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

International audience

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.

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.

A solution to the state estimation problem of systems with unmeasurable non-zero mean inputs/disturbances, which do not satisfy the disturbance decoupling conditions, is given using the Kalman filtering and Bayesian estimation theory. The proposed estimation algorithm, named Supervisory Kalman Filter (SKF), consists of a Kalman filter with an extra update step which is inspired by the particle filtering technique. The extra step, called supervisory layer, numerically solves the measurement equations for the portion of the state vector that cannot be estimated by the Kalman filter. First, it produces N randomly generated state vectors, the particles, which are distributed based on the Kalman filter’s last updated estimate. Then, the estimated measurement vector associated with each particle is compared to the actual measurement vector to identify the particle’s probability to be a solution. Finally, a so-called resampling stage is implemented to refine the particles with higher likelihoods. The effectiveness of the SKF is demonstrated by comparing its estimation performance with that of the standard Kalman Filter and the particle filter for a vehicle state estimation problem. The estimation results confirm that the SKF precisely estimates those states of the vehicle that cannot be estimated by either the Kalman filter or the particle filter, regardless of the unknown disturbances from the road. The filtering methodology offered in the article has a potential to improve performance of the systems presented in the patents WO2011115960, WO2010024751, US8073528, and US20110299730. Keywords: Bayesian state estimation, disturbance decoupling conditions, inputs/disturbances, particle filtering, state estimation, systems with unknown kalman filtering, unscented kalman filtering

State estimation with a destination constraint using pseudo-measurements

State estimation is the basic problem in every area of science and engineering. For the state estimation problem, Kalman filter is the generally used technique when the system is linear. Various derivatives of Kalman filter are proposed earlier for non-linear systems, i.e. Extended Kalman Filter and Unscented Kalman Filter. But, there is a need of tuning in these estimation techniques and therefore the tuning of process and measurement noise covariance matrices is required. Earlier, the different optimization techniques are used for the tuning of Extended Kalman Filter like Genetic Algorithm, Human Opinion Dynamics based Optimization and Particle Swarm Optimization. In this paper, Hybrid HOD-GSA has been proposed for the tuning of Extended Kalman Filter and also to solve the trapping problem of GSA. Then, the results taken from Hybrid HOD-GSA are compared with the results taken from Human Opinion Dynamics and Particle Swarm Optimization in terms of accuracy, error rate, standard deviation and convergence.

State estimation for complex networks with randomly occurring coupling delays

On the use of shape constraints for state estimation in reaction 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.

This paper proposes a new way of using particle filters for set-membership state estimation problems. For nonlinear state estimation problems, stochastic particle filters have been proposed which maintain a large number of solution candidates by using Monte-Carlo simulation. Set-membership approach for state estimation is an alternative method that is, in certain situations, more suitable than stochastic models. However, parametric modeling of the set-membership description (e.g. ellipsoidal approximation) is not easy even for linear models, and it often yields sets that are much larger than true uncertainty sets. We show that particle filters that utilize Monte-Carlo simulation are more suitable for set-membership approach for nonlinear models

Decomposition of the Estimation and Control Problems in Power Systems

This paper considers the estimation problem of traffic state in freeway networks in light of the unknown input filtering (UIF) framework. The freeway traffic flow is modeled as a dynamic stochastic nonlinear system and is based on a recently developed speed-extended cell-transmission model of freeway traffic. Since the environmental conditions on a freeway may change over time, model parameters estimation is also considered. It is shown that the dual state and parameter estimation problem can be solved by applying the UIF to a nonlinear system with unknown inputs. Recently, a nonlinear version of the extended recursive three-step filter, named as the NERTSF, was employed to solve the problem. However, a numerical approximation method is used to calculate the model partial derivatives. To relax that restriction, in this paper a derivative-free versions of the NERTSF is further proposed to solve the addressed estimation problem of the freeway traffic flow.

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.

State estimation of complex-valued neural networks with leakage delay: A dynamic event-triggered approach

In this paper, the state estimation problem is investigated for a class of nonlinear stochastic systems with mixed time delays under event-triggered scheme. Based on the proposed event-triggered scheme, the new measurement is sampled and then transmitted to the estimator only when the difference between the current measurement and the previous transmitted measurement is greater than a fixed threshold, thereby reducing the network traffic. Attention is focused on the design problem of the estimator such that the estimation error in mean square is bounded and then minimized. A novel Lyapunov function is employed to derive sufficient condition for the existence of the desired state estimators whose gain matrix can be computed by solving a convex problem. The gain matrix of the estimator is characterized in order that an asymptotic bound of the estimation error is found and subsequently minimized. A numerical simulation is provided to illustrate the effectiveness of the results.