Introduction

The study and design of physical systems can be carried out using empirical methods. We can apply various signals to a physical system and measure its responses. If the performance is not satisfactory, we can adjust some of its parameters or connect to it a compensator to improve its performance. This approach relies heavily on past experience and is carried out by trial and error, and it has succeeded in designing many physical systems.

Empirical methods may become unworkable if physical systems are complex, or they may become too expensive or too dangerous to be experimented on. In these cases, analytical methods become indispensable. The analytical study of physical systems consists of four parts: modeling, development of mathematical descriptions, analysis, and design. We briefly introduce each of these tasks.

The distinction between physical systems and models is important in engineering. For example, circuits or control systems studied in any textbook are models of physical systems. A resistor with a constant resistance is a model; it will burn out if the applied voltage is over a limit. This power limitation is often disregarded in its analytical study. An inductor with a constant inductance is again a model; in reality, the inductance may vary with the amount of current flowing through it. Modeling is a very important problem, for the success of the design depends upon whether the physical system is properly modeled.

A physical system may have different models, depending on the questions asked. It may also be modeled differently in different operational ranges. For example, an electronic amplifier is modeled differently at high and low frequencies. A spaceship can be modeled as a particle in investigating its trajectory; however, it must be modeled as a rigid body in maneuvering. A spaceship may even be modeled as a flexible body when it is connected to a space station. In order to develop a suitable model for a physical system, a thorough understanding of the physical system and its operational range is essential. In this text, we will call a model of a physical systems simply a system. Thus a physical system is a device or a collection of devices existing in the real world; a system is a model of a physical system.

Once a system (or model) is selected for a physical system, the next step is to apply various physical laws to develop mathematical equations to describe the system. For example, we apply Kirchhoff's voltage and current laws to electrical systems and Newton's law to mechanical systems. The equations that describe systems may assume many forms; they may be linear equations, nonlinear equations, integral equations, difference equations, differential equations or others. Depending on the problem under study, one form of equation may be preferable to another in describing the same system. In conclusion, a system may have different mathematical-equation descriptions, just as a physical system may have many different models.

After a mathematical description is obtained, we then carry out analyses— quantitative and qualitative. In quantitative analysis, we are interested in responses of systems excited by certain inputs. In qualitative analysis, we are interested in general properties of systems, such as stability, controllability, and observability. Qualitative analysis is very important, because design techniques may often evolve from this study.

If the response of a system is unsatisfactory, the system must be modified. In some cases, this can be achieved by adjusting some parameters of the system; in other cases, compensators must be introduced. Note that the design is carried out on the model of the physical system. If the model is properly chosen, then the performance of the physical system should be improved by introducing the required adjustments or compensators. If the model is poor, then the performance of the physical system may not improve and the design is useless. Selecting a model that is close enough to a physical system and yet simple enough to be studied analytically is the most difficult and important problem in system design.