Riccati-like Difference Equations Research Articles

Robust Kalman filtering for two-dimensional systems with multiplicative noises and measurement degradations: The finite-horizon case

Abstract In this paper, robust Kalman filtering problem is investigated for a class of two-dimensional (2-D) shift-varying uncertain systems with both additive and multiplicative noises over a finite horizon. The measurement outputs suffer from randomly occurring degradations obeying certain probabilistic distributions, and the norm-bounded parameter uncertainties enter into both the state and the output matrices. The main aim of this paper is to design a robust Kalman filter such that, in the presence of parameter uncertainties and degraded measurements, certain upper bound of the generalized estimation error variance is locally minimized in the trace sense at each shift step. Recursion of the generalized estimation error variances for the addressed 2-D system is first established via the introduction of a 2-D identity quadratic filter, based on which an upper bound of the generalized estimation error variance is obtained. Subsequently, such an upper bound is minimized in the trace sense by properly designing the filter parameters. The design scheme of the robust Kalman filter is presented in terms of two Riccati-like difference equations which can be recursively computed for programmed applications. Finally, a numerical example is provided to demonstrate effectiveness of the proposed filter design method.

Event-triggered resilient filtering with measurement quantization and random sensor failures: Monotonicity and convergence

This paper is concerned with the remote state estimation problem for a class of discrete-time stochastic systems. An event-triggered scheme is exploited to regulate the sensor-to-estimator communication in order to preserve limited network resources. A situation is considered where the sensors are susceptible to possible failures and the signals are quantized before entering the network. Furthermore, the resilience issue for the filter design is taken into account in order to accommodate the possible gain variations in the course of filter implementation. In the simultaneous presence of measurement quantizations, sensor failures and gain variations, an event-triggered filter is designed to minimize certain upper bound of the covariance of the estimation error in terms of the solution to Riccati-like difference equations. Further analysis demonstrates the monotonicity of the minimized upper bound with respect to the value of thresholds. Subsequently, a sufficient condition is also established for the convergence of the steady-state filter. A numerical example is presented to verify the effectiveness of the proposed filtering algorithm.

Resilient State Estimation for 2-D Time-Varying Systems With Redundant Channels: A Variance-Constrained Approach.

This paper investigates the state estimation problem for a class of 2-D time-varying systems with error variance constraints, where the implemented estimator gain is subject to stochastic perturbations. Redundant channels are utilized as a protocol to strengthen the transmission reliability and the channels' packet dropout rates are described by mutually uncorrelated Bernoulli distributions. The objective of the addressed problem is to design a resilient estimator such that an upper bound on the estimation error variance is first guaranteed and then minimized at each time step, where the considered gain perturbations are characterized by their statistical properties. By employing the induction method and the variance-constrained approach, an upper bound on the estimation error variance is first constructed by means of the solutions to two Riccati-like difference equations and, subsequently, a locally minimal upper bound is achieved by appropriately designing the gain parameter. Then, an effective algorithm is proposed for designing the desired estimator, which is in a recursive form suitable for online applications. Finally, a numerical simulation is provided to demonstrate the usefulness of the proposed estimation scheme.

Krein-space based robust [formula omitted] fault estimation for two-dimensional uncertain linear discrete time-varying systems

In this study, the robust H∞ fault estimation problem for two-dimensional linear time-varying systems with norm-bounded unknown input, measurement noise, and time-varying process uncertainty is investigated. By introducing an equivalent auxiliary system and a new certain indefinite quadratic form performance function, the system uncertainty can be appropriately considered into the new performance function and the fault estimator design is converted to the minimization problem of a quadratic form. Based on the partially equivalence property between the deterministic quadratic form problem and the Krein space estimation theory, the two-dimensional H∞ fault estimation problem can be solved via signal deconvolution in Krein space. Through employing projection operation and Riccati-like difference equation in two dimensions, both the recursive form fault estimator and the explicit condition for existence of the estimator are derived. One Darboux equation example is provided to illustrate the effectiveness of the proposed fault estimator.

Recursive Filtering for Time-Varying Systems with Random Access Protocol

This paper is concerned with the recursive filtering problem for a class of networked linear time-varying systems subject to the scheduling of the random access protocol (RAP). The communication between the sensor nodes and the remote filter is implemented via a shared network. For the purpose of preventing the data from collisions, only one sensor node is allowed to get access to the network at each time instant. The transmission order of sensor nodes is orchestrated by the RAP scheduling, under which the selected nodes obtaining access to the network could be characterized by a sequence of independent and identically-distributed variables. The aim of the addressed filtering problem is to design a recursive filter such that the filtering error covariance could be minimized by properly designing the filter gain at each time instant. The desired filter gain is calculated recursively by solving two Riccati-like difference equations. Furthermore, the boundedness issue of the corresponding filtering error covariance is investigated. Sufficient conditions are obtained to ensure the lower and upper bounds of the filtering error covariance. Two illustrative examples are given to demonstrate the correctness and effectiveness ofour developed recursive filtering approach.

State Estimation for Stochastic Complex Networks With Switching Topology

This technical note studies the state estimation problem for stochastic complex networks with switching topology. A set of Bernoulli random variables are used to describe the switching behavior of the network. By using the structure of the extended Kalman filter (EKF), a recursive estimator is developed for each node to guarantee an optimized upper bound on the state estimation error covariance despite the switching coupling and linearization errors. Compared with the existing results on state estimation for stochastic complex networks, a distinct feature for the proposed estimator is that the estimation errors for all nodes are not formulated in an augmented vector so that the gain matrix can be determined separately for each node by solving two Riccati-like difference equations. Based on the stochastic analysis theory, it is shown that the estimation error is bounded in mean square under certain conditions. Simulation results are provided to verify the effectiveness of the proposed estimator.

An event-triggered approach to robust recursive filtering for stochastic discrete time-varying spatial-temporal systems

In this paper, the robust recursive filtering problem is investigated for a class of uncertain stochastic discrete time-varying spatial-temporal systems with an event-based communication mechanism. The system under consideration includes a set of sensors located at specified points where each sensor can only receive the measurement output from the system at corresponding position. In order to reduce the communication cost, an event-triggered mechanism is utilized to decide when a certain measurement output should be transmitted from sensors to the centralized filter. By re-organizing the state variables, the uncertain spatial-temporal system is first transformed into an ordinary uncertain difference dynamic system. The aim of this paper is to design a filter such that, for all possible parameter uncertainties, the filtering error covariance is guaranteed to have an upper bound which is then minimized at each time-instant under the event-triggered mechanism. By using the matrix derivation approach, the desired filter gain is obtained in terms of the solution to certain Riccati-like difference equations. Finally, a numerical example is employed to demonstrate the effectiveness of the filtering scheme proposed.

State estimation for nonlinearly coupled complex networks with application to multi-target tracking

This paper is concerned with the state estimation problem for a class of nonlinearly coupled complex networks with random coupling strengths. The coupling strengths are assumed to be chosen from a set of uniform distributions with non-negative means. The estimator is developed for each node by using the structure of the extended Kalman filter, where the gain matrix is determined by optimizing an upper bound matrix for the estimation error covariance despite the linearization errors and coupling terms. It is shown that the gain matrix can be derived separately for each node by solving two Riccati-like difference equations. By using the stochastic analysis technique, a sufficient condition is established which guarantees the boundedness of the estimation errors. As an application, we show how the nonlinearly coupled complex networks can be used to describe the target motions with interacting behaviors and the proposed estimator can be used to derive the state estimates of the targets. A numerical example is provided to verify the effectiveness of the proposed estimator.

Distributed extended Kalman filter with nonlinear consensus estimate

This paper is concerned with the distributed filtering problem for discrete-time nonlinear systems over a sensor network. In contrast with the distributed filters with linear consensus estimate, a distributed extended Kalman filter (EKF) is developed with nonlinear consensus estimate. Specifically, a new nonlinear consensus protocol with polynomial form is proposed to generate the consensus estimate. By using the variance-constrained approach, the Kalman gain matrix is determined for each node to guarantee an optimized upper bound on the state estimation error covariance despite consensus terms and linearization errors. It is shown that the Kalman gain matrix can be derived by solving two Riccati-like difference equations. The effectiveness of the proposed filter is evaluated on an indoor localization of a mobile robot with visual tracking systems.

Recursive filtering for communication-based train control systems with packet dropouts

Abstract Accurate information about the train position and velocity is critically important for Communication-based Train Control (CBTC) systems. However, it is practically difficult to obtain the precise information of such information due mainly to the “inaccurate measurements” induced by the measurement noises and the “unreliable communication” caused by the wireless train-ground communication. In this paper, a recursive filtering algorithm is proposed to generate the estimates of the train position and velocity for CBTC systems subject to the measurement noise and packet dropouts. Firstly, the dynamics of a train is modeled based on the Newton’s motion equation. Then, a Bernoulli distributed sequence is introduced to describe the packet dropout phenomenon of the wireless communication. The purpose of the problem addressed is to design a recursive filter such that there exists an upper bound for the filtering error covariance. Subsequently, such an upper bound is minimized by properly designing the filter parameter recursively. The desired filter parameter is obtained by solving two Riccati-like difference equations that are of a recursive form suitable for online applications. Finally, an illustrative example is given to show the effective of the proposed filter design scheme.

SNR Analysis for Linear Quadratic Control Over Gaussian Channels with Feedback

This paper investigates signal-to-noise ratio (SNR) required for achieving desired LQR performance in control over Gaussian channels with feedback. The SNR required in control under the presence of Gaussian disturbance is described using the solution of a Riccati equation based on a technique of averaging the solution of a Riccati difference equation. The minimality of the SNR presented is guaranteed by evaluating information transmitted through the channel. The SNR required in control under the absence of disturbance is also investigated for the control law whose data transmission strategy is modified to be time-varying is described using the solution of a Riccati-like difference equation. The resultant performance level is ensured to be bounded based on a similar averaging technique.

A Variance-Constrained Approach to Recursive Filtering for Nonlinear 2-D Systems With Measurement Degradations.

This paper is concerned with the recursive filtering problem for a class of nonlinear 2-D time-varying systems with degraded measurements over a finite horizon. The phenomenon of measurement degradation occurs in a random way depicted by stochastic variables satisfying certain probabilities distributions. The nonlinearities under consideration are dealt with through the Taylor expansion, where the high-order terms of the linearization errors are characterized by norm-bounded parameter uncertainties. The objective of the addressed problem is to design a filter which guarantees an upper bound of the estimation error variance and subsequently minimizes such a bound with the desired gain parameters. By means of mathematical induction, an upper bound is first derived for the estimation error variance by constructing two sets of Riccati-like difference equations, and then the obtained bound is minimized by properly selecting the filter parameter at each time step. Both the minimal upper bound and the desired filter parameter are suitable for recursive online computation. Furthermore, the effect of the stochastic measurement degradation on the filtering performance is discussed. Finally, a simulation example is presented to demonstrate the effectiveness of the designed filter.

Event-based distributed recursive filtering for state-saturated systems with redundant channels

In this paper, the event-based distributed recursive filtering problem is investigated for a class of discrete-time state-saturated systems subject to random occurring nonlinearities and measurement losses over the wireless sensor network. In the addressed measurement model, the sensors are assumed to have redundant communication channels that are helpful in increasing the probability of successfully delivering the measurements. At each intelligent node, the local estimation is obtained based on its own measurement and those transmitted from its neighbors according to the sensor topology. In order to reduce the bandwidth consumption and estimator update frequencies, an event-based signal transmission strategy is employed as opposed to the traditional time-based one. An upper bound for the estimation error covariance is constructed at each time step, which is shown to be the solution of a Riccati-like difference equation. Subsequently, the estimator parameter is designed to minimize such an upper bound. Moreover, the performance of the proposed estimator is analyzed by discussing how the packet losses of the measurements affect the obtained upper bound of the error covariance. Finally, a numerical simulation is exploited to show the effectiveness of the proposed distributed filter design algorithm.

Distributed consensus extended Kalman filter: a variance‐constrained approach

This study is concerned with the distributed state estimation problem for non-linear systems over sensor networks. By using the strategy of consensus on prior estimates, a distributed extended Kalman filter (EKF) is developed for each node to guarantee an optimised upper bound on the state estimation error covariance despite consensus terms and linearisation errors. The Kalman gain matrix is derived for each node by solving two Riccati-like difference equations. It is shown that the estimation error is bounded in mean square under certain conditions. The effectiveness of the proposed filter is evaluated on an indoor localisation of a mobile robot with visual tracking systems.

Event-based input and state estimation for linear discrete time-varying systems

ABSTRACTIn this paper, the joint input and state estimation problem is considered for linear discrete-time stochastic systems. An event-based transmission scheme is proposed with which the current measurement is released to the estimator only when the difference from the previously transmitted one is greater than a prescribed threshold. The purpose of this paper is to design an event-based recursive input and state estimator such that the estimation error covariances have guaranteed upper bounds at all times. The estimator gains are calculated by solving two constrained optimisation problems and the upper bounds of the estimation error covariances are obtained in form of the solution to Riccati-like difference equations. Special efforts are made on the choices of appropriate scalar parameter sequences in order to reduce the upper bounds. In the special case of linear time-invariant system, sufficient conditions are acquired under which the upper bound of the error covariance of the state estimation is asymptomatically bounded. Numerical simulations are conducted to illustrate the effectiveness of the proposed estimation algorithm.

Non-augmented state estimation for nonlinear stochastic coupling networks

This paper considers the state estimation problem for discrete-time nonlinear stochastic coupling networks. A non-augmented filter is designed for each node to guarantee an optimized upper bound on the state estimation error covariance matrix despite nodes coupling as well as the linearization errors. Compared with the existing augmented filter, the cross-covariance matrices between coupling nodes are not required to be computed and the gain matrix can be obtained separately for each node by solving two Riccati-like difference equations.

Variance-Constrained State Estimation for Nonlinearly Coupled Complex Networks.

This paper studies the state estimation problem for nonlinearly coupled complex networks. A variance-constrained state estimator is developed by using the structure of the extended Kalman filter, where the gain matrix is determined by optimizing an upper bound matrix for the estimation error covariance despite the linearization errors and coupling terms. Compared with the existing estimators for linearly coupled complex networks, a distinct feature of the proposed estimator is that the gain matrix can be derived separately for each node by solving two Riccati-like difference equations. By using the stochastic analysis techniques, sufficient conditions are established which guarantees the state estimation error is bounded in mean square. A numerical example is provided to show the effectiveness and applicability of the proposed estimator.

Event-based recursive filtering for time-delayed stochastic nonlinear systems with missing measurements

In this paper, the recursive filtering is investigated for a type of time-delayed stochastic nonlinear systems with event-based communication protocols and missing measurements. A varying condition threshold in event triggering protocols is introduced to better match up with the system dynamic performance. The purpose of this paper is to develop an easy-implemented recursive algorithm with consideration of linearization errors, time-delays, packet losses as well as adopted communication protocols. By carrying out some elaborate mathematical operation, the desired parameters can be derived by means of the solutions to two Riccati-like difference equations. Such parameters can effectively suppress the trace of filtering error covariances and therefore the developed filtering algorithm is suboptimal. Finally, an illustrative example is presented to show the effectiveness of the developed filtering algorithm.

On scheduling of deception attacks for discrete-time networked systems equipped with attack detectors

This paper is concerned with the attack scheduling of deception attacks for discrete-time systems with attack detection. Specifically, for a class of Kalman filters with χ2 detectors, an attacker with the given attack task needs to decide how many the maximum number is or how much the attack probability is for different kinds of attack scenarios (i.e. consecutive deception attacks or randomly launched deception attacks). Firstly, in light of the property of χ2 distribution, the predictions of detection probabilities under attacks are calculated in mean-square sense for these two attack scenarios. Then, in terms of predicted probabilities, the deception attack schemes are designed in the framework of Kalman filtering. For the case of consecutive attacks, the maximum number of attacks is determined via recursive Riccati-like difference equations. For the case of randomly launched deception attacks, the desired attack probability is obtained by solving a Riccati-like equation. Finally, a numerical example on target tracking is presented to demonstrate the effectiveness of the proposed suboptimal attack schemes.

Recursive state estimation for discrete time-varying stochastic nonlinear systems with randomly occurring deception attacks

This paper is concerned with the state estimation problem for a class of discrete time-varying stochastic nonlinear systems with randomly occurring deception attacks. The stochastic nonlinearity described by statistical means which covers several classes of well-studied nonlinearities as special cases is taken into discussion. The randomly occurring deception attacks are modelled by a set of random variables obeying Bernoulli distributions with given probabilities. The purpose of the addressed state estimation problem is to design an estimator with hope to minimize the upper bound for estimation error covariance at each sampling instant. Such an upper bound is minimized by properly designing the estimator gain. The proposed estimation scheme in the form of two Riccati-like difference equations is of a recursive form. Finally, a simulation example is exploited to demonstrate the effectiveness of the proposed scheme.