Gaussian sum high-order CKF for dynamic state estimation

As high-dimensional nonlinear systems, some state variables of fermentation processes are unmeasurable, and nonlinear state estimation such as the high-degree cubature Kalman filter can be used to obtain the estimations of these variables. However, the high-degree cubature Kalman filter may suffer filter failure when dealing with high-dimensional systems. To solve the problem that the numerical stability of the high-degree cubature Kalman filter will decline with the increasement of dimension, an improved high-degree cubature Kalman filter is proposed. In this paper, arbitrary degree cubature rule is given and the improved high-degree cubature Kalman filter algorithm based on diagonalization of matrix is derived. Furthermore, the implementation process of the improved algorithm is constructed in industrial yeast fermentation process to demonstrate the performance of using the improved high-degree cubature Kalman filter.

Nonlinear dynamic state estimation in instrumented structures with conditionally linear Gaussian substructures

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

International audience

Adaptive Gaussian sum squared-root cubature Kalman filter with split-merge scheme for state estimation

Chapter 8 - Dynamic state estimation in electric power systems

We investigated the vertical variability of the thermal/dynamical structure of the North and tropical Atlantic ocean from a set of historical hydrographic data [Reynaud et al., 1998]. The analysis was performed in 10° geographic bins in terms of vertical empirical orthogonal functions (EOFs) in isopycnal coordinates as first proposed by Gavart and De Mey [1997] (hereinafter referred to as GDM97). In the isopycnal coordinate system the state vector is made up of isopycnal displacements (η) and the compensated potential temperature along isopycnals (θ). GDM97 showed that in the Azores‐Madeira region the isopycnal EOFs consistently help to better separate physical processes in the water column and to better project the dynamic height anomaly downward in comparison with the classical EOF analysis in depth coordinates. In the present study, we extended their work geographically and the connection with state estimation and data assimilation was more formalized. From our results, smooth spatial variations of isopycnal EOFs were found in the North Atlantic and confirmed the large‐scale relevance of isopycnal EOFs in the ocean. North of 20°N the dominant isopycnal EOF was associated with a quasihomogeneous vertical displacement of the isopycnal surfaces (η ≈ 1) with a quasiconservation of compensated potential temperature on those surfaces (θ ≈ 0). That mode was strongly modified by thermocline‐intensified baroclinic effects along the path of the Gulf Stream up to 40°W. Some physical interpretations of the vertical EOFs were also suggested in several areas as far as the isopycnal formulation was concerned (e.g., some thermal fronts, the Mediterranean outflow, water lenses, or “meddies”). This work confirmed the validity of the ideas of GDM97 over the entire North Atlantic ocean. The potential use of the isopycnal EOFs in state estimation problems was discussed. They are possible candidates to reduce the order of state estimation problems provided that they have good observability properties. These observability properties were studied in the case of satellite altimetry; that is, we examined how the dominant isopycnal EOFs were connected to sea‐level anomalies. In most bins it was found that the order of an estimation problem could be truncated to one (primarily north of 20°N) or two modes. The dominant EOF was found to account for most of the surface dynamic height variability with a rather normally distributed error of only a few centimeters. Dynamically more robust results were also presented, performing a univariate isopycnal analysis in η, leaving coherent thermal effects out. As a result, the dominant character of mode 1 was further enhanced everywhere. These univariate isopycnal EOFs can be used for purely dynamical state estimation. More generally, we believe that our results can be used to set up reduced‐order data assimilation schemes in the North Atlantic or in other regions with good historical data.

Parallel Kalman filter group integrated particle filter method for the train nonlinear operational status high-precision estimation under non-Gaussian environment

Nonlinear filtering algorithm is the key technology for dealing with complex systems in sensor data processing. To improve the filtering accuracy of the nonlinear filtering algorithm in the non‐Gaussian case, an improved version of the Gaussian sum algorithm, the Gaussian sum adaptive sparse grid quadrature filter (GSASQF), is proposed. The proposed algorithm overcomes the challenges by introducing the Gaussian sum principle, which converts the non‐Gaussian state and noise in the system into the form of weighted sum of Gaussian components. Based on the Bayesian filtering framework, a three‐level sparse grid sampling rule is introduced, with the sparse grid orthogonal filtering algorithm serving as the sub‐filter. By determining the sampling point parameters, the filtering process for each combination of Gaussian components is implemented, thereby ensuring the filtering accuracy of each group. In addition, in combination with the ideal of data‐driven, the weight of each Gaussian component combination is adaptively updated inversely by the values of the sensor measurement, which improves the global filtering accuracy of nonlinear system under non‐Gaussian noise. The combination of these three improvements enables high‐precision filtering of non‐Gaussian non‐linear systems. Theoretical analysis and simulation confirm that the proposed GSASQF algorithm provides advantages in filtering accuracy for nonlinear non‐Gaussian filtering problems.

Static state estimators have been in use in control centres for over four decades now. Recent availability of synchronized phasor measurements for bus voltages and line currents allows development of dynamic state estimators which can track the frequency and synchronous machine state variables and complement the information provided by static state estimators on the network states. This chapter provides a brief overview of the static state estimation problem followed by an introduction to dynamic state estimation. The problem of dynamic state estimation is formulated and a dynamic state estimator whose implementation lends itself to large size power grids is described. The chapter also contains a discussion of observability analysis for dynamic state estimators which presents additional challenges due to the time varying and nonlinear nature of the system and measurement equations.

Smart grid is built by combination of electric and information technologies and achieves the two-way interaction between power utilization and power generation. Unfortunately, new security threats appears together with cyber-physical communication systems. In order to properly monitor power network, an effective cyber attack detection and state estimation method are required to know attack and system states. This article considers the problem of robust grid state estimation and suggests a technique for distributed state estimation in power networks. First, the distribution power system incorporating multiple synchronous generators is modeled as a state-space framework, where attack occurs in measurements. Basically, the false data injection attacks can interfere with state estimation process by tampering with sensor measurements. Using mean squared error principle, the distributed dynamic state estimation algorithm is designed where local and neighboring gains are obtained using optimal filter and graph theory. For local gain computation, the attack parameter is obtained using the Bayesian learning process. The convergence condition of the proposed approach is derived. Extensive simulation results show that the proposed approach is able to estimate the system state within a short period of time. Hopefully, the proposed methodology can be used to tolerate the cyber attacks for improving the confidence of the grid state estimation process.

State estimation for complex networks with randomly occurring coupling delays

A novel set-membership-based smoothing method for state estimation using the optimal bounding ellipsoid (OBE) algorithms is presented. OBE filters have been proven to be effective in state estimation problems with unknown but bounded errors. Compared with filtering methods, smoothing methods provide a much more accurate and reliable state estimate because observations beyond the current estimation time are used. The new method is a Rauch-Tung-Striebel (RTS)-type smoother which employs both forward and backward passes to estimate the system state. The forward pass is performed using the OBE filter, while the backward pass maintains the fundamental spirit of OBE algorithm in the backward direction. The minimum-volume and minimum-trace bounding ellipsoids containing the feasible state set are derived from this algorithm. Simulation results show the performance of the proposed smoother is superior to both the traditional OBE filter and Kalman filter for state estimation.

State estimation is a core objective in cyber-physical systems. In the state estimation problem over linear systems, the Kalman filter is the standard solution. The filter is the format on which the solutions to subsequent estimation problems are based. Among these problems are the estimation problem in the presence of packet drops and estimation problem involving event-based triggers. We study in this paper both phenomena simultaneously. In an attempt to find the Kalman-like filter, which proves the Gaussianity of the state and offers a set of update equations, our paper shows that no such filter exists. More precisely, one cannot find an event-based trigger such that under possible packet drops, the state variable remains a Gaussian variable. This conclusion can be reasonably extended to a more general setting.

In this paper, we consider the problem of state estimation through observations possibly corrupted with both bad data and additive observation noises. A mixed $\ell_1$ and $\ell_2$ convex programming is used to separate both sparse bad data and additive noises from the observations. Through using the almost Euclidean property for a linear subspace, we derive a new performance bound for the state estimation error under sparse bad data and additive observation noises. Our main contribution is to provide sharp bounds on the almost Euclidean property of a linear subspace, using the "escape-through-a-mesh" theorem from geometric functional analysis. We also propose and numerically evaluate an iterative convex programming approach to performing bad data detections in nonlinear electrical power networks problems.