Nonlinear Moving Horizon State Estimation

MHE is an optimization based strategy for state estimation that explicitly allows for nonlinear models and inequality constraints. In this work we investigate strategies to guarantee the stability of moving horizon estimation (MHE). We begin our discussion by analyzing the stability of the abstract MHE problem. Sufficient conditions for asymptotic stability are established. Using forward dynamic programming to analyze MHE, we introduce the fundamental concept of arrival cost to bound the size of the estimation problem. The key result is if one can construct a global lower bound for the arrival cost, then asymptotic stability is guaranteed. In the second part of our discussion, we propose a strategy that circumvents the need for a global lower bound for the arrival cost. In particular, by including bounds to guarantee the existence of a decreasing nominal sequence, we circumvent the need for a global lower bound for the arrival cost. This result is significant, because, in general, we cannot calculate nor bound the arrival cost. Notable exceptions are linear systems, where the Kalman filter covariance generates a global lower bound for the arrival cost.

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

On the stability of certain systems under resonance conditions

This paper considers a general nonlinear econometric model framework that contains a large class of estimators defined as solutions to optimization problems. For this framework we derive several asymptotically equivalent forms of a test statistic for the local (in a way made precise in the paper) multivariate nonlinear inequality constraints testH:h(β) ≥ 0 versusK: β ∈RK. We extend these results to consider local hypotheses tests of the formH:h1(β) ≥ 0 andh2(β) = 0 versusK: β ∈RK. For each test we derive the asymptotic distribution for any size test as a weighted sum of χ2-distributions. We contrast local as opposed to global inequality constraints testing and give conditions on the model and constraints when each is possible. This paper also extends the well-known duality results in testing multivariate equality constraints to the case of nonlinear multivariate inequality constraints and combinations of nonlinear inequality and equality constraints.

Computation of arrival cost for moving horizon estimation via unscented Kalman filtering

In the last decade, moving horizon estimation (MHE) has emerged as a powerful technique for tackling the problem of estimating the state of a dynamic system in the presence of nonlinearities and disturbances. MHE is based on the idea of minimizing an estimation cost function defined on a sliding window composed of a finite number of time stages. The cost function is usually made up of two contributions: a prediction error computed on a recent batch of inputs and outputs; an arrival cost that serves the purpose of summarizing the past data. However, the diffusion of such techniques has been hampered by: i) the difficulty in choosing the arrival cost so as to ensure stability of the overall estimation scheme; ii) the request of an adequate computational effort on line. In this paper, both problems are addressed and possible solutions are proposed. First, by means of a novel stability analysis, it is constructively shown that under very general observability conditions a quadratic arrival cost is sufficient to ensure the stability of the estimation error provided that the weight matrix is adequately chosen. Second, a novel approximate MHE algorithm is proposed that is based on nonlinear programming sensitivity calculations. The approximate MHE algorithm has the same stability properties of the optimal one which make the overall approach suitable to be applied in real settings. Preliminary simulation results confirm the effectiveness of proposed method.

Within the Moving Horizon Estimation (MHE) formulation, the contributions of measurements outside the horizon are summarized by the arrival cost term. We propose a gradient condition for the arrival cost which ensures that the state estimate does not change if it perfectly explains new measurements. Furthermore, the updated arrival cost locally approximates the arrival cost, which would be obtained by an exact arrival cost update, if the gradient condition is satisfied. Based on the proposed gradient condition, we derive formulas for updating quadratic arrival cost terms from one MHE iteration to the next.

In this work, the asymptotic stability bounds are identified for a class of linear quasi-periodic dynamical systems with stochastic parametric excitations and nonlinear perturbations. The application of a Lyapunov–Perron (L-P) transformation converts the linear part of such systems to a linear time-invariant form. In the past, using the Infante’s approach for linear time-invariant systems, stability theorem and corollary were derived and demonstrated for time periodic systems with variation in stochastic parameters. In this study, the same approach is extended toward linear quasi-periodic with stochastic parameter variations. Furthermore, the Lyapunov’s direct approach is employed to formulate the stability conditions a for quasi-periodic system with nonlinear perturbations. If the nonlinearities satisfy a bounding condition, sufficient conditions for asymptotic stability can be derived for such systems. The applications of stability theorems are demonstrated with practical examples of commutative and noncommutative quasi-periodic systems.

Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by dX(t)=A(ξ(t))X(t)dt+H(ξ(t))X(t)dw(t), where ξ(t) is a finite‐valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a sufficient asymptotic stability condition.

This paper derives several properties unique to nonlinear model hypothesis testing problems involving linear or nonlinear inequality constraints in the null or alternative hypothesis. The paper is organized around a lemma which characterizes the set containing the least favorable parameter value for a nonlinear model inequality constraints hypothesis test. We then present two examples which illustrate several implications of this lemma. We also discuss the impact of these properties on the empirical implementation and interpretation of these test procedures.

A secondary controller which uses a number of Annular Momentum Control Devices (AMCD's) for enhancement of structural damping in large flexible space structures is investigated. The closed-loop secondary controller is shown to be equivalent to a certain closed-loop optimal linear regulator, and the necessary and sufficient conditions for asymptotic stability are obtained. Sufficient conditions for asymptotic stability are also obtained which are much less restrictive than those obtained in an earlier paper. Design of a robust stable primary controller is also discussed.

In this article, using the method of stochastic Lyapunov functions (SLF), we obtain new algebraic criteria and sufficient conditions for the asymptotic stability with probability 1 for the solutions of a system of linear stationary difference equations with random (of the type of a vector "white" sequence of random variables) coefficients, which are discrete analogs of the conditions established earlier by several authors for stochastic Ito equations with continuous time. We assume that in the absence of parametric random perturbations, an unperturbed determined system of difference equations is asymptotically Lyapunov stable (the matrix A of the system is convergent). The conditions are formulated either in terms of the negative definiteness of certain matrix relations, involving the matrix A and the matrices of the parametric perturbations Bi, i = i, 2 .... , r (sufficient conditions, which in many cases are nearly necessary), or in terms of the existence of a positive-definite solution H of the Sylvester matrix algebraic equation e

In this paper we consider the asymptotic stability of linear interval time-delay systems on the base of using Lyapunov's direct method and methods of interval analysis. A sufficient condition of asymptotic stability is obtained using the concept of Lyapunov-Krasovsky functional.

A otimização em tempo real (RTO - Real-time Optimization) é um procedimento que permite aumentar a lucratividade em operações industriais pela busca das condições ótimas de operação considerando aspectos econômicos e de processo durante a produção. Desde a década de 70, diversos modos de implementação dessa ferramenta foram propostos, sendo o método de dois passos por adaptação de parâmetros do modelo (MPA -Model Parameter Adaptation) um dos mais utilizados industrialmente. Apesar disso, a MPA utiliza informações em estado estacionário para a resolução da RTO e isso pode trazer desvantagens na aplicação e na busca do ponto ótimo em sistemas dinâmicos. Recentemente a otimização em tempo real por adaptação persistente de parâmetros (ROPA - Real-time Optimization with Persistent Adaptation) foi proposta como uma nova metodologia de RTO para contornar essas desvantagens. Utilizando uma abordagem híbrida que considera informações dos estados estacionário e transiente, este método apresentou resultados promissores, porém esses resultados foram apenas em sistemas simulados, sem a aplicação em sistemas físicos. Além disso há poucos estudos sobre as diferentes abordagens relacionadas a aplicação dessa metodologia, como o uso de diferente estimadores. Visando complementar o desenvolvimento da ROPA, um equipamento experimental que simula a operação de Gas Lift, extração por injeção de gás comum em plataformas de petróleo offshore, foi utilizado para aplicar a ROPA. O experimento teve por objetivo avaliar o comportamento da ROPA em um sistema real e comparar seus resultados em relação à MPA e em relação a si próprio com o uso de dois estimadores diferentes, o filtro de Kalman estendido (EKF - Extended Kalman Filter), único estimador utilizado na ROPA até o momento, e o estimador de horizonte móvel (MHE - Moving Horizon Estimation). Para isto foi feita a modelagem do experimento, já que a ROPA e a MPA são baseadas no uso de modelos, e avaliada a adequação do modelo ao equipamento experimental. A MPA e a ROPA (utilizando o EKF como estimador de parâmetros) foram implementadas, utilizando o modelo desenvolvido, e os resultados da otimização comparados. Na comparação a ROPA teve um lucro de 1% maior em relação à MPA e teve boa performance na otimização durante períodos transientes. Em seguida seria feito a comparação do EKF e do MHE na ROPA, porém foi feito um estudo anterior para verificar a estimação do MHE no modelo proposto. Nesse estudo avaliou-se a influência de uma parte do equacionamento do MHE chamado de custo de chegada (arrival cost) e verificou-se que esse termo é importante para obter uma estimação adequada dos parâmetros do modelo. Após o estudo foi realizada a comparação da ROPA com o EKF e com o MHE e a ROPA-EKF teve o lucro 1% maior em relação à ROPA-MHE mas ambos tiveram performances semelhantes na estimação. Em conclusão, a ROPA foi bem sucedida quando aplicada em um equipamento físico, a ROPA teve resultados melhores quando comparado à MPA, principalmente em momentos não estacionários, e a ROPA mostrou-se flexível no uso de outros estimadores.

We propose a novel method to compute the arrival cost for the moving horizon estimator. The choice of the arrival cost is an important challenge and is known to have significant influence on the performance of the estimator. Most common approaches are based on implementing a complementary extended Kalman filter to propagate an approximate measure of the uncertainty. Our approach is based on the probabilistic interpretation of the moving horizon estimator and its analogy to the maximum a posteriori estimator. We derive a method to directly obtain the required uncertainties from the Hessian of the moving horizon estimation objective function. We showcase our novel approach with the challenging visual-inertial sensor fusion problem that commonly arises in visual navigation systems. The estimation performance is significantly better compared to our previous results based on the extended Kalman filter. Additionally, the proposed algorithm calibrates the inertial sensor online and is immediately ready for operation.