Optimal filtering for linear systems with state and observation delays

In this paper, the optimal filtering problem for linear systems with state and observation delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate, error variance, and various error covariances. As a result, the optimal estimate equation similar to the traditional Kalman–Bucy one is derived; however, it is impossible to obtain a system of the filtering equations, that is closed with respect to the only two variables, the optimal estimate and the error variance, as in the Kalman–Bucy filter. The resulting system of equations for determining the filter gain matrix consists, in the general case, of an infinite set of equations. It is however demonstrated that a finite set of the filtering equations, whose number is specified by the ratio between the current filtering horizon and the delay values, can be obtained in the particular case of equal or commensurable (τ=qh, q is natural) delays in the observation and state equations. In the example, performance of the designed optimal filter for linear systems with state and observation delays is verified against the best Kalman–Bucy filter available for linear systems without delays and two versions of the extended Kalman–Bucy filter for time delay systems. Copyright © 2005 John Wiley & Sons, Ltd.     

Robust H2 filtering of linear systems with time delays

The problem of robust H₂ estimation of a combination of states of a stationary linear system with time delays is considered. Since the problem is infinite dimensional in nature, an attempt is being made to develop finite dimensional methods that will guarantee a preassigned estimation accuracy. The approach of minimizing the trace of a matrix that overbounds the exact covariance of the estimation error is considered. Sufficient conditions are given in the form of linear matrix inequalities (LMIs). The results are illustrated by a numerical example. Copyright © 2003 John Wiley & Sons, Ltd.     

Optimal filtering for linear systems with state and multiple observation delays

This article solves the optimal filtering problem for linear systems with state and multiple observation delays. The optimal estimate equation similar to the traditional Kalman–Bucy one is derived, and the system of equations for determining the filter gain matrix consists of an infinite set of equations. It is then demonstrated that a finite set of the filtering equations can be obtained in case of commensurable delays. In the example, the designed optimal filter is compared to the traditional Kalman–Bucy filter.     

Optimal filtering for networked systems with Markovian communication delays

This paper is concerned with optimal filter problems for networked systems with random transmission delays, while the delay process is modeled as a multi-state Markov chain. By defining a delay-free observation sequence, the optimal filter problems are transformed into ones of the Markov jumping parameter system. We first present an optimal Kalman filter, which is with time-varying, path-dependent filter gains, and the number of the paths grows exponentially in time delay. Thus an alternative optimal Markov jump linear filter is presented, in which the filter gains just depend on the present value of the Markov chain. Further, an optimal filter with constant-gains is developed, the existence condition for the stabilizing solutions to the filter is given, and it can be shown that the proposed Markov jump linear filter converges to the constant-gain filter under appropriate assumptions.     

Robust H∞ deconvolution filtering for uncertain singular Markovian jump systems with time‐varying delays

The problem of H_∞ deconvolution filter design for a class of singular Markovian jump systems with time‐varying delays and parameter uncertainties is considered in this paper. By constructing a more comprehensive stochastic Lyapunov‐Krasovskii functional, novel delay‐dependent conditions are established to guarantee the filtering error system is not only stochastically admissible, but also satisfies a prescribed H_∞‐norm level for all admissible uncertainties. The desired filter parameters can be obtained by solving a set of strict linear matrix inequalities. Two examples and an electrical RLC circuit example are employed to verify the effectiveness and usefulness of the proposed methods in the paper. Copyright © 2015 John Wiley & Sons, Ltd.     

Refined discretized Lyapunov functional method for systems with multiple delays

The previously proposed discretized Lyapunov functional method for systems with multiple delay is refined using variable elimination and Jensen inequality. The resulting new stability criterion is simpler. Numerical examples indicate that the new method is much less conservative for a given discretization mesh. Copyright © 2003 John Wiley & Sons, Ltd.     

Delay‐range‐dependent robust H∞ filtering for uncertain systems with interval time‐varying delays

This paper is concerned with the problem of delay‐range‐dependent robust H_∞ filtering for systems with time‐varying delays in a range. The aim of this problem is to design a filter such that, for all admissible uncertainties, the filtering error system is robustly asymptotically stable with a prescribed H_∞ level. The desired filter can be constructed by solving a set of linear matrix inequalities (LMIs). An illustrative numerical example is provided to demonstrate the effectiveness of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

On stability of linear time-delay systems with multiple delays

In this article, delay-dependent stability criteria for linear retarded and neutral systems with multiple delays are proposed by employing the Lyapunov-Krasovskii functional approach and integral inequality. By the N-segmentation of delay length, we obtain more relaxed results on the delay bounds which guarantee the asymptotic stability of the linear retarded and neutral systems with multiple delays. Simulation results show that the stability boundary calculated by the proposed stability criteria are very near to the real stability boundary of the example systems. Moreover, it is shown that the proposed stability criteria are less conservative than several other existing criteria.     

Optimal full-Order filtering for discrete-time systems with random measurement delays and multiple packet dropouts

This paper is concerned with the estimation problem for discrete-time stochastic linear systems with possible single unit delay and multiple packet dropouts. Based on a proposed uncertain model in data transmission, an optimal full-order filter for the state of the system is presented, which is shown to be of the form of employing the received outputs at the current and last time instants. The solution to the optimal filter is given in terms of a Riccati difference equation governed by two binary random variables. The optimal filter is reduced to the standard Kalman filter when there are no random delays and packet dropouts. The steady-state filter is also investigated. A sufficient condition for the existence of the steady-state filter is given. The asymptotic stability of the optimal filter is analyzed.     

Robust distributed H∞ filtering over an uncertain sensor network with multiple fading measurements and varying sensor delays

In this paper, the problem of robust distributed H_∞ filtering is investigated for state‐delayed discrete‐time linear systems over a sensor network with multiple fading measurements, random time‐varying communication delays, and norm‐bounded uncertainties in all matrices of the system. The diagonal matrices, whose elements are individual independent random variables, are utilized to describe the multiple fading measurements. Furthermore, the Bernoulli‐distributed white sequences are introduced to model the random occurrence of time‐varying communication delays. In the proposed filtering approach, the stability of the estimation error system is first shown by the Lyapunov stability theory and the H_∞ performance is then achieved using a linear matrix inequality method. Finally, two numerical examples are given to show the effectiveness and performance of the proposed approach.     

Reachable set estimation for discrete‐time linear systems with time delays

This paper is concerned with the reachable set estimation problem for discrete‐time linear systems with multiple constant delays and bounded peak inputs. The objective is to check whether there exists a bounded set that contains all the system states under zero initial conditions. First, delay‐dependent conditions for the solvability of the addressed problem are derived by employing a novel Lyapunov–Krasovskii functional. The obtained conditions are expressed in terms of matrix inequalities, which are linear when only one scalar variable is fixed. On the basis of these conditions, an ellipsoid containing the reachable set of the considered system is obtained. An approach for determining the smallest ellipsoid is also provided. Second, the approach and results developed in the first stage are generalized to the case of systems with polytopic parameter uncertainties, and delay‐dependent conditions are given in the form of relaxed matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.     

Integral sliding mode design for robust filtering and control of linear stochastic time‐delay systems

This paper presents an integral sliding mode technique robustifying the optimal controller for linear stochastic systems with input and observation delays, which is based on integral sliding mode compen‐sation of disturbances. The general principles of the integral sliding mode compensator design are modified to yield the basic control algorithm oriented to time‐delay systems, which is then applied to robustify the optimal controller. As a result, two integral sliding mode control compensators are designed to suppress disturbances in state and observation equations, respectively, from the initial time moment. Moreover, it is shown that if certain matching conditions hold, the designed compensator in the state equation can simultaneously suppress observation disturbances, as well as the designed compensator in the observation equation can simultaneously suppress state disturbances. The obtained robust control algorithm is verified by simulations in the illustrative example, where the compensator in the observation equation provides simultaneous suppression of state and observation disturbances. Copyright © 2005 John Wiley & Sons, Ltd.     

Mean square stability of linear stochastic neutral‐type time‐delay systems with multiple delays

This paper studies mean square exponential stability of linear stochastic neutral‐type time‐delay systems with multiple point delays by using an augmented Lyapunov‐Krasovskii functional (LKF) approach. To build a suitable augmented LKF, a method is proposed to find an augmented state vector whose elements are linearly independent. With the help of the linearly independent augmented state vector, the constructed LKF, and properties of the stochastic integral, sufficient delay‐dependent stability conditions expressed by linear matrix inequalities are established to guarantee the mean square exponential stability of the system. Differently from previous results where the difference operator associated with the system needs to satisfy a condition in terms of matrix norms, in the current paper, the difference operator only needs to satisfy a less restrictive condition in terms of matrix spectral radius. The effectiveness of the proposed approach is illustrated by two numerical examples.     

Admissible consensus analysis for high‐order linear singular swarm systems with multiple time‐varying delays and topology variances

Admissible consensus analysis problems for high‐order linear time‐invariant singular swarm systems with multiple time delays and time‐varying interaction topologies are investigated. First, necessary and sufficient conditions for admissible consensus are presented, and admissible consensus problems are transformed into admissible problems of multiple lower dimensional singular systems. Then, on the basis of the first equivalent form, explicit expressions of consensus functions are given, where the impacts of initial states of agents and protocols, time delays and topology variances are determined, respectively. Moreover, it is shown that if interaction topologies are balanced, then swarm systems with the same initial states but different interaction topologies and different time delays have an identical consensus function. Finally, numerical simulations are given to demonstrate theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal linear filtering and extrapolation of realizations of a vector random function observed with errors

A solution to an optimal linear filtering and extrapolation problem is proposed. The investigation object is a realization of multidimensional random function, which is measured with random errors. The solution method does not put any essential constraints on the investigated random fimction, measurment error characteristics, and on the character of relations between them. The solution is based on Pugachov’s canonical decomposition of the investigated random function, and its form is convenient for use with a computer. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 125–132, July–August, 2000.     

Optimal linear estimators for systems with multiple random measurement delays and packet dropouts

This article is concerned with the optimal linear estimation problem for linear discrete-time stochastic systems with possible multiple random measurement delays and packet dropouts, where the largest random delay is limited within a known bound and packet dropouts can be infinite. A new model is constructed to describe the phenomena of multiple random delays and packet dropouts by employing some random variables of Bernoulli distribution. By state augmentation, the system with random delays and packet dropouts is transferred to a system with random parameters. Based on the new model, the least mean square optimal linear estimators including filter, predictor and smoother are easily obtained via an innovation analysis approach. The estimators are recursively computed in terms of the solutions of a Riccati difference equation and a Lyapunov difference equation. A sufficient condition for the existence of the steady-state estimators is given. An example shows the effectiveness of the proposed algorithms.     

Finite‐time boundedness and L2‐gain analysis for switched positive linear systems with multiple time delays

In this paper, we study the finite‐time boundedness, stabilization, and L₂‐gain for switched positive linear systems (SPLS) with multiple time delays. Using multiple linear copositive Lyapunov functions, sufficient conditions in terms of linear matrix inequalities are obtained for the problems of finite‐time boundedness and stabilization and the design of state feedback controllers for SPLS. Under asynchronous switching, L₂‐gain analysis is developed for SPLS under the constraint of average dwell time. Numerical examples are given to illustrate our theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Decentralized tracking control for linear uncertain interconnected systems with time delays

This paper investigates the robust output tracking problem for a class of large‐scale linear uncertain systems with interactions and time delays. Time delays exist in both states and controls. Using the Riccati equation, a procedure for determining decentralized linear control laws is presented such that the closed‐loop system asymptotically tracks the reference output and rejects any constant but unknown disturbances. The main feature of this approach is that the uncertain systems may contain time delays in both states and controls as well as in interactions between subsystems. A numerical example is included to show the results. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Exponential H∞ filtering for switched linear systems with interval time‐varying delay

This paper deals with the problem of exponential H_∞ filtering for a class of continuous‐time switched linear system with interval time‐varying delay. The time delay under consideration includes two cases: one is that the time delay is differentiable and bounded with a constant delay‐derivative bound, whereas the other is that the time delay is continuous and bounded. Switched linear filters are designed to ensure that the filtering error systems under switching signal with average dwell time are exponentially stable with a prescribed H_∞ noise attenuation level. Based on the free‐weighting matrix approach and the average dwell technology, delay‐dependent sufficient conditions for the existence of such a filter are derived and formulated in terms of linear matrix inequalities (LMIs). By solving that corresponding LMIs, the desired filter parameterized matrices and the minimal average dwell time are obtained. Finally, two numerical examples are presented to demonstrate the effectiveness of the developed results. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimal unbiased filtering via linear matrix inequalities

Solutions to the optimal H_∞ and L₂−L_∞ unbiased reduced-order filtering problems are obtained in terms of linear matrix inequalities (LMIs). The order of the optimal filter is equal to the number of measurements. Both continuous- and discrete-time results are presented. An explicit parametrization of all optimal unbiased filters is provided in terms of a free contractive matrix.