1
Introduction

1.1 Preliminary Considerations

A control system may be defined as a collection of interconnected components that can be made to achieve a desired response in the face of external disturbances. The "desired response" could be the tracking of a specified dynamic trajectory, in which case the control system takes the form of a servomechanism. An example of this type of control system is a robot arm that is programmed to grasp some object and to move it to a specified location. There is a second class of control system termed the regulator, for which the "desired response" is to maintain a certain physical quantity within specified limits. A simple example of this kind of control system is the thermostat.

There are two basic ways in which a control system can be made to operate. In open-loop mode, the response of the system is determined only by the controlling input(s). As an example, let us suppose that we wish to control the temperature of a room in winter with the use of a fan-heater that heats up and circulates the air within the room. By setting the temperature control to "medium," for instance, we should be able to get the room temperature to settle down to an agreeable level during the morning hours. However, as the day progresses and the external environment becomes warmer, the room temperature also will rise, because the rate at which heat is added by the fan-heater exceeds the rate at which heat is dissipated from the room. Conversely, when night sets in and the external temperature falls, the temperature in the room will decrease below the desired level unless the heater setting is raised. This is a fundamental limitation of open-loop control systems. They can perform satisfactorily as long as the external conditions do not affect the system much. The simple example we have described may be considered a physical analog of thermoregulatory control in poikilothermic or "cold-blooded" animals. The design of the thermoregulatory processes in these animals do not allow core body temperature to be maintained at a level independent of external conditions; as a consequence, the animal's metabolism also becomes a function of external temperature.

Coming back to the example of the heating system, one way to overcome its limitation might be to anticipate the external changes in temperature and to "preprogram" the temperature setting accordingly. But how would we know what amounts of adjustment are required under the different external temperature conditions? Furthermore, while the external temperature generally varies in a roughly predictable pattern, there will be occasions when this pattern is disrupted. For instance, the appearance of a heavy cloud cover during the day could limit the temperature increase that is generally expected. These problems can be eliminated by making the heater "aware" of changes in the room temperature, thereby allowing it to respond accordingly. One possible scheme might be to measure the room temperature, compare the measured temperature with the desired room temperature, and adjust the heater setting in proportion to the difference between these two temperatures. This arrangement is known as proportional feedback control. There are, of course, other control strategies that make use of the information derived from measurements of the room temperature. Nevertheless, there is a common feature in all these control schemes: They all employ feedback. The great mathematician-engineer, Norbert Wiener (1961), characterized feedback control as "a method of controlling a system by reinserting into it the results of its past performance." In our example, the system output (the measured room temperature) is "fed back" and used to adjust the input (fan speed). As a consequence, what we now have is a control system that operates in closed-loop mode, which also allows the system to be self-regulatory. This strategy of control is ubiquitous throughout Nature: The physiological analog of the simple example we have been considering is the thermoregulatory control system of homeothermic or "warm-blooded" animals. However, as we will demonstrate throughout this book, the exact means through which closed-loop control is achieved in physiological systems invariably turns out to be considerably more complicated than one might expect.

1.2 Historical Background

The concept of physiological regulation dates back to ancient Greece (~500 BC), where the human body was considered a small replica of the universe. The four basic elements of the universe - air, water, fire, and earth - were represented in the body by blood, phlegm, yellow bile, and black bile, respectively. The interactions among pairs of these elements produced the four irreducible qualities of wetness, warmth, dryness, and cold. It was the harmonious balance among these elements and qualities that led to the proper functioning of the various organ systems. The Greek physician, Galen (about second century AD), consolidated these traditional theories and promoted a physiological theory that was largely held until the end of the sixteenth century. Similar concepts that developed alongside the Taoist school of thought may be traced back to the third century BC in ancient China. Here, the universe was composed of five agents (Wu Xing): wood, fire, earth, metal, and water. These elements interacted with one another in two ways - one was a productive relationship, in which one agent would enhance the effects of the other; the other was a limiting or destructive relationship whereby one agent would constrain the effects of the other. As in the Graeco-Roman view, health was maintained by the harmonious balancing of these agents with one another (Unschuld, 1985).

The notion of regulatory control clearly persisted in the centuries that followed, as the writings of various notable physiologists such as Boyle, Lavoisier, and Pflüger demonstrate. However, this concept remained somewhat vague until the end of the nineteenth century when French physiologist Claude Bernard thought about self-regulation in more precise terms. He noted that the cells of higher organisms were always bathed in a fluid medium, for example, blood or lymph, and that the conditions of this environment were maintained with great stability in the face of disturbances to the overall physiology of the organism. The maintenance of these relatively constant conditions was achieved by the organism itself. This observation so impressed him that he wrote: "It is the fixity of the 'milieu interieur' which is the condition of free and independent life." He added further that "all the vital mechanisms, however varied they may be, have only one object, that of preserving constant the conditions of life in the internal environment." In the earlier half of this century, Harvard physiologist Walter Cannon (1939) refined Bernard's ideas further and demonstrated systematically these concepts in the workings of various physiological processes, such as the regulation of adequate water and food supply through thirst and hunger sensors, the role of the kidneys in regulating excess water, and the maintenance of blood acid-base balance. He went on to coin the word homeostasis to describe the maintenance of relatively constant physiological conditions. However, he was careful to distinguish the second part of the term, that is, "stasis," from the word "statics," since he was well aware that although the end result was a relatively unchanging condition, the coordinated physiological processes that produce this state are highly dynamic.

Armed with the tools of mathematics, Wiener in the 1940s explored the notion of feedback to a greater level of detail than had been done previously. Mindful that most physiological systems were nonlinear, he laid the foundation for modeling nonlinear dynamics from a Volterra series perspective. He looked into the problem of instability in neurological control systems and examined the connections between instability and physiological oscillations. He coined the word "cybernetics" to describe the application of control theory to physiology, but with the passage of time, this term has come to take on a meaning more closely associated with robotics. The race to develop automatic airplane, radar, and other military control systems during the Second World War provided a tremendous boost to the development of control theory. In the post-war period, an added catalyst for even greater progress was the development of digital computers and the growing availability of facilities for the numerical solution of the complex control problems. Since then, research on physiological control systems has become a field of study on its own, with major contributions coming from a mix of physiologists, mathematicians, and engineers. These pioneers of "modern" physiological control systems analysis include Adolph (1961), Grodins (1963), Clynes and Milsum (1970), Milhorn (1966), Milsum (1966), Bayliss (1966), Stark (1968), Riggs (1970), Guyton et al. (1973), and Jones (1973).

1.3 Systems Analysis: Fundamental Concepts

Prior to analyzing or designing a control system, it is useful to define explicitly the major variables and structures involved in the problem. One common way of doing this is to construct a block diagram. The block diagram captures in schematic form the relationships among the variables...