Fading structural memory

Perfectly elastic deformation and perfectly viscous flow are idealizations that are approximately realized in some limiting conditions. In general the condensed matter has a fading structural memory, and the velocity with which a system that has been perturbed forgets the configuration that it had in the past roughly defines its solid or liquid nature. In ordinary liquids, molecular... 

In the last two centuries, a lot of attempts and discussion have been made on the elucidation and development of the various constitutive models of liquids. Some of the theoretical models that can be mentioned here are Boltzmann, Maxwell (UCM, LCM, COM, 1PM), Voight or Kelvin, Jeffrey, Reiner-Rivelin, Newton, Oldroyd, Giesekus, graded fluids, composite fluids, retarded fluids with a strong backbone and fading memory, and so on. Further and deeper knowledge related to the physical and mathematical consequences of the structural models of liquids and of the elasticity of liquids can be found in Ref. [6]. 

Eqs. (7)-(10) constitute the structures of the fading memory least-squares filter for state estimation, which are in the recursive form. 

The optimal Kalman filter is derived under the optimality criterion of least-mean-square error of the state, while the fading memory least-squares filter (FMLS) is under least-squares error of the measurement. However, it is interesting to see that they turn out to have the same form of structures. Examing Eqs. (7)-(10), it is found that and in FMLS are equivalent to the a priori state error covariance and the posterior error covariance Pjt, respectively, in the Kalman filter. The formula for Pj i is... 

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