*Department of Information Engineering and Process Control:*

- Optimal Control Design
- Model Predictive Control
- Dynamic Optimisation
- Modelling and Control of Chemical Reactors, Biochemical Reactors, Distillation Columns and Heat Exchangers
- Control Engineering Education
- Information Technologies
- Distributed and Decentralized Optimization
- Guaranteed Parameter Estimation
- Machine Learning in Process Control

*Department of Mathematics:*

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.

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.

Research is focused on the control of the system in distributed and decentralized way, in order to decrease computational burden per calculation unit or increase privacy of each node in network. This approach can be also helpful to find the global optimum of non-convex optimization problems.

The quality of the results of model-based optimization and control strongly depends on the accuracy of the models employed. It is essential that the predictions of variables that are considered in the optimization problem, e.g. product quality parameters, are accurate. The quality of the models can be improved by online adaptation of crucial parameters via robust state and parameter estimation schemes. In this respect, we pursue a guaranteed parameter estimation approach to obtain robust estimates of uncertain parameters while avoiding unreliable approximations that are associated with classical estimation approaches.

Machine learning is attracting huge interest not only in academia but also in the industry. The primary aim of this research is to study the application of machine learning approaches to enhance and design controllers of various nature and structure.

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.