Estimation Error Covariance Research Articles

Stochastic important‐data‐based attack model and defense strategies for cyber‐physical system: A data‐driven method

AbstractThis article investigates the security ensured state estimation problem for cyber‐physical system via a data‐driven method. First, based on the fact that different packets possess varying a degree of significance, that is, some packets play more important roles in the state estimation than others, a novel stochastic important‐data‐based (IDB) attack mechanism is constructed from the attacker's perspective, which can focus on attacking the important packets thus is expected to achieve more destructiveness. Second, as a countermeasure to the proposed IDB attack, a new data‐driven compensation method is proposed, for the first attempt, to compensate for the attack effect and enhance the estimation quality. The designed defense strategy has the following two advantages: (1) only system input and output data are utilized to establish the novel estimator, without knowing the actual system model knowledge, and (2) by constructing a data‐driven output predictor to compensate for the data loss, the accuracy of the state estimation can be efficiently improved. With the aid of the least squares technique and completing square technique, a minimum upper bound matrix for the estimation error covariance is obtained by properly designing the estimator gain. Finally, an illustrative example is given to highlight the destructiveness of the designed stochastic IDB attack and the effectiveness of the proposed novel data‐based compensation method.

Multi-sensor information fusion filtering for delayed nonlinear stochastic singular systems with packet disorders

The fusion filtering problem is investigated for a class of time-varying nonlinear singular systems with packet disorders (PDs). The key idea is to transfer the original singular systems with multiple sensors to full-order augmented nonsingular systems by utilizing the matrix transformation method. The phenomenon of PDs accompanied with random transmission delays (RTDs) is considered, which exists in the channel from sensor to filter. A set of independent and identically distributed random variables obeying known probability distribution are introduced to model the RTDs. A centralized fusion filter is designed, which employs the rounding function of the mathematical expectation of the RTDs. Furthermore, the nonlinear function is linearized by utilizing the method of Taylor series expansion. Then, the desired upper bounds of the estimation error covariance are guaranteed and then locally minimized through appropriately designing the corresponding filter gains. Finally, a numerical example with comparisons is carried out to verify the feasibility of the proposed fusion filtering method.

Collaborative Parameter Estimation of Multiple Unmanned Surface Vessels: A Robust Distributed Estimator-Based Approach

In this paper, the collaborative parameter estimation of multiple unmanned surface vessels (USVs) with model structure uncertainties is studied. The considered parameter estimation problem is first converted into a distributed state and parameter joint estimation problem. Then, the robust distributed estimator is constructed to handle the inevitable model structure uncertainties, and the upper bounds of prediction and estimation error covariance matrices are derived respectively. Additionally, the upper bound of the estimation error covariance matrix is minimized by designing appropriate estimator gains. Finally, the advantages of the proposed distributed parameter estimation approach from the perspectives of information interaction and consideration of model structure uncertainties are verified via simulations and practical experiments.

Event-Triggered Reverse Attacks on Remote State Estimation

Security issues in cyber-physical systems have attracted increasing attention in recent years. In this paper, a security problem in a remote estimation application is considered, where an attacker tries to degrade estimation performance via malicious attacks. In our scenario, a smart sensor transmits its innovation to a remote estimator with a residue-based false data detector. Instead of assuming that the attacker launches a consecutive man-in-the-middle attack, we consider the case with intermittent attacks. Reverse attack is a simple attack strategy, which enables an attacker to change signs of the innovation sequences. Therefore, owing to the above reasons, an event-triggered reverse attack strategy is proposed to degrade system performance. First, the stealthiness of the event-triggered linear deception attack strategy is studied. Then, the evolution of the estimation error covariance is computed and the reverse attack is proven to be the optimal linear deception attack. We demonstrate that our event-triggered reverse attack is more destructive than a random reverse attack. Furthermore, a convex optimization problem is established to design event-triggered parameters. Comparisons of attack strategies are provided in a numerical example to validate the superiority of the reverse attack.

Adaptive STCPF for UCA pose estimation with improved GA-BPNN and multiple fading factors

In response to the challenge of pose estimation accuracy for unmanned carrier-based aircraft in complex maritime landing environments, we propose an adaptive strong tracking cubature particle filtering method based on an improved Genetic Algorithm-Based Backpropagation Neural Network (GA-BPNN) and multiple fading factors. Despite the effectiveness of strong tracking cubature Kalman filtering (STCKF) in adjusting the covariance of prediction estimation errors using fading factors, it still has limitations in terms of model correction and parameter estimation. Addressing this issue, our research delves deeper into the role of fading factors in adjusting process noise covariance and introduces two methods to maintain covariance matrix symmetry: direct matrix solving and trace matrix solving. This novel approach directly implements real-time dynamic feedback correction of system model parameters within the filtering framework, effectively resolving filtering divergence issues while maintaining a high level of conceptual clarity and interpretability. To further enhance the algorithm’s performance, we introduce an improved neural network algorithm to compensate for linear approximation errors in fading factor calculations, thereby improving filtering accuracy. Moreover, considering the limitations of cubature Kalman filtering when dealing with non-Gaussian systems, we integrate the adaptive STCKF based on the improved GA-BPNN into particle filtering as an importance function, aiming to enhance non-Gaussian processing performance and alleviate particle degeneracy issues. In conclusion, both simulation and real-world data tests confirm the superiority of our research approach.

Estimation-free Prediction Algorithms

For Time-varying, Time-invariant, and steady-state systems, Kalman Filter can be implemented as a prediction algorithm, since it produces the state prediction and the corresponding prediction error covariance matrix via the state estimation and the corresponding estimation error covariance matrix. Lainiotis Filter is equivalent to Kalman Filter and can be used to compute the prediction. In this paper, for Time-varying, Time-invariant and steady state systems, estimation-free Prediction Algorithms are derived via Kalman and Lainiotis filters; they are equivalent and compute iteratively the prediction and the corresponding prediction error covariance matrix. The estimation and the corresponding estimation error covariance matrix are not needed and are not computed. The proposed estimation-free prediction algorithms are faster than the Kalman filter.

On Information Fusion in Optimal Linear FDI Attacks Against Remote State Estimation

This paper studies the problem of false data injection (FDI) attacks against remote state estimation. The scenario that malicious attackers can intercept original data packets and also eavesdrop on some side information of system states with extra sensors is considered. To clarify a counter-intuitive issue in existing work, a different innovation-based linear attack policy fusing all available information is proposed. First, the evolution of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> estimation error covariance with FDI attacks is derived. Then, explicit solutions of optimal stealthy attack coefficients are obtained without solving optimization problems numerically. The condition under which there exist multiple optimal attacks is analyzed. Additionally, an easy-to-check criterion for comparing two information fusion methods in scalar systems is given. Simulation results show that, compared with existing work, the proposed attack strategy can completely deceive the anomaly detector and cause more severe performance degradation in remote state estimation.

Encryption-Decryption-Based State Estimation With Multirate Measurements Against Eavesdroppers: A Recursive Minimum-Variance Approach

This paper is concerned with the secure state estimation problem for a networked system with multi-rate measurements. The measurement data are transmitted to the estimator over a communication network, where an eavesdropper could intercept the transmitted information and generate the state estimates based on the overheard signals. In order to protect the privacy of the system state from information leakage, a novel encryptor is proposed to encrypt the measurement signals into codewords before being transmitted, and a decryptor is then adopted at the user side to decrypt the received codewords into the desired measurement signals, based on which a recursive state estimator is developed whose parameter is calculated by solving several coupled difference equations. Here, the secret key for decryption is available to the user but is unknown to the eavesdropper. The encryptor and the estimator are co-designed such that 1) at the user side, the estimation error covariance has an upper bound that is subsequently minimized in the sense of the matrix trace, where the uniform boundedness of the error covariance is also examined; and 2) at the eavesdropper side, the corresponding state estimation error is divergent. Two simulation examples are provided to demonstrate the effectiveness of our developed secure state estimation scheme.

Distributed Anti-Eavesdropping Fusion Estimation Under Energy Constraints

This paper studies the distributed fusion estimation problem with energy-constrained sensors in the presence of eavesdroppers, where smart sensors send their local estimates to a remote fusion center (FC). To enhance privacy level, a novel encryption strategy is proposed by establishing an optimization problem, where the optimization objective is constructed by maximizing a combination of the fusion terminal estimation error covariance and the cost of encryption process in finite time domain. Meanwhile, the established problem is decomposed into several independent sub-optimization problems under a relaxation condition, and a sufficient condition that depends on the system parameters is derived such that the sub-optimization problems have optimal solutions. In this case, an optimal encryption strategy is designed with an analytical solution. Finally, a simulation example is given to show the effectiveness of the proposed methods.

A distributed diffusion Kalman filter with event‐triggered mechanism and guaranteed stability

AbstractIn this article, a distributed diffusion Kalman filtering algorithm with event‐triggered communication (DDKF‐E) is studied for discrete‐time nonlinear systems. According to the event‐triggered communication protocol, the data between sensors and estimators are transmitted only when the predefined conditions are satisfied. Considering the characteristic of event‐triggered method and truncated error by linearization, an upper bound of the estimation error covariance matrix is obtained by using the variance‐constrained method. The Kalman gain is designed to minimize the upper bound and then two Riccati equations are obtained. Furthermore, the stochastic stability theory is used to prove the stability of DDKF‐E, and it is derived that the estimation error of DDKF‐E is exponentially bounded in mean square. Finally, numerical simulations validate the effectiveness of the DDKF‐E algorithm.

Remote state estimator design under the round‐Robin protocol

SummaryThis paper focuses on the remote estimation problem for cyber‐physical systems under the round‐Robin protocol. A round‐Robin protocol‐based Kalman filter is deployed to estimate the real system states on the remote side. When sensor measurements are transmitted by communication networks, the round‐Robin protocol with the zero‐input‐holder strategy is adopted to determine which sensor measurements are transmitted, reducing the communication burden and risk of being attacked. Both the protocol‐based open‐loop and closed‐loop estimation scenarios are considered. The delay resulting from adopting the protocol with the zero‐input‐holder strategy is handled effectively. Corresponding filtering algorithms are proposed, and the estimation error covariance is proved to be converged. Finally, in addition to the effectiveness of the proposed round‐Robin protocol‐based filtering algorithms, simulation results are given to demonstrate that the performance of the proposed filtering algorithms can match that of the classical Kalman filter.

Using the Commercial GNSS RO Spire Data in the Neutral Atmosphere for Climate and Weather Prediction Studies

Recently, the NOAA has included GNSS (Global Navigation Satellite System) Radio Occultation (RO) data as one of the crucial long-term observables for weather and climate applications. To include more GNSS RO data in its numerical weather prediction systems, the NOAA Commercial Weather Data Pilot program (CWDP) started to explore the commercial RO data available on the market. After two rounds of pilot studies, the CWDP decided to award the first Indefinite Delivery Indefinite Quantity (IDIQ) contract to GeoOptics and Spire Incs. in 2020. This study examines the quality of Spire RO data products for weather and climate applications. Spire RO data collected from commercial CubeSats are carefully compared with data from Formosa Satellite Mission 7–Constellation Observing System for Meteorology, Ionosphere, and Climate-2 (COSMIC-2), the fifth-generation European Centre for Medium-Range Weather Forecasts (ECMWF) atmospheric reanalysis (ERA5), and high-quality radiosonde data. The results demonstrate that, despite their generally lower Signal-Noise-Ratio (SNR), Spire RO data show a pattern of lowest penetration height similar to that of COSMIC-2. The Spire and COSMIC-2 penetration heights are between 0.6 and 0.8 km altitude over tropical oceans. Although using different GNSS RO receivers, the precision of Spire STRATOS receivers is of the same quality as those of the COSMIC-2 TriG (Global Positioning System—GPS, GALILEO, and GLObal NAvigation Satellite System—GLONASS) RO Receiver System (TGRS) receivers. Furthermore, the Spire and COSMIC-2 retrieval accuracies are quite comparable. We validate the Spire temperature and water vapor profiles by comparing them with collocated radiosonde observation (RAOB) data. Generally, over the height region between 8 km and 16.5 km, the Spire temperature profiles match those from RS41 RAOB very well, with temperature biases of &lt;0.02 K. Over the height range from 17.8 to 26.4 km, the temperature differences are ~−0.034 K, with RS41 RAOB being warmer. We also estimate the error covariance matrix for Spire, COSMIC-2, and KOMPSAT-5. The results show that the COSMIC-2 estimated error covariance values are slightly more significant than those from Spire over the oceans at the mid-latitudes (45°N–30°N and 30°S–45°S), which may be owing to COSMIC-2 SNR being relatively lower at those latitudinal zones.

Event-Based Optimal Stealthy False Data-Injection Attacks Against Remote State Estimation Systems.

Security is a crucial issue for cyber-physical systems, and has become a hot topic up to date. From the perspective of malicious attackers, this article aims to devise an efficient scheme on false data-injection (FDI) attacks such that the performance on remote state estimation is degraded as much as possible. First, an event-based stealthy FDI attack mechanism is introduced to selectively inject false data while evading a residual-based anomaly detector. Compared with some existing methods, the main advantage of this mechanism is that it decides when to launch the FDI attacks dynamically according to real-time residuals. Second, the state estimation error covariance of the compromised system is used to evaluate the performance degradation under FDI attacks, and the larger the state estimation error covariance, the more the performance degradation. Moreover, under attack stealthiness constraints, an optimal strategy is presented to maximize the trace of the state estimation error covariance. Finally, simulation experiments are carried out to illustrate the superiority of the proposed method compared with some existing ones.

Optimal transmission strategy for multiple Markovian fading channels: Existence, structure, and approximation

This paper investigates the optimal transmission strategy for remote state estimation over multiple Markovian fading channels. A smart sensor is used to obtain a local state estimate of a system, and transmits it to a remote estimator. A new transmission strategy is proposed by co-designing the channel allocation and the transmission power control. The co-designing problem is modeled as a constrained Markov decision process (CMDP) to minimize the expected average estimation error covariance subject to the energy constraint over an infinite horizon. The CMDP is then relaxed as an unconstrained Markov decision process (UMDP) using the Lagrange multiplier method. Sufficient conditions for the existence of the optimal stationary policy for the UMDP are established to obtain the optimal transmission strategy. The structure of the optimal transmission power control policy for the UMDP with discounted cost is also elucidated. Taking account of the discrete-continuous hybrid action space, a parameterized deep Q-network (P-DQN) algorithm is employed to obtain an approximate optimal policy for the UMDP. Finally, a moving vehicle example is introduced to illustrate the effectiveness of the developed methods.

How far can the statistical error estimation problem be closed by collocated data?

Abstract. Accurate specification of the error statistics required for data assimilation remains an ongoing challenge, partly because their estimation is an underdetermined problem that requires statistical assumptions. Even with the common assumption that background and observation errors are uncorrelated, the problem remains underdetermined. One natural question that could arise is as follows: can the increasing amount of overlapping observations or other datasets help to reduce the total number of statistical assumptions, or do they introduce more statistical unknowns? In order to answer this question, this paper provides a conceptual view on the statistical error estimation problem for multiple collocated datasets, including a generalized mathematical formulation, an illustrative demonstration with synthetic data, and guidelines for setting up and solving the problem. It is demonstrated that the required number of statistical assumptions increases linearly with the number of datasets. However, the number of error statistics that can be estimated increases quadratically, allowing for an estimation of an increasing number of error cross-statistics between datasets for more than three datasets. The presented generalized estimation of full error covariance and cross-covariance matrices between datasets does not necessarily accumulate the uncertainties of assumptions among error estimations of multiple datasets.

Challenges for Inline Observation Error Estimation in the Presence of Misspecified Background Uncertainty

Abstract For data assimilation to provide faithful state estimates for dynamical models, specifications of observation uncertainty need to be as accurate as possible. Innovation-based methods based on Desroziers diagnostics, are commonly used to estimate observation uncertainty, but such methods can depend greatly on the prescribed background uncertainty. For ensemble data assimilation, this uncertainty comes from statistics calculated from ensemble forecasts, which require inflation and localization to address under sampling. In this work, we use an ensemble Kalman filter (EnKF) with a low-dimensional Lorenz model to investigate the interplay between the Desroziers method and inflation. Two inflation techniques are used for this purpose: 1) a rigorously tuned fixed multiplicative scheme and 2) an adaptive state-space scheme. We document how inaccuracies in observation uncertainty affect errors in EnKF posteriors and study the combined impacts of misspecified initial observation uncertainty, sampling error, and model error on Desroziers estimates. We find that whether observation uncertainty is over- or underestimated greatly affects the stability of data assimilation and the accuracy of Desroziers estimates and that preference should be given to initial overestimates. Inline estimates of Desroziers tend to remove the dependence between ensemble spread–skill and the initially prescribed observation error. In addition, we find that the inclusion of model error introduces spurious correlations in observation uncertainty estimates. Further, we note that the adaptive inflation scheme is less robust than fixed inflation at mitigating multiple sources of error. Last, sampling error strongly exacerbates existing sources of error and greatly degrades EnKF estimates, which translates into biased Desroziers estimates of observation error covariance. Significance Statement To generate accurate predictions of various components of the Earth system, numerical models require an accurate specification of state variables at our current time. This step adopts a probabilistic consideration of our current state estimate versus information provided from environmental measurements of the true state. Various strategies exist for estimating uncertainty in observations within this framework, but are sensitive to a host of assumptions, which are investigated in this study.

Encoding–decoding-based fusion estimation with filter-and-forward relays and stochastic measurement delays

This paper addresses the encoding–decoding-based fusion estimation problem for a class of systems with signal relays and stochastic measurement delays. A set of Bernoulli distributed random variables is used to model the randomly occurring measurement delays. To enhance the system performance and extend the transmission distance, a filter-and-forward relay is adopted in the transmission link. To handle the limited bandwidth of the communication channels, an encoding–decoding scheme based on a probabilistic quantizer is proposed for the sensor–relay and relay–estimator channels. For each relay, a local filter is constructed with a proper filter gain at each time step with aim to minimize an upper bound of the local estimation error covariance. The effects of the encoding–decoding scheme on the error covariance are also analyzed quantitatively. The estimates obtained from the local filters are then fused at the remote estimator using the covariance intersection fusion strategy. Finally, a target tracking simulation example is presented to demonstrate the effectiveness of the proposed framework.

Dynamic event-based forecasting-aided state estimation for active distribution systems subject to limited communication resource

In this paper, the dynamic event-based forecasting-aided state estimation (FASE) method is developed to deal with the state estimation problem of the active distribution system (ADS) subject to communication constraints and non-linear measurements. The proposed method first constructs a state-space model of the ADS to describe system state time evolution. Secondly, to use communication resources more effectively, the dynamic event-triggered scheme (ETS) is exploited to schedule the data transmission. Aiming at the problem of the ADS in the presence of the non-linear measurement, the Gaussian integral is approximated by the spherical cubature rule to obtain the mean and covariance of the state variables after non-linear transformation. Moreover, the upper bound of the estimation error covariance containing non-triggering errors is derived, and then minimized by suitably designing filter gain, thus developing the dynamic event-triggered cubature Kalman filter (DET-CKF) algorithm to perform state estimation for ADSs. Finally, a series of simulation experiments are conducted to verify the effectiveness of the developed FASE method.

Multiple description encoding-decoding-based resilient filtering for complex networks under the round-Robin protocol

The problem of distributed filtering for a class of discrete-time nonlinear complex networks subject to the network bandwidth limitation is investigated. In order to reduce the network transmission burden, the round-robin protocol is employed. Under which, a quantization-based encoding-decoding mechanism is introduced to protect the security of measurements. Taking into account the quality of transmission data packets, the multiple description coding scheme is proposed. In order to describe the behavior of the packet dropouts induced by DoS attackers, two independent Bernoulli random variables are presented. Based on the round-robin protocol and multiple description encoding-decoding scheme, a distributed recursive filtering is derived to estimate the states of complex networks. Furthermore, the sufficient conditions to obtain the upper bound of estimation error covariance are given. Finally, an example is presented to certify the effectiveness of the proposed distributed recursive filtering.

OPTIMAL RECURSIVE IDENTIFICATION TECHNIQUES FROM QUANTIZED OUTPUTS

Recursive identification algorithms for the parameter estimation of linear systems from multilevel quantized outputs are introduced in this paper. The proposed algorithms are proved to be optimal in the sense that they minimize the a posteriori parameter estimation error covariance matrix. Numerical simulations are carried out to illustrate their performance for different quantization steps and Signal-to-Noise Ratios, as well as their capability to track time varying parameters.