state and/or output, while the outputs are something of interest that comes out of the system, often depending on the system’s state and/or on the inputs. A simple example of a system could be a buffer-tank with an inflow and an outflow. The tank wall is the boundary, the inflow is the input, the outflow is the output, and the level in the tank could be its inner state (Fig. 9).

Figure 9 A buffer tank could be regarded as a system with the inflow as input, the outflow as output, and the tank-level as an internal state of the system. A system can in itself contain other sub-systems. The buffer-tank in the example above could be a sub-system of a larger system, a food-processing plant for example. Representing a process as a system is useful in many ways. First of all it forces one to think about how to classify various process parameters. Which parameters are considered important for the process? Which parameter is to be controlled? Secondly it sets up a good framework for developing a mathematical model of the process, which is a crucial part of control theory. Most systems can be said to map a certain set of inputs u t , to a certain set of outputs y t , by applying an operator H to the inputs, that is, y t =H {u t}. This operator is often modelled mathematically by use of differential equations or difference equations. If the system is static, the current outputs only depend on the current input to the system. That is, in a static system the outputs are not dependent on anything that have happened previously in the system. A dynamic system is a system in which the outputs depend on the current inputs as well as the previous inputs. The state of the system could be described as the amount of information needed to remember what happened to the system previously. A system can have a number of properties which could be related to the mathematical model. One such property is linearity. A linear system is scalable and obeys the superposition principle. Assume that y 1 t= H {u1 t } and y 2 t =H {u 2 t} is true. If the system is linear then the following must also be true:  y 1 t y 2 t =H { u1 t  u 2 t }. Another important property of systems is time-invariance. A time-invariant system does not explicitly depend on time; if the output y t is produced by input u t , and the system is timeinvariant, then a time shift in the in-signal u t should result in an equal time shift in the outsignal y t . A system that is both linear and time-invariant is called a LTI-system (Linear Time-Invariant). The mathematical properties of these systems are useful when developing controllers and a number of theorems regarding popular control-strategies are only valid for LTI-systems. 14

/Olle_Trollberg