This specialisation belongs to the Master's Systems Control.

## What is Control Theory?

In this specialisation the main focus is laid upon the study of the core of the Systems and Control discipline: to mathematically describe a system, to study its properties and to adapt it such that it behaves in a desired way (the control problem). So you will concentrate on mathematical theory, although practical problems are always nearby, for inspiration and application. As a student, you will be able to tailor the programme to address your own individual interests and needs.

A mathematical model of a system should reflect its main features. It may be represented by difference or differential equations, but also by inequalities, algebraic equations, and logical constraints.

System models are linear or non-linear. Linear models are relatively simple and convenient. However, if non-linearities play a prominent role, then important system properties may be missed by a linear model. Take as an example an autopilot for an airplane. If it should keep the plane at a fixed heading, speed and height, then its design can be based on a linear model. If it should control take-off and landing, then non-linear models are necessary. For the design of an autopilot for the airplane, we can use a model, which is based on switching between a finite number of linear models. This yields so-called hybrid models and there are many challenging questions for this class of systems.

Simple system models can be studied analytically by working out mathematical equations. Simulation of the mathematical model, however, is necessary for the analysis of more complicated systems. Think for instance of a robot, or think of a big airplane or an economical system. However, always checks using simplified analytical models are necessary: simulations alone cannot prove the correctness of the underlying system models.

By choosing and controlling inputs or, more general, by imposing additional constraints on some of the variables, the system may be influenced so as to obtain certain desired behaviour. This is the control problem. For the case of a simple linear systems, control strategies have been well developed. However, consider for example the problem of controlling the temperature in a bulk storage room. The model is given by a partial differential equation describing the temperature distribution in the storage room. A cooling device can be switched on and off (non-linearity) to control the product temperature (which reflects a continuum of quantities to control). Recently, research has been completed to design a controller that robustly controls the switching times.

## How is Control Theory used?

Researchers in Control Theory today develop the theories and strategies that will be used in the applications of tomorrow. But from society there is a constant pressure to invent the required theories for necessary applications today. Therefore control theory specialists also may work in larger teams working at complicated applications to provide required control solutions “on the fly”.