Models of Dynamical Systems

Abstract

Dynamical systems can be modelled from different viewpoints. This chapter summarises the main notions. Each of the succeeding chapters uses one of these models for fault diagnosis and fault-tolerant control. Fundamental NotionsFault-tolerant control is based on models. These models have to describe the nominal as well as the faulty system. The following introduces the different models which can be used for fault-tolerant control, starting with the definition of a system as a set of interconnected components, and introducing faults as events which prevent the system components to perform the function they have been designed for. Dynamical systems.A system is a set of interconnected components. Each of the components has been chosen (or designed) by the system engineer so as to achieve some function of interest. A function describes what the design engineer expects the component to perform, independently of how it is performed. A component performs some function because it has been designed so as to exploit some physical principles, which in general are expressed by some relationships between the time evolution of some system variables. Such relationships are called constraints, and the time evolution of a variable is called its trajectory.The components are interconnected by energy or information flows. Energy flows characterise physical systems, which are called "process". Information flows characterise information and control systems.To illustrate these notions, consider for example a tank. "Storage", which is the function classically associated with it, refers to a special operating mode in which the input and the output flows are both equal to zero. In that mode, the mass in the tank stays constant, which indeed justifies the "storage" denomination. However, many different functions could be assigned to a tank, for example, the decoupling (smoothing) of the output flow from some variations of the input flow. This example shows that the notion of function is not univoque, unless the function is understood through the mathematical expression of the constraint that it introduces. In the tank