Data assimilation for marine monitoring and prediction: The MERCATOR operational assimilation systems and the MERSEA developments

During the past 15 years, a number of initiatives have been undertaken at national level to develop ocean forecasting systems operating at regional and/or global scales. The co-ordination between these efforts has been organized internationally through the Global Ocean Data Assimilation Experiment (GODAE). The French MERCATOR project is one of the leading participants in GODAE. The MERCATOR systems routinely assimilate a variety of observations such as multi-satellite altimeter data, sea-surface temperature and in situ temperature and salinity profiles, focusing on high-resolution scales of the ocean dynamics. The assimilation strategy in MERCATOR is based on a hierarchy of methods of increasing sophistication including optimal interpolation, Kalman filtering and variational methods, which are progressively deployed through the Syst`eme d’Assimilation MERCATOR (SAM) series. SAM-1 is based on a reduced-order optimal interpolation which can be operated using ‘altimetry-only’ or ‘multi-data’ set-ups; it relies on the concept of separability, assuming that the correlations can be separated into a product of horizontal and vertical contributions. The second release, SAM-2, is being developed to include new features from the singular evolutive extended Kalman (SEEK) filter, such as three-dimensional, multivariate error modes and adaptivity schemes. The third one, SAM-3, considers variational methods such as the incremental four-dimensional variational algorithm. Most operational forecasting systems evaluated during GODAE are based on least-squares statistical estimation assuming Gaussian errors. In the framework of the EU MERSEA (Marine EnviRonment and Security for the European Area) project, research is being conducted to prepare the next-generation operational ocean monitoring and forecasting systems. The research effort will explore nonlinear assimilation formulations to overcome limitations of the current systems. This paper provides an overview of the developments conducted in MERSEA with the SEEK filter, the Ensemble Kalman filter and the sequential importance re-sampling filter.

Robustness in partial pole placement

The robustness of state feedback solutions to the problem of partial pole placement obtained by a new projection procedure is examined. The projection procedure gives a reduced-order pole assignment problem. It is shown that the sensitivities of the assigned poles in the complete closed-loop system are bounded in terms of the sensitivities of the assigned reduced-order poles, and the sensitivities of the unaltered poles are bounded in terms of the sensitivities of the corresponding open-loop poles. If the assigned poles are well-separated from the unaltered poles, these bounds are expected to be tight. The projection procedure is described in [3], and techniques for finding robust (or insensitive) solutions to the reduced-order problem are given in [1], [2].

Balance model for equatorial long waves

Geophysical fluid models often support both fast and slow motions. As the dynamics are often dominated by the slow motions, it is desirable to filter out the fast motions by constructing balance models. An example is the quasi geostrophic (QG) model, which is used widely in meteorology and oceanography for theoretical studies, in addition to practical applications such as model initialization and data assimilation. Although the QG model works quite well in the mid-latitudes, its usefulness diminishes as one approaches the equator. Thus far, attempts to derive similar balance models for the tropics have not been entirely successful as the models generally filter out Kelvin waves, which contribute significantly to tropical low-frequency variability. There is much theoretical interest in the dynamics of planetary-scale Kelvin waves, especially for atmospheric and oceanic data assimilation where observations are generally only of the mass field and thus do not constrain the wind field without some kind of diagnostic balance relation. As a result, estimates of Kelvin wave amplitudes can be poor. Our goal is to find a balance model that includes Kelvin waves for planetary-scale motions. Using asymptotic methods, we derive a balance model for the weakly nonlinear equatorial shallow-water equations. Specifically we adopt the ‘slaving’ method proposed by Warn et al. (Q. J. R. Meteorol. Soc., vol. 121, 1995, pp. 723–739), which avoids secular terms in the expansion and thus can in principle be carried out to any order. Different from previous approaches, our expansion is based on a long-wave scaling and the slow dynamics is described using the height field instead of potential vorticity. The leading-order model is equivalent to the truncated long-wave model considered previously (e.g. Heckley & Gill, Q. J. R. Meteorol. Soc., vol. 110, 1984, pp. 203–217), which retains Kelvin waves in addition to equatorial Rossby waves. Our method allows for the derivation of higher-order models which significantly improve the representation of Rossby waves in the isotropic limit. In addition, the ‘slaving’ method is applicable even when the weakly nonlinear assumption is relaxed, and the resulting nonlinear model encompasses the weakly nonlinear model. We also demonstrate that the method can be applied to more realistic stratified models, such as the Boussinesq model.

An observer design procedure for a class of nonlinear time-delay systems

This paper considers a class of uncertain, nonlinear differential state delayed control systems and presents a reduced-order observer design procedure to asymptotically estimate any vector state functionals. The method proposed involves decomposition of the delayed portion of the system into two parts: a matched and mismatched part. Provided that the rank of the mismatched part is less than the number of the outputs, a reduced-order linear functional observer, with any prescribed stability margin, can be constructed by using a simple procedure. A numerical example is given to illustrate the new design procedure and its features.

Partial-state observers for nonlinear systems

This note deals with the design of reduced-order observers for a class of nonlinear systems. The order reduction of the observer is achieved by only estimating a required partial set of the state vector. Necessary and sufficient conditions are derived for the existence of reduced-order observers. An observer design procedure based on linear matrix inequalities is given. A numerical example is given to illustrate the design method.

State and input simultaneous estimation for a class of time-delay systems with uncertainties

This brief addresses the problem of estimation of both the states and the unknown inputs of a class of systems that are subject to a time-varying delay in their state variables, to an unknown input, and also to an additive uncertain, nonlinear disturbance. Conditions are derived for the solvability of the design matrices of a reduced-order observer for state and input estimation, and for the stability of its dynamics. To improve computational efficiency, a delay-dependent asymptotic stability condition is then developed using the linear matrix inequality formulation. A design procedure is proposed and illustrated by a numerical example.

On the existence and design of functional observers for linear systems

Darouach [1] recently derived necessary and sufficient conditions for the existence and stability of functional observers with an order, p, equals to the dimension m of the vectors to be estimated. In general, these conditions are difficult to satisfy and when they are not, the only available option is to increase the order of the functional observers. This note presents new conditions for the existence of a general pth-order functional observer. Systematic procedures for the synthesis of reduced-order functional observers are given. A numerical example is given to illustrate the design procedures.

Design of linear functional observers for time-delay systems of the neutral-type

This paper presents the design of reduced-order linear functional observers for a class of linear time-delay systems of the neutral-type. The type of the observer proposed in this paper is without internal delay and its order is the same as the number of linear functions to be estimated. First, conditions for the existence of the reduced-order functional observers that are capable of asymptotically estimating any given function of the state vector are derived. Then, based on the newly derived existence conditions, a procedure is given for the determination of the observer parameters. The results derived in this paper include a range of linear systems and extend some existing results of linear functional observers to linear neutral delay systems. A numerical example is given to illustrate the design procedure.

Disturbance decoupled observers for systems with unknown inputs

This note deals with the design of reduced-order disturbance decoupled scalar functional observers for linear systems with unknown inputs. Based on a parametric approach, existence conditions are derived and a design procedure for finding reduced-order scalar functional observers is given. The derived existence conditions are relaxed and the procedure can find first-order disturbance decoupled scalar functional observers for some cases where the number of unknown inputs is more than the number of outputs. Also, the observer matching condition, which is the necessary requirement for the design of state observers for linear systems with unknown inputs, is not required. Numerical examples are given to illustrate the attractiveness of the proposed design method.

Partial state and unknown input estimation for time-delay systems

This article considers the problem of estimating a partial set of the state vector and/or unknown input vector of linear systems driven by unknown inputs and time-varying delay in the state variables. Three types of reduced-order observers, namely, observers with delays, observers without internal delays and delay-free observers are proposed in this article. Existence conditions and design procedures are presented for the determination of parameters for each case of observers. Numerical examples are presented to illustrate the design procedures.

Design of distributed functional observers for interconnected time-delay systems

This paper considers the design of reduced-order distributed functional observers for interconnected linear systems with the presence of time delays in the interconnections. Unlike observers which consider only the ideal non-delayed output information transfer, the proposed observer is capable of dealing with delayed output information from geographically separated subsystems. It is shown that by accepting measurement data from other subsystems, the conditions under which an observer exists can be made less conservative. Existence conditions, systematic and a straightforward procedure for the synthesis of the observers are given along with numerical examples illustrating the effectiveness and simplicity of the design algorithm.

Incremental criterion prediction of personality facets over factors: obtaining unbiased estimates and confidence intervals

 Many researchers have argued that higher order models of personality such as the Five Factor Model are insufficient, and that facet-level analysis is required to better understand criteria such as well-being, job performance, and personality disorders. However, common methods in the extant literature used to estimate the incremental prediction of facets over factors have several shortcomings. This paper delineates these shortcomings by evaluating alternative methods using statistical theory, simulation, and an empirical example. We recommend using differences between Olkin-Pratt adjusted r-squared for factor versus facet regression models to estimate the incremental prediction of facets and present a method for obtaining confidence intervals for such estimates using double adjusted-. r-squared bootstrapping. We also provide an R package that implements the proposed methods.

Global finite-time stabilisation for a class of stochastic nonlinear systems by output feedback

In this paper, the problem of global finite-time stabilisation by output feedback is considered for a class of stochastic nonlinear systems. First, based on homogeneous systems theory and the adding a power integrator technique, a homogeneous reduced order observer and control law are constructed in a recursive manner for the nominal system. Then, the homogeneous domination approach is used to deal with the nonlinearities in drift and diffusion terms; it is shown that the proposed output-feedback control law can guarantee that the closed-loop system is global finite-time stable in probability. Finally, simulation examples are carried out to demonstrate the effectiveness of the proposed control scheme.

A functional observer based fault detection technique for dynamical systems

This paper presents a functional observer based fault detection method. The fault detection is achieved using a functional observer based fault indicator that asymptotically converges to a fault indicator that can be derived based on the nominal system. The asymptotic value of the proposed fault indicator is independent of the functional observer parameters and also the convergence rate of the fault indicator can be altered by choosing appropriate functional observer parameters. The advantage of using this new method is that the observed system is not necessarily needed to be observable; therefore, the proposed fault detection technique is also applicable for systems where state observers cannot be designed; moreover, the functional observer fault detection scheme is always of reduced order in comparison to a state observer based scheme.

Design of state observers for interconnected time-delay systems via a coordinate transformation approach

This paper considers the design of state observers for interconnected time-delay systems using a coordinate transformation method. Through such a transformation, the system that has interconnection and state delays is metamorphosed into a new system that injects time-delay information into its input and output terms, before reintroducing them back into the latter system, effectively coupling the delay terms into the IO injection terms and eliminating the delay values from the state variables. Next, full-order and reduced-order observers are designed based on the transformed system. Finally, the observed states of the transformed system that correspond to the original system is used to deduce the estimates of the original system. A numerical example is provided of an interconnected time-delay system.