A fractional representation approach to the robust regulation problem for SISO systems

The purpose of this article is to develop a new approach to the robust regulation problem for plants which do not necessarily admit coprime factorizations. The approach is purely algebraic and allows us dealing with a very general class of systems in a unique simple framework. We formulate the famous internal model principle in a form suitable for plants defined by fractional representations which are not necessarily coprime factorizations. By using the internal model principle, we are able to give necessary and sufficient solvability conditions for the robust regulation problem and to parameterize all robustly regulating controllers.Comment: 13 pages, 1 figure, to appear in Systems & Control Letter

About Dynamical Systems Appearing in the Microscopic Traffic Modeling

Motivated by microscopic traffic modeling, we analyze dynamical systems which have a piecewise linear concave dynamics not necessarily monotonic. We introduce a deterministic Petri net extension where edges may have negative weights. The dynamics of these Petri nets are well-defined and may be described by a generalized matrix with a submatrix in the standard algebra with possibly negative entries, and another submatrix in the minplus algebra. When the dynamics is additively homogeneous, a generalized additive eigenvalue may be introduced, and the ergodic theory may be used to define a growth rate under additional technical assumptions. In the traffic example of two roads with one junction, we compute explicitly the eigenvalue and we show, by numerical simulations, that these two quantities (the additive eigenvalue and the growth rate) are not equal, but are close to each other. With this result, we are able to extend the well-studied notion of fundamental traffic diagram (the average flow as a function of the car density on a road) to the case of two roads with one junction and give a very simple analytic approximation of this diagram where four phases appear with clear traffic interpretations. Simulations show that the fundamental diagram shape obtained is also valid for systems with many junctions. To simulate these systems, we have to compute their dynamics, which are not quite simple. For building them in a modular way, we introduce generalized parallel, series and feedback compositions of piecewise linear concave dynamics.Comment: PDF 38 page

Duality and separation theorems in idempotent semimodules

We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert's projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and half-spaces over the max-plus semiring.Comment: 24 pages, 5 Postscript figures, revised (v2

Purity filtration of multidimensional linear systems

International audienceIn this paper, we show how the purity filtration of a finitely presented module, associated with a multidimensional linear system, can be explicitly characterized by means of classical concepts of module theory and homological algebra. Our approach avoids the use of sophisticated homological algebra methods such as spectral sequences used in [3], [4], [5], associated cohomology used in [9], and Spencer cohomology used in [12], [13]. It allows us to develop efficient implementations in the PURITYFILTRATION and AbelianSystems packages. The purity filtration gives an intrinsic classification of the torsion elements of the module by means of their grades, and thus a classification of the autonomous elements of the multidimensional linear system by means of their codimensions. The results developed here are used in [16] to determine an equivalent block-triangular linear system of the multidimensional linear system formed by equidimensional diagonal blocks. This equivalent linear system highly simplifies the computation of a Monge parametrization of the original linear system

Grade filtration of linear functional systems

The grade filtration of a finitely generated left module M over an Auslander regular ring D is a built-in classification of the elements of M in terms of their grades (or their (co)dimensions if D is also a Cohen-Macaulay ring). In this paper, we show how grade filtration can be explicitly characterized by means of elementary methods of homological algebra. Our approach avoids the use of sophisticated methods such as bidualizing complexes, spectral sequences, associated cohomology, and Spencer cohomology used in the literature of algebraic analysis. Efficient implementations dedicated to the computation of grade filtration can then be easily developed in the standard computer algebra systems (see the Maple package PurityFiltration and the GAP4 package AbelianSystems). Moreover, this characterization of grade filtration is shown to induce a new presentation of the left D-module M which is defined by a block-triangular matrix formed by equidimensional diagonal blocks. The linear functional system associated with the left D-module M can then be integrated in cascade by successively solving inhomogeneous linear functional systems defined by equidimensional homogeneous linear systems of increasing dimension. This equivalent linear system generally simplifies the computation of closed-form solutions of the original linear system. In particular, many classes of underdetermined/overdetermined linear systems of partial differential equations can be explicitly integrated by the packages PurityFiltration and AbelianSystems, but not by computer algebra systems such as Maple.La filtration par grade d'un module à gauche M finiment engendré sur un anneau Auslander-régulier D est une classification intrinsèque des éléments de M en fonction de leurs grades (ou de leurs (co)dimensions si D est aussi un anneau de Cohen-Macaulay). Dans ce papier, nous montrons comment la filtration par grade peut être explicitement caractérisée au moyen de techniques élémentaires d'algèbre homologique. Notre approche évite l'utilisation de techniques sophistiquées telles que les complexes bidualisants, les suites spectrales, la cohomologie associée et la cohomologie de Spencer utilisées dans la littérature d'analyse algébrique. Des implantations efficaces dédiées au calcul de la filtration par grade peuvent alors être facilement développées dans les systèmes standards de calcul formel (voir le package PurityFiltration de Maple et le package AbelianSystems de GAP4). De plus, cette caractérisation de la filtration par grade induit une nouvelle présentation du D-module à gauche M qui est définie par une matrice triangulaire par blocs formée de blocs diagonaux équidimensionnels. Le système linéaire fonctionnel associé au D-module à gauche M peut alors être intégré en cascade par la résolution successive de systèmes linéaires fonctionnels inhomogènes définis par des systèmes linéaires homogènes équidimensionnels de dimension croissante. Ce système linéaire équivalent simplifie généralement le calcul des solutions sous formes closes du système linéaire originel. En particulier, de nombreux systèmes linéaires sur-déterminés/sous-déterminés d'équations aux dérivées partielles peuvent être explicitement intégrés au moyen des packages PurityFiltration et AbelianSystems, alors qu'ils ne peuvent l'être par des systèmes de calcul formel tels que Maple

Serre's reduction of linear partial differential systems with holonomic adjoints

Given a linear functional system (e.g., ordinary/partial differential systems, differential time-delay systems, difference systems), Serre's reduction aims at finding an equivalent linear functional system which contains fewer equations and fewer unknowns. The purpose of this paper is to study Serre's reduction of underdetermined linear systems of partial differential equations with either polynomial, formal power series or analytic coefficients and with holonomic adjoints in the sense of algebraic analysis. We prove that these linear partial differential systems can be defined by means of a single linear partial differential equation. In the case of polynomial coefficients, we give an algorithm to compute the corresponding equation.Etant donné un système fonctionnel linéaire (e.g., système d'équations différentielles ordinaires, système d'équations aux dérivées partielles, système d'équations différentielles à retard, système d'équations aux différences), la réduction de Serre a pour but de trouver un système fonctionnel linéaire équivalent contenant moins d'équations et d'inconnues. L'objectif de ce papier est l'étude de la réduction de Serre des systèmes linéaires sous-déterminés d'équations aux dérivées partielles à coefficients polynomiaux, séries formelles ou séries localement convergentes, dont les adjoints sont holonomes au sens de l'analyse algébrique. Nous prouvons que de tels systèmes peuvent être définis par une seule équation aux dérivées partielles. Dans le cas des coefficients polynomiaux, nous donnons un algorithme permettant de calculer l'équation correspondante

Serre's reduction of linear systems of partial differential equations with holonomic adjoints

Given a linear functional system (e.g., ordinary/partial di erential system, di erential time-delay system, di erence system), Serre's reduction aims at nding an equivalent linear functional system which contains fewer equations and fewer unknowns. The purpose of this paper is to study Serre's reduction of underdetermined linear systems of partial di erential equations with either polynomial, formal power series or analytic coe cients and with holonomic adjoints in the sense of algebraic analysis. We prove that these linear partial di erential systems can be de ned by means of a single linear partial di erential equation. In the case of polynomial coe cients, we give an algorithm to compute the corresponding equation

A prospective study on the role of immunotherapy in metastatic cancer patients at Combined Military Hospital Dhaka

Background: Immunotherapy is a treatment that uses a person’s immune system to fight cancer. Immunotherapy can boost or change how the immune system works so it can find and attack cancer cells. Among several cancers, metastatic cancer causes high mortality, and immunotherapies are expected to be effective in the prevention and treatment of metastatic cancer patients. In Bangladesh, we do have not enough research-based information regarding the role of immunotherapy in treating metastatic cancer patients. This study aimed to assess the role of immunotherapy in treating metastatic cancer patients. Methods: This prospective observational study was conducted in combined military hospital Dhaka, Bangladesh during the period from 26 March 2021 to 21 July 2022. In total 19 patients with metastatic cancer were enrolled in this study as study subjects. Proper written consent was taken from all the participants before data collection. Two (02) different outcomes were studied in this study; progression-free survival (PFS) and side effect percentages. A predesigned questionnaire was used in data collection. All data were processed, analyzed, and disseminated by using MS excel and SPSS version 23 program as per necessity. Results: In using pembrolizumab, side effects, fatigue, nausea, and decreased appetite were found 43%, 22%, and 20% lesser respectively than chemotherapy which was noticeable. In using nivolumab, as a side effect, skin rash was found 66% lesser than chemotherapy. Besides this, itching, face swelling, and apnea was found 33% lesser. On the other hand, in using atezolizumab, as side effects, swelling of arms and constipation were found 66% lesser, and itching, as well as apnea, was found 33% lesser than that in chemotherapy. At the 6-month follow-up we observed that in the nivolumab and atezolizumab treated groups 66% of cases survived separately whereas, in pembrolizumab treated group, 61% survived. Conclusions: In this study in all treatment groups, side effects were found as lesser than that in using chemotherapy. No major complication of any patients was observed in this study. So, can conclude that immune checkpoint inhibitors (ICIs) are better choice for metastatic diseases and ICIs exert lesser side effects than conventional chemotherapies

Symbolic methods for developing new domain decomposition algorithms

The purpose of this work is to show how algebraic and symbolic techniques such as Smith normal forms and Gröbner basis techniques can be used to develop new Schwarz-like algorithms and preconditioners for linear systems of partial differential equationsL'objet de ce travail est de monter comment les techniques algébriques et symboliques telles que les formes normales de Smith et les techniques de bases de Gröbner peuvent être utilisées pour développer de nouveaux algorithmes de type Schwarz et des préconditionneurs pour les systèmes linéaires d'équations aux dérivées partielles