obviously satisfied, since the first term becomes hrge as K --f 0 and E -+ (a'b), which is finite. So the criterion reduces to (with k(n) = K k ( n ) so t,hat 0 < k(n) 5 k ) t.he following:NON replacing W*(z) by LW(z) and dividing throughout by IpI 2) =kAk(n -1) 5 26k(.n) and denohing CY/]^] = 9 2 0, (22) becomes 11 e 7 K 2) 9 2 0, + -+ 2E sgn p > 0;and correspondingly, (15) becomes A k ( n -1) sgn p 2 Fmh 6k(n)Putting r/6 = X, (23) and ( 2 4 ) can be written as 1)Chen's criterion [4] for LTV cases can be obtained by putting e = 0, X = 1, 6 = l/q, and using only the positive sign in (26).
REFERERCES[I] G. P. Szego. "On the absolute stability of sampled-data syst.ems," Proc. .Vat. Acad. Sci., vol. 50, ~p.~p58-560, 1963: [a] E. I. Jury and B. ' X , Lee, On the stablhty of a certain class of nonlinear samnled-dat.a srstems." IEEE Trans. Automd. contr.. vol. AC-9. nD. 51-61, Jan.<1964. [3] Pa. Z. Tsypkin, "Absolute stabi1it.y of nonlinear automatic sampled-det.a systems," Automat. Remote Conlr. (USSR), vol. 25, pp. 1030-1036, ,l964,. [1] C. T. Chen, "On the stability of sampled-data feedback systems m t h timevarying pain," I B E E Trans. Autamat. Conir. (Corresp.), vol. AG11, pp. . _ _ 151 K. 5. Narendra and J. H. Taylor, "Liapunov functions for nonlinear t i m e 'varying systems,"Inform. Contr.. vol. 1: : pp. 3i8-393, 1968. [61 J. B. Pearson. Jr. and J. E. Gibson, On the asympt.otic stability of a class of saturating'sampled-data systems," I E E E Trans. A p p l . Ind., vol. 83, [ i ] G. P. SzegS and R.,,E. Kalman. "A1xolut.e stability of a system of finit.0 Time-Varying Nonlinear Systems e 6X RIIN-YEN WU Absfracf-Sufficient conditions for the stability of a class of multi-(25b) plicative time-varging as well as time-invariant nonlinear systems 3) A k ( n -1) ~g n p 5 6Fmin R(n) which are but of the theorem \<
