Riccati-like Difference Equations Research Articles

Linear quadratic optimal control of stochastic 2-D Roesser models

This paper investigates the linear quadratic (LQ) optimal control problem for the stochastic two-dimensional (2-D) systems governed by Roesser models with multiplicative noise. The main contribution is to give the necessary and sufficient optimality condition by proposing a set of novel forward and backward stochastic partial difference equations (FBSPDE), and to further present the explicitly optimal feedback control laws on the finite horizon and on the infinite horizon based on the Riccati-like difference equations and the algebraic equation, respectively. Several numerical simulations are provided to illustrate the performance of the designed controllers.

Encryption–decryption-based state estimation for nonlinear complex networks subject to coupled perturbation

This paper discusses the encryption-decryption-based state estimation (EDBSE) issue for coupled perturbation complex networks (CPCNs) in the framework of the Kalman-type filtering scheme. A uniform distributed random variable is employed to characterize the coupled perturbation among different network units. A uniform-quantization-dependent encryption-decryption (UQDED) scheme is considered here to orchestrate the transmitted data. A novel EDBSE approach is developed such that the upper bounds of prediction error (PE) covariance (PEC) and estimation error (EE) covariance (EEC) can be derived by resolving Riccati-like difference equations and the estimation parameter (EP) is determined by minimizing the trace of the upper bound of EEC. Furthermore, a uniformly bounded condition is elaborated to evaluate the algorithm performance of EDBSE. Finally, an illustrative example is conducted to verify the validity of the introduced EDBSE method.

Token-Bucket-Protocol-Based Recursive Remote State Estimation for Complex Networks Under Amplify-and-Forward Relays.

This article is concerned with a recursive remote estimation problem for a class of nonlinear complex networks subject to the token bucket protocol (TBP) and amplify-and-forward (AF) relays. The influence of the TBP is considered, for the first time, in the context of networked state estimation, where the token consumptions are modeled in a stochastic manner, so as to describe the potential size variability of transmitted measurement signals. Once processed by the TBP, the signals, with stochastic channel coefficients, are transmitted to the remote estimator via AF relays, where a failure in transmission under the TBP may occur due to insufficient tokens in the bucket. An extended-Kalman-filter-based novel recursive estimator is proposed, and by solving Riccati-like difference equations, an upper bound of prediction/estimation error covariance is determined and further minimized through the design of an appropriate estimator gain. The impact of the TBP on estimation performance is also investigated. Some numerical simulations are presented to demonstrate the effectiveness of the proposed estimator and the effects of the TBP.

Quadratic filtering for discrete time-varying non-Gaussian systems under binary encoding schemes

This paper is concerned with the recursive quadratic filtering problem for a class of linear discrete-time systems subject to non-Gaussian noises. Considering its robustness against channel noises, the binary encoding scheme is utilized in the process of data transmission from sensors to the filter. Under such a scheme, the original signal is first encoded into a bit string, and then transmitted via memoryless binary symmetric channels (with certain crossover probabilities). Subsequently, the received bit string is recovered by a decoder at the receiver end. The primary purpose of this paper is to design a recursive quadratic filter for the underlying non-Gaussian systems with a minimized upper bound on the filtering error covariance. For this purpose, an augmented system is first constructed by aggregating the original vectors and their second-order Kronecker powers. Accordingly, an upper bound on the filtering error covariance is obtained in the form of solutions to certain Riccati-like difference equations, and the obtained bound is then minimized by properly choosing the filter parameter. Moreover, sufficient conditions are established to guarantee the boundedness of filtering error covariance. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed quadratic filtering algorithm.

Event-Triggered Recursive State Estimation for Stochastic Complex Dynamical Networks Under Hybrid Attacks.

In this article, the event-based recursive state estimation problem is investigated for a class of stochastic complex dynamical networks under cyberattacks. A hybrid cyberattack model is introduced to take into account both the randomly occurring deception attack and the randomly occurring denial-of-service attack. For the sake of reducing the transmission rate and mitigating the network burden, the event-triggered mechanism is employed under which the measurement output is transmitted to the estimator only when a preset condition is satisfied. An upper bound on the estimation error covariance on each node is first derived through solving two coupled Riccati-like difference equations. Then, the desired estimator gain matrix is recursively acquired that minimizes such an upper bound. Using the stochastic analysis theory, the estimation error is proven to be stochastically bounded with probability 1. Finally, an illustrative example is provided to verify the effectiveness of the developed estimator design method.

Remote Estimation for Energy Harvesting Systems Under Multiplicative Noises: A Binary Encoding Scheme With Probabilistic Bit Flips

In this article, we investigate the state estimation problem for a class of networked systems with energy harvesting technologies, where the sensor is capable of replenishing energy from the environment. The underlying system is subject to both additive and multiplicative stochastic noises, and the measurement is transmitted to the remote estimator only when the current energy storage is larger than the transmission energy consumption. A binary encoding scheme is utilized in the communication process, under which the measurements are quantized into a bit string, transmitted via memoryless binary symmetric channels with certain probabilistic bit flips, and recovered at the receiver. A minmax robust estimator is designed to minimize the worst-case covariance of the estimation error in terms of the solutions to Riccati-like difference equations. Furthermore, the influence of the length of bit stream on the transmission rate and the estimation performance is discussed, and conditions guaranteeing the boundedness of the proposed estimator are provided. Finally, a numerical example is exploited to verify the effectiveness of the results.

Quantized recursive filtering for networked systems with stochastic transmission delays

This paper investigates the recursive filtering problem for a class of networked systems subject to the uniform quantization effects and stochastic transmission delays. The system output is quantized according to a uniform quantization mechanism, and then sent to the remote filter via a communication network undergoing stochastic transmission delays (which are modeled by a sequence of independent and identically distributed variables). To deal with the stochastic transmission delays, an indicator function is delicately designed to ensure that the filtering process is implemented based on the quantized measurement with the newest timestamp available for the filter. With the aid of the indicator function, a free-delay system is obtained by using the augmented system method. The aim of this paper is to design a Kalman-type filter for the augmented system such that an upper bound of the filtering error covariance is guaranteed and minimized. With the aid of the stochastic analysis method, the desired upper bound of the filtering error covariance is derived by recursively solving two Riccati-like difference equations. Then, the upper bound is minimized by properly selecting the filter parameters. Finally, a numerical example is provided to illustrate the validity of the developed filtering scheme.

Recursive Filtering for Time-Varying Discrete Sequential Systems Subject to Deception Attacks: Weighted Try-Once-Discard Protocol

In this article, recursive filtering is investigated for time-varying discrete sequential systems (DSSs) under weighted try-once-discard (WTOD) protocols, which are employed to govern the access authorization of a shared network in order to remit the communication burden. A transmission model, dependent on a Bernoulli distributed white sequence, is developed to describe the phenomenon of deception attacks. In light of the adopted protocol and the attack model, a recursive algorithm with the form of Riccati-like difference equations is developed to optimize the filtering performance in the mean square sense. Furthermore, by resorting to the mathematical induction, the convergence of the proposed recursive algorithm is discussed profoundly. Finally, a simulation example is presented to verify the availability of the designed recursive filter.

Partial-nodes-based state estimation for linear complex networks with randomly occurring sensor delay and stochastic coupling strength

This paper is concerned with the partial-nodes-based state estimation (PNBSE) problem for a class of time-varying complex networks with randomly occurring sensor delay (ROSD) and stochastic coupling strength (SCS). We assume that only partial outputs of nodes can be measured and utilized in the estimation algorithm design. The ROSD is expressed by a set of Bernoulli distributed random variables and the occurrence probabilities are certain. Moreover, the SCS is represented by a set of random variables which obeys the uniform distribution. The aim of this paper is to design a recursive state estimator based on the framework of the Kalman filter, where the optimization problem of the upper bound of the estimation error covariance (EEC) is discussed by employing the variance-constrained method. It can be seen that the gain matrix of each node can be obtained by solving two Riccati-like difference equations. In addition, the performance of the PNBSE method is characterized by discussing the relationship between the occurrence probabilities of ROSD and the upper bound of the EEC, where the related mathematical proof is provided. Finally, some simulations are given to show the validity and correctness of the presented PNBSE approach.

Linear Optimal Estimation for Discrete-time and Continuous-time Systems with Multiple Measurement Delays

In this paper, we investigate the linear optimal estimation problems of discrete-time and continuous-time systems with multiple state delays in measurements. For discrete-time systems, we obtain the linear optimal estimation of state by direct calculation of optimal gain in terms of the solution to a retarded Riccati-like difference equation instead of a group of Riccati difference equations. For continuous-time systems, we also obtain the analytical expression of linear optimal estimation without resorting to Riccati partial differential equations. All the Riccati equations are of the same dimension as the system to be estimated and the computational cost is much saved. Infinite horizon case is also studied by stability analysis. Kalman filter can be recovered from our result when delays disappear. A numerical example is provided to demonstrate the results.

Recursive Filtering With Measurement Fading: A Multiple Description Coding Scheme

In this article, the recursive filtering problem is studied for a class of discrete-time nonlinear stochastic systems subject to fading measurements. In order to facilitate the data transmission in a resource-constrained communication network, the multiple description coding scheme is adopted to encode the fading measurements into two descriptions with the identical importance. Two independent Bernoulli distributed random variables are introduced to govern the occurrences of the packet dropouts in two channels from the encoders to the decoders. The channel fading phenomenon is characterized by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> th-order Rice fading model whose coefficients are mutually independent random variables obeying certain probability distributions. The purpose of the problem addressed is to design a recursive filter, such that in the simultaneous presence of the stochastic noises, the channel fading and the data coding–decoding mechanism, an upper bound of the filtering error variance is obtained and then minimized at each time step. In virtue of the Riccati difference equation technique and the stochastic analysis approach, the explicit form of the desired filter parameters is derived by solving a sequence of coupled algebraic Riccati-like difference equations. Finally, a simulation experiment is provided to show the applicability of the developed filtering scheme.

An event-triggered recursive state estimation approach for time-varying nonlinear complex networks with quantization effects

This paper proposes a novel robust extended Kalman filter (REKF) for a class of discrete time-varying nonlinear complex networks with event-triggered communication and quantization effects. In this approach, it is assumed that each sensor transmits data to its estimator over two redundant communication channels. Before the data from each sensor is sent to its estimator, an event-triggered strategy is employed to reduce the wastage of communication bandwidth and unnecessary executions. Afterwards, a logarithmic quantizer is implemented to quantize data. The estimator parameters for each node are derived separately into two Riccati-like difference equations so that the achieved upper bound is minimized by the proposed estimator parameters. Finally, the simulation results are included to illustrate the performance of the derived filtering method.

A mean-field formulation for the mean-variance control of discrete-time linear systems with multiplicative noises

ABSTRACT This paper considers the stochastic optimal control of a multi-period mean-variance trade-off performance criterion with and without constraints for discrete-time linear systems subject to multiplicative noises. We adopt a mean-field approach to tackle the problem and obtain a solution for the unconstrained case in terms of a Riccati-like difference equation. From this general result, we obtain a sufficient condition for a closed-form solution for one of the constrained problems considered in the paper. When particularised to the portfolio selection problem, we show that our results retrieve some of the results available in the literature. We conclude the paper by illustrating the obtained optimal controls with a multi-period portfolio selection problem where we minimise the sum of the mean-variance trade-off costs of a portfolio against a benchmark along the time.

Extended Kalman filtering subject to random transmission delays: Dealing with packet disorders

This paper studies the extended Kalman filtering problem for a class of nonlinear discrete-time systems with random transmission delays (RTDs) and RTD-induced packet disorders. The relationship between the RTDs and the resulting packet disorders is discussed. The bounded RTDs, which take place in the sensor-to-filter channel, are modeled as independent and identically distributed random variables obeying a certain probability distribution. A novel filter structure is proposed that utilizes an integer-valued function of the mathematical expectation of the RTDs so as to compensate the RTD-induced effects. Under the proposed extended Kalman filter, an upper bound for the filtering error covariance is derived by solving two Riccati-like difference equations, and subsequently minimized (in the sense of trace) by appropriately designing the filter gains. A numerical simulation is provided to verify the validity of the developed filter design scheme.

Event-Based Distributed Adaptive Kalman Filtering With Unknown Covariance of Process Noises

In this article, the distributed adaptive Kalman filtering is investigated for discrete-time stochastic nonlinear systems with gain perturbation as well as unknown covariance of process noises. For the adopted event-triggered communication scheduling, a distributed Kalman filter with an event timestamp is first constructed to effectively fuse the information from neighbors and itself while guaranteeing the unbiasedness. In light of stochastic analysis, the desired filter gain, achieving the suboptimality of filtering performance, is obtained recursively by solving two optimization issues with the form of Riccati-like difference equations. With the help of the fashionable weighted fusion conception combined with the well-known law of large numbers, a recursive estimation of process noise covariance is derived step by step and consequently suits for online computation. Finally, the effectiveness of the proposed filtering scheme is verified via a “lineland” system model.

Robust Finite-Horizon Filtering for 2-D Systems With Randomly Varying Sensor Delays

This paper is concerned with the robust finite-horizon filter design problem for a class of two-dimensional (2-D) time-varying systems with norm-bounded parameter uncertainties and incomplete measurements. The incomplete measurements cover randomly occurring sensor delays and missing measurements that are presented in a unified form by resorting to a stochastic Kronecker delta function. The occurrences of the sensor delays and missing measurements are governed by stochastic variables with known probability distributions. The main aim of the addressed problem is to design a recursive filter with appropriate gain parameters that ensure the local minimum of certain upper bound on the estimation error variance at each time instant. With the aid of the inductive approach and the 2-D Riccati-like difference equations, one of the first few attempts is made to tackle the robust filter design problem for 2-D uncertain systems with random sensor delays over a finite horizon. Sufficient conditions are provided for the existence of an upper bound on the estimation error variance, an algorithm is then developed to derive such an upper bound, and finally the desired filter is designed to minimize the obtained upper bound. The filter design procedure is of a recursive form that facilitates the online calculation. A numerical simulation is carried out to show the effectiveness of the developed filtering scheme.

Distributed Resilient Filtering for Power Systems Subject to Denial-of-Service Attacks

This paper addresses the distributed resilient filtering problem for a class of power systems subject to denial-of-service (DoS) attacks. A novel distributed filter is first constructed to practically reflect the impact from both cyber-attacks and gain perturbations. For all possible occurrence of DoS attacks and gain perturbations, an upper bound of filtering error covariance is derived by resorting to some typical matrix inequalities. Furthermore, the desired filter gain relying on the solution of two Riccati-like difference equations is obtained with the help of the gradient-based approach and the mathematical induction. The developed algorithm with a recursive form is independent of the global information and thus satisfies the requirements of scalability and distributed implementation online. Finally, a benchmark simulation test is exploited to check the usefulness of the designed filter.

Event-Triggered Recursive Filtering for Shift-Varying Linear Repetitive Processes.

This paper addresses the recursive filtering problem for shift-varying linear repetitive processes (LRPs) with limited network resources. To reduce the resource occupancy, a novel event-triggered strategy is proposed where the concern is to broadcast those necessary measurements to update the innovation information only when certain events appear. The primary goal of this paper is to design a recursive filter rendering that, under the event-triggered communication mechanism, an upper bound (UB) on the filtering error variance is ensured and then optimized by properly determining the filter gains. As a distinct kind of 2-D systems, the LRPs are cast into a general Fornasini-Marchesini model by using the lifting technique. A new definition of the triggering-shift sequence is introduced and an event-triggered rule is then constructed for the transformed system. With the aid of mathematical induction, the filtering error variance is guaranteed to have a UB which is subsequently optimized with appropriate filter parameters via solving two series of Riccati-like difference equations. Theoretical analysis further reveals the monotonicity of the filtering performance with regard to the event-triggering threshold. Finally, an illustrative simulation is given to show the feasibility of the designed filtering scheme.

&lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$H_{\infty}$ &lt;/tex-math&gt; &lt;/inline-formula&gt; Fault Estimation for 2-D Linear Discrete Time-Varying Systems Based on Krein Space Method

This paper addresses the finite horizon ${H_\infty }$ fault estimation problem for 2-D linear discrete time-varying systems with bounded unknown input and measurement noise. The main contribution of this paper is the ${H_\infty }$ fault estimator for 2-D systems with a necessary and sufficient existence condition. By introducing a partially equivalent stochastic dynamic system in Krein space, the necessary and sufficient condition for the existence of the ${H_\infty }$ fault estimator is derived based on innovation analysis and projection formula in Krein space. Then, the solution of the estimator is achieved by means of a Riccati-like difference equation for 2-D systems. Finally, a thermal process example is given to demonstrate the effectiveness of the proposed method.

Target Tracking for Wireless Localization Systems With Degraded Measurements and Quantization Effects

In this paper, the target tracking problem is investigated for a class of wireless localization systems with degraded measurements and quantization effects. The measurement degradations induced by unreliable channels are characterized by a random parameter matrix, and the quantization effects are modeled by sector-bounded uncertainties. A recursive filtering algorithm is proposed in order to track the plant states as accurately as possible. An upper bound on the filtering error covariance is first derived and such an upper bound is then minimized by properly designing the filter gain at each sampling instant. The desired filter parameters are obtained by solving two sets of Riccati-like difference equations that are in a recursive form suitable for online application. Finally, a simulation experiment is carried out for the scenario of personnel localization/tracking problems in a mine industrial site and the effectiveness of the proposed filter method is demonstrated.