If all of the eigenvalues of the Hadamard product [A.sub.d] [??] E are located at the left-hand plane, then the S-system in (1) is stable, wherein the steady state parameter
E is the outer product of the equilibrium flux vector [[bar.V].sup.+] and equilibrium state inverse vector [[??].sub.d]; i.e., E = [[bar.V].sup.+] [cross product] [[??].sub.d] where the vector [[??].sub.d] = [1 / [[bar.x].sub.1]] with i = 1, ..., n, and [[bar.x].sub.i]] denotes the dependent variable [x.sub.i] in equilibrium.
Depending on the actual engineering needs, online data filtering is mainly performed on coordinates ([x.sub.e], [y.sub.e], [z.sub.e]) in this article, so x = [[[x.sub.e] [y.sub.e] [z.sub.e]].sup.T] is selected as the state parameter
, and the measurement parameters are the measured coordinates' information.
One can infer that such a relation bears similarity to the pressure term with the contribution from perfect fluid (p = [omega][rho]), where [omega] is the equation of state parameter
for perfect fluid.
Vertical dashed lines mark important points where state parameter
lines intersect or change direction.
where X is the predicted state parameter
vector; [PHI] is the state transition matrix; X is the estimated state parameter
vector; [R.sub.[bar.X]] is the predicted state covariance matrix; [R.sub.X] is the posteriori covariance matrix of the state parameters
; Q is the covariance matrix of the processing noise; K is the gain matrix of the Kalman filter; H is the coefficient matrix of the observation equation; P is the observation weight matrix; R is the observation noise variance matrix, such that R = [P.sup.-1] ; and V is the predicted parameter vector, also called the innovation vector.
s - state parameter
in the external part of network;
Several researchers try to correlate index properties of granular soils with critical state parameter
including fines content , void ratio , and liquidity index .
 and Gajo and Muir Wood  developed the above equation to include the effect of state parameter
[psi] = e - [e.sub.cr], in which e is the void ratio and [e.sub.cr] is the magnitude of the void ratio on the critical-state line, as follows:
For example, Been & Jefferies (1985) proposed using state parameter
Y defined as the difference between the initial void ratio of the sandy soil sample and the void ratio at the CSL for the same effective mean pressure p'.