A.I. Matasov

Neutral Stochastic Differential Delay Equations with Markovian Switching

Abstract Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see Kolmanovskii, V.B. and Nosov, V.R., Stability and Periodic Modes of Control Systems with Aftereffect; Nauka: Moscow, 1981 and Mao X., Stochastic Differential Equations and Their Applications; Horwood Pub.: Chichester, 1997). Given that many systems are often subject to component failures or repairs, changing subsystem interconnections and abrupt environmental disturbances etc., the structure and parameters of underlying NSDDEs may change abruptly. One way to model such abrupt changes is to use the continuous‐time Markov chains. As a result, the underlying NSDDEs become NSDDEs with Markovian switching which are hybrid systems. So far little is known about the NSDDEs with Markovian switching and the aim of this paper is to close this gap. In this paper we will not only establish a fundamental theory for such systems but also discuss some important properties of the solutions e.g. boundedness and stability.

Estimators for uncertain dynamic systems

Preface. 1. Guaranteed Parameter Estimation. 2. Guaranteed Estimation in Dynamic Systems. 3. Kalman Filter in Guaranteed Estimation Problem. 4. Stochastic Guaranteed Estimation Problem. 5. Estimation Problems in Systems with Aftereffect. Bibliography. Index.

The Kalman-Bucy filter accuracy in the guaranteed parameter estimation problem with uncertain statistics

The guaranteed parameter estimation problem for a linear dynamical system with uncertain disturbance statistics is considered. The estimate of the Kalman-Bucy filter nonoptimality degree is constructed. It is based on the duality theory. This estimate is determined by the Kalman-Bucy filter characteristics only and can be calculated without knowledge of the accurate solution of the optimal guaranteed estimation problem. >

The Kalman-Bucy filter accuracy in the guaranteed estimation problem

The optimal guaranteed a priori estimation problem is considered. The Kalman-Bucy filter is used for the approximate solution of this problem. An analytical estimate for the nonoptimality degree of the Kalman-Bucy filter is obtained. This estimate is determined solely by the Kalman-Bucy filter characteristics. Thus the Kalman-Bucy filter efficiency can be established without accurate solution of the difficult optimal guaranteed estimation problem.<<ETX>>

Mean-square filtering problem for discrete Volterra equations

Abstract The mean-square filtering problem for the discrete Volterra equations is a nontrivial task due to an enormous amount of operations required for the implementation of optimal filter. A difference equation of a moderate dimension is chosen as an approximate model for the original system. Then the reduced Kalman filter can be used as an approximate but efficient estimator. Using the duality theory of convex variational problems, a level of nonoptimality for the chosen filter is obtained. This level can be efficiently computed without exactly solving the full filtering problem.

The Kalman-Bucy filter in the guaranteed estimation problem

The optimal guaranteed a priori estimation problem is considered. The Kalman-Bucy filter is used for the approximate solving of this problem. The analytical estimate for the nonoptimality degree of the Kalman-Bucy filter is obtained. This estimate is determined by the Kalman-Bucy filter characteristics only. Thus the Kalman-Bucy filter efficiency can be established without accurate solving of the difficult optimal guaranteed estimation problem. >

Weight and Time Recursions in Dynamic State Estimation Problem With Mixed-Norm Cost Function

The mixed-norm cost functions arise in many applied optimization problems. As an important example, we consider the state estimation problem for a linear dynamic system under a nonclassical assumption that some entries of state vector admit jumps in their trajectories. The estimation problem is solved by means of mixed l1/l2-norm approximation. This approach combines the advantages of the well-known quadratic smoothing and the robustness of the least absolute deviations method. For the implementation of the mixed-norm approximation, a dynamic iterative estimation algorithm is proposed. This algorithm is based on weight and time recursions and demonstrates the high efficiency. It well identifies the rare jumps in the state vector and has some advantages over more customary methods in the typical case of a large amount of measurements. Nonoptimality levels for current iterations of the algorithm are constructed. Computation of these levels allows to check the accuracy of iterations.

State Estimation via l1-norm Approximation: Application to Inertial Navigation

Abstract The problem of the detection of jumps in the biases of the sensors of a strapdown inertial navigation system at bench tests is considered. This problem is set as a state estimation problem for a dynamic system under the presence of outliers in object disturbances. The estimation method is based on l 1 -norm approximation and is robust to abrupt changes in signals.

Constructive filtering algorithms for delayed systems with uncertain statistics

A stochastic optimal guaranteed estimation problem for dynamic delayed systems with uncertain statistics is considered. The solution of this problem reduces to a complex nonsmooth extremal problem. To obtain an approximate solution, the nonsmooth problem is replaced by a smooth one. Constructive filtering algorithms are obtained from an approximate solution of the smooth problem under the assumption that the delay is small in comparison with the observation time. Estimates for the nonoptimality levels of the proposed filtering algorithms are derived.

On a Boundary Value Problem in the Filtering Theory for Discrete Volterra Equations

Abstract A boundary value problem that arises in the filtering theory for discrete Volterra equations is considered. An important dependence between primal and adjoint variables is obtained.