Distributed State Estimation for Nonlinear Systems with Unknown Parameters

Robust unscented Kalman filter with adaptation of process and measurement noise covariances

This paper addresses the problem of state estimation for nonlinear systems by means of the unscented Kalman filter (UKF). Compared to the traditional extended Kalman filter, the UKF does not require the local linearization of the system equations used in the propagation stage. Important results using the UKF have been reported recently but in every case the system equations used by the filter were considered known. Not only that, such models are usually considered to be differential equations, which requires that numerical integration be performed during the propagation phase of the filter. In this paper the dynamical equations of the system are taken to be difference equations--thus avoiding numerical integration--and are built from data without prior knowledge. The identified models are subsequently implemented in the filter in order to accomplish state estimation. The paper discusses the impact of not knowing the exact equations and using data-driven models in the context of state and joint state-and-parameter estimation. The procedure is illustrated by means of examples that use simulated and measured data.

Over the last 20-30 years, the extended Kalman filter (EKF) has become the algorithm of choice in numerous nonlinear estimation and machine learning applications. These include estimating the state of a nonlinear dynamic system as well estimating parameters for nonlinear system identification (eg, learning the weights of a neural network). The EKF applies the standard linear Kalman filter methodology to a linearization of the true nonlinear system. This approach is sub-optimal, and can easily lead to divergence. Julier et al. (1997), proposed the unscented Kalman filter (UKF) as a derivative-free alternative to the extended Kalman filter in the framework of state estimation. This was extended to parameter estimation by Wan and Van der Merwe et al., (2000). The UKF consistently outperforms the EKF in terms of prediction and estimation error, at an equal computational complexity of (OL/sup 3/)/sup l/ for general state-space problems. When the EKF is applied to parameter estimation, the special form of the state-space equations allows for an O(L/sup 2/) implementation. This paper introduces the square-root unscented Kalman filter (SR-UKF) which is also O(L/sup 3/) for general state estimation and O(L/sup 2/) for parameter estimation (note the original formulation of the UKF for parameter-estimation was O(L/sup 3/)). In addition, the square-root forms have the added benefit of numerical stability and guaranteed positive semi-definiteness of the state covariances.

Nonlinear state estimation is a pre-requisite for advanced process control and fault diagnosis tasks. In literature, various recursive nonlinear filtering techniques have been proposed and used for state estimation of nonlinear systems. Over the last few years, Moving Horizon Estimation (MHE) is increasingly being used for state estimation of nonlinear systems. Moving horizon estimation works with a window of data and hence requires additional online computation compared to recursive nonlinear filters. However, MHE performs both smoothing and filtering and thus has the potential to obtain more accurate state estimates as compared to the recursive filters. Most of the available comparisons of MHE with recursive filters are based on simulation case studies where the true states and parameters, as well as noise processes are exactly known. In this work, we apply MHE to a three-tank experimental setup and compare its performance with various nonlinear filters available in literature such as Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Gaussian Sum EKF (GSEKF). We present some of the challenges in experimental implementation of state estimation approaches. These include presence of unknown disturbances, non-whiteness of noise signals as well as lack of accurate measurements. We then discuss the approach followed for obtaining model parameters and noise characterization to make the models amenable for filter implementation. It is found that EKF, GSEKF and UKF perform as well as MHE as far as accuracy is concerned, but require significantly lower computational efforts.

Extended Kalman filter (EKF) has been widely accepted as an important tool in the state estimation of nonlinear systems. However, defects of EKF such as biased estimation, computational complexity and inefficacy when analytical derivatives (Jacobians or Hessians) can not be calculated also affect its application in many fields. In this paper, unscented Kalman filter (UKF) was employed as an alternative to estimate the vehicle velocity and wheel angular speed of the anti-lock braking system (ABS). Simulation results show that the UKF can provide more accurate and efficient performance than the EKF for the state estimation.

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This manuscript deals with parametric and state estimation problem of discrete-time nonlinear polynomials systems. In this respect, the aim is to design a Recursive State and Parametric Estimation (RSPE) algorithm, based on least square principle, polynomial adjustable models and Kalman filter theory. Our simulation results show the validity of the proposed algorithm.

Guest Editorial: Situational awareness of integrated energy systems

Simultaneous state and parameter estimation is essential for control system design and dynamic modeling of physical systems. This capability provides critical real-time insight into system behavior, supports the discovery of underlying mechanisms, and facilitates adaptive control strategies. Surveyed in this review paper are two classes of state and parameter estimation methods: Kalman Filters and Luenberger Observers. The Kalman Filter framework, including its major variants such as the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), Cubature Kalman Filter (CKF), and Ensemble Kalman Filter (EnKF), has been widely applied for joint and dual estimation in linear and nonlinear systems under uncertainty. In parallel, Luenberger observers, typically used in deterministic settings, offer alternative approaches through high-gain, sliding mode, and adaptive observer structures. This review focuses on the theoretical foundations, algorithmic developments, and application domains of these methods and provides a comparative analysis of their advantages, limitations, and practical relevance across diverse engineering scenarios.

Image reconstruction algorithm of electromagnetic tomography based on fractional Kalman filter

An extended model-based observer for state estimation in nonlinear hysteretic structural systems

This paper investigates the application of multisensor data fusion (MSDF) technique to enhance the state estimation of a nonlinear plant. The proposed method is based on Kalman filters approach to improve the state estimation obtained by the novel adaptive unscented Kalman filter (AUKF). The common trend for the KF implementation assumes pre-specified fixed distribution matrices for both process and measurement noises. Here, however, the variance matrices for both process and measurement noise signals are assumed unknown a priori and thus incrementally estimated and updated using a sliding time window paradigm within which an estimation of the noise variance is calculated and adaptively updated each time the window is shifted forward. The proposed methodology is tested on a simulated continuous stirred tank reactor (CSTR) problem to estimate 4 states of this nonlinear plant. The simulation results demonstrate the superiority of the suggested method in state estimation compared with a previously reported approach.

A new framework is presented for understanding how a nonnormal probability density function (pdf) may affect a state estimate and how one might usefully exploit the nonnormal properties of the pdf when constructing a state estimate. A Bayesian framework is constructed that naturally leads to an expansion of the expected forecast error in a polynomial series consisting of powers of the innovation vector. This polynomial expansion in the innovation reveals a new view of the geometric nature of the state estimation problem. It is shown that this expansion in powers of the innovation provides a direct relationship between a nonnormal pdf describing the likely distribution of states and a normal pdf determined by powers of the forecast error. One implication of this perspective is that when state estimation is performed on a nonnormal pdf it leads to state estimates based on the mean to be nonlinear functions of the innovation. A direct relationship is shown between the degree to which the state estimate varies with the innovation and the moments of the distribution. These and other implications of this new view of ensemble state estimation in nonlinear systems are illustrated in simple scalar systems as well as on the Lorenz attractor.

The problem of tracking epidemic spreading on networks is relevant to the control of morbidity. The transmission dynamics of an epidemic can be described by a simple compartmental model. Specifically, the estimation of epidemic spreading on networks can be achieved by a nonlinear Kalman filter, which is a tool for state estimation of nonlinear systems. In this article, epidemic spreading on networks is described by compartmental models, such as susceptible-infected-susceptible, susceptible-infected-recovered, and susceptible-infected-recovered-susceptible models. The dynamics of epidemic spreading on homogeneous networks, including Erdös-Renyi network, Watts-Strogatz network, and Newman-Watts network, are estimated by several nonlinear Kalman filters, including extended Kalman filter, unscented Kalman filter, and third-degree and fifth-degree cubature Kalman filters. The performance comparison in terms of accuracy and stability forms a guideline of utilizing nonlinear Kalman filters for tracking epidemic spreading. The theoretical analysis has been validated through numerical experiments.

An Improved Gaussian Sum Unscented Kalman Filter