State Estimation Problem For Systems Research Articles

On Kalman filtering with linear state equality constraints

This article is concerned with the state estimation problem for linear systems with linear state equality constraints. We re-examine constrained Kalman filter variations and propose an alternative derivation of the optimal constrained Kalman filter for time variant systems. This results in an oblique state projection that gives the smallest error covariance. A simple example illustrates the performance of the different Kalman filters.

Attack-Resilient <inline-formula> <tex-math notation="LaTeX">$\mathcal H_2$</tex-math> </inline-formula>, <inline-formula> <tex-math notation="LaTeX">$\mathcal H_\infty$</tex-math> </inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$\ell _1$</tex-math> </inline-formula> State Estimator

This paper considers the secure state estimation problem for noisy systems in the presence of sparse sensor integrity attacks. We show a fundamental limitation, that is, $2\rho$ -detectability is necessary for achieving bounded estimation errors, where $\rho$ is the number of attacks. This condition is weaker than the $2\rho$ -observability condition typically assumed in the literature. Conversely, we propose a real-time state estimator that achieves the fundamental limitation. The proposed state estimator is inspired by robust control and fault detection and isolation, that is, it consists of local Luenberger estimators, local residual detectors, and a global fusion process. We show its performance guarantees for $\mathcal H_2$ , $\mathcal H_\infty$ , and $\ell _1$ systems. Finally, numerical examples show that it has relatively low estimation errors among existing algorithms and average computation time for systems with a sufficiently small number of compromised sensors.

Robust event-triggered state estimation: A risk-sensitive approach

In this work, we investigate a robust event-triggered remote state estimation problem for linear Gaussian systems with a stochastic event-triggering condition. The reference measure approach is used to obtain a robust event-triggered estimate that minimizes the so-called risk-sensitive criterion, which refers to the expectation of the exponential of the sum of the squared estimation error. We introduce the reference measure, under which, the measurements are identically independently distributed (i.i.d.) and independent of the states, and propose a map to link the “real-world” measure to the reference measure so that the recursions of the information states under the reference measure can be obtained. Based on these results, the risk-sensitive criteria are reformulated under the reference measure and closed-form expressions of the risk-sensitive event-triggered posterior and prior estimates are presented, which are shown to evolve in simple recursive Kalman-like structures. Moreover, two sufficient stability conditions for the proposed estimators are given, where the first requires the solution of a time-varying Riccati equation to be positive-definite and satisfy a specific inequality, which can be further extended to the scenario when the weighting matrix in the risk-sensitive criterion is time-variant; the second gives the range of values of the risk-sensitive parameter and covariance of the initial state for which the proposed estimators are stable. Comparative simulation results demonstrate that the proposed risk-sensitive event-triggered estimator is more robust to model uncertainties compared with a typical minimum mean squared error (MMSE) estimator with stochastic event-triggered sensor scheduling.

On robust Kalman filter for two-dimensional uncertain linear discrete time-varying systems: A least squares method

The robust Kalman filter design problem for two-dimensional uncertain linear discrete time-varying systems with stochastic noises is investigated in this study. First, we prove that the solution to a certain deterministic regularized least squares problem constrained by the nominal two-dimensional system model is equivalent to the generalized two-dimensional Kalman filter. Then, based on this relationship, the robust state estimation problem for two-dimensional uncertain systems with stochastic noises is interpreted as a deterministic robust regularized least squares problem subject to two-dimensional dynamic constraint. Finally, by solving the robust regularized least squares problem and using a simple approximation, a recursive robust two-dimensional Kalman filter is determined. A heat transfer process serves as an example to show the properties and efficacy of the proposed filter.

Enhancing Observability in Distribution Grids Using Smart Meter Data

Due to limited instrumentation, distribution grids are currently challenged by observability issues. On the other hand, smart meter data are communicated to utility operators from nodes with renewable generation and elastic demand. This paper employs grid data from metered buses toward inferring the underlying grid state. System nodes are partitioned into buses with time-varying injections that are assumed metered, and buses with relatively stationary conventional loads that are non-metered. Exploiting the variability at metered buses and the stationarity of conventional loads, the novel idea here is to solve the non-linear power flow (PF) equations jointly over consecutive time instants. By putting forth a coupled formulation of the PF problem (CPF), grid states can be recovered by metering fewer buses. An intuitive and easily verifiable rule pertaining to the locations of (non-)metered buses on the physical grid is shown to be a necessary and sufficient criterion for local observability in radial networks. To account for noisy smart meter readings, a coupled power system state estimation (CPSSE) problem is further devised. Both CPF and CPSSE tasks are tackled via augmented semi-definite program relaxations. The observability criterion along with the CPF and CPSSE solvers are numerically corroborated using randomly generated and actual solar and load data on the IEEE 34-bus benchmark feeder.

Parameter selection method for support vector machine based on adaptive fusion of multiple kernel functions and its application in fault diagnosis

A new model parameter selection method for support vector machine based on adaptive fusion of multiple kernel functions is proposed in this paper. Characteristics of local kernels, global kernels, mixtures of kernels and multiple kernels were analyzed. Fusion coefficients of the multiple kernel function, kernel function parameters and regression parameters are combined to form the parameters of the state vector. Thus, the model selection problem is transformed into a nonlinear system state estimation problem. Then, we use a fifth-degree cubature Kalman filter to estimate the parameters. In this way, we realize adaptive selection of the multiple kernel function weighted coefficient, the kernel parameters and the regression parameters. A simulation experiment was performed to interpret the PE process for fault diagnosis.

Unbiased Minimum Variance Estimation for Discrete-Time Systems with Measurement Delay and Unknown Measurement Disturbance

This paper addresses the state estimation problem for stochastic systems with unknown measurement disturbances whose any prior information is unknown and measurement delay resulting from the inherent limited bandwidth. For such complex systems, the Kalman-like one-step predictor independent of unknown measurement disturbances is designed based on the linear unbiased minimum variance criterion and the reorganized innovation analysis approach. One simulation example shows the effectiveness of the proposed algorithms.

State estimation under non-Gaussian Lévy and time-correlated additive sensor noises: A modified Tobit Kalman filtering approach

Abstract The Tobit Kalman filter (TKF) is a powerful tool in solving the state estimation problem for linear systems with censored measurements. This paper is concerned with the Tobit Kalman filtering problem for discrete time-varying systems subject to non-Gaussian Levy and time-correlated additive measurement noises. By referencing to the measurement differencing method, the time-correlation of the measurement noises is transformed into the cross-correlation between the equivalent measurement noise and the process noise. Then, by resorting to the Levy-Ito theorem, the non-Gaussian Levy measurement noises are transformed into equivalent Gaussian noises with unknown covariances. Based on the transformed Gaussian measurement noises, a modified recursive TKF is designed where the unknown noise covariances are carefully calculated. Simulation results are provided to illustrate the effectiveness of the proposed filter.

State estimation for bilinear impulsive control systems under uncertainties

The state estimation problem for uncertain impulsive control systems with a special structure is considered. The initial states are taken to be unknown but bounded with given bounds. We assume here that the coefficients of the matrix included in the differential equations are not exactly known, but belong to the given compact set in the corresponding space. We present here algorithms that allow to find the external ellipsoidal estimates of reachable sets for such bilinear impulsive uncertain systems.

Remote Nonlinear State Estimation With Stochastic Event-Triggered Sensor Schedule.

This paper concentrates on the remote state estimation problem for nonlinear systems over a communication-limited wireless sensor network. Because of the non-Gaussian property caused by nonlinear transformation, the unscented transformation technique is exploited to obtain approximate Gaussian probability distributions of state and measurement. To reduce excessive data transmission, uncontrollable and controllable stochastic event-triggered scheduling schemes are developed to decide whether the current measurement should be transmitted. Compared with some existing deterministic event-triggered scheduling schemes, the newly developed ones possess a potential superiority in maintaining Gaussian property of innovation process. Under the proposed schemes, two nonlinear state estimators are designed based on the unscented Kalman filter. Stability and convergence conditions of these two estimators are established by analyzing behaviors of estimation error and error covariance. It is shown that an expected compromise between communication rate and estimation quality can be achieved by properly tuning event-triggered parameter matrix. Numerical examples are provided to testify the validity of the proposed results.

Discrete-time state estimation for stochastic polynomial systems over polynomial observations

This paper presents a solution to the mean-square state estimation problem for stochastic nonlinear polynomial systems over polynomial observations confused with additive white Gaussian noises. The solution is given in two steps: (a) computing the time-update equations and (b) computing the measurement-update equations for the state estimate and error covariance matrix. A closed form of this filter is obtained by expressing conditional expectations of polynomial terms as functions of the state estimate and error covariance. As a particular case, the mean-square filtering equations are derived for a third-degree polynomial system with second-degree polynomial measurements. Numerical simulations show effectiveness of the proposed filter compared to the extended Kalman filter.

Finite time attack detection and supervised secure state estimation for CPSs with malicious adversaries

In this paper, we investigate the finite time attack detection and secure state estimation problem for linear cyber physical systems (CPS) with s out of p sensors being attacked. By exploring the distinct properties of the undetectable and indistinguishable attacks to a CPS, novel explicit criteria on the detectability and distinguishability of the CPS under attack are proposed in terms of the r-resilient observability. This enables us to explicitly characterize the undetectable and/or indistinguishable attacks to the CPS. Then based on such characterization, we propose a finite time detector to solve the attack detection problem and a finite time alarm event driven supervised estimator to solve the secure state estimation problem, separately. Theoretical analysis shows that, the proposed detector and estimator can detect attacks and estimate the state within a prescribed finite time, respectively. Finally, some numerical simulations on an unmanned ground vehicle as an illustration of attack detection and secure state estimation is proposed to verify the effectiveness of the proposed schemes.

Conic Relaxations for Power System State Estimation With Line Measurements

This paper deals with the nonconvex power system state estimation (PSSE) problem, which plays a central role in the monitoring and operation of electric power networks. Given a set of noisy measurements, PSSE aims at estimating the vector of complex voltages at all buses of the network. This is a challenging task due to the inherent nonlinearity of power flows (PFs), for which the existing methods lack guaranteed convergence and theoretical analysis. Motivated by these limitations, we propose a novel convexification framework for the PSSE problem using semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations. We first study a related PF problem as the noiseless counterpart, which is cast as a constrained minimization program by adding a suitably designed objective function. We study the performance of the proposed framework in the case where the set of measurements includes: 1) nodal voltage magnitudes and 2) branch active PFs over at least a spanning tree of the network. It is shown that the SDP and SOCP relaxations both recover the true PF solution as long as the voltage angle difference across each line of the network is not too large (e.g., less than $90^{\circ }$ for lossless networks). By capitalizing on this result, penalized SDP and SOCP problems are designed to solve the PSSE, where a penalty based on the weighted least absolute value is incorporated for fitting noisy measurements with possible bad data. Strong theoretical results are derived to quantify the optimal solution of the penalized SDP problem, which is shown to possess a dominant rank-one component formed by lifting the true voltage vector. An upper bound on the estimation error is also derived as a function of the noise power, which decreases exponentially fast as the number of measurements increases. Numerical results on benchmark systems, including a 9241-bus European system, are reported to corroborate the merits of the proposed convexification framework.

Guaranteed State Estimation for Nonlinear Discrete-Time Systems via Indirectly Implemented Polytopic Set Computation

This paper proposes a new set-membership technique to implement polytopic set computation for nonlinear discrete-time systems indirectly. The proposed set-membership technique is applied to solve the guaranteed state estimation problem for nonlinear discrete-time systems with a bounded description of noises and parameters. A common practice for this problem in the literature is to search an optimal zonotope to bound the intersection of the evolved uncertain state trajectory and the region of the state space consistent with the observed output at each observation update. The new approach keeps the polytopic set resulting from the intersection intact and computes the evolution of this intact polytopic set for the next time step through representing the polytopic set exactly by the intersection of zonotopes. Such an approach avoids the overapproximation bounding process at each observation update, and thus, a more accurate state estimation can be obtained. An illustrative example is provided to demonstrate the effectiveness of the proposed guaranteed state estimation via indirectly implemented polytopic set computation.

State estimation of neutral Markovian jump systems: A relaxed L–K functional approach

This paper investigates the state estimation problem of uncertain neutral delay systems with Markovian jumping parameters. First, a novel Lyapunov–Krasovskii (L–K) functional containing the interconnected information between neutral and discrete delay is proposed. Then, based on Jensen’s integral and Wirtinger-based inequality, the obtained results are improved by relaxing the positive-definiteness restrictions on Lyapunov matrices. Third, the state estimators are designed to guarantee the asymptotic stability of the error state system. Finally, numerical examples are provided to show the effectiveness of the proposed results.

Kalman Filter Reinforced by Least Mean Square for Systems with Unknown Inputs

This paper addresses the state estimation problem of linear discrete-time time-varying stochastic systems with unknown inputs (UIs). It is shown that the globally optimal unbiased minimum-variance filters may not satisfy the minimum-variance property, and hence they cannot eliminate noises appropriately. If this is the case, the well-known Kalman filter may give a better solution, which however may also not be the best one due to that the imbedded unknown input model may not be practical. To remedy the filtering degradation problem, a robust filter named as the KFLMS, which has good noise rejection property for such systems, is developed in this paper, where the UI estimates are obtained by using least mean square algorithm and the state estimation is achieved via the previous proposed two-stage Kalman filtering approach. Numerical examples are provided to show the effectiveness of the proposed results. Specifically, simulation results illustrate the goodness of the new method in the sense of lower root mean square error and better noise rejection property.

On stability and convergence of optimal estimation for networked control systems with dual packet losses without acknowledgment

This paper studies the optimal state estimation problem for networked control systems with control and observation packet losses but without packet acknowledgment (ACK). The packet ACK is a signal sent by the actuator to inform the estimator whether control packets are lost or not. Systems with packet ACK are usually called transmission control protocol (TCP)-like systems, and those without ACK are named user datagram protocol (UDP)-like systems. For UDP-like systems, the optimal estimator is derived and it is consisted of an exponentially increasing number of terms. By developing an auxiliary estimator, it is shown that there exists a critical observation packet arrival rate determining the stability of the expected EC (EEC), and it is identical to its counterpart for TCP-like systems. It is revealed that whether there is packet ACK or not has no effect on the stability of the EEC. Furthermore, under some conditions the EEC converges exponentially.

Adaptive state estimation for cyber physical systems under sparse attacks

This paper investigates the state estimation problem for cyber physical systems under sparse attacks. Firstly, the fundamental state estimation problem is transferred to an optimization problem with a unique solution. Secondly, an adaptive estimation method for sparse attacks is proposed, which convergence property is well proved. The advantage of proposed method is that the step-size can be adaptively adjusted based on the dynamic estimation errors. Therefore, the computing time is less than some existing methods while guaranteeing the desired performance. Then, a suitable state feedback is designed to improve the computing speed while enhancing the resiliency for the destroyed system. Finally, the speed performance and accuracy of proposed algorithm are verified by two numerical examples.

OBSERVABILITY ANALYSIS AND STATE ESTIMATION APPROACH OF POWER SYSTEM

This paper introduces a new approach to solve the problem of power system state estimation and check the observability of the measurement scheme. The state estimation is a powerful technique for system monitoring and control. Through a numerical method for observability analysis in power system state estimation based on weighted least square (WLS), the system state will be implemented and the measurement scheme will be judged. In addition to that, investigation for the effects of the measurements redundancy on the accuracy of the estimated variables is also discussed. The technique is applied to IEEE 30 bus system. Test results show the effectiveness of the proposed method to reach the best of the state estimator solution.

Extending the common information model for smart grids operational computations based on bus-branch models

In the modern smart grids (SGs), the heterogeneity of elements makes their interoperability difficult. In order to facilitate the integration and the interaction between different types of elements, the International Electrotechnical Commission (IEC) has introduced some standards (e.g., CIM, SCL), aimed at defining a common language for the communication among the different elements of a power system. However, without a harmonisation of these standards, the development and the implementation of systems and applications will result in a noticeable amount of single engineering design schemes. An ontology-based approach could collect knowledge from different applications, bridging the gap of harmonisation among such models. This paper deals with the implementation of a CIM ontology based on a bus-branch model that aims to support the power system state estimation problem. Resulting ontology has been instantiated by using a case study on a real power flow problem and evaluated by applying some well-known metrics.