Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data

While nonlinear stochastic partial differential equations arise naturally in spatiotemporal modeling, inference for such systems often faces two major challenges: sparse noisy data and ill-posedness of the inverse problem of parameter estimation. To overcome the challenges, we introduce a strongly regularized posterior by normalizing the likelihood and by imposing physical constraints through priors of the parameters and states. We investigate joint parameter-state estimation by the regularized posterior in a physically motivated nonlinear stochastic energy balance model (SEBM) for paleoclimate reconstruction. The high-dimensional posterior is sampled by a particle Gibbs sampler that combines MCMC with an optimal particle filter exploiting the structure of the SEBM. In tests using either Gaussian or uniform priors based on the physical range of parameters, the regularized posteriors overcome the ill-posedness and lead to samples within physical ranges, quantifying the uncertainty in estimation. Due to the ill-posedness and the regularization, the posterior of parameters presents a relatively large uncertainty, and consequently, the maximum of the posterior, which is the minimizer in a variational approach, can have a large variation. In contrast, the posterior of states generally concentrates near the truth, substantially filtering out observation noise and reducing uncertainty in the unconstrained SEBM

An Improved Particle Filtering-based Approach for Health Prediction and Prognosis of Nonlinear Systems

Health monitoring of nonlinear systems is broadly concerned with the system health tracking and its prediction to future time horizons. Estimation and prediction schemes constitute as principle components of any health monitoring technique. Particle filter (PF) represents a powerful tool for performing state and parameter estimation as well as prediction of nonlinear dynamical systems. Estimation of the system parameters along with the states can yield an up-to-date and reliable model that can be used for long-term prediction problems through utilization of particle filters. This feature enables one to deal with uncertainty issues in the resulting prediction step as the time horizon is extended. Towards this end, this paper presents an improved method to achieve uncertainty management for long-term prediction of nonlinear systems by using particle filters. In our proposed approach, an observation forecasting scheme is developed to extend the system observation profiles (as time-series) to future time horizon. Particles are then propagated to future time instants according to a resampling algorithm instead of considering constant weights for the particles propagation in the prediction step. The uncertainty in the long-term prediction of the system states and parameters are managed by utilizing dynamic linear models for development of an observation forecasting scheme. This task is addressed through an outer adjustment loop for adaptively changing the sliding observation injection window based on the Mahalanobis distance criterion. Our proposed approach is then applied to predicting the health condition as well as the remaining useful life (RUL) of a gas turbine engine that is affected by degradations in the system health parameters. Extensive simulation and case studies are conducted to demonstrate and illustrate the capabilities and performance characteristics of our proposed and developed schemes

Reduced-order Unscented Kalman Filtering with application to parameter identification in large-dimensional systems

See also erratum DOI:10.1051/cocv/2011001International audienceWe propose a general reduced-order filtering strategy adapted to Unscented Kalman Filtering for any choice of sampling points distribution. This provides tractable filtering algorithms which can be used with large-dimensional systems when the uncertainty space is of reduced size, and these algorithms only invoke the original dynamical and observation operators, namely, they do not require tangent operator computations, which of course is of considerable benefit when nonlinear operators are considered. The algorithms are derived in discrete time as in the classical UKF formalism - well-adapted to time discretized dynamical equations - and then extended into consistent continuous-time versions. This reduced-order filtering approach can be used in particular for the estimation of parameters in large dynamical systems arising from the discretization of partial differential equations, when state estimation can be handled by an adequate Luenberger observer inspired from feedback control. In this case, we give an analysis of the joint state-parameter estimation procedure based on linearized error, and we illustrate the effectiveness of the approach using a test problem inspired from cardiac biomechanics

Adaptive Observer Design for Nonlinear Interconnected Systems by the Application of LaSalle's Theorem

In this article, a class of nonlinear interconnected systems with uncertain time varying parameters (TVPs) is considered. Both the interconnections and the isolated subsystems are nonlinear. The differences between the unknown TVPs and their corresponding nominal values are assumed to be bounded where the nominal value is not required to be known. A dynamical system is proposed and then, the error systems between the original interconnected system and the designed dynamical system are analysed. A set of conditions is developed such that the augmented systems formed by the error dynamical systems and the designed adaptive laws are uniformly ultimately bounded. Specifically, the state observation errors are asymptotically convergent to zero based on the LaSalle's Theorem while the parameter estimation errors are uniformly ultimately bounded, and the classical condition of persistent excitation is not required. A case study on a coupled inverted pendulum system is presented to demonstrate the developed methodology, and simulation shows that the proposed approach is effective and practicable

Health Monitoring of Nonlinear Systems with Application to Gas Turbine Engines

Health monitoring and prognosis of nonlinear systems is mainly concerned with system health tracking and its evolution prediction to future time horizons. Estimation and prediction schemes constitute as principal components of any health monitoring framework. In this thesis, the main focus is on development of novel health monitoring techniques for nonlinear dynamical systems by utilizing model-based and hybrid prognosis and health monitoring approaches. First, given the fact that particle filters (PF) are known as a powerful tool for performing state and parameter estimation of nonlinear dynamical systems, a novel dual estimation methodology is developed for both time-varying parameters and states of a nonlinear stochastic system based on the prediction error (PE) concept and the particle filtering scheme. Estimation of system parameters along with the states generate an updated model that can be used for a long-term prediction problem. Next, an improved particle filtering-based methodology is developed to address the prediction step within the developed health monitoring framework. In this method, an observation forecasting scheme is developed to extend the system observation profiles (as time-series) to future time horizons. Particles are then propagated to future time instants according to a resampling algorithm in the prediction step. The uncertainty in the long-term prediction of the system states and parameters are managed by utilizing dynamic linear models (DLM) for development of an observation forecasting scheme. A novel hybrid architecture is then proposed to develop prognosis and health monitoring methodologies for nonlinear systems by integration of model-based and computationally intelligent-based techniques. Our proposed hybrid health monitoring methodology is constructed based on a framework that is not dependent on the structure of the neural network model utilized in the implementation of the observation forecasting scheme. Moreover, changing the neural network model structure in this framework does not significantly affect the prediction accuracy of the entire health prediction algorithm. Finally, a method for formulation of health monitoring problems of dynamical systems through a two-time scale decomposition is introduced. For this methodology the system dynamical equations as well as the affected damage model, are investigated in the two-time scale system health estimation and prediction steps. A two-time scale filtering approach is developed based on the ensemble Kalman filtering (EnKF) methodology by taking advantage of the model reduction concept. The performance of the proposed two-time scale ensemble Kalman filters is shown to be more accurate and less computationally intensive as compared to the well-known particle filtering approach for this class of nonlinear systems. All of our developed methods have been applied for health monitoring and prognosis of a gas turbine engine when it is affected by various degradation damages. Extensive comparative studies are also conducted to validate and demonstrate the advantages and capabilities of our proposed frameworks and methodologies

Completely Recursive Least Squares and Its Applications

The matrix-inversion-lemma based recursive least squares (RLS) approach is of a recursive form and free of matrix inversion, and has excellent performance regarding computation and memory in solving the classic least-squares (LS) problem. It is important to generalize RLS for generalized LS (GLS) problem. It is also of value to develop an efficient initialization for any RLS algorithm. In Chapter 2, we develop a unified RLS procedure to solve the unconstrained/linear-equality (LE) constrained GLS. We also show that the LE constraint is in essence a set of special error-free observations and further consider the GLS with implicit LE constraint in observations (ILE-constrained GLS). Chapter 3 treats the RLS initialization-related issues, including rank check, a convenient method to compute the involved matrix inverse/pseudoinverse, and resolution of underdetermined systems. Based on auxiliary-observations, the RLS recursion can start from the first real observation and possible LE constraints are also imposed recursively. The rank of the system is checked implicitly. If the rank is deficient, a set of refined non-redundant observations is determined alternatively. In Chapter 4, base on [Li07], we show that the linear minimum mean square error (LMMSE) estimator, as well as the optimal Kalman filter (KF) considering various correlations, can be calculated from solving an equivalent GLS using the unified RLS. In Chapters 5 & 6, an approach of joint state-and-parameter estimation (JSPE) in power system monitored by synchrophasors is adopted, where the original nonlinear parameter problem is reformulated as two loosely-coupled linear subproblems: state tracking and parameter tracking. Chapter 5 deals with the state tracking which determines the voltages in JSPE, where dynamic behavior of voltages under possible abrupt changes is studied. Chapter 6 focuses on the subproblem of parameter tracking in JSPE, where a new prediction model for parameters with moving means is introduced. Adaptive filters are developed for the above two subproblems, respectively, and both filters are based on the optimal KF accounting for various correlations. Simulations indicate that the proposed approach yields accurate parameter estimates and improves the accuracy of the state estimation, compared with existing methods

Inference in Nonlinear Systems with Unscented Kalman Filters

An increasing number of scientific disciplines, most notably the life sciences and health care, have become more quantitative, describing complex systems with coupled nonlinear di↵erential equations. While powerful algorithms for numerical simulations from these systems have been developed, statistical inference of the system parameters is still a challenging problem. A promising approach is based on the unscented Kalman filter (UKF), which has seen a variety of recent applications, from soft tissue mechanics to chemical kinetics. The present study investigates the dependence of the accuracy of parameter estimation on the initialisation. Based on three toy systems that capture typical features of real-world complex systems: limit cycles, chaotic attractors and intrinsic stochasticity, we carry out repeated simulations on a large range of independent data instantiations. Our study allows a quantification of the accuracy of inference, measured in terms of two alternative distance measures in function and parameter space, in dependence on the initial deviation from the ground truth