Comparing Kalman Filters and Observers for Power System Dynamic State Estimation with Model Uncertainty and Malicious Cyber Attacks

Kalman filters and observers are two main classes of dynamic state estimation (DSE) routines. Power system DSE has been implemented by various Kalman filters, such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF). In this paper, we discuss two challenges for an effective power system DSE: (a) model uncertainty and (b) potential cyber attacks. To address this, the cubature Kalman filter (CKF) and a nonlinear observer are introduced and implemented. Various Kalman filters and the observer are then tested on the 16-machine, 68-bus system given realistic scenarios under model uncertainty and different types of cyber attacks against synchrophasor measurements. It is shown that CKF and the observer are more robust to model uncertainty and cyber attacks than their counterparts. Based on the tests, a thorough qualitative comparison is also performed for Kalman filter routines and observers.Comment: arXiv admin note: text overlap with arXiv:1508.0725

On Differential Privacy and Traffic State Estimation Problem for Connected Vehicles

This letter focuses on the problem of traffic state estimation for highway networks with junctions in the form of on- and off-ramps while maintaining differential privacy of traffic data. Two types of sensors are considered, fixed sensors such as inductive loop detectors and connected vehicles which provide traffic density and speed data. The celebrated nonlinear second-order Aw-Rascle- Zhang (ARZ) model is utilized to model the traffic dynamics. The model is formulated as a nonlinear state-space difference equation. Sensitivity relations are derived for the given data which are then used to formulate a differentially private mechanism which adds a Gaussian noise to the data to make it differentially private. A Moving Horizon Estimation (MHE) approach is implemented for traffic state estimation using a linearized ARZ model. MHE is compared with Kalman Filter variants namely Extended Kalman Filter, Ensemble Kalman Filter and Unscented Kalman Filter. Several research and engineering questions are formulated and analysis is performed to find corresponding answers.Comment: TO APPEAR IN THE 61ST IEEE CONFERENCE ON DECISION AND CONTROL (CDC), CANCUN, MEXICO, DECEMBER 2022. arXiv admin note: text overlap with arXiv:2209.0284

Extended Kalman Filter on SE(3) for Geometric Control of a Quadrotor UAV

An extended Kalman filter (EKF) is developed on the special Euclidean group, SE(3) for geometric control of a quadrotor UAV. It is obtained by performing an extensive linearization on SE(3) to estimate the state of the quadrotor from noisy measurements. Proposed estimator considers all the coupling effects between rotational and translational dynamics, and it is developed in a coordinate-free fashion. The desirable features of the proposed EKF are illustrated by numerical examples and experimental results for several scenarios. The proposed estimation scheme on SE(3) has been unprecedented and these results can be particularly useful for aggressive maneuvers in GPS denied environments or in situations where parts of onboard sensors fail.Comment: arXiv admin note: text overlap with arXiv:1304.6765, arXiv:1411.298

Asymptotically Optimal Nonlinear Filtering

In this note we present a computationally simple algorithm for non-linear filtering. The algorithm involves solving, at a given point in state space, an algebraic Riccati equation. The coefficients of this equation vary with the given point in state space. We investigate conditions under which the state estimate given by this algorithm converges asymptotically to the first order minimum variance estimate given by the extended Kalman filter. We also investigate conditions for determining a region of stability for the filter given by this algorithm. The analysis is based on stable manifold theory and Hamilton-Jacobi-Bellman (HJB) equations. The motivation for introducing HJB equations is given by reference to the maximum likelihood approach to deriving the extended Kalman filter

A second order minimum-energy filter on the special orthogonal group

Abstract— This work documents a case study in the application of Mortensen’s nonlinear filtering approach to invariant systems on general Lie groups. In this paper, we consider the special orthogonal group SO(3) of all rotation matrices. We identify the exact form of the kinematics of the minimumenergy (optimal) observer on SO(3) and note that it depends on the Hessian of the value function of the associated optimal control problem. We derive a second order approximation of the dynamics of the Hessian by neglecting third order terms in the expansion of the dynamics. This yields a Riccati equation that together with the optimal observer equation form a second order minimum-energy filter on SO(3). The proposed filter is compared to the multiplicative extended Kalman filter (MEKF), arguably the industry standard for attitude estimation, by means of simulations. Our studies indicate superior transient and asymptotic tracking performance of the proposed filter as compared to the MEKF

A hybrid EKF and switching PSO algorithm for joint state and parameter estimation of lateral flow immunoassay models

This is the post-print version of the Article. The official published can be accessed from the link below - Copyright @ 2012 IEEEIn this paper, a hybrid extended Kalman filter (EKF) and switching particle swarm optimization (SPSO) algorithm is proposed for jointly estimating both the parameters and states of the lateral flow immunoassay model through available short time-series measurement. Our proposed method generalizes the well-known EKF algorithm by imposing physical constraints on the system states. Note that the state constraints are encountered very often in practice that give rise to considerable difficulties in system analysis and design. The main purpose of this paper is to handle the dynamic modeling problem with state constraints by combining the extended Kalman filtering and constrained optimization algorithms via the maximization probability method. More specifically, a recently developed SPSO algorithm is used to cope with the constrained optimization problem by converting it into an unconstrained optimization one through adding a penalty term to the objective function. The proposed algorithm is then employed to simultaneously identify the parameters and states of a lateral flow immunoassay model. It is shown that the proposed algorithm gives much improved performance over the traditional EKF method.This work was supported in part by the International Science and Technology Cooperation Project of China under Grant 2009DFA32050, Natural Science Foundation of China under Grants 61104041, International Science and Technology Cooperation Project of Fujian Province of China under Grant 2009I0016

A Quick Guide for the Iterated Extended Kalman Filter on Manifolds

The extended Kalman filter (EKF) is a common state estimation method for discrete nonlinear systems. It recursively executes the propagation step as time goes by and the update step when a set of measurements arrives. In the update step, the EKF linearizes the measurement function only once. In contrast, the iterated EKF (IEKF) refines the state in the update step by iteratively solving a least squares problem. The IEKF has been extended to work with state variables on manifolds which have differentiable $\boxplus$ and $\boxminus$ operators, including Lie groups. However, existing descriptions are often long, deep, and even with errors. This note provides a quick reference for the IEKF on manifolds, using freshman-level matrix calculus. Besides the bare-bone equations, we highlight the key steps in deriving them.Comment: 2 pages excluding reference

Various Ways to Compute the Continuous-Discrete Extended Kalman Filter

International audienceThe Extended Kalman Filter (EKF) is a very popular tool dealing with state estimation. Its continuous-discrete version (CD-EKF) estimates the state trajectory of continuous-time nonlinear models, whose internal state is described by a stochastic differential equation and which is observed through a noisy nonlinear form of the sampled state. The prediction step of the CD-EKF leads to solve a differential equation that cannot be generally solved in a closed form. This technical note presents an overview of the numerical methods, including recent works, usually implemented to approximate this filter. Comparisons of theses methods on two different nonlinear models are finally presented. The first one is the Van der Pol oscillator which is widely used as a benchmark. The second one is a neuronal population model. This more original model is used to simulate EEG activity of the cortex. Experiments showed better stability properties of implementations for which the positivity of the prediction matrix is guaranteed