Quasi-optimal assimilation of satellite retrievals for incremental 4D-Var

An interface between satellite retrievals and the incremental version of the four-dimensional variational assimilation scheme is developed, making full use of the information content of satellite measurements. In this paper, expressions for the function that calculates simulated observations from model states (called “observation operator”), together with its tangent linear version and adjoint, are derived. Results from our work can be used for implementing a quasi-optimal assimilation of satellite retrievals (e.g., of atmospheric trace gases) in operational meteorological centres.

Relationship between åström control and the kalman linear regulator — Caines revisited

The relationship between minimum variance and minimum expected quadratic loss feedback controllers for linear univariate discrete-time stochastic systems is reviewed by taking the approach used by Caines. It is shown how the two methods can be regarded as providing identical control actions as long as a noise-free measurement state-space model is employed.

Neurofuzzy control using Kalman filtering state feedback with coloured noise for unknown non-linear processes

This paper presents a controller design scheme for a priori unknown non-linear dynamical processes that are identified via an operating point neurofuzzy system from process data. Based on a neurofuzzy design and model construction algorithm (NeuDec) for a non-linear dynamical process, a neurofuzzy state-space model of controllable form is initially constructed. The control scheme based on closed-loop pole assignment is then utilized to ensure the time invariance and linearization of the state equations so that the system stability can be guaranteed under some mild assumptions, even in the presence of modelling error. The proposed approach requires a known state vector for the application of pole assignment state feedback. For this purpose, a generalized Kalman filtering algorithm with coloured noise is developed on the basis of the neurofuzzy state-space model to obtain an optimal state vector estimation. The derived controller is applied in typical output tracking problems by minimizing the tracking error. Simulation examples are included to demonstrate the operation and effectiveness of the new approach.

Dynamic integrated system optimization and parameter estimation for discrete time optimal control of nonlinear systems

An algorithm for solving nonlinear discrete time optimal control problems with model-reality differences is presented. The technique uses Dynamic Integrated System Optimization and Parameter Estimation (DISOPE), which achieves the correct optimal solution in spite of deficiencies in the mathematical model employed in the optimization procedure. A version of the algorithm with a linear-quadratic model-based problem, implemented in the C+ + programming language, is developed and applied to illustrative simulation examples. An analysis of the optimality and convergence properties of the algorithm is also presented.

A new approach for the solution of the optimal set-point tracking problem for nonlinear systems

In this paper, a discrete time dynamic integrated system optimisation and parameter estimation algorithm is applied to the solution of the nonlinear tracking optimal control problem. A version of the algorithm with a linear-quadratic model-based problem is developed and implemented in software. The algorithm implemented is tested with simulation examples.

Optimal control of nonlinear differential algebraic equation systems

A novel iterative procedure is described for solving nonlinear optimal control problems subject to differential algebraic equations. The procedure iterates on an integrated modified linear quadratic model based problem with parameter updating in such a manner that the correct solution of the original non-linear problem is achieved. The resulting algorithm has a particular advantage in that the solution is achieved without the need to solve the differential algebraic equations . Convergence aspects are discussed and a simulation example is described which illustrates the performance of the technique. 1. Introduction When modelling industrial processes often the resulting equations consist of coupled differential and algebraic equations (DAEs). In many situations these equations are nonlinear and cannot readily be directly reduced to ordinary differential equations.

Semi-dwarfing (Rht-B1b) improves nitrogen-use efficiency in wheat, but not at economically optimal levels of nitrogen availability

A UK field experiment compared a complete factorial combination of three backgrounds (cvs Mercia, Maris Huntsman and Maris Widgeon), three alleles at the Rht-B1 locus as Near Isogenic Lines (NILs: rht-B1a (tall), Rht-B1b (semi-dwarf), Rht-B1c (severe dwarf)) and four nitrogen (N) fertilizer application rates (0, 100, 200 and 350 kg N/ha). Linear+exponential functions were fitted to grain yield (GY) and nitrogen-use efficiency (NUE; GY/available N) responses to N rate. Averaged over N rate and background Rht-B1b conferred significantly (P<0.05) greater GY, NUE, N uptake efficiency (NUpE; N in above ground crop / available N) and N utilization efficiency (NUtEg; GY / N in above ground crop) compared with rht-B1a and Rht-B1c. However the economically optimal N rate (Nopt) for N:grain price ratios of 3.5:1 to 10:1 were also greater for Rht-B1b, and because NUE, NUpE and NUtE all declined with N rate, Rht-Blb failed to increase NUE or its components at Nopt. The adoption of semi-dwarf lines in temperate and humid regions, and the greater N rates that such adoption justifies economically, greatly increases land-use efficiency, but not necessarily, NUE.

A tutorial on pseudospectral methods for computational optimal control

[English] This paper is a tutorial introduction to pseudospectral optimal control. With pseudospectral methods, a function is approximated as a linear combination of smooth basis functions, which are often chosen to be Legendre or Chebyshev polynomials. Collocation of the differential-algebraic equations is performed at orthogonal collocation points, which are selected to yield interpolation of high accuracy. Pseudospectral methods directly discretize the original optimal control problem to recast it into a nonlinear programming format. A numerical optimizer is then employed to find approximate local optimal solutions. The paper also briefly describes the functionality and implementation of PSOPT, an open source software package written in C++ that employs pseudospectral discretization methods to solve multi-phase optimal control problems. The software implements the Legendre and Chebyshev pseudospectral methods, and it has useful features such as automatic differentiation, sparsity detection, and automatic scaling. The use of pseudospectral methods is illustrated in two problems taken from the literature on computational optimal control. [Portuguese] Este artigo e um tutorial introdutorio sobre controle otimo pseudo-espectral. Em metodos pseudo-espectrais, uma funcao e aproximada como uma combinacao linear de funcoes de base suaves, tipicamente escolhidas como polinomios de Legendre ou Chebyshev. A colocacao de equacoes algebrico-diferenciais e realizada em pontos de colocacao ortogonal, que sao selecionados de modo a minimizar o erro de interpolacao. Metodos pseudoespectrais discretizam o problema de controle otimo original de modo a converte-lo em um problema de programa cao nao-linear. Um otimizador numerico e entao empregado para obter solucoes localmente otimas. Este artigo tambem descreve sucintamente a funcionalidade e a implementacao de um pacote computacional de codigo aberto escrito em C++ chamado PSOPT. Tal pacote emprega metodos de discretizacao pseudo-spectrais para resolver problemas de controle otimo com multiplas fase. O PSOPT permite a utilizacao de metodos de Legendre ou Chebyshev, e possui caractersticas uteis tais como diferenciacao automatica, deteccao de esparsidade e escalonamento automatico. O uso de metodos pseudo-espectrais e ilustrado em dois problemas retirados da literatura de controle otimo computacional.

Robust pole assignment in linear state feedback

Numerical methods are described for determining robust, or well-conditioned, solutions to the problem of pole assignment by state feedback. The solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized. It is shown that for these solutions, upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized. A measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed-loop poles, and hence the suitability of the given poles for assignment.

Optimal growth strategies when mortality and production rates are size-dependent

Pontryagin's maximum principle from optimal control theory is used to find the optimal allocation of energy between growth and reproduction when lifespan may be finite and the trade-off between growth and reproduction is linear. Analyses of the optimal allocation problem to date have generally yielded bang-bang solutions, i.e. determinate growth: life-histories in which growth is followed by reproduction, with no intermediate phase of simultaneous reproduction and growth. Here we show that an intermediate strategy (indeterminate growth) can be selected for if the rates of production and mortality either both increase or both decrease with increasing body size, this arises as a singular solution to the problem. Our conclusion is that indeterminate growth is optimal in more cases than was previously realized. The relevance of our results to natural situations is discussed.

Robust pole assignment and optimal stability margins

A robust pole assignment by linear state feedback is achieved in state-space representation by selecting a feedback which minimises the conditioning of the assigned eigenvalues of the closed-loop system. It is shown here that when this conditioning is minimised, a lower bound on the stability margin in the frequency domain is maximised.

An Earth Observation Land Data Assimilation System (EO-LDAS)

Current methods for estimating vegetation parameters are generally sub-optimal in the way they exploit information and do not generally consider uncertainties. We look forward to a future where operational dataassimilation schemes improve estimates by tracking land surface processes and exploiting multiple types of observations. Dataassimilation schemes seek to combine observations and models in a statistically optimal way taking into account uncertainty in both, but have not yet been much exploited in this area. The EO-LDAS scheme and prototype, developed under ESA funding, is designed to exploit the anticipated wealth of data that will be available under GMES missions, such as the Sentinel family of satellites, to provide improved mapping of land surface biophysical parameters. This paper describes the EO-LDAS implementation, and explores some of its core functionality. EO-LDAS is a weak constraint variational dataassimilationsystem. The prototype provides a mechanism for constraint based on a prior estimate of the state vector, a linear dynamic model, and EarthObservationdata (top-of-canopy reflectance here). The observation operator is a non-linear optical radiative transfer model for a vegetation canopy with a soil lower boundary, operating over the range 400 to 2500 nm. Adjoint codes for all model and operator components are provided in the prototype by automatic differentiation of the computer codes. In this paper, EO-LDAS is applied to the problem of daily estimation of six of the parameters controlling the radiative transfer operator over the course of a year (> 2000 state vector elements). Zero and first order process model constraints are implemented and explored as the dynamic model. The assimilation estimates all state vector elements simultaneously. This is performed in the context of a typical Sentinel-2 MSI operating scenario, using synthetic MSI observations simulated with the observation operator, with uncertainties typical of those achieved by optical sensors supposed for the data. The experiments consider a baseline state vector estimation case where dynamic constraints are applied, and assess the impact of dynamic constraints on the a posteriori uncertainties. The results demonstrate that reductions in uncertainty by a factor of up to two might be obtained by applying the sorts of dynamic constraints used here. The hyperparameter (dynamic model uncertainty) required to control the assimilation are estimated by a cross-validation exercise. The result of the assimilation is seen to be robust to missing observations with quite large data gaps.

Sea surface temperature from a geostationary satellite by optimal estimation

Optimal estimation (OE) is applied as a technique for retrieving sea surface temperature (SST) from thermal imagery obtained by the Spinning Enhanced Visible and Infra-Red Imager (SEVIRI) on Meteosat 9. OE requires simulation of observations as part of the retrieval process, and this is done here using numerical weather prediction fields and a fast radiative transfer model. Bias correction of the simulated brightness temperatures (BTs) is found to be a necessary step before retrieval, and is achieved by filtered averaging of simulations minus observations over a time period of 20 days and spatial scale of 2.5° in latitude and longitude. Throughout this study, BT observations are clear-sky averages over cells of size 0.5° in latitude and longitude. Results for the OE SST are compared to results using a traditional non-linear retrieval algorithm (“NLSST”), both validated against a set of 30108 night-time matches with drifting buoy observations. For the OE SST the mean difference with respect to drifter SSTs is − 0.01 K and the standard deviation is 0.47 K, compared to − 0.38 K and 0.70 K respectively for the NLSST algorithm. Perhaps more importantly, systematic biases in NLSST with respect to geographical location, atmospheric water vapour and satellite zenith angle are greatly reduced for the OE SST. However, the OE SST is calculated to have a lower sensitivity of retrieved SST to true SST variations than the NLSST. This feature would be a disadvantage for observing SST fronts and diurnal variability, and raises questions as to how best to exploit OE techniques at SEVIRI's full spatial resolution.

Optimal estimation of sea surface temperature from split-window observations

Optimal estimation (OE) improves sea surface temperature (SST) estimated from satellite infrared imagery in the “split-window”, in comparison to SST retrieved using the usual multi-channel (MCSST) or non-linear (NLSST) estimators. This is demonstrated using three months of observations of the Advanced Very High Resolution Radiometer (AVHRR) on the first Meteorological Operational satellite (Metop-A), matched in time and space to drifter SSTs collected on the global telecommunications system. There are 32,175 matches. The prior for the OE is forecast atmospheric fields from the Météo-France global numerical weather prediction system (ARPEGE), the forward model is RTTOV8.7, and a reduced state vector comprising SST and total column water vapour (TCWV) is used. Operational NLSST coefficients give mean and standard deviation (SD) of the difference between satellite and drifter SSTs of 0.00 and 0.72 K. The “best possible” NLSST and MCSST coefficients, empirically regressed on the data themselves, give zero mean difference and SDs of 0.66 K and 0.73 K respectively. Significant contributions to the global SD arise from regional systematic errors (biases) of several tenths of kelvin in the NLSST. With no bias corrections to either prior fields or forward model, the SSTs retrieved by OE minus drifter SSTs have mean and SD of − 0.16 and 0.49 K respectively. The reduction in SD below the “best possible” regression results shows that OE deals with structural limitations of the NLSST and MCSST algorithms. Using simple empirical bias corrections to improve the OE, retrieved minus drifter SSTs are obtained with mean and SD of − 0.06 and 0.44 K respectively. Regional biases are greatly reduced, such that the absolute bias is less than 0.1 K in 61% of 10°-latitude by 30°-longitude cells. OE also allows a statistic of the agreement between modelled and measured brightness temperatures to be calculated. We show that this measure is more efficient than the current system of confidence levels at identifying reliable retrievals, and that the best 75% of satellite SSTs by this measure have negligible bias and retrieval error of order 0.25 K.

Exact Search Directions for Optimization of Linear and Nonlinear Models Based on Generalized Orthonormal Functions

A novel technique for selecting the poles of orthonormal basis functions (OBF) in Volterra models of any order is presented. It is well-known that the usual large number of parameters required to describe the Volterra kernels can be significantly reduced by representing each kernel using an appropriate basis of orthonormal functions. Such a representation results in the so-called OBF Volterra model, which has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a nonlinear static mapping given by the Volterra polynomial series. Aiming at optimizing the poles that fully parameterize the orthonormal bases, the exact gradients of the outputs of the orthonormal filters with respect to their poles are computed analytically by using a back-propagation-through-time technique. The expressions relative to the Kautz basis and to generalized orthonormal bases of functions (GOBF) are addressed; the ones related to the Laguerre basis follow straightforwardly as a particular case. The main innovation here is that the dynamic nature of the OBF filters is fully considered in the gradient computations. These gradients provide exact search directions for optimizing the poles of a given orthonormal basis. Such search directions can, in turn, be used as part of an optimization procedure to locate the minimum of a cost-function that takes into account the error of estimation of the system output. The Levenberg-Marquardt algorithm is adopted here as the optimization procedure. Unlike previous related work, the proposed approach relies solely on input-output data measured from the system to be modeled, i.e., no information about the Volterra kernels is required. Examples are presented to illustrate the application of this approach to the modeling of dynamic systems, including a real magnetic levitation system with nonlinear oscillatory behavior.