Guaranteed Nonlinear State Estimator for Cooperative Systems

This paper is about state estimation for continuous-time nonlinear models, in a context where all uncertain variables can be bounded. More precisely, cooperative models are considered, i.e., models that satisfy some constraints on the signs of the entries of the Jacobian of their dynamic equation. In this context, interval observers and a guaranteed recursive state estimation algorithm are combined to enclose the state at any given instant of time in a subpaving. The approach is illustrated on the state estimation of a waste-water treatment process.     

Geometric Ergodicity for Piecewise Contracting Processes with Applications for Tropical Stochastic Lattice Models

Stochastic lattice models are increasingly prominent as a way to capture highly intermittent unresolved features of moist tropical convection in climate science and as continuum mesoscopic models in material science. Stochastic lattice models consist of suitably discretized continuum partial differential equations interacting with Markov jump processes at each lattice site with transition rates depending on the local value of the continuum equation; they are a special case of piecewise deterministic Markov processes but often have an infinite state space and unbounded transition rates. Here a general theorem on geometric ergodicity for piecewise deterministic contracting processes is developed with full generality to apply to stochastic lattice models. A highly nontrivial application to the stochastic skeleton model for the Madden‐Julian oscillation (Thual et al., 2013) is developed here where there is an infinite state space with unbounded and also degenerate transition rates. Geometric ergodicity for the stochastic skeleton model guarantees exponential convergence to a unique invariant measure that defines the statistical tropical climate of the model. Another application of the general framework is developed here for stochastic lattice models designed to capture intermittent fluctuation in the simplest tropical climate models. Other straightforward applications to models motivated by material science are mentioned briefly here. © 2016 Wiley Periodicals, Inc.     

Online affine model identification of nonlinear processes using a new adaptive neuro-fuzzy approach

This paper presents a new online identification algorithm to drive an adaptive affine dynamic model for nonlinear and time-varying processes. The new algorithm is devised on the basis of an adaptive neuro-fuzzy modeling approach. Two adaptive neuro-fuzzy models are sequentially identified on the basis of the most recent input-output process data to realize an online affine-type model. A series of simulation test studies has been conducted to demonstrate the efficient capabilities of the proposed algorithm to automatically identify an online affine-type model for two highly nonlinear and time-varying continuous stirred tank reactor (CSTR) benchmark problems having inherent non-affine dynamic model representations. Adequacy assessments of the identified models have been explored using different evaluation measures, including comparison with an adaptive neuro-fuzzy inference system (ANFIS) as the pioneering and the most popular adaptive neuro-fuzzy system with powerful modeling features.     

Numerical computation of the value function of optimally controlled stochastic switching processes by multi-grid techniques

By the dynamic programming principle the value function of an optimally controlled stochasticswitching process can be shown to satisfy a boundary value problem for a fully nonlinear second-order elliptic differential equation of Hamilton-Jacobi-Bellman (HJB-) type. For the numerical solution of that HJB-equation we present a multi-grid algorithm whose main features arethe use of nonlinear Gauss-Seidel iteration in the smoothing process and an adaptive local choice of prolongations and restrictions in the coarse-to-fine and fine-to-coarse transfers. Local convergence is proved by combining nonlinear multi-grid convergence theory and elementarysubdifferential calculus. The efficiency of the algorithm is demonstrated for optimal advertising in stochastic dynamic sales response models of Vidale-Wolfe type.     

Yang-Baxter algebra and generation of quantum integrable models

We discover an operator-deformed quantum algebra using the quantum Yang-Baxter equation with the trigonometric R-matrix. This novel Hopf algebra together with its q→1 limit seems the most general Yang-Baxter algebra underlying quantum integrable systems. We identify three different directions for applying this algebra in integrable systems depending on different sets of values of the deforming operators. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, and the associated Lax operators generate and classify them in a unified way. Variable values yield a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 470–485, June, 2007.     

Spatiotemporal Models for Gaussian Areal Data

We introduce a class of spatiotemporal models for Gaussian areal data. These models assume a latent random field process that evolves through time with random field convolutions; the convolving fields follow proper Gaussian Markov random field (PGMRF) processes. At each time, the latent random field process is linearly related to observations through an observational equation with errors that also follow a PGMRF. The use of PGMRF errors brings modeling and computational advantages. With respect to modeling, it allows more flexible model structures such as different but interacting temporal trends for each region, as well as distinct temporal gradients for each region. Computationally, building upon the fact that PGMRF errors have proper density functions, we have developed an efficient Bayesian estimation procedure based on Markov chain Monte Carlo with an embedded forward information filter backward sampler (FIFBS) algorithm. We show that, when compared with the traditional one-at-a-time Gibbs sampler, our novel FIFBS-based algorithm explores the posterior distribution much more efficiently. Finally, we have developed a simulation-based conditional Bayes factor suitable for the comparison of nonnested spatiotemporal models. An analysis of the number of homicides in Rio de Janeiro State illustrates the power of the proposed spatiotemporal framework.

Supplemental materials for this article are available online in the journal’s webpage.     

Adaptive Inverse Control Using a Gradient-Projection Optimization Technique

As an application of an optimization technique, a gradient-projection method is employed to derive an adaptive algorithm for updating the parameters of an inverse which is designed to cancel the effects of actuator uncertainties in a control system. The actuator uncertainty is parametrized by a set of unknown parameters which belong to a parameter region. A desirable inverse is implemented with adaptive estimates of the actuator parameters. Minimizing an estimation error, a gradient algorithm is used to update such parameter estimates. To ensure that the parameter estimates also belong to the parameter region, the adaptive update law is designed with parameter projection. With such an adaptive inverse, desired control system performance can be achieved despite the presence of the actuator uncertainties.     

Bayesian and Maximum Likelihood Estimation for Gaussian Processes on an Incomplete Lattice

This article proposes a new approach for Bayesian and maximum likelihood parameter estimation for stationary Gaussian processes observed on a large lattice with missing values. We propose a Markov chain Monte Carlo approach for Bayesian inference, and a Monte Carlo expectation-maximization algorithm for maximum likelihood inference. Our approach uses data augmentation and circulant embedding of the covariance matrix, and provides likelihood-based inference for the parameters and the missing data. Using simulated data and an application to satellite sea surface temperatures in the Pacific Ocean, we show that our method provides accurate inference on lattices of sizes up to 512 × 512, and is competitive with two popular methods: composite likelihood and spectral approximations.     

Algorithm for the continuous estimation of a disturbance in a stochastic differential equation

Based on the approach of the theory of dynamic inversion, the problem of continuous estimation of an unknown deterministic disturbance in an Ito stochastic differential equation is investigated with the use of inaccurate measurements of the current phase state. An auxiliary model equation with a control approximating the unknown input is derived. The suggested solution algorithm is constructive; an estimate for its convergence rate is written explicitly.     

Parameter estimation of partially observed continuous time stochastic processes via the EM algorithm

An algorithm is presented for the problem of maximum likelihood (ML) estimation of parameters of partially observed continuous time random processes. This algorithm is an extension of the EM algorithm [3] used in the time series literature, and preserves its main features. It is then applied to the problem of parameter estimation of continuous time, finite state or infinite state (diffusions) Markov processes observed via a noisy sensor. The algorithm in general involves iterations of non-linear smoothing with known parameters and then a non-stochastic maximization. For special cases, including linear models and AR/ARMA processes observed in white noise, each iteration is easily performed with finite dimensional filters. Finally, the algorithm is applied to parameter estimation of “randomly slowly varying” linear systems observed in white noise, and explicit results are derived.     

Statistical inference for misspecified ergodic Lévy driven stochastic differential equation models

We consider the estimation problem of misspecified ergodic Lévy driven stochastic differential equation models based on high-frequency samples. We utilize a widely applicable and tractable Gaussian quasi-likelihood approach which focuses on mean and variance structure. It is shown that the Gaussian quasi-likelihood estimators of the drift and scale parameters still satisfy polynomial type probability estimates and asymptotic normality at the same rate as the correctly specified case. In their derivation process, the theory of extended Poisson equation for time-homogeneous Feller Markov processes plays an important role. Our result confirms the reliability of the Gaussian quasi-likelihood approach for SDE models.     

Adaptive Wavelet Solution to the Stokes Problem

This paper deals with the design and analysis of adaptive wavelet method for the Stokes problem. First, the limitation of Richardson iteration is explained and the multiplied matrix M0 in the paper of Bramble and Pasciak is proved to be the simplest possible in an appropiate sense. Similar to the divergence operator, an exact application of its dual is shown; Second, based on these above observations, an adaptive wavelet algorithm for the Stokes problem is designed. Error analysis and computational complexity are given; Finally, since our algorithm is mainly to deal with an elliptic and positive definite operator equation, the last section is devoted to the Galerkin solution of an elliptic and positive definite equation. It turns out that the upper bound for error estimation may be improved.     

Bayesian nonparametric estimation of first passage distributions in semi‐Markov processes

Bayesian nonparametric (BNP) models provide a flexible tool in modeling many processes. One area that has not yet utilized BNP estimation is semi‐Markov processes (SMPs). SMPs require a significant amount of computation; this, coupled with the computation requirements for BNP models, has hampered any applications of SMPs using BNP estimation. This paper presents a modeling and computational approach for BNP estimation in semi‐Markov models, which includes a simulation study and an application of asthma patients' first passage from one state of control to another.     

Iterative Algorithms for Multiscale State Estimation, Part 1: Concepts

The objective of the present investigation is to explore the potential of multiscale refinement schemes for the numerical solution of dynamic optimization problems arising in connection with chemical process systems monitoring. State estimation is accomplished by the solution of an appropriately posed least-squares problem. To offer at any instant of time an approximate solution, a hierarchy of successively refined problems is designed using a wavelet-based Galerkin discretization. In order to fully exploit at any stage the approximate solution obtained also for an efficient treatment of the arising linear algebra tasks, we employ iterative solvers. In particular, we will apply a nested iteration scheme to the hierarchy of arising equation systems and adapt the Uzawa algorithm to the present context. Moreover, we show that, using wavelets for the formulation of the problem hierarchy, the largest eigenvalues of the resulting linear systems can be controlled effectively with scaled diagonal preconditioning. Finally, we deduce appropriate stopping criteria and illustrate the characteristics of the solver with a numerical example.     

Approximation of the distribution of convergence times for stochastic global optimisation

How long should we run a stochastic global optimisation algorithm such as simulated annealing? How should we tune such an algorithm? This paper proposes an approach to the study of these questions through successive approximation of a generic stochastic global optimisation algorithm with a sequence of stochastic processes, culminating in a backtracking adaptive search process. Our emerging understanding of backtracking adaptive search can thus be used to study the original algorithm. The first approximation, the averaged range process, has the same expected number of iterations to convergence as the original process.     

Revised simplex algorithm for finite Markov decision processes

We introduce a revised simplex algorithm for solving a typical type of dynamic programming equation arising from a class of finite Markov decision processes. The algorithm also applies to several types of optimal control problems with diffusion models after discretization. It is based on the regular simplex algorithm, the duality concept in linear programming, and certain special features of the dynamic programming equation itself. Convergence is established for the new algorithm. The algorithm has favorable potential applicability when the number of actions is very large or even infinite.     

Adaptive control of three continuous-time portfolio and consumption models

An economic application of adaptive control is presented using three continuous time portfolio and consumption models that are natural generalizations of a model of Merton. In these models of the wealth of an individual investor, it is assumed that the various parameters are deterministic functions of time or stochastic processes. An adaptive control problem arises for each of these models when it is assumed that the average return rate of the risky asset, which is either a deterministic function or a stochastic process, is not observed. For these models, a recursive family of estimators of the average return rate of the risky asset is given based on the observations of the wealth. These estimates are used in the control of the wealth equation.This research was partially supported by NSF Grant No. ECS-84-03286-A01 and by University of Kansas General Research Allocation No. 3806-XO-0038.     

Multivariate sensitivity analysis for dynamic models with both random and random process inputs

Dynamic models with both random and random process inputs are frequently used in engineering. However, sensitivity analysis (SA) for such models is still a challenging problem. This paper, therefore, proposes a new multivariate SA technique to aid the safety design of these models. The new method can decompose the SA of dynamic models into a series of SA of their principle components based on singular value decomposition, which will make the SA of dynamic models much more efficient. It is shown that the effect of both random and random process inputs on the uncertainty of dynamic output can be measured from their effects on both the distributions and directions of the principle components, based on which the individual sensitivities are defined. The generalized sensitivities are then proposed to synthesize the information that is spread between the principal components to assess the influence of each input on the entire uncertainty of dynamic output. The properties of the new sensitivities are derived and an efficient estimation algorithm is proposed based on unscented transformation. Numerical results are discussed with application to a hydrokinetic turbine blade model, where the new method is compared with the existing variance-based method.