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International audience; This article presents a discrete-time identification method able to characterize multiscale systems. It belongs to the pseudo-linear regression class, and uses a model parameterization established on generalized bases of orthonormal transfer functions. An analysis of the estimated parameters accuracy is provided, and in order to tend towards statistical efficiency, an iterative procedure of the basis poles selection is proposed. Simulation examples show the interest of the approach.

International audience; This paper proposes an improved version of closed-loop output-error identification algorithms, where the predictor is established on a generalized basis of orthonormal transfer functions. It is shown that the selection of the basis poles impacts the convergence conditions and the bias distribution of the schemes.These algorithms present several advantages: They are able to identify in closed-loop fast sampled systems, stiff systems (with modes spread over three decades or more), and reduced order models. Moreover, they are suitable for unstable systems or controllers. A simulation example shows the effectiveness of this approach. These algorithms can be employed in an open-loop context by using a straightforward simplification.

The notion of a differential manifold proposed by Riemann in his inaugural lecture in 1854 is a generalization of curves in the plane, or curves or surfaces in everyday space; these curves and surfaces (extensively studied by Gauss, who supervised Riemann’s thesis), and more generally manifolds, are assumed to be regular. For a curve, being regular means having a tangent at every point; for a surface, it means having a tangent plane at every point. Locally, each curve “looks like” its tangent and each surface “looks like” its tangent plane. Manifolds can be studied locally using charts in the same way that we study regions of our planet geographically. It was Gauss who suggested charts to study curves and surfaces locally. We cannot represent the whole Earth with a single chart unless we are willing to accept significant deformations of some of its regions, for example the poles in the classical Mercator projection. Hence, we require more than one chart; these charts are assembled into an atlas.

Tensor calculus was the culmination of pioneering work by B. Christoffel and G. Ricci, in 1869 and 1887–1896, respectively. It reached maturity in a joint publication by Ricci-Curbastro and Levi-Civita in 1900. It is difficult to overstate its importance as a field – general relativity could not exist without it. Tensor calculus plays an essential role in every area of physics; it is also crucial for continuum mechanics and many other engineering sciences, and it, of course, lies at the heart of differential geometry. Section 4.2 of this chapter is purely algebraic (with some algebraic topology in section 4.2.6). The mathematical objects that physicists call tensors are more precisely tensor fields; they are only introduced in section 4.3. Historically, “tensors” were presented to students as exotic mathematical millipedes of the form tj1, …, jpi1, …, ip (or more briefly) that behave in a certain way under change of coordinates (see [4.3, 4.4]). This stemmed from the decision to only define the components of the tensor (see [4.2]). As a result, students could complete an entire course of tensor calculus without finding a satisfactory answer to the question: “But what actually is a tensor?” We will, of course, choose a different approach by defining tensor fields before attempting to study them.

In 1998, A. Karimi and I.D. Landau published in the journal "Systems and Control letters" an article entitled "Comparison of the closed-loop identification methods in terms of bias distribution". One of its main purposes was to provide a bias distribution analysis in the frequency domain of closed-loop output error identification algorithms that had been recently developed. The expressions provided in that paper are only valid for prediction error identification methods (PEM), not for pseudo-linear regression (PLR) ones, for which we give the correct frequency domain bias analysis, both in open- and closed-loop. Although PLR was initially (and is still) considered as an approximation of PEM, we show that it gives better results at high frequencies.

The most essential groundwork for the notion of “abstract measure” and the theory of integration was laid by E. Borel and H. Lebesgue in the early 20th century. Abstract measures are founded upon the notion of measurable sets (sets equipped with a σ -algebra

(I) The problem of “solving algebraic equations by radicals” (where by algebraic equations we mean those of the form f ( x ) = 0, for f a non-zero polynomial with rational coefficients) was one of the most important topics in mathematics from the earliest periods of antiquity until the 19th century. The Babylonians, and later the Greeks, already knew how to solve quadratic equations, although their formulas were more complex than those we use today, since their notation was inferior and they lacked the concepts of zero and negative numbers. Cubic equations were solved by S. del Ferro (around 1515) and N. Tartaglia (whose contributions were published in 1545 by G. Cardano and are often incorrectly attributed to the latter); quartic equations were solved by L. Ferrari using a method that was also published in Cardano’s Ars Magna together with the method proposed by Tartaglia. In 1576, R. Bombelli compiled a summary of all of this work using simpler notation in Algebra . However, all efforts to solve quintic equations were unsuccessful from the end of the 16th century until the early 19th century. J.-L. Lagrange was arguably the first to recognize the underlying reasons in his memoirs from 1770–1771. Finally, P. Ruffini attempted to show that the general quintic equation cannot be solved by radicals in a series of dense and controversial memoirs gradually published between 1799 and 1813. One proof that was entirely correct but weighed down by long calculations was provided by N. Abel in 1824; in 1826, he showed.

In the 1930s and 1940s, after the publication of the book by S. Banach, the mathematical community might have been forgiven for thinking that functional analysis would henceforth essentially be set in stone, given the natural framework provided by normed vector spaces (and their completions: Banach spaces). But the holomorphic functions on an open set in the complex plane are just one example of a topogical vector space that is non-normable despite having the structure of a Frechet-Montel space. The general theory of topological vector spaces was outlined by A. Kolmogorov and J. von Neumann in 1935, then completed in 1945-1946 by the fundamental contributions of G. Mackey on locally convex spaces. The significance of these generalizations became clear with the advent of distribution theory, developed by L. Schwartz between 1945 and 1950 following precursory work by S. Sobolev and S. Bochner; this new theory propelled a previously unheard of type of topological vector space into the spotlight. Dieudonne and Schwartz showed in an article published in 1949 that these spaces are in fact strict inductive limits of Frechet spaces. The groundwork for this article had been laid by a summary published in 1942 by Dieudonne on duality in locally convex spaces, as well as the contributions by Mackey mentioned above. The article by Dieudonne-Schwartz was completed in the early 1950s by A. Grothendieck: generalization of the open mapping theorem and the closed graph theorem, notions of a Schwartz space and of a nuclear space, and general theory of tensor products of topological vector spaces (which we sadly do not have the space to discuss here beyond a few basic ideas). Other generalizations of the open mapping theorem and the closed graph theorem can also be found in the literature, notably the ones derived by V. Ptak. For an overview of these results, readers can refer to (Volume II).

International audience; We investigate one-dimensional 'generalized convolution behaviors' (gen. beh.) that comprise differential and delay-differential behaviors in particular. We thus continue work of, for instance, Breth'e, van Eijndhoven, Fliess, Gluesing-Luerssen, Habets, Loiseau, Mounier, Rocha, Vettori, Willems, Yamamoto, Zampieri of the last twenty-five years. The signal space for these behaviors is the space E of smooth complex-valued functions on the real line. The ring of operators is the commuta-tive integral domain E' of distributions with compact support with its convolution product that acts on E by a variant of the convolution product and makes it an E'-module. Both E and E' carry their standard topologies. Closed E'-submodules of finite powers of E were introduced and studied by Schwartz already in 1947 under the name 'invariant varieties' and are called gen. beh. here. A gen. beh. is called a behavior if it can be described by finitely many convolution equations. The ring E' is not noetherian and therefore the standard algebraic arguments from one-dimensional differential systems theory have to be completed by methods of topological algebra. Standard constructions like elimination or taking (closed) images of behaviors may lead to gen. beh. and therefore the consideration of the latter is mandatory. It is not known whether all gen. beh. are indeed behaviors, but we show that many of them are, in particular all autonomous ones. The E'-module E is neither injective nor a cogenerator and, in particular, does not admit elimination in Willems' sense. But the signal submodule PE of all polynomial-exponential signals is injective for finitely generated modules and thus admits elimination. This is a useful replacement and approximation of the injectivity of E since the polynomial-exponential part of any gen. beh. is dense in it. We also describe a useful replacement of the cogenerator property and thus establish a strong relation between convolution equations and their solution spaces. Input/output structures of gen. beh. exist and are used to prove that also many nonautonomous generalized behaviors are indeed behaviors. The E’-torsion elements of E, i.e., the smooth functions which satisfy at least one nonzero convolution equation, are called ’mean-periodic functions’ and were studied by many outstanding analysts. Their results are significant for gen. beh..