*Department of Information Engineering and Process Control:*

- Modelling and Simulation
- System Identification
- Optimal Control Design
- Adaptive Control
- Neural Networks and Fuzzy Control
- Model Predictive Control
- Dynamic Optimisation
- Modelling and control of chemical reactors, biochemical reactors, distillation columns and heat exchangers
- Control Engineering Education
- Information technologies

*Department of Mathematics:*

Modelling and simulation play an important role in the investigation of static and dynamic properties of chemical processes, units and systems. Most chemical systems are strongly non-linear and their simulation is necessary for the control design as well as for the investigation of the overall control systems. The main aim of the research is to develop program packages for modelling and simulation of various kinds of models. During the last year a package for PC in Simulink and C-language was created.

System identification deals with problem of the parameter estimation of static or dynamic systems from observed input-output data. Among many topics of system identification, the following areas have been investigated in this project:

- nonparametric methods, correlation and spectral analysis
- recursive identification of transfer functions of continuous-time systems, Z-transform discrete-time models and delta-transform discrete-time models
- identification in closed-loop

A program package IDTOOL has been developed for Simulink. This toolbox implements recursive LS algorithm LDDIF and provides blocks for continuous and discrete time parameter estimation.

The main aim of this area is to develop a package of algorithms and program implementation of various known control design for a given plant. The research interests include single input-single output systems as well as multivariable dynamic systems. Control design covers strategies in discrete-time and continuous-time formulation. A program package is created in Matlab and Simulink environment.

Most of technological plants exhibit non-linear behaviour. To apply a successful control design to practical problems is a substantial effort. The processes are known to be modelled and controlled with serious difficulties caused by their non-linear behaviour, high order dynamics, and tendency to instability. Many of industrial processes must be considered as multivariable systems. In a great deal of available control design techniques it is often necessary to carry out the steps of modelling, identification and control design. Theory and implementation of adaptive control in technological systems have been the long-time research topics. The activities in the adaptive control have been concentrated to three main areas as follows:

- self-tuning control - characterised by repeating parameter estimation and control design
- model reference adaptive control based on the Lyapunov method
- decentralised adaptive control

The aim of this research is to investigate fuzzy controllers based on genetic algorithms, two-layer hierarchical control structures for biochemical systems, integrated optimising algorithms for higher layers of hierarchical control structures, artificial neural-network models obtained by back-propagation for specified biochemical systems, design of a robust long-range constrained predictive control algorithms on the basis of ANN involving a stochastic approximation training algorithm, and development of a control system for our laboratory fermenter.

Model Predictive control (MPC) has been successful not only in academia but in industrial process applications as well. Its main drawbacks are the stability problems. The aim of this research is to enhance the basic input-output predictive methods. The problem is solved by means of the Youla-Kučera parameterisation of all stabilising controllers. Both finite and infinite horizon formulations are handled. Another approach is to assume that the loop is already controlled by a linear controller and to find the minimum number of control, or tracking error steps that leads to stable closed-loop behaviour. In all cases, it can be shown that the minimum number of steps is closely related to the number of unstable poles/zeros of the plant. Another area of research is development of new methods for explicit model predictive control. In this approach, the optimal solution to the given MPC problem is obtained for all admissible initial conditions by employing parametric programming methods. The resulting optimal feedback law is then represented by a look-up table, which allows for real-time implementation of MPC to processes with rapid sampling.

Increased quality requirements in chemical and petrochemical industries call for more complicated and sophisticated control strategies. Moreover, there is a need to know the achievable limits of performance and speed of transient behaviour of processes. Optimal control theory is able to provide responses to these questions. In this research, changeover problems in multicomponent distillation, waste-water treatment are studied.

The research of all research groups is focused on modelling and control of various types of chemical and biochemical processes.

Research in this domain focuses on application of information technologies in control education. This covers interactive on-line blocks and automatic generation of testing problems. The current research involves personification of students problems.

Research in this domain is oriented to:

- application of information technologies for data treatment and visualisation
- development of static and dynamic web pages not olny for purposes of measurement and control but for general information treatment
- automatic data acquisition from various internet sources

Open Source solutions are applied:: web, mail, smb servers, databases (MySQL), programming tools (PHP, JavaScript) on operating systems GNU/Linux, FreeBSD, Solaris.

Network Science is an interdisciplinary area that deals with the structure and dynamics of real networks with applications in social sciences, electrical engineering, bioinformatics, security, etc. In this area, research is focused on algorithmic and structural aspects of networks (e.g. estimations of the size of a dominating set, networks structure comparison by the graphlet method). Testing is conducted on the real data of the protein-protein interactions networks.

Spectral graph theory constitutes an important sub-discipline of algebraic graph theory. Its significance consists in pulling together methods of advanced linear algebra, group theory and combinatorics to the study of properties of graphs and discrete structures in general. In addition to this, it turns out that spectral graph theory has found numerous applications, most notably in chemistry where spectra of graphs representing molecules carry information about their energy and hence stability. Numerous methods have been developed for investigation of graph spectra; a recent one consists in graph inversion, an area the members of the team have been active in recent years.

Research in this domain focuses on fuzzy sets and interval-valued fuzzy sets, theory of aggregation operators, measure and integral theory, copulas, triangular norms and conorms. Theoretical results are applied in multi-criteria decision making, image processing, and artificial intelligence.

Basic research in number theory is immensely important not only for the theory itself but also for every area of mathematics. The results of this theory bring applications to different areas of scientific and social life. It is useful to deal with the special properties of individual types of numbers (divisibility, different expansion of real numbers, etc.), each numerical theoretical result being a potential contribution to solving basic problems and hypotheses such as problems with Fermat numbers, Mersen numbers, the existence of prime numbers of different shapes (e.g. n!+1), perfect numbers, the existence of sequences consisting of prime numbers, express rational numbers by prime numbers, etc., Catalan's, Goldbach's and Waring's hypothesis, and others.

Probabilistic number theory also includes the theory of distribution of sequences, which is a key area of mathematical analysis. Specially, the theory of distribution of modulo 1, it is an important part of number theory with many applications in mathematics and also in other disciplines. The most important case is the theory of uniform distribution of modulo 1, which is the theoretical basis of the Quasi-Monte Carlo method known as an effective tool for solving the whole spectrum of difficult problems in different areas of human knowledge. For small dimensions, the estimation of quasi-Monte Carlo integration error by using Koksma-Hawka's theorem is better than the estimation of error for Monte Carlo method. The aim is to find an error estimation that could be used for large dimensions (for example, in the insurance exist integrals for dimension m = 360).

Below the density on the set of natural numbers, we mean every non-negative measure. The basic densities are asymptotic density, Buck density, uniform density, logarithmic density, weight density, densities assigned to summation methods, especially densities to matrix summation methods are known. Using density we can define special types of convergence, so called convergence according to the ideal containing null-sets for a given density. To asymptotic density, we receive statistical convergence, even density, even convergence, For the general ideal of subsets of the set of natural numbers, we receive so called I-convergence introduced by P. Kostyrko, T. Šalát and W. Wilczinski (2000-2001), that is equivalent to convergence according dual filter. Another type of convergence is almost convergence, strong - Ceasaro convergence and so on. Special convergences have been shown to have various applications in Number Theory and Mathematical Analysis. Inclusions, their metric and topological properties are studied among sets of such converging sequences.

The notion of continuity is a nondetachable part of our life, there is no scientific area in which the term would not appear. The notion continuity and its various generalized types, as quasi-continuity, somewhat continuity, almost continuity, almost quasi-continuity, and others have a number of applications in mathematics itself, whether we consider it for the functions of one or more variables (finitely or infinitely many variables). It is known that if we consider the functions of several variables then the separate quasi-continuity at each variable gives the quasi-continuity, the same statement does not apply to continuity. Although each continuous function is quasi-continuous (the opposite does not apply). There is a natural question, whether there is a stronger type of continuity for functions of several variables, such that its separate continuity would guarantee the continuity of a given function. It is necessary to find a given type of continuity and to examine its properties and to compare it with the notion of a strong separate connection. In the final dimension, the strong separate continuity implies the continuity of the function of several variables.