Quantifying uncertainty in state and parameter estimation

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015

Most data based state and parameter estimation methods require suitable initial values or guesses to achieve convergence to the desired solution, which typically is a global minimum of some cost function. Unfortunately, however, other stable solutions (e.g., local minima) may exist and provide suboptimal or even wrong estimates. Here we demonstrate for a 9-dimensional Lorenz-96 model how to characterize the basin size of the global minimum when applying some particular optimization based estimation algorithm. We compare three different strategies for generating suitable initial guesses and we investigate the dependence of the solution on the given trajectory segment (underlying the measured time series). To address the question of how many state variables have to be measured for optimal performance, different types of multivariate time series are considered consisting of 1, 2, or 3 variables. Based on these time series the local observability of state variables and parameters of the Lorenz-96 model is investigated and confirmed using delay coordinates. This result is in good agreement with the observation that correct state and parameter estimation results are obtained if the optimization algorithm is initialized with initial guesses close to the true solution. In contrast, initialization with other exact solutions of the model equations (different from the true solution used to generate the time series) typically fails, i.e. the optimization procedure ends up in local minima different from the true solution. Initialization using random values in a box around the attractor exhibits success rates depending on the number of observables and the available time series (trajectory segment).

Applied Mathematics and Computation, 2010

We introduce an adaptive learning rules for estimating all unknown parameters of delay dynamical system using a scalar time series. Sufficient condition for synchronization is derived using Krasovskii-Lyapunov theory. This scheme is highly applicable in secure communication since multiple messages can be transmitted through multiple parameter modulations. One of the advantage of this method is that parameter estimation is also possible even when only one time series of the transmitter is available. We present numerical examples for Mackey-Glass system with periodic delay time which are used to illustrate the validity of this scheme. The corresponding numerical results and the effect of external noise are also studied.

Physical Review E, 2011

Investigation of observability properties of nonlinear dynamical systems aims at giving a hint on how much dynamical information can be retrieved from a system using a certain measuring function. Such an investigation usually requires the knowledge of the system equations. This paper addresses the challenging problem of investigating observability properties of a system only from recorded data. From previous studies it is known that phase spaces reconstructed from poor observables are characterized by local sharp pleatings, local strong squeezing of trajectories and global inhomogeneity. A statistic is then proposed to quantify such properties of poor observability. Such a statistic was computed for a number of bench models for which observability studies had been previously performed. It was found that the statistic proposed in this paper, estimated exclusively from data, correlates generally well with observability results obtained using the system equations. It is possible to arrive at the same order of observability among the state variables using the proposed statistic even in the presence of noise with standard deviation as high as 10% of the data. The paper includes the application of the proposed statistic to sunspot time series.

—The problem of state estimation for nonlinear systems with unknown state or measurement delays is still an open problem. In this paper we consider the case of measurement delay and propose an approach that combines a delay identifier with a suitable high-gain observer in order to achieve simultaneous estimation of state and delay. We provide sufficient conditions that guarantee the exponential convergence to zero of the errors, globally with respect to the system variables and locally with respect to the delay estimation. We validate the method through an example concerning population models.

10th International Conference on Information Sciences, Signal Processing and their Applications, ISSPA 2010, 2010

Sensor fusion algorithms often assume perfect time synchronization of the sensor clocks. In a practical sensoractuator setup this is often difficult to achieve which in turn can give rise to systematic errors in the sensor fusion. In this article we suggest how the effect of the synchronization error from an unknown static or slowly varying measurement time-delays in a nonlinear state space system can be handled by linearizing the measurement equation in time. Based on the linearization an augmented system is constructed from which the system states and the delays can be jointly estimated. Expressions for the system, measurement, and covariance matrices of the augmented system are derived. Finally, the feasibility of the suggested approach is demonstrated by an example and a Monte-Carlo simulation.

We present an augmented continuous-discrete extended Kalman filter (EKF), capable of mitigating the effects of bad data and (random) measurement delay. We use an innovations-based Fisher-type scheme to identify and remove occurrences of corrupted sensor observations. We rely on time-stamp technology to accurately determine the duration of delay experienced by each received measurement packet. The length of delay is then used to construct a "delay-aware" optimum MMSE state estimate. A Flywheel Energy Storage System (FESS) example is used to demonstrate the high quality of dynamic state estimation achieved with the proposed method in the presence of bad sensor samples and delayed measurement packets.

We propose the original methods for reconstructing model delay-differential equations from chaotic time series for various classes of time-delayed feedback systems including: i) scalar time-delay systems with arbitrary nonlinear function, ii) high-order time-delay systems, iii) systems with several coexisting delays, and iv) coupled time-delay systems. These methods are based on the statistical analysis of time intervals between extrema in the time series of time-delay systems and the projection of infinite-dimensional phase space of these systems to suitably chosen low-dimensional subspaces. The methods allow one to recover the delay times, the nonlinear functions, and the parameters characterizing the inertial properties of the systems and to define the a priori unknown order of a time-delay system. In the case of coupled time-delay systems the methods are able to define also the type, strength, and direction of coupling and can be used for the analysis of unidirectional and mutual coupling of time-delay systems for a wide range of the coupling coefficients variation. The proposed methods are efficient for the analysis of short time series under sufficiently high levels of noise. The methods are successfully applied to recovery of standard time-delay systems from their simulated time series corrupted with noise and to modeling various electronic oscillators with delayed feedback from their experimental time series. The proposed methods are applied to the problem of hidden message extraction in the communication systems with nonlinear mixing of information signal and chaotic signal of a time-delay system. Different ways for encryption and decryption of information in these communication schemes are investigated. Using both numerical and experimental data we obtained a high quality of the information signal extraction from the transmitted signal for different message signals and different configurations of the chaotic transmitter with a priori unknown parameters.

Probabilistic Engineering Mechanics, 2006

The focus of this paper is Bayesian state and parameter estimation using nonlinear models. A recently developed method, the particle filter, is studied that is based on stochastic simulation. Unlike the well-known extended Kalman filter, the particle filter is applicable to highly nonlinear models with non-Gaussian uncertainties. Recently developed techniques that improve the convergence of the particle filter simulations are introduced and discussed. Comparisons between the particle filter and the extended Kalman filter are made using several numerical examples of nonlinear systems. The results indicate that the particle filter provides consistent state and parameter estimates for highly nonlinear models, while the extended Kalman filter does not.

2008

In the nonlinear prediction of scalar time series, the common practice is to reconstruct the state space using time-delay embedding and apply a local model on neighborhoods of the reconstructed space. The method of false nearest neighbors is often used to estimate the embedding dimension. For prediction purposes, the optimal embedding dimension can also be estimated by some prediction error minimization criterion. We investigate the proper state space reconstruction for multivariate time series and modify the two abovementioned criteria to search for optimal embedding in the set of the variables and their delays. We pinpoint the problems that can arise in each case and compare the state space reconstructions (suggested by each of the two methods) on the predictive ability of the local model that uses each of them. Results obtained from Monte Carlo simulations on known chaotic maps revealed the non-uniqueness of optimum reconstruction in the multivariate case and showed that predicti...

2003

Abstract In this paper, we present results of uncertain state estimation of systems that are monitored with limited accuracy. For these systems, the representation of state uncertainty as confidence intervals offers significant advantages over the more traditional approaches with probabilistic representation of noise.

AIAA Guidance, Navigation, and Control (GNC) Conference, 2013

Nonlinear estimation techniques are often used to estimate constant and time-varying parameters. The purpose of this paper is to use illustrative examples to compare the accuracy of several estimation techniques (the extended Kalman filter, the unscented Kalman filter, and the ensemble adjustment Kalman filter) along with retrospective cost model refinement. Both constant and time-varying examples are considered. Each algorithm is tuned to illustrate its capabilities for the given examples.

2008 American Control Conference, 2008

This paper addresses the problem of multiple parameter estimation in dynamical systems, where the solution algorithm is built upon the principles of extracting statistical information contents or patterns in the framework of Symbolic Domain Filtering. The proposed algorithm has been tested for estimation of two slowly varying parameters in an active electronic system that is constructed in the classical Duffing equation setting.

2007

Abstract: The state estimation of nonlinear systems with delayed measurements is investigated in this paper. The proposed approach is based on the representation of the nonlinear system by a decoupled multiple model that, to our knowledge, has not been investigated extensively. This multiple model approach offers an interesting alternative to the classically used multiple model known as Takagi-Sugeno multiple model.

1998

Most traditional methods for extracting the relationships between two time series are based on cross-correlation. In a non-linear non-stationary environment, these techniques are not su cient. We show in this paper how to use hidden Markov models (HMMs) to identify the lag (or delay) between di erent variables for such data. We rst present a method using maximum likelihood estimation and propose a simple algorithm which is capable of identifying associations between variables. We also adopt an information-theoretic approach and develop a novel procedure for training HMMs to maximise the mutual information between delayed time series. Both methods are successfully applied to real data: we show that HMMs are capable of modelling the oil drilling process and that they outperform existing methods for computing a crucial parameter, namely the lag for return.

International Journal of Bifurcation and Chaos, 2004

We review the problem of estimating parameters and unobserved trajectory components from noisy time series measurements of continuous nonlinear dynamical systems. It is first shown that in parameter estimation techniques that do not take the measurement errors explicitly into account, like regression approaches, noisy measurements can produce inaccurate parameter estimates. Another problem is that for chaotic systems the cost functions that have to be minimized to estimate states and parameters are so complex that common optimization routines may fail. We show that the inclusion of information about the time-continuous nature of the underlying trajectories can improve parameter estimation considerably. Two approaches, which take into account both the errors-in-variables problem and the problem of complex cost functions, are described in detail: shooting approaches and recursive estimation techniques. Both are demonstrated on numerical examples.