A comparison of two methods for developing the linearization of a shallow-water model

We develop the linearization of a semi-implicit semi-Lagrangian model of the one-dimensional shallow-water equations using two different methods. The usual tangent linear model, formed by linearizing the discrete nonlinear model, is compared with a model formed by first linearizing the continuous nonlinear equations and then discretizing. Both models are shown to perform equally well for finite perturbations. However, the asymptotic behaviour of the two models differs as the perturbation size is reduced. This leads to difficulties in showing that the models are correctly coded using the standard tests. To overcome this difficulty we propose a new method for testing linear models, which we demonstrate both theoretically and numerically. © Crown copyright, 2003. Royal Meteorological Society

Predictive control using feedback linearization based on dynamic neural models

This paper presents a hybrid control strategy integrating dynamic neural networks and feedback linearization into a predictive control scheme. Feedback linearization is an important nonlinear control technique which transforms a nonlinear system into a linear system using nonlinear transformations and a model of the plant. In this work, empirical models based on dynamic neural networks have been employed. Dynamic neural networks are mathematical structures described by differential equations, which can be trained to approximate general nonlinear systems. A case study based on a mixing process is presented.

Input constraints handling in an MPC/feedback linearization scheme

The combination of model predictive control based on linear models (MPC) with feedback linearization (FL) has attracted interest for a number of years, giving rise to MPC+FL control schemes. An important advantage of such schemes is that feedback linearizable plants can be controlled with a linear predictive controller with a fixed model. Handling input constraints within such schemes is difficult since simple bound contraints on the input become state dependent because of the nonlinear transformation introduced by feedback linearization. This paper introduces a technique for handling input constraints within a real time MPC/FL scheme, where the plant model employed is a class of dynamic neural networks. The technique is based on a simple affine transformation of the feasible area. A simulated case study is presented to illustrate the use and benefits of the technique.

Input/output linearization using dynamic recurrent neural networks

This paper brings together two areas of research that have received considerable attention during the last years, namely feedback linearization and neural networks. A proposition that guarantees the Input/Output (I/O) linearization of nonlinear control affine systems with Dynamic Recurrent Neural Networks (DRNNs) is formulated and proved. The proposition and the linearization procedure are illustrated with the simulation of a single link manipulator.

Dynamic input/output linearization using recurrent neural networks

A dynamic recurrent neural network (DRNN) is used to input/output linearize a control affine system in the globally linearizing control (GLC) structure. The network is trained as a part of a closed loop that involves a PI controller, the goal is to use the network, as a dynamic feedback, to cancel the nonlinear terms of the plant. The stability of the configuration is guarantee if the network and the plant are asymptotically stable and the linearizing input is bounded.

Stable linearization using multilayer neural networks

The main limitation of linearization theory that prevents its application in practical problems is the need for an exact knowledge of the plant. This requirement is eliminated and it is shown that a multilayer network can synthesise the state feedback coefficients that linearize a nonlinear control affine plant. The stability of the linearizing closed loop can be guaranteed if the autonomous plant is asymptotically stable and the state feedback is bounded.

Real-time application of a constrained predictive controller based on dynamic neural networks with feedback linearization

This paper describes an experimental application of constrained predictive control and feedback linearisation based on dynamic neural networks. It also verifies experimentally a method for handling input constraints, which are transformed by the feedback linearisation mappings. A performance comparison with a PID controller is also provided. The experimental system consists of a laboratory based single link manipulator arm, which is controlled in real time using MATLAB/SIMULINK together with data acquisition equipment.

Optimal linearization trajectories for tangent linear models

We examine differential equations where nonlinearity is a result of the advection part of the total derivative or the use of quadratic algebraic constraints between state variables (such as the ideal gas law). We show that these types of nonlinearity can be accounted for in the tangent linear model by a suitable choice of the linearization trajectory. Using this optimal linearization trajectory, we show that the tangent linear model can be used to reproduce the exact nonlinear error growth of perturbations for more than 200 days in a quasi-geostrophic model and more than (the equivalent of) 150 days in the Lorenz 96 model. We introduce an iterative method, purely based on tangent linear integrations, that converges to this optimal linearization trajectory. The main conclusion from this article is that this iterative method can be used to account for nonlinearity in estimation problems without using the nonlinear model. We demonstrate this by performing forecast sensitivity experiments in the Lorenz 96 model and show that we are able to estimate analysis increments that improve the two-day forecast using only four backward integrations with the tangent linear model. Copyright © 2011 Royal Meteorological Society

Perturbation, extraction and refinement of invariant pairs for matrix polynomials

Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a first-order perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures.

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.

Gaussian quadrature 4D-VAR

A new incremental four-dimensional variational (4D-Var) data assimilation algorithm is introduced. The algorithm does not require the computationally expensive integrations with the nonlinear model in the outer loops. Nonlinearity is accounted for by modifying the linearization trajectory of the observation operator based on integrations with the tangent linear (TL) model. This allows us to update the linearization trajectory of the observation operator in the inner loops at negligible computational cost. As a result the distinction between inner and outer loops is no longer necessary. The key idea on which the proposed 4D-Var method is based is that by using Gaussian quadrature it is possible to get an exact correspondence between the nonlinear time evolution of perturbations and the time evolution in the TL model. It is shown that J-point Gaussian quadrature can be used to derive the exact adjoint-based observation impact equations and furthermore that it is straightforward to account for the effect of multiple outer loops in these equations if the proposed 4D-Var method is used. The method is illustrated using a three-level quasi-geostrophic model and the Lorenz (1996) model.

On the generalized eigenvalue problem for the Rossby wave vertical velocity in the presence of mean flow and topography

In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ulti- mately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a non- linear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should first solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter.