This paper presents some fundamental insights into observer design for
the class of Lipschitz nonlinear systems. The stability of the nonlin
ear observer for such systems is not determined purely by the eigenval
ues of the linear stability matrix, The correct necessary and sufficie
nt conditions on the stability matrix that ensure asymptotic stability
of the observer are presented, These conditions are then reformulated
to obtain a sufficient condition for stability in terms of the eigenv
alues and the eigenvectors of the Linear stability matrix. The eigenva
lues have to be located sufficiently far out into the left half-plane,
and the eigenvectors also have to be sufficiently well-conditioned fo
r ensuring asymptotic stability. Based on these results, a systematic
computational algorithm is then presented for obtaining the observer g
ain matrix so as to achieve the objective of asymptotic stability.