Uncertainty Modelling in Time-Delay Systems: Parametric vs. Unstructured Approach RADEK MATUŠŮ, ROMAN PROKOP, LIBOR PEKAŘ Department of Automation and Control Engineering Faculty of Applied Informatics Tomas Bata University in Zlín nám. T. G. Masaryka 5555, 760 01 Zlín CZECH REPUBLIC {rmatusu; prokop; pekar}@fai.utb.cz Abstract: - This contribution is focused on comparison of two basic approaches to uncertainty modelling and corresponding robust stability analyses for a system with uncertain time-delay. A paper bleaching process, described both as a system with parametric uncertainty and in the form of unstructured multiplicative uncertainty model, is considered as a testing plant. The robust stability of closed control loop with selected controller and appropriate uncertain model of controlled plant is verified and obtained results are compared. Key-Words: - Uncertainty Modelling, Time-Delay Systems, Parametric Uncertainty, Unstructured Uncertainty, Robust Stability Analysis 1 Introduction The whole classical control theory as well as many contemporary methods use some form of mathematical model of a controlled system for a controller design. The crucial problem, however, is that assumed ideal mathematical model, due to many reasons, practically never exactly matches the real behaviour of the plant. One of possible approaches how to overcome this discrepancy grounds in utilization of an uncertain model and subsequent robust controller design. There are two principal ways of uncertainty modelling in the literature [1], [2], [3] – parametric or unstructured approach. Both of them have their advantages and drawbacks. Consequently, each of approaches is more suitable for different situations. This contribution presents the comparison of uncertainty modelling and subsequent robust stability analyses for a fist order system with uncertain time-delay term. The tests are performed by means of the simulation examples with a paper bleaching process [4]. 2 Uncertainty Modelling The introductory part has already foreshadowed that difference between real process and its mathematical model is the fundamental and omnipresent control problem. For example, the parameters of controlled plant need not to be known exactly or they can be even time- variant (however, only “slowly” from the robust control point of view). Then, nonlinearity in controlled system can be neglected and consequently discrepancy could originate in linear approximation in given operational point. Or a simplified model can be intentionally used instead of originally very complex system (e.g. caused by neglecting the fast dynamic effects due to system order reduction, assumption of a distributed-parameter system as a lumped-parameter one, or time-delay neglect) because of easier calculations. In robust control, respecting these factors in mathematical description leads to the use of uncertain model. In other words, not only one nominal model, but the whole family of models given by some neighborhood of the nominal one is defined. The “size” of this neighborhood can be described in two main ways – as a parametric or unstructured uncertainty. The combination of both main methods is also possible. Then one speaks about mixed uncertainty. The real parametric uncertainty is utilized if the structure of system is known but its actual physical parameters are not. On the contrary, unstructured uncertainty does not require even knowledge of structure (order) of model. Parametric uncertainty is defined through intervals which the imprecisely known parameters lie within. The unstructured uncertainty description is based on restriction of the area of possible appearance of frequency characteristics. However, the terminology used in this paper is not the one and only possible. The scientific literature presents also different nomenclatures, e.g. structured (=parametric) vs. nonparametric (=unstructured) or possibly parametric vs. dynamic, which are subsequently divided into unstructured and structured (with different meaning than in the previous case). Thus one has to be careful about the terminology of each author. This paper Recent Researches in Automatic Control ISBN: 978-1-61804-004-6 310