ERCIM News No.40 - January 2000

## Control and System Theory: Introduction

by Jan H. van Schuppen

Control is used to effectively operate machines and computers. The general public is not fully aware of the way technological products are critically dependent on feedback control for their operation. A description of several examples of control follows.

In telephone switches there is admission control for every arriving call. If there were no admission control during the periods of overload (think of TV games in which the public is asked to dial the studio), the performance would decline dramatically to a few percentages of the theoretical capacity or even to zero.

Dynamic speed limits are displayed on motorways in major urban areas of Europe to smooth the traffic flow. The control algorithms for these limits are based on a mathematical model for traffic flow and on control theory. In a juice processing plant control is used to determine the starting time and the starting conditions of all operations. Conditions to be checked include the availability of raw materials and of resources, and the proper functioning of machines (breakdowns of machines are a major problem). New procedures for air traffic control current under development in Europe and the U.S.A. are based on control theory. The mathematical model for this is based on the laws of mechanics and on the ordering of discrete operations.

Standards for levels of toxic substances are set by national agencies after experimentation with animals and with human beings. The algorithms used for estimation of concentration levels in the different organs and of parameters in the mathematical models are based on system theory and system identification. Control of waste water treatment plants is based on control theory and system identification. Predictions of weather over a horizon of several days, of air pollution concentrations, and of sea water levels at coastal areas are all based on the Kalman filter developed as part of control theory.

In the examples mentioned, control in the form of feedback is used. Sensors provide information about physical conditions or about the logical state of the system. This information is then used to adjust the input to the engineering system such that a controlled variable will stay near a set point or will follow a reference trajectory. Control design requires the formulation of a mathematical model often in the form of a (partial) differential equation, or an automaton, or a combination of these as in a hybrid system. Control theory then provides a procedure for the construction of a control law. The control law specifies which input value to use for every state of the system.

Control theory as a scientific subject has a long history, see the keynote by Vladimír Kucera. Subjects of mathematics used in control and system theory include (partial) differential equations, functional analysis, linear algebra, numerical linear algebra, differential geometry, and algebraic geometry. Computer science subjects include automata theory, Petri nets, computation and complexity, and real-time operating systems.

The motivation for control theory shifts with the development of technology and with the needs of society. Early motivating engineering systems were: amplifiers for radios, radar detection equipment, aircraft, and aerospace vehicles. Recently control has been applied to the following technological products, areas, and services and more of this is likely to follow: cars, motorway networks, air traffic control, communication networks, manufacturing, macroeconomic control, mathematical finance, public health, the life sciences, environmental protection, weather prediction, and climate Modelling.

The research program for control and system theory in the coming decade includes the following topics. Control theory for discrete-event systems, for hybrid systems, for particular classes of nonlinear systems, for systems described by partial differential equations, and for particular classes of stochastic systems. Realization and identification is likely to develop with algebra because of the availability of symbolic computation algorithms.

Control of nonlinear systems and of systems described by partial differential equations is likely to develop for mechanical and other physical systems. Control of discrete-event systems and of hybrid systems is motivated by the use of computers for control of engineering systems. This subject will be influenced by the developments in theoretical computer science, in particular by computation, complexity, automata theory, and algebra and coalgebra.

The papers of this Special Theme Section are structured as follows. The first four articles concern concrete engineering problems for which control and system theory are used. Papers on control theory follow next. Signal processing is based on methods from system theory and two papers describe applications. Control theory is based on system theory, in particular on realization and system identification. Several papers describe how system theory has been or may be used for engineering control and signal processing problems.