# Algorithms and Numerical Issues in the Behavioral Approach

Ricardo Zavala Yoe

Onderzoeksoutput

## Samenvatting

Professionals involved in systems and control theory and control engineering formulate equations which describe a physical system for all time. This fact is referred to as modelling . The better the model (the equations obtained) the better the knowledge we get from the system under study. Using these model we can modify the original system if we are not satisfied with its performance (as most of the times happens). This process is called controlling a system. Since the model of a system is given in terms of differential or difference equations, designing a controller means finding a new set of equations which "combined" in some way with the model of the given system produces a desired performance. Such a specification is established by the designer (a systems and control theorist or a control engineer). There has been a paradigmatic point of view during the controller design stage: the cause - effect point of view (referred to as input - output approach). Although at first sight this approach might look logical, this is not always the case. It is well known that there exist systems for which this way of modelling is not suitable. We describe some of these in the thesis. They illustrate the necessity of having a wider point of view. Roughly speaking, such a wider point of view for modelling and control systems is offered by the so called Behavioral Approach for systems and control. Here, we consider that our systems are representation - free structures and as such we are not restricted to any other paradigm, rather we give our point of view as an alternative. We consider a system as an entity that is described by its trajectories (solutions to differential or difference equations) over time. In this way we do not need to be model structure dependent. This is an issue in our field because we can consider traditional results in our area as particular cases of our more general approach. Although this Behavioral Approach is well established, the main contribution done in this thesis is - roughly speaking - providing a description of all controllers that satisfy a certain performance. This is called a parametrization of all controllers. On the other hand, since the Behavioral Approach is intended to be a tool for modelling and control of physical systems, a natural and key issue is the numerical implementation of the obtained models and controls from this viewpoint. Particularly, controllers are implemented via algorithms which end up with actual computational implementations simulating the controlled system (the modified system which satisfies our new requirements). This is remarkable because we might not have the real, actual physical structure at hand and one way to simulate it is by means of programs which run in a digital computer. In this thesis we study a well known technique to design polynomial control algorithms (neither implemented in the computer before nor reported about its consequences). As a matter of fact, we show that this technique does not work correctly in actual numerical implementations. In contrast, we offer improved options to do this. This way is simpler and easier to implement in the computer. We tested this result in many examples to support our claim. We consider this fact as key during this work. The algorithms were implemented in MATLAB and SCILAB.
Originele taal-2 English Doctor of Philosophy Praagman, Cornelis, SupervisorTrentelman, Harry L., Supervisor, Externe Persoon s.n. 9036725607, 9036725615 Published - 2006