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Chapter 1. Introduction to Feedback and Control

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Chapter 1. Introduction to Feedback and Control Chapter 1 Introduction Feedback is a central feature of life. The process of feedback governs how we grow, respond to stress and challenge, and regulate factors such as body temperature, blood pressure, and cholesterol level. The mechanisms operate at every level, from the interaction of proteins in cells to the interaction of organisms in complex ecologies. Mahlon B. Hoagland and B. Dodson, from The Way Life Works, 1995 [3]. In this chapter we provide an introduction to the basic concept of feedback and the related engineering discipline of control. 1.1 What is Feedback? The term feedback is used to refer to a situation in which two (or more) dynamical systems are connected together such that each system influences the other and their dynamics are thus strongly coupled. Simple causal rea- soning about such a system is di±cult because the ¯rst system influences the second and the second system influences the ¯rst, leading to a circular argument. This makes reasoning based on cause and e®ect tricky and it is necessary to analyze the system as a whole. A consequence of this is that the behavior of a feedback system is often counterintuitive and therefore necessary to resort to formal methods to understand them. An early example of a feedback system is the centrifugal governor, in which the shaft of a team engine is connected to a flyball mechanism that is itself connected to the throttle of the steam engine, as illustrated in Fig- ure 1.1. 1 The system is designed so that as the speed of the engine increases 1The centrifugal governer is often called the \Watt governor", because James Watt 1 2 CHAPTER 1. INTRODUCTION (a) (b) Figure 1.1: The centrifugal governor (a), developed in the 1780s, was an enabler of the successful Watt steam engine (b), which fueled the indus- trial revolution. Figures courtesy Richard Adamek (copyright 1999) and Cambridge University. (perhaps due to a lessening of the load on the engine), the flyballs spread apart and a linkage causes the throttle on the steam engine to be closed. This in turn slows down the engine, which causes the flyballs to come back together. When properly designed, the flyball governor maintains a constant speed of the engine, roughly independent of the loading conditions. Feedback has many interesting properties that can be exploited in de- signing systems. As in the case of the flyball governor, feedback can make a system very resilient towards external influences. It is possible to create linear behavior out of nonlinear components. More generally, feedback al- lows a system to be very insensitive both to external disturbances and to variations in its individual components. Feedback has one major disadvan- tage: it may create dynamic instabilities in a system, causing oscillations or even runaway behavior. It is for this reason that a substantial portion of the study of feedback systems is devoted to developing an understanding of dynamics and mastery of techniques in dynamical systems. Feedback systems are ubiquitous in both natural and engineered systems. Homeostasis in biological systems maintains thermal, chemical, and biolog- ical conditions through feedback. Global climate dynamics depend on the feedback interactions between the atmosphere, oceans, land, and the sun. popularized its use on steam engines. However, contrary to common belief, James Watt did not invent the flyball governer itself. 1.2. WHAT IS CONTROL? 3 Ecologies are ¯lled with examples of feedback, resulting in complex interac- tions between animal and plant life. The dynamics of economies are based on the feedback between individuals and corporations through markets and the exchange of goods and services. 1.2 What is Control? The term \control" has many meanings and often varies between communi- ties. In this book, we de¯ne control to be the use of algorithms and feedback in engineered systems. Thus, control includes such examples as feedback loops in electronic ampli¯ers, set point controllers in chemical and materi- als processing, “fly-by-wire" systems on aircraft, and even router protocols that control tra±c flow on the Internet. Emerging applications include high con¯dence software systems, autonomous vehicles and robots, real-time re- source management systems, and biologically engineered systems. At its core, control is an information science, and includes the use of information in both analog and digital representations. A modern controller senses the operation of a system, compares that against the desired behavior, computes corrective actions based on a model of the system's response to external inputs, and actuates the system to e®ect the desired change. This basic feedback loop of sensing, computation, and actuation is the central concept in control. The key issues in designing con- trol logic are ensuring that the dynamics of the closed loop system are stable (bounded disturbances give bounded errors) and that dynamics have the de- sired behavior (good disturbance rejection, fast responsiveness to changes in operating point, etc). These properties are established using a variety of modeling and analysis techniques that capture the essential physics of the system and permit the exploration of possible behaviors in the presence of uncertainty, noise, and component failures. A typical example of a modern control system is shown in Figure 1.2. The basic elements of of sensing, computation, and actuation are clearly seen. In modern control systems, computation is typically implemented on a digital computer, requiring the use of analog-to-digital (A/D) and digital- to-analog (D/A) converters. Uncertainty enters the system through noise in sensing and actuation subsystems, external disturbances that a®ect the underlying system physics, and uncertain dynamics in the physical system (parameter errors, unmodeled e®ects, etc). Control engineering relies on and shares tools from physics (dynamics and modeling), computer science (information and software) and operations 4 CHAPTER 1. INTRODUCTION noise external disturbances noise Output § Actuators System Sensors § Plant D/A Computer A/D Controller operator input Figure 1.2: Components of a modern control system. research (optimization and game theory), but it is also di®erent from these subjects, in both insights and approach. A key di®erence with many scienti¯c disciplines is that control is fun- damentally an engineering science. Unlike natural science, whose goal is to understand nature, the goal of engineering science is to understand and de- velop new systems that can bene¯t mankind. Typical examples are systems for transportation, electricity, communication and entertainment that have contributed dramatically to the comfort of life. While engineering originally emerged as traditional disciplines such as mining, civil, mechanical, elec- trical and computing, control emerged as a systems discipline around 1950 and cut across these traditional disciplines. The pinnacle of achievement in engineering science is to ¯nd new systems principles that are essential for dealing with complex man-made systems. Feedback is such a principle and it has had a profound impact on engineering systems. Perhaps the strongest area of overlap between control and other disci- plines is in modeling of physical systems, which is common across all areas of engineering and science. One of the fundamental di®erences between control-oriented modeling and modeling in other disciplines is the way in which interactions between subsystems (components) are represented. Con- trol relies on input/output modeling that allows many new insights into the behavior of systems, such as disturbance rejection and stable interconnec- tion. Model reduction, where a simpler (lower-¯delity) description of the dynamics is derived from a high ¯delity model, is also very naturally de- scribed in an input/output framework. Perhaps most importantly, model- 1.3. CONTROL SYSTEM EXAMPLES 5 ing in a control context allows the design of robust interconnections between subsystems, a feature that is crucial in the operation of all large, engineered systems. Control share many tools with the ¯eld of operations research. Opti- mization and di®erential games play central roles in each, and both solve problems of asset allocation in the face of uncertainty. The role of dynamics and interconnection (feedback) is much more ingrained within control, as well as the concepts of stability and dynamic performance. Control is also closely associated with computer science, since virtually all modern control algorithms are implemented in software. However, con- trol algorithms and software are very di®erent from traditional computer software. The physics (dynamics) of the system are paramount in analyzing and designing them and their (hard) real-time nature dominates issues of their implementation. From a software-centric perspective, an F-16 is simply another peripheral, while from a control-centric perspective, the computer is just another implementation medium for the feedback law. Neither of these are adequate abstractions, and this is one of the key areas of current research in the ¯eld. 1.3 Control System Examples Control systems are all around us in the modern technological world. They maintain the environment, lighting, and power in our buildings and factories, they regulate the operation of our cars, consumer electronics, and man- ufacturing processes, they enable our transportation and communications systems, and they are critical elements in our military and space systems. For the most part, they are hidden from view, buried within the code of processors, executing their functions accurately and reliably. Nevertheless, their existence is a major intellectual and engineering accomplishment that is still evolving and growing, promising ever more important consequences to society. 1.3.1 Early Examples The proliferation of control in engineered systems has occurred primarily in the latter half of the 20th Century. There are some familiar exceptions, such as the Watt governor described earlier and the thermostat (Figure 1.3a), designed at the turn of the century to regulate temperature of buildings. The thermostat, in particular, is often cited as a simple example of feed- back control that everyone can understand.
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