In this paper we show how to completely and exactly decompose the optimal Kalman filter of stochastic systems in multimodeling form in terms of one pure-slow and two pure-fast, reduced-order, independent, Kalman filters. The reduced-order Kalman filters are all driven by the system measurements. This leads to a parallel Kalman filtering scheme and removes ill-conditioning of the original full-order singularly perturbed Kalman filter. The results obtained are valid for steady state. In that direction, the corresponding algebraic filter Riccati equation is completely decoupled and solved in terms of one pure-slow and two pure fast, reduced-order, independent, algebraic Riccati equations. A nonsingular state transformation that exactly relates the state variables in the original and new coordinates (in which the required decomposition is achieved) is also established. The eighth order model of a passenger car under road disturbances is used to demonstrate efficiency of the proposed filtering technique. [S0022-0434(00)01703-2]

In this paper we show how to exactly decompose the algebraic Riccati equations of deterministic multimodeling in terms of one pure-slow and two pure-fast algebraic Riccati equations. The algebraic Riccati equations obtained are of reduced-order and nonsymmetric. However, their ( ) perturbations (where = and , are small positive singular perturbation parameters) are symmetric. The Newton method is perfectly suited for solving the nonsymmetric reduced-order pure-slow and pure-fast algebraic Riccati equations since excellent initial guesses are available from their ( ) perturbed reduced-order symmetric algebraic Riccati equations that can be solved rather easily. The proposed decomposition scheme might facilitates new approaches to mutimodeling control problems that are conceptually simpler and numerically more efficient than the ones previously used.

This paper considers the relative entropy between the conditional distribution and an incorrectly initialized ®lter for the estimation of one component of a Markov process given observations of the second component. Using the Markov property, we ®rst establish a decomposition of the relative entropy between the measures on observation path space associated to di¨erent initial conditions. Using this decomposition, it is shown that the relative entropy of the optimal ®lter relative to an incorrectly initialized ®lter is a positive supermartingale. By applying the decomposition to signals observed in additive, white noise, a relative entropy bound is obtained on the integrated, expected, mean square di¨erence between the optimal and incorrectly initialized estimates of the observation function.

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In this paper, the finite-time optimal control problem for time-invariant linear singularly perturbed systems is considered. The reduced-order pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed differential matrix Riccati equation of dimension n1 + n2 into the regular differential matrix Riccati equation pure-slow of dimension n1 and the stiff differential matrix Riccati equation pure-fast of dimension n2. A formula is derived that produces the solution of the original singularly perturbed matrix Riccati differential equation in terms of solutions of the pure-slow and pure-fast reduced-order differential matrix Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear systems independently in pure-slow and pure-fast time scales. An example for a catalytic fluid reactor model has been include to demonstrate the utility of the method.