Comparison of standard and modified recursive state estimation techniques for nonlinear systems

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.

This paper presents a Hybrid Particle Filter based RNN method for state estimation of non-linear dynamical system with knowledge of its input and output measurements. Particle filters are sequential Monte Carlo methods based on point mass (or particle) representations of probability densities, which is used to train Recurrent Neural Networks for estimation problems. The performance this method is compared with EKF based estimation and RNN based estimation. An Induction motor is considered as typical non-linear system and is implemented in MATLAB environment.

This paper studies the state estimation problem for nonlinear systems with missing measurements and filter‐and‐forward relays. By introducing stochastic variables obeying the Bernoulli distribution, the phenomenon of randomly occurring missing measurements is taken into consideration. The filter‐and‐forward relay is implemented in the channel from the sensor to the remote estimator, enhancing the quality of signal transmission. A matrix‐inequality‐based approach is proposed to design the filter in the filter‐and‐forward relay under which the filtering error in the relay is ensured to be ultimately bounded. Subsequently, a recursive estimator is built by employing the extended Kalman filter method. An upper bound on the estimation error covariance is derived by solving Riccati‐like difference equations and is then minimized by choosing an appropriate estimator gain. Finally, the validity of the estimation method is verified through two examples.

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.

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

This paper is concerned with the state estimation problem for a class of discrete time-varying stochastic nonlinear systems with randomly occurring deception attacks. The stochastic nonlinearity described by statistical means which covers several classes of well-studied nonlinearities as special cases is taken into discussion. The randomly occurring deception attacks are modelled by a set of random variables obeying Bernoulli distributions with given probabilities. The purpose of the addressed state estimation problem is to design an estimator with hope to minimize the upper bound for estimation error covariance at each sampling instant. Such an upper bound is minimized by properly designing the estimator gain. The proposed estimation scheme in the form of two Riccati-like difference equations is of a recursive form. Finally, a simulation example is exploited to demonstrate the effectiveness of the proposed scheme.

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.

We present a distributed observer for nonlinear systems, which can be used to estimate the state of a central plant with unknown parameters using a network of sensors that measure noisy linear combinations of the state. Our state estimator is based on the unscented Kalman filter and the dual Kalman filter method, which runs two Kalman filters in parallel for both state and parameter estimation. This allows us to improve estimates of the central plants parameters in order to improve state estimates, and vice versa. Furthermore, using results from stochastic stability analysis, we prove that under certain conditions the expected value of the proposed algorithm's estimation error is bounded. This result also shows us that as t goes to infinity, the error bound scales as a function of the largest eigenvalue of the sensor noise covariance matrix. Lastly, we demonstrate the ability of our state estimator to accurately track the state of nonlinear systems through simulations.

The extended fuzzy Kalman filter (EFKF) is developed recently and used for state estimation of the nonlinear systems with uncertainty. Based on extension of the orthogonality principle and the extended fuzzy Kalman filter, an improved fuzzy Kalman filters (IFKF) is proposed in this paper, which is more applicable and can deal with the state estimation of the nonlinear systems better than the EFKF. A simulation study is provided to verify the efficiency of the proposed method.

This paper investigated the problem of state and fault estimation for nonlinear discrete time systems in presence of unknown disturbances. A novel unbiased minimum variance filter (UMVF) is derived by reconstructing the non linear version of NUMV filter. In this work we assume that no prior knowledge about the dynamic of the disturbance and the fault are known. In this paper we considers that the fault affects both the system state and measurement equations, but the disturbance affects only the system state. The NUMV filter presented in this paper is an extension of the filter presented in [11]. The efficacy of the proposed filter is demonstrated by two simulation examples.

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

This paper presents new estimation methods for discrete fractional-order state-space systems with coloured measurement noise. A novel approach is proposed to convert a fractional system with coloured measurement noise to a system with white measurement noise in which the process and measurement noises are correlated with each other. In this paper, two new Kalman filter algorithms for fractional-order linear state-space systems with coloured measurement noise, as well as a new extended Kalman filter algorithm for state estimation in nonlinear fractional-order state-space systems with coloured measurement noise, are proposed. The accuracy of the equations and relations is confirmed in several theorems. The validity and effectiveness of the proposed algorithms are verified by simulation results and compared with previous work. Results show that for linear and nonlinear fractional-order systems with coloured noise, the proposed methods are more accurate than conventional methods regarding estimation error and estimation error covariance. Simulation results demonstrate that the proposed algorithms can accurately perform estimation in fractional-order systems with coloured measurement noise.

Image reconstruction algorithm of electromagnetic tomography based on fractional Kalman filter

We present a novel observer design for a class of single-output nonlinear systems with Markov jumps. The Markov jump process interferes with a deterministic nonlinear dynamics at random times and retains its state for a certain amount of time (dwell time). The estimation process is reset at these random times, depending on the reset values of the state process, and then evolves as a deterministic estimate of the state process itself. The novelty is given by the reset mechanism adopted for the estimation process itself, depending on the reset values of the state process. We prove that, as long as the mathematical expectation of the dwell times has a positive lower bound and the transition rate of the jump process at the first exit time out of any point is bounded, the state estimation error of the switching dynamics asymptotically converges to zero with probability one. The state estimate over each dwell time is designed using the novel technique of "output immersion".

State estimation scheme for nonlinear dynamical systems based on the stochastic approximation