There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Convex Analysisの二,三の進展について

We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.

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Flatness and defect of non-linear systems: introductory theory and examples

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

We are concerned in this article with the control of wheeled mobile robots, which constitute a class of nonholonomic mechanical systems. More precisely, we are interested in solving the problem of tracking with stability of a reference trajectory, by means of linearizing ''static'' and ''dynamic'' state feedback laws. We give conditions to avoid possible singularities of the feedback laws.

Control of nonholonomic wheeled mobile robots by state feedback linearization

We present a strict Lyapunov function for hyperbolic systems of conservation laws that can be diagonalized with Riemann invariants. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. It is shown that the derived boundary control allows to guarantee the local convergence of the state towards a desired set point. Furthermore, the control can be implemented as a feedback of the state only measured at the boundaries. The control design method is illustrated with an hydraulic application, namely the level and flow regulation in an horizontal open channel

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A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws

http://www.gipsa-lab.fr/~christophe.prieur/Papers/automatica03.pdf

Boundary feedback control in networks of open channels

We give a new sufficient condition on the boundary conditions for the exponential stability of one-dimensional nonlinear hyperbolic systems on a bounded interval. Our proof relies on the construction of an explicit strict Lyapunov function. We compare our sufficient condition with other known sufficient conditions for nonlinear and linear one-dimensional hyperbolic systems.

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Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems

The issue of the accuracy of areal rainfall estimation is discussed through a case study on a catchment area in the Ckvennes region of France. The basic tool for the analysis is a ''scaled estimation error variance'' which is computed from a scaled climatological variogram model of the rainfall field. We show how this variance provides a theoretical criterion to compare the accuracy that can be expected with three linear estimators (Thiessen, spline. and kriging) for various networks' densities. To support the methodology an experimental validation of the scald estimation error variance computation is per. formed, using "reference" areal rainfall values computed with a very high density network. 5Good estimates of mean areal rainfall are needed as inputs to hydrologic models. When based on ground measurements, their accuracy depends on the spatial variability of the rainfall process and on the rain gage networkdensity. Accuracy is of particular importance when areal rainfall estimates must be computed in real time as inputs to runoff forecasting models. The high cost of equipping and maintaining the required telemetering facilities, especially in mountainous regions, often results in low-density observation networks. Making the most of a given network (whether telemetered or not) is a prime concern shared by all operators and hydrologic designers. Consequently, the interpolation of data collected from scattered rain gages has long been an important research topic in hydrology. Creutin and Obled [1982] have compared the performances of commonly used linear estimators. Their study shows that for low to medium density networks sophisticated methods (e.g., spline surface fitting or kriging) give better results than simpler conventional methods (e.g., Thiessen polygons or arithmetic means). Furthermore, various advantages and disadvantages are outlined for each method, providing a basis for evaluating their applicability to specific hydrologic problems. More recently, a similar comparison was applied to annual rainfalls by Tabios and Salas [1985], yielding results consistent with those of Creutin and Obled. However, it is often more useful (but also more difficult) for a hydrologist to evaluate the error involved in areal rather than in point rainfall estimation. The theoretical error variance of any linear areal rainfall estimator can obviously be computed under the assumption that the observations are a realization of a random field with a given covariance model. However, a major difficulty arises in validating the calculated theoretical variance from experimental data, since no direct

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Georges Bastin

Papers

Control of nonholonomic wheeled mobile robots by state feedback linearization

A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws

Boundary feedback control in networks of open channels

Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems

On the Accuracy of Areal Rainfall Estimation - a Case-study