Mathematical Model Developments

Abstract

To integrate control theory in engineering practice, a bridge between real-world systems and abstract mathematical systems theory must be built. For example, applying the control theory to analyze and regulate in the desired manner the energy or information flows, the designer is confronted with the need to find adequate mathematical models of the phenomena and design controllers. Mathematical models can be found using basic physical laws. In particular, in electrical, mechanical, fluid, or thermal systems, the mechanism of storing, dissipating, transforming, and transferring energies are analyzed. We will use the Lagrange equations of motion, as well as the Kirchhoff and Newton laws to illustrate the model developments. The real-world systems integrate many components and subsystems. One can reduce interconnected systems to simple, idealized subsystems (components). However, this idealization, in most cases, is unpractical. For example, one cannot study electric motors without studying devices to be actuated, and to control electric motors, power amplifiers must be integrated as well. That is, electromechanical systems integrate mechanical systems, electromechanical motion devices (actuators and sensors), and power converters. Analyzing power converters, the designer studies switching devices (transistors or thyristors), drivers, circuits, filters, and so forth. The primary objective of this chapter is to illustrate how one can develop mathematical models of dynamic systems using basic principles and laws. Through illustrative examples, differential equations will be found to model dynamic systems. A functional block diagram of the controlled (closed-loop) dynamic systems is illustrated in Figure 2.1.1.