Uncertain Systems

The first step in most applications of mathematics is to determine a mathematical model for the system under investigation. The model may be used in a number of different ways. For example, a mathematical and computational analysis of the model often leads to a better understanding of the real physical system it represents. From a more practical viewpoint the model can be used to make predictions about the future behaviour of the system, or to design algorithms of automatic control which ensure that the system behaves in some desirable fashion. However, in each of these applications it is of fundamental importance to keep in mind that the model is only a model, its behaviour and that of the real system might be quite different. The origins and causes of this possible discrepancy are many and in the systems theory literature are collectively referred to as

model uncertainties

:

Parameter uncertainty. The model may depend on some physical parameters which are not known precisely.

Imperfect knowledge of the dynamics. There may be nonlinear and/or time-varying effects which are not known accurately.

Unknown inputs and neglected dynamics. A system is usually in dynamic interaction with its environment and it is often not clear where the boundary of the system should be drawn. Uncertainties arise if parts of the real system dynamics are not accounted for in the model and if the inputs to the system from the environment are not accurately known.

Model simplification. Although an accurate complex model of the real physical system may be available, it is often necessary to simplify this for the purpose of analysis and design. E.g. nonlinearities and time-variations are neglected, infinite dimensional systems are replaced by finite dimensional ones and sometimes further model reduction techniques are used to reduce the dimension of the system.

Discretization and Rounding Errors. If simulations are carried out on a computer, discretization methods must be applied and rounding errors are introduced which will lead to unknown nonlinear model perturbations.