Multidimensional Systems Theory Research Articles

Optimal water allocation integrated with water supply, replenishment, and spill in the in-series reservoir based on an improved decomposition and dynamic programming aggregation method

Abstract In humid regions with the monsoon climate, seasonal water shortages and water spills occur alternately because of uneven temporal and spatial distributions of water resources. An optimization model for the in-series reservoir (ISR) with replenishment pumping stations was developed to obtain the minimum annual sum of water shortage and systematically considered the reservoir operation rule of water spill and replenishment. This model features multiple dimensions; dynamic programming (DP) may cause a ‘curse of dimensions’, while the decomposition-coordination method has difficulty in judging logic conditions in the reservoir operation rules. So, an improved decomposition and DP aggregation (DDPA) method was proposed. The proposed model and the method were applied to a real case in the humid region of southern China. Compared to a conventional scheduling method, the water supply was increased by 0.8% and replenishment was reduced by 2.5%. Moreover, a comparison between DDPA and six heuristic algorithms was discussed. All heuristic algorithms' objective function values only obtained local optimal solutions, and the water shortage of the system was 0.12–20.5%. The obtained results demonstrated that DDPA was the better choice for highly complex multi-reservoir systems. The proposed optimization algorithm enriched the optimization theory of multi-dimensional and multi-variable complex systems.

Development of a Control System for a Technological Line for the Production of Products from Foamed Polysty

Purpose of research. When designing new and modernizing existing control systems for the polymer drawing process, an important role is played by solving the problems of creating mathematical or simulation models and synthesizing control algorithms that reduce the share of manual labor. The article provides an overview of approaches to the development and modeling of automated control systems for extrusion production lines for the production of polymer products. The possibility of developing and modeling a multidimensional control system for the process of stretching foamed polystyrene based on the Matlab mathematical analysis package is considered. The relevance of the research topic is related to the trend of growth in the consumption market and the implementation of management in the context of creating more complex forms of packaging from polymers.Methods. For the development of the control system and its analysis, the fundamentals of the theory of multidimensional systems in control problems, the theory of plastics processing, methods of computer simulation of control systems were applied.Results. The article presents an algorithm for controlling the molding process and its implementation in the Simulink environment. Graphs of changes in the pulling speed and winding speed, changes in the thickness and tension of the web over time are presented and analyzed. A scheme of process line automation based on the developed algorithm is presented.Conclusion. The results of testing the model show that the use of mathematical modeling of the control system for stretching the expanded polystyrene web is possible to solve the problem of increasing production efficiency on existing equipment, which will bring their characteristics closer to new production lines, leaving the mechanical part unchanged. The significance of the work is due to the fact that the use of modern control systems can improve the quality of manufactured products, reduce the consumption of raw materials. Materials and research methods can be applied in the development of projects for other installations, for example, in the production of an optical sheet.

Extension of the Bessmertnyĭ Realization Theorem for Rational Functions of Several Complex Variables

We prove a realization theorem for rational functions of several complex variables which extends the main theorem of Bessmertnyĭ (in: Alpay, Gohberg, Vinnikov (eds) Interpolation Theory, Systems Theory and Related Topics. Operator Theory Advances and Applications vol 134. Birkhauser Verlag, Basel, pp 157–185, 2002). In contrast to Bessmertnyĭ’s approach of solving large systems of linear equations, we use an operator theoretical approach based on the theory of Schur complements. This leads to a simpler and more “natural” construction to solving the realization problem as we need only apply elementary algebraic operations to Schur complements such as sums, products, inverses, and compositions. A novelty of our approach is the use of Kronecker product as opposed to the matrix product in the realization problem. As such our synthetic approach leads to a solution of the realization problem that has potential for further extensions and applications within multidimensional systems theory especially for those linear models associated with electric circuits, networks, and composites.

On Serre Reduction of Multidimensional Systems

Serre reduction of a system plays a key role in the theory of Multidimensional systems, which has a close connection with Serre reduction of polynomial matrices. In this paper, we investigate the Serre reduction problem for two kinds of nD polynomial matrices. Some new necessary and sufficient conditions about reducing these matrices to their Smith normal forms are obtained. These conditions can be easily checked by existing Gröbner basis algorithms of polynomial ideals.

Improper integrals for stability theory of multidimensional phase systems

Improper integrals for stability theory of multidimensional phase systems

Parametric Connotation and Theoretical Extension of Parametric Architecture Aesthetics

Parametric Design has aroused the innovation in the architectural form creation, space conception and technology conception as a sort of new thinking mode. Coming along with this situation, architectural aesthetics has already started a new step in the influence of parametric design. Using the basic principles of philosophy, aesthetics, information technology, from the Multidimensional Vision systems theory, information theory, behaviour psychology, philosophical thinking and logical induction dual perspective of parametric architectural beauty philosophical connotation, aesthetic principles, aesthetic value and artistic regularity analysis, so as to construct a theoretical framework parameterized Aesthetics. Proposed integrity and process, interaction with equality and pluralism and individuality is an important aesthetic value principle parametric architectural aesthetics. And from complex parametric building phenomenon, summed parametric architectural aesthetics content evolution include: aesthetic dimension in the form of expansion of the connotation of spatial variation in the type of aesthetic and technical aesthetics extend and so on. Expanding the creative vision of today's architectural and aesthetic theory could provide new ways of thinking, and provide a theoretical basis for aesthetic parametric building program evaluation.

Obituary: Alfred Leo Maria Fettweis (1926–2015)

Alfred Fettweis, a giant in the fields “circuits and systems” and “multidimensional systems theory”, passed away on August 20, 2015 in Bochum, Germany. Our community has lost a pioneer in the field who always had been closely connected to our journal “Multidimensional Systems and Signal Processing”, for which he served as a founding board member from 1990 to 1993. It is only a sad coincidence that the sequence of his many personal contributions to MSSP will end with a joint paper together with one of us in this issue. Alfred Fettweis had close scientific cooperation and many exchange of ideas with Nirmal K. Bose, the founding editor of this journal, and his advice with respect to the evolution of MSSP had been always appreciated. Born on November 27, 1926 in Eupen, Belgium, Alfred Fettweis received the degrees of Ingenieur Civil Electricien and Docteur en Sciences Appliquees from Catholic University of Louvain, Belgium, in 1951 and 1963, respectively. During the time between 1951 and 1963 he was employed as an engineer at the International Telephone and Telegraph Corporation (ITT) in Belgium and in the USA (1954–1956)—an experience from which he accrued valuable practical skill and experience. This phase of his professional life in an industrial environment strongly influenced his later career as a scientist. He always emphasized that scientific results, including even theoretical ones, should have a practical impact.

Special issue on symbolic methods in multidimensional systems theory

In the seventies, the study of transfer matrices of linear time-invariant systems led to the development of the polynomial approach (Kailath, Rosenbrock, et al.). In this approach, univariate polynomial matrices play a central role (e.g., Hermite and Smith normal forms, Bezout equations etc.). When generalizing linear systems given by ordinary differential/difference equations to differential time-delay systems, systems with parameters, 2-D or 3-D filters and circuits, one had to face the case of systems described by means of matrices with entries in multivariate polynomial rings. These new systems were called 2-D or 3-D and, more generally, multidimensional systems (Bose, Lin, Pommaret, Oberst, Youla, Wood, et al.). For such systems, no normal forms such as Hermite and Smith forms exist. In order to handle these problems, the concept of Grobner bases (developed by Buchberger) was introduced in multidimensional systems theory. In many ways, the computation of these bases can be seen as an extension of both Gaussian elimination and the Euclidean algorithm. Grobner bases have been implemented in nearly all modern computer algebra systems. Another difficulty with multidimensional systems is the lack of a canonical first order representation. Several models (Roesser, Fornasini-Marchesini, reduced Spencer form etc.) have been studied extensively. The geometric approach to systems and control (Basile,Marro, Wonham, et al.) has recently been extended to these classes of multidimensional systems yielding promising results. The reduction to first order representations (realization problem) is of great importance to many applications of multidimensional systems theory such as uncertainty modeling for robust control. Special computer algebra packages have been developed for the effective manipulation of linear fractional transformations. In his pioneering work, Kalman developed a module-theoretic approach to the realization problem. Independently, the Japanese school of Sato, Kashiwara, and Kawai proposed an

Editorial — Leonid Pavlovich Shilnikov

This special issue presents a selection of papers from the conference “Dynamics, Bifurcations and Strange Attractors” dedicated to the memory of Leonid Pavlovich Shilnikov (1934–2011) to commemorate his contributions to the theory of dynamical systems and bifurcations. The conference was held at the Lobachevsky State University of Nizhny Novgorod, Russia, on 1–5 July 2013. The conference was attended by 155 participants from all over the world, who contributed to the three focal topics: bifurcations and strange attractors; dynamical systems with additional structures (Hamiltonian, time-reversible, etc.); applications of dynamical systems. The topics were chosen in confluence with pivotal contributions by L. P. Shilnikov to the fields. The speakers presented their current research and outlined future directions in both theory and frontier applications. The organizers of the conference are grateful to its sponsors: Russian Foundation of Basic Research, D. Zimin’s Russian Charitable Foundation “Dynasty,” R&D company Mera-NN, and K. V. Kirsenko (Russia), as well as Office of Naval Research (USA) and its officers, Drs. M. Harper (UK) and M. Shlesinger (USA). We thank the Editors-in-Chief of the International Journal of Bifurcations and Chaos: Ron Chen and Leon Chua for having the proceedings published here. L. P. Shilnikov served on the Editorial Board of the journal from the time it was founded. Our dear friend, mentor and fellow researcher, L. P. Shilnikov conceptualized the theory of global bifurcations of high-dimensional systems and was one of the founders of the mathematical theory of dynamical chaos. He built a profound research school in the city of Nizhny Novgorod (Gorky formerly) — the Shilnikov School that continues to this day. His works greatly influenced the overall development of the mathematical theory of dynamical systems as well as nonlinear dynamics, in general. Shilnikov’s findings have been included in most textand reference books, and are used worldwide by mathematics students and nonlinear dynamists to study the qualitative theory of dynamical systems and chaos. The elegance and completeness of his results let them reach “the heart of the matter,” and provide applied researchers with an in-depth mathematical understanding of the outcomes of natural experiments. The popularity and appreciation are due to the “living classic” status attained by Professor Shilnikov over several decades of his life through continuous hard work on bifurcation theory of multidimensional dynamical systems, mathematical chaos theory and theory of strange attractors. L. P. Shilnikov was born in Kotelnich, Kirov region of Russia on December 17, 1934. After graduating from a local high school in 1952, he became a student in the Department of Physics and Mathematics at Gorky State University. After graduation in 1957, he continued his PhD studies at the same university. He defended his PhD thesis “On birth of stable periodic orbits from singular trajectories” in 1962, it focused on the multidimensional generalization of basic homoclinic bifurcations, which were originally discovered and studied for systems on a plane by A. A. Andronov and E. A. Leontovich in the early 1930s.

The Lin-Bose Problem

This brief studies prime factorization problems of a multivariate polynomial matrix. We mainly investigate the Lin-Bose problem proposed by Lin and Bose in the multidimensional system theory context. A simple proof to the Lin-Bose problem is presented in a much easier way to understand. As a by-product, we obtain some properties on modules generated by all the rows of a matrix.

Special issue on Multidimensional systems theory and control

Multidimensional systems and signal processing is, todaymore than ever before, a very lively research area, which attracts a great deal of attention and research interest, both in the control and signal processing communities. Indeed, an increasingly wider research community devotes much of its efforts to the solution of control, estimation and signal processing problems involving multidimensional systems. This is undoubtedly motivated by the fact that this area plays a crucial role in a variety of applications, ranging from remote sensing for environmental monitoring and geological mapping, to medical imaging and the automatic control of industrial processes. The impact that this topic has shown in recent years in events such as the International Symposium on Mathematical Theory of Networks and Systems (MTNS), and the importance of other events entirely devoted to this topic, such as the International Workshop onMultidimensional Systems (nDS), are a clear evidence of the increasing interest that this area attracts within the control, signal processing and mathematics communities. The broad range of application areas for multidimensional systems and signal processing, the great need to tackle unsolved fundamental mathematical problems within this area, and the ever increasing capacity of computing technologies, provide substantial motivation for further development of the underlying mathematical tools. Motivated by these facts, this Special Issue aims at providing an overview of several recent developments in this wide area. The contributions selected for this Special Issue clearly display the broad scope of multidimensional systems, from open mathematical

Finite spectrum assignment of systems with general delays

This article presents a new framework for dealing with pure and distributed delays simultaneously and impartially as general delays in algebraic finite spectrum assignment. The framework based on an important property common to general delays gives a sufficient condition for finite spectrum assignability of scalar systems with several general delays and a design method of a controller achieving finite spectrum assignment. Especially in algebraic control theory for delay systems, pure and distributed delays have never been dealt with simultaneously and impartially. The framework is significantly important also for multidimensional system theory because its potential applicability to the discussion on systems with general delays, i.e. the possibility that especially distributed delay operators are used as variables for describing multidimensional systems, is clarified for the first time.

The Fast Fourier Transform

Fast Fourier transforms (FFTs) are fast algorithms, i.e., of low complexity, for the computation of the discrete Fourier transform (DFT) on a finite abelian group. They are among the most important algorithms in applied and engineering mathematics and in computer science, in particular for one‐ and multidimensional systems theory and signal processing. We give a relatively short survey of the FFT for arbitrary finite abelian groups, cyclic or not, with complete and partially novel proofs, the main distinction being explicit induction formulas for the FFT in all cases which generalize the original FFT‐algorithm due to Cooley and Tukey and, much earlier, to Gauß. We believe that our approach has didactic advantages over the usual ones. We also present the application of the FFT to fast convolution algorithms, and the so‐called number theoretic transforms over finite coefficient rings. We do not treat those algorithms which decrease the multiplicative complexity at the expense of many more rational linear combinations, which in this context are considered costless, nor do we discuss the DFT for nonabelian finite groups.

Modeling and analysis of hierarchical cellular networks with bidirectional overflow and take-back strategies under generally distributed cell residence times

The rapid growth of wireless services and mobile users drives a great interest in cellular networks with a hierarchical structure. Hierarchical cellular networks (HCNs) can provide high system capacity, efficient channel utilization and inherent load-balancing capability. In this paper, we develop an analytical model and a performance analysis method for a two-layer HCN with bidirectional overflow and take-back strategies. Mobile users are divided into two classes. The call requests (including new and handoff calls) of fast and slow users are preferably assigned to the macrolayer and microlayer, respectively. A call from a fast user or slow user can overflow to its non-preferable layer if there is no channel available. The successful overflow call can be taken back to its preferable layer if a channel becomes available. Since the commonly used exponentially distributed assumption for cell residence time and then the channel occupancy time does not hold for emerging mobile networks, we model various cell residence times by general distributions to adapt to more flexible mobility environments. The channel occupancy times are derived in terms of the Laplace transforms of various cell residence times. The handoff rates, overflow rates and take-back rates of each layer are also derived in terms of the new call arrival rates and related probabilities. The stationary probabilities (and then the performance measures) are determined on the basis of the theory of multi-dimensional loss systems.

On multivariate polynomial matrix factorization problems

This paper studies the multivariate polynomial matrix factorization problems which have applications to multidimensional systems theory and signal processing. We first extract an algorithm from Pommaret's proof of the Lin-Bose conjecture. Then we simplify our algorithm, and prove a theorem which gives a sufficient and necessary condition for a multivariate polynomial matrix to have an minor left prime (MLP) factorization. Examples are given to illustrate the effectiveness of this algorithm. Our results hold for any coefficient field and thus have a wide range of applications.

On Lin–Bose problem

This paper study generalized Serre problem proposed by Lin and Bose in multidimensional system theory context [Multidimens. Systems and Signal Process. 10 (1999) 379; Linear Algebra Appl. 338 (2001) 125]. This problem is stated as follows. Let F∈ A l× m be a full row rank matrix, and d be the greatest common divisor of all the l× l minors of F. Assume that the reduced minors of F generate the unit ideal, where A= K[ x 1,…, x n ] is the polynomial ring in n variables x 1,…, x n over any coefficient field K. Then there exist matrices G∈ A l× l and F 1∈ A l× m such that F= GF 1 with det G= d and F 1 is a ZLP matrix. We provide an elementary proof to this problem, and treat non-full rank case.

Digital sound synthesis of string instruments with the functional transformation method

The theory of multidimensional continuous and discrete systems is applied to derive a parametric description of musical sounds from a physical model of real or virtual string instruments. The mathematical representation of this model is given by a partial differential equation for a vibrating string. Suitable functional transformations with respect to time and space turn this partial differential equation into a multidimensional transfer function. It is the starting point for the derivation of a discrete-time system by classical analog to discrete transformations. The coefficients of this discrete model depend explicitly on the geometric properties and material constants of the underlying physical model. This ensures a meaningful behaviour of the discrete system under varying conditions and allows for an intuitive control by the user. Furthermore, the performance of real-time implementations is discussed. Finally, several extensions of this synthesis method for computer music applications are presented.

A Coordinate-Free Approach to the Method of Moments in the Theory of Multidimensional Stochastic Systems

A coordinate-free approach to the theory of the distribution moments of random variables was considered. The main relations for the moments were obtained using the theory of symmetric polylinear forms. Problems of polynomial approximation of the distributions of the state vector of multidimensional stochastic systems were discussed.

Controllable and Autonomous nD Linear Systems

The theory of multidimensional systems suffers in certain areas from a lack of development of fundamental concepts. Using the behavioural approach, the study of linear shift-invariant nD systems can be encompassed within the well-established framework of commutative algebra, as previously shown by Oberst. We consider here the discrete case. In this paper, we take two basic properties of discrete nD systems, controllability and autonomy, and show that they have simple algebraic characterizations. We make several non-trivial generalizations of previous results for the 2D case. In particular we analyse the controllable--autonomous decomposition and the controllable subsystem of autoregressive systems. We also show that a controllable nD subsystem of (k^q)^{(\Ints^n)} is precisely one which is minimal in its transfer class.

A formal theory of matrix primeness

Primeness of nD polynomial matrices is of fundamental importance in multidimensional systems theory. In this paper we define a quantity which describes the "amount of primeness" of a matrix and identify it as the concept of grade in commutative algebra. This enables us to produce a theory which unifies many existing results, such as the Bezout identities and complementation laws, while placing them on a firm algebraic footing. We also present applications to autonomous systems, behavioural minimality of regular systems, and transfer matrix factorization.