2. Models and Dynamics

The models that form the basis for the design of adaptive control algorithms considered throughout the book for distributed collector solar fields are described. The aim is not to describe in detail the model of a specific plant but instead to describe reduced complexity models (including partial differential equations or linear or nonlinear finite dimensional state-space models) that provide insight into the structure of the dominant dynamics of distributed collector solar fields and may serve as basis for controller design. Hence, we start by a hyperbolic partial differential equation satisfied by the temperature distribution along the field, which is obtained from the principle of conservation of energy. Using either finite differences or orthogonal collocation, this infinite dimensional model is then approximated by a finite dimensional bilinear state-space model, from which linearized state-space models are also obtained. A number of plants that can be represented by models that are similar to distributed collector solar fields, such as glass tube manufacturing or a moisture control system, are described in order to prepare the ground for illustrating, in subsequent chapters, how to use in more general applications the adaptive control algorithms considered in the book.