Obstructions to Genericity in Study of Parametric Problems in Control Theory

We investigate systems of equations, involving parameters from the point of view of both control theory and computer algebra. The equations might involve linear operators such as partial (q-)differentiation, (q-)shift, (q-)difference as well as more complicated ones, which act trivially on the parameters. Such a system can be identified algebraically with a certain left module over a non-commutative algebra, where the operators commute with the parameters. We develop, implement and use in practice the algorithm for revealing all the expressions in parameters, for which e.g. homological properties of a system differ from the generic properties. We use Groebner bases and Groebner basics in rings of solvable type as main tools. In particular, we demonstrate an optimized algorithm for computing the left inverse of a matrix over a ring of solvable type. We illustrate the article with interesting examples. In particular, we provide a complete solution to the "two pendula, mounted on a cart" problem from the classical book of Polderman and Willems, including the case, where the friction at the joints is essential . To the best of our knowledge, the latter example has not been solved before in a complete way.Comment: 20 page

Exact linear modeling using Ore algebras

Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas the operators with varying coefficients were addressed only recently. This question can be tackled either using Gr\"obner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods to compute "most powerful unfalsified models" (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples

Floor 3d advertizing in Russia

Today the possibilities of technical progress in advertizing are extending so that, commercials are involved in the production of dinosaurs, aliens and many other beings. All this was presented to us by 3D-animation effects and, of course, special programs, thanks to which these effects are created. Advertising is the most visible form of marketing. It is one of the most effective marketing tools you can use to build a share of the prospect’s mind. If you know exactly what you want from your advertising, where to direct your message, and how to express what you want your audience to know, your advertising will be effective. The purpose of the article consists is to show the distribution and the use of the floor 3D advertising in Russia and to show you how it can be used with spectacular results in your advertising campaigns.Возможности технического прогресса в рекламе сегодня расширились настолько, что при производстве роликов могут быть задействованы динозавры, инопланетяне и многие другие чуждые реальной действительности существа и объекты. Все это нам подарили 3D-анимационные эффекты и, конечно, специальные программы, благодаря которым эти самые эффекты создаются.Цель статьи состоит в том, чтобы показать распределение и использование напольной 3D рекламы в России и показать, как это может использоваться с захватывающими результатами на рекламных кампаниях. Реклама является наиболее заметной формой маркетинга. Она является одним из наиболее эффективных маркетинговых инструментов, которые можно использовать для создания перспективы. Если вы точно знаете, что вы хотите получить от вашей рекламы, куда направить свое сообщение, и как сказать, что вы хотите, чтобы ваша аудитория знала, ваша реклама будет эффективной

Introduction to Systems and Control Theory

These lecture notes give a completely self-contained introduction to the control theory of linear time-invariant systems. No prior knowledge is requried apart from linear algebra and some basic familiarity with ordinary differential equations. Thus, the course is suited for students of mathematics in their second or third year, and for theoretically inclined engineering students. Because of its appealing simplicity and elegance, the behavioral approch has been adopted to a large extend. A short list of recommended text books on the subject has been added, as a suggestion for further reading

Algebraic Systems Theory

Control systems are usually described by differential equations, but their properties of interest are most naturally expressed in terms of the system trajectories, i.e., the set of all solutions to the equations. This is the central idea behind the so-called "behavioral approach" to systems and control theory. On the other hand, the manipulation of linear systems of differential equations can be formalized using algebra, more precisely, module theory and homological methods ("algebraic analysis"). The relationship between modules and systems is very rich, in fact, it is a categorical duality in many cases of practical interest. This leads to algebraic characterizations of structural systems properties such as autonomy, controllability, and observability. The aim of these lecture notes is to investigate this module-system correspondence. Particular emphasis is put on the application areas of one-dimensional rational systems (linear ODE with rational coefficients), and multi-dimensional constant systems (linear PDE with constant coefficients)