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In this paper we propose a novel formulation of the predictor used in open-loop recursive identification algorithms. The predicted output is expressed by means of an orthogonal Laguerre transfer functions basis. This predictor representation presents many advantages: It makes it possible to identify robustly oversampled systems without any bias in low frequency, and to obtain relevant reduced order models. The Laguerre pole plays the role of a tuning parameter enabling the selection of the best approximation frequency area. The proposed schemes address both output error and ARMAX systems. Simulation and experimental results show all the practical benefits provided by these algorithms.

It is a well-known fact that modern linear algebra is the fruit of the labor of countless different authors. The notion of vector , characterized by n components with respect to a given basis, was proposed by A. Cayley (1850), while during the same period H. Grassmann and W. Hamilton developed a more “geometric” perspective, avoiding the use of coordinates (see, for example, Grassmann’s relation).

Category theory was introduced by S. Eilenberg and S. MacLane in an article published in 1945, which also axiomatized the notions of functor and natural transformation . MacLane wrote that the notion of category was defined in order to define the notion of functor, which was in turn defined in order to define the notion of natural transformation. The archetypal example of a natural (or canonical ) isomorphism is the one that identifies finite-dimensional vector spaces with their biduals. The notion of structure gradually began to emerge at the end of the 19th Century, and was fully formalized in Elements de mathematique by N. Bourbaki (vector space structures, topological spaces, etc.). A vector space, for example, is a set equipped with a vector space structure. This “structuralist” perspective adopted by Bourbaki is based on set theory. In the category theory, however, the objects of a category are not always sets, and consequently the morphisms are not always mappings. A functor F F is a “large function” from a category C C to a category D D that sends each object A in C to an object F(A) F A in D D and each morphism f : A → B in C C to a morphism ℱ(f):ℱ(A)→ℱ(B) ℱ f : ℱ A → ℱ B in D D .

The elementary algebraic structures gradually emerged throughout the 19th Century. The first such structures were groups, which were already implicit in the work performed by C.F. Gauss, J.-L. Lagrange and A.-L. Cauchy but only truly initiated by E. Galois, and fields, which were formalized by the successors of Gauss. Lattices and fields were introduced by R. Dedekind (1871), and commutative rings were introduced by D. Hilbert (1897). Division rings, the first example of which was given by W. Hamilton (1843), were not systematically studied until E. Steinitz (1910). The work performed on these topics was summarized in the classical book by B.L. van der Waerden following the lectures given by E. Noether and E. Artin between 1926 and 1928. This book was regularly updated until 1967 over the course of seven editions. Many mathematicians (including O. Ore, N. Jacobson, I. Kaplansky and P.M. Cohn) later contributed to the development of the theory of non-commutative rings. The theory of determinants gradually began to emerge during the second half of the 18th Century, with G. Cramer, E. Bezout and P.-S. Laplace, well before the concept of matrices was introduced. Methods for calculating these determinants were systematically developed around 1815 by Cauchy, with further progress around 1850 by J. Sylvester and A. Cayley. However, a definitive formalization was not found until L. Kronecker and K. Weierstrass in 1870–1880, even though the notion of matrix was introduced in 1850 by J. Sylvester and the notion of the rank of a matrix was proposed by F. Frobenius shortly thereafter.

International audience; Independent roots of a polynomial Independent solutions of differential equations Galois/Picard-Vessiot extensions a b s t r a c t In this paper, poles and zeros are defined for linear time-varying systems using suitable ground field extensions. The definitions of the system poles, transmission poles, invariant zeros, hidden modes, etc, are given in an intrinsic module-based framework and are consistent in the sense that the poles are connected to the stability of the system and the zeros to the zeroing of the output for non zero inputs. In particular, it is proved that the necessary and sufficient condition for a continuous-time system to be exponentially stable is similar to the well-known condition in the time-invariant case.

Robust feedback linearization of nonlinear systems without full state information is studied in this paper. The state is reconstructed from the output and the input by a linear observer combined with a diffeomorphism. It is shown that a separation principle holds, i.e., the estimated state can be used for performing robust feedback linearization, which exactly transforms the nonlinear system into a linear system equal to its linear approximation around a nominal operating point. Local stability of the resulting closed loop is proved by theoretical arguments and its robustness is illustrated through an example.

Two complete and intrinsic definitions of linear systems are existing: (i) in the behavioral approach , a linear system is the kernel B of a matrix-valued operator R in a power of a signal space W; (ii) in the module theoretic setting , a linear system is the cokernel M of R (thus, M is a finitely presented module over a ring of operators). These two formulations have connections and under certain conditions the knowledge of M is equivalent to that of B. The minimal conditions under which such an equivalence exists are investigated in this paper and the case of convolution systems is especially considered.

In this paper it is given a definition of poles of linear discrete-and continuous-time varying systems in terms of differential modules over non commutative rings. Necessary and sufficient conditions for stability are deduced in the particular case of systems with no multiple poles.

International audience; We consider discrete behaviors with varying coefficients. Our results are new also for one-dimensional systems over the time-axis of natural numbers and for varying coefficients in a field, we derive the results, however, in much greater generality: Instead of the natural numbers we use an arbitrary submonoid N of an abelian group, for instance the standard multidimensional lattice of r-dimensional vectors of natural numbers or integers. We replace the base field by any commuta-tive self-injective ring F, for instance a direct product of fields or a quasi-Frobenius ring or a finite factor ring of the integers. The F-module W of functions from N to F is the canonical discrete signal module and is a left module over the natural associated noncommutative ring A of difference operators with variable coefficients. Our main result states that this module is injective and therefore satisfies the fundamental principle: An inhomogeneous system of linear difference equations with variable coefficients has a solution if and only if the right side satisfies the canoni-cal compatibility conditions. We also show that for the typical cases of partial difference equations and in contrast to the case of constant coefficients the A-module W is not a cogenerator. We also generalize the standard one-dimensional theory for periodic coefficients to the multidimensional situation by invoking Morita equivalence. AMS-classification: 93C05, 93C55, 16D50, 39A06, 39A14

International audience; MSC: 93D20 93C15 93B25 34D05 34D20 Keywords: Exponential stability Differential behavior Time-varying Duality Serre category We develop a new approach to exponential stability of linear time-varying (LTV) differential behaviors that is analogous to that in our paper on exponential stability of discrete LTV behaviors (H. Bourlès et al., 2015 [5]). Stability theory for differential state space systems with smooth coefficients is an important subject in the literature. For differential LTV behaviors with arbitrary smooth coefficients there is no reasonable stability theory. Therefore we restrict the smooth varying coefficients to functions that are defined by means of locally convergent Puiseux series. All rational functions are of this type. We introduce a new kind of behaviors and prove a module-behavior duality for these. We define a new notion of weak exponential stability (w.e.s.) of a behavior B and its associated finitely generated (f.g.) module M and show that the w.e.s. modules and behaviors are closed under isomor-phisms, subobjects, factor objects and extensions. The standard uniform exponential stability of state space equations is not preserved under behavior isomorphisms and unsuitable for a behavioral theory. In the main result we assume a nonzero f.g. torsion module M and its associated autonomous behavior B. Such a module may be regular or irregular singular according to the Galois theory of differential equations. If it is nonzero and regular singular it is never w.e.s. For irregular singular M we characterize w.e.s. of most B algebraically and constructively.

In this note, a robust nonlinear controller for a nonlinear system subject to model uncertainties is proposed. Such a controller associates a "robust feedback linearization" with a robust linear H/sub /spl infin// controller. The robust feedback linearization exactly transforms the nonlinear system into a linear system equal to the linear approximation of the original nonlinear system around a nominal operating point. The robustness of the resulting overall nonlinear controller is proved by theoretical arguments and illustrated through an application example, the control of a magnetic bearing system.

This paper presents a robust controller designed to address both local and interarea oscillations. It is an extension of the "desensitized four-loop regulator" (DFLR) that has been developed by Electricite/spl acute/ de France. The suggested controller is a second-order state-space regulator which will be called the "extended DFLR" (EDFLR). As the DFLR, the EDFLR can be converted to an AVR+PSS structure. The performance of the EDFLR has been extensively tested using modal analysis and time-domain simulations of a large-scale model of the European power system.

This paper presents a physical interpretation of two state feedback controllers for damping power system electromechanical oscillations. They have been developed by Electricite/spl acute/ de France (EDF). The first one is called the desensitized four loop regulator (DFLR) and it is designed to damp local electromechanical oscillations. It is a robust controller which offers good performance despite the variations of the generator operating conditions. The second controller is called the extended desensitized four loop regulator (EDFLR) and it is designed to address both local and interarea oscillations. The physical interpretation is accomplished converting the state feedback scheme to the standard structure formed by an automatic voltage regulator (AVR) plus a power system stabilizer (PSS). Two widely used PSS design methods based on eigenvalue sensitivities and frequency response are reviewed to obtain the interpretation. The DFLR can be interpreted as a controller which provides the suitable phase compensation according to these two PSS design methods over a wider frequency range. The EDFLR can be interpreted as a controller which maximizes its robustness under uncertainties at both PSS output and the input of the plant.

The exact model-matching problem is formulated and solved for linear time-varying systems. The condition for the existence of a proper solution, which is well known in the time-invariant case, is proven here to still be valid in the time-varying case. The properness is characterized using the Smith-MacMillan form at infinity, recently defined by the authors for the transfer matrices with time-varying coefficients.

A robust coordinated AVR+PSS, called the "desensitized Four Loop regulator", has been designed for the French power system and is currently being implemented. It is shown here that it is possible to transform the structure of this controller into any standard IEEE AVR+PSS structure. The AVR+PSSs obtained in this manner are efficient for damping all kinds of electromechanical oscillations of a large scale power system (including inter-area oscillations). This approach is combined with modal analysis techniques. Its efficiency has been verified on several large scale systems and is illustrated here on a modified version of the New England system. The obtained AVR+PSSs have a good robustness against significant changes in the characteristics of the network.

The notions of poles and zeros at infinity and their relations are extended to the case of linear continuous time-varying systems. This study is based on the notion of a "newborn systems which is, in a mathematical point of view, a graded module extension over the noncommutative ring of differential operators. It is proved to be a relevant generalization to the time-varying case of the equivalence class, for the so-called "restricted equivalence" of Rosenbrock's polynomial matrix descriptions. The authors' approach is intrinsic and unifies the definitions previously given in the literature in the time-invariant case.

Reachability, observability, controllability and reconstructibibty of linear Periodic Discrete-Time (LPDT) systems are defined here in the context of module theory. We describe after, the two possible structures of the generalized state space representation according to the LPDT system to be regular or descrptor. The no-equivalence between canonical and minimal realization for LPDT (and consequently for discrete time-varying) systems is clearly proved. Periodic structural invariants of LPDT system (such as poles, zeros…) and their link with invariants factors (when they exist) of periodic matrices is sketched in the appendix.

A robust coordinated AVR/PSS, called the "desensitized four loops regulator" (DFLR) has been recently designed using the "desensitized control" theory. Its synthesis method (which can be applied to problems other than excitation control) and its structure are detailed in this paper. It is a state feedback controller including an integrator on the regulation error. This structure, although well known in the control community, is not usual for the excitation control of generators (except in France). Hence, in order to make this approach more widely applicable in an industrial point of view, this controller is placed, using approximations, in the standard AVR+PSS structure IEEE ST1A+PSS1A. The resulting AVR+PSS is proven to have good performance, in particular for damping inter-area oscillations.

The structural properties of linear periodic discrete-time systems are analyzed in the periodic polynomial representation. It is shown that the classical polynomial approach for linear time-invariant systems can be extended to periodic systems. New definitions and properties are given in terms of skew polynomial rings and periodic difference algebra. Necessary and sufficient conditions for the characterization of reachability, controllability, observability and reconstructibility are given in this framework.

A robust continuous governor control for small-signal and transient stability is proposed. The turbine under consideration is a steam turbine with two sections and is of the reheat type. The controller can be viewed as including the speed droop governor and the fast-valving; the latter is not standard, in that it generates a continuous control. Hence, the disadvantages of the (usual) bang-bang fast-valving are avoided. The control signal is generated by two controllers, with the first one for small disturbances and the second one for large disturbances. Our approach is innovative in that both controllers have the same structure and only differ in the tuning of their parameters. They generate a continuous control and are switched with a bumpless transition to ensure the continuity of the overall control. The most important nonlinearity of the turbine is taken into account, and a feedback linearization is made-but in a somewhat unusual manner to obtain a robust controller. A linear quadratic control is also used to ensure that the steady-state objectives are met and that good transients are obtained. Simulations with the EUROSTAG software show improved efficiency with the proposed controller compared with the current one, especially in the case of a short-circuit, an islanding or a temporary loss of synchronism of the machine considered; the current controller has a standard structure, including a bang-bang fast-valving.

Several kinds of finite poles and zeros are defined and their relationships are studied in an intrinsic manner. Some of these definitions are more precise than the usual ones (all kinds of multiplicities of an input-output decoupling zero are defined here). In addition, we believe that our approach clarifies those notions, and it makes it possible to very concisely establish fundamental relationships between various poles and zeros. Some new relationships are derived. The notions of input-output stability and transfer stability are also introduced. Several examples illustrate the theory.

The interconnection of linear systems may exhibit disturbing phenomena, like hidden modes or a lack of control variables. These features, which cannot be taken into account by the transfer function approach, are here explained by utilizing the language of modules, which is strongly related to Willems' behavorial approach. Several examples are carefully discussed.

Resume Une nouvelle sorte de stabilite entree-sortie est definie sur la base d'un espace de Sobolev. Des versions locales du theoreme des petits gains et du theoreme de passivite sont etablies dans ce formalisme.

Resume Les poles et les zeros sont definis, et les relations qu'ils verifient sont etudiees, dans une "approche invariante". De plus, une nouvelle relation est etablie.

In this paper, a robust linear voltage controller is proposed to improve power system stability and performance under varying operating conditions. This controller includes an automatic voltage regulator and power system stabilizer. It limits the variations of the terminal voltage, damps electromechanical oscillations and increases the excitation during and after a short circuit in all operating points. The design method chosen to improve the parametric robustness is desensitivity. It is the first time that a parametric robust design method has been applied to resolve the problem of voltage regulation. The method's principles, when the parameter is a vector, are presented. The desensitized controller has a simple structure and therefore can easily be implemented. The results obtained with simulation software and with the Electricite de France Transient Network Analyser show the good performance and the advantages of the controller with respect to various disturbances. This method can be applied to ail kinds of generators, and is particularly well suited in the case of turbogenerators working under very different operating conditions. >

A linear system has been defined as a module, or roughly speaking, as a set of variables which satisfy a set of differential and algebraic equations. This definition has a strong connection with the construction of the model of a dynamic systems in practice. As a matter of fact, one of the most common way of building models for dynamic systems is the so-called analytical modeling which consists of writing the equations of the basic principles of physics in accordance to the nature of the system (electrical energy conservation laws, Lagrange formalism for mechanical systems, thermodynamics heat exchange laws, etc.). Examples for this way of doing are the models presented in Chapter 8.

Many engineering problems need a time-varying modeling and some examples are shown in this section. The most common class of LTV systems is the one of periodic systems. They are encountered in signal processing and communications as, for example, filters which incorporate modulators in the signal path (see, e.g., [252] in which spectral characterizations of linear periodically time-varying systems are used to the study of gated phased-locked loops), as well as in control.

We recall here some known facts about the stability of LTV systems given by their state representation $$ \dot{x}(t)=A(t)x(t)+B(t)u(t) $$

The model-matching problem consists in assigning the transfer matrix of a system. Many control problems reduce to model-matching and several approaches have been developed. The model-matching can be exact when a given transfer matrix is assigned or approximate when the design is such that the transfer matrix of the controlled system gets a specified structure (as is the case for decoupling control) or gets as close as possible to a reference transfer matrix. In the latter case, the distance between the two transfers is expressed in terms of a chosen norm of the difference between the two transfer matrices like, e.g., the ||.|| ∞ norm for the H ∞ control. The exact model-matching (EMM) consists in precisely assigning all entries of the transfer matrix to given transfer functions. It can be done by feedforward (pre or postcompensation) or by feedback (static or dynamic).

Here we present the general mathematical notions necessary for the sequel. We begin with categories. For the applications to control theory, abelian categories of modules (1.2.4.3) are the most important. Therefore, in a first reading, beginners may limit themselves to that case. Functors (Subsect. 1.2.3) will be used throughout the book.

The topic of this chapter is the structure at infinity of discrete and continuous linear time-varying systems in a unified approach. When interconnecting two subsystems, ”impulsive motions” may arise. In the continuous-time case, those impulsive motions are linear combinations of the Dirac distribution δ and its derivatives; they were first studied by Verghese [346]. In the discrete-time case, those impulsive behaviors are backward solutions with finite support; they were revealed by Lewis [216]. The space spanned by all impulsive motions of a system is called its ”impulsive behavior” and is denoted as \( \mathfrak{B}_{\infty }\). The structure of this impulsive behavior must be studied, and, for the integrity of the system resulting from the interconnection, all impulsive motions must be avoided.

Let K be a field, and let \(f\left( X\right) \in \mathbf{K}\left[ X \right] \) be a polynomial with coefficients in K. In general, the equation \(f\left( x\right) =0\) has solutions in an extension of K, not in K itself. For example, if K = ℚ , the equation x2 + 2 = 0 has no solution in ℚ , it has in ℂ its full set of solutions \(\left\{ i\sqrt{2},-i\sqrt{2}\right\} \), and these solutions are already in the extension \( \mathbb{Q} \left( i,\sqrt{2}\right) \subsetneq \mathbb{C} \) of ℚ . The study of algebraic field extensions is the Galois theory, recalled in Section 4.2.

Section 2.2 contains present basic facts on rings and modules. Bimodules are also introduced, for they are very useful to study the properties of modules of homomorphisms and of tensor products.

The pole placement consists of tuning the natural response of a given system. The input-output representation is convenient for engineers and thus widely used in industrial applications. For this reason, the output feedback problem is of particular interest. It is addressed here using the transfer matrix formalism given in Section 5.5.2 and the stability characterization by the notions of poles introduced in Chapter 6. We consider continuous-time systems (i.e., \(\partial\) stands exclusively for d/dt) which are supposed controllable and observable. These results allow the closed-loop control of LTV systems but also to drive nonlinear systems around a given trajectory.

Historically, a linear system was first considered as a ”black box” excited by inputs, generating outputs, and characterized by a transfer matrix [22]. In 1960, Kalman founded the theory of state-space systems [171], inaugurating the era of ”modern control theory”. This approach is however limited by several facts.

Stability of an LTV system can be evaluated from the stability of its autonomous part. This is shown in Chapter 12 where an analytic approach for stability of the LTV systems is given. However, stability can be studied using the poles of the system. This is the direction followed in the present section.

International audience

In this paper we revisit the synthesis of the Power System Stabilizers (PSSs) following some recent objectives and constraints imposed by the evolution of the European interconnected power system. Some principles of approaches used in an independent and even concurrent way are combined and enriched to provide a coordinated way to adjust the parameters of standard (IEEE-type) PSS loops of several machines in a large-scale interconnected system in order to simultaneously reach damping objectives for several oscillatory modes of different natures, i.e., both local and inter-area ones. The trade-off performance/robustness is the central point of the approach since, on the one hand, as the European synchronous zone is being continuously expanded, approximative models should be used in order to keep the size of the problem noncritical from the computational point of view. On the other hand, the increasing size of the system leads to the slipping to lower values of the frequency of the inter-area modes. They are thus further from the local modes which stay at higher frequencies and this makes the coordination of the damping actions for both classes more difficult. Full nonlinear validation simulations are run on a realistic large-scale model of the interconnected European power system.