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Introduction to the

Mathematical Theory of

Systems and Control

Plant Controller

Jan Willem Polderman

Jan C. Willems

Preface

The purpose of this preface is twofold. Firstly, to give an informal historical introduction to the subject area of this book, Systems and Control, and secondly, to explain the philosophy of the approach to this subject taken in this book and to outline the topics that will be covered.

A brief history of systems and control

Control theory has two main roots: regulation and trajectory optimization. The first, regulation, is the more important and engineering oriented one. The second, trajectory optimization, is mathematics based. However, as we shall see, these roots have to a large extent merged in the second half of the twentieth century.

The problem of regulation is to design mechanisms that keep certain to-be- controlled variables at constant values against external disturbances that act on the plant that is being regulated, or changes in its properties. The system that is being controlled is usually referred to as the plant, a passe- partout term that can mean a physical or a chemical system, for example. It could also be an economic or a biological system, but one would not use the engineering term “plant” in that case.

Examples of regulation problems from our immediate environment abound. Houses are regulated by thermostats so that the inside temperature remains constant, notwithstanding variations in the outside weather conditions or changes in the situation in the house: doors that may be open or closed, the

Preface xi

FIGURE P.1. Fly ball governor.

speed changes that would naturally occur in the prime mover when there was a change in the load, which occurred, for example, when a machine was disconnected from the prime mover. Watt’s fly-ball governor achieved this goal by letting more steam into the engine when the speed decreased and less steam when the speed increased, thus achieving a speed that tends to be insensitive to load variations. It was soon realized that this adjustment should be done cautiously, since by overreacting (called overcompensation), an all too enthusiastic governor could bring the steam engine into oscil- latory motion. Because of the characteristic sound that accompanied it, this phenomenon was called hunting. Nowadays, we recognize this as an instability due to high gain control. The problem of tuning centrifugal gov- ernors that achieved fast regulation but avoided hunting was propounded to James Clerk Maxwell (1831–1870) (the discoverer of the equations for electromagnetic fields) who reduced the question to one about the stability of differential equations. His paper “On Governors,” published in 1868 in the Proceedings of the Royal Society of London, can be viewed as the first mathematical paper on control theory viewed from the perspective of reg- ulation. Maxwell’s problem and its solution are discussed in Chapter 7 of this book, under the heading of the Routh-Hurwitz problem.

The field of control viewed as regulation remained mainly technology driven during the first half of the twentieth century. There were two very important developments in this period, both of which had a lasting influence on the field. First, there was the invention of the Proportional–Integral–Differential (PID) controller. The PID controller produces a control signal that consists of the weighted sum of three terms (a PID controller is therefore often called a three-term controller). The P-term produces a signal that is proportional to the error between the actual and the desired value of the to-be-controlled variable. It achieves the basic feedback compensation control, leading to a control input whose purpose is to make the to-be-controlled variable in-

xii Preface

crease when it is too low and decrease when it is too high. The I-term feeds back the integral of the error. This term results in a very large correction signal whenever this error does not converge to zero. For the error there hence holds, Go to zero or bust! When properly tuned, this term achieves robustness, good performance not only for the nominal plant but also for plants that are close to it, since the I-term tends to force the error to zero for a wide range of the plant parameters. The D-term acts on the derivative of the error. It results in a control correction signal as soon as the error starts increasing or decreasing, and it can thus be expected that this antic- ipatory action results in a fast response. The PID controller had, and still has, a very large technological impact, particularly in the area of chemical process control. A second important event that stimulated the development

ground

ampli- V fier

Vin Vout

μVout μ = (^) R 1 R+^1 R 2

R 1

R 2

FIGURE P.2. Feedback amplifier.

of regulation in the first half of the twentieth century was the invention in the 1930s of the feedback amplifier by Black. The feedback amplifier (see Figure P.2) was an impressive technological development: it permitted sig- nals to be amplified in a reliable way, insensitive to the parameter changes inherent in vacuum-tube (and also solid-state) amplifiers. (See also Exer- cise 9.3.) The key idea of Black’s negative feedback amplifier is subtle but simple. Assume that we have an electronic amplifier that amplifies its input voltage V to Vout = KV. Now use a voltage divider and feed back μVout to the amplifier input, so that when subtracted (whence the term negative feedback amplifier) from the input voltage Vin to the feedback amplifier, the input voltage to the amplifier itself equals V = Vin − μVout. Combining

xiv Preface

B

A

FIGURE P.3. Brachystochrone.

B

A

FIGURE P.4. Cycloid.

Jakob, Leibniz, de l’Hˆopital, Tschirnhaus, and Newton. Newton submit- ted his solution anonymously, but Johann Bernoulli recognized the culprit, since, as he put it, ex ungue leonem: you can tell the lion by its claws. The brachystochrone turned out to be the cycloid traced by a point on the cir- cle that rolls without slipping on the horizontal line through A and passes through A and B. It is easy to see that this defines the cycloid uniquely (see Figures P.3 and P.4). The brachystochrone problem led to the development of the Calculus of Variations, of crucial importance in a number of areas of applied mathe- matics, above all in the attempts to express the laws of mechanics in terms of variational principles. Indeed, to the amazement of its discoverers, it was observed that the possible trajectories of a mechanical system are precisely those that minimize a suitable action integral. In the words of Legendre, Ours is the best of all possible worlds. Thus the calculus of variations had far-reaching applications beyond that of finding optimal paths: in certain applications, it could also tell us what paths are physically possible. Out of these developments came the Euler–Lagrange and Hamilton equations as conditions for the vanishing of the first variation. Later, Legendre and Weierstrass added conditions for the nonpositivity of the second variation, thus obtaining conditions for trajectories to be local minima. The problem of finding optimal trajectories in the above sense, while ex- tremely important for the development of mathematics and mathematical physics, was not viewed as a control problem until the second half of the twentieth century. However, this changed in 1956 with the publication of Pontryagin’s maximum principle. The maximum principle consists of a very

Preface xv

general set of necessary conditions that a control input that generates an optimal path has to satisfy. This result is an important generalization of the classical problems in the calculus of variations. Not only does it allow a much larger class of problems to be tackled, but importantly, it brought forward the problem of optimal input selection (in contrast to optimal path selection) as the central issue of trajectory optimization.

Around the same time that the maximum principle appeared, it was realized that the (optimal) input could also be implemented as a function of the state. That is, rather than looking for a control input as a function of time, it is possible to choose the (optimal) input as a feedback function of the state. This idea is the basis for dynamic programming, which was formulated by Bellman in the late 1950s and which was promptly published in many of the applied mathematics journals in existence. With the insight obtained by dynamic programming, the distinction between (feedback based) regulation and the (input selection based) trajectory optimization became blurred. Of course, the distinction is more subtle than the above suggests, particularly because it may not be possible to measure the whole state accurately; but we do not enter into this issue here. Out of all these developments, both in

PLANT

CONTROLLER

FEEDBACK

Actuators Sensors

exogenous inputs to-be-controlled outputs

outputs

measured inputs

control

FIGURE P.5. Intelligent control.

the areas of regulation and of trajectory planning, the picture of Figure P. emerged as the central one in control theory. The basic aim of control as it is generally perceived is the design of the feedback processor in Figure P.5. It emphasizes feedback as the basic principle of control: the controller accepts the measured outputs of the plant as its own inputs, and from there, it computes the desired control inputs to the plant. In this setup, we consider the plant as a black box that is driven by inputs and that produces outputs. The controller functions as follows. From the sensor outputs, information is obtained about the disturbances, about the actual dynamics of the plant if these are poorly understood, of unknown parameters, and of the internal state of the plant. Based on these sensor observations, and on the control

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it showed how to compute the feedback control processor of Figure P.5 in order to achieve optimal disturbance attenuation. In this result the plant is assumed to be linear, the optimality criterion involves an integral of a quadratic expression in the system variables, and the disturbances are modeled as Gaussian stochastic processes. Whence the terminology LQG problem. The LQG problem, unfortunately, falls beyond the scope of this introductory book. In addition to being impressive theoretical results in their own right, these developments had a deep and lasting influence on the mathematical outlook taken in control theory. In order to emphasize this, it is customary to refer to the state space theory as modern control theory to distinguish it from the classical control theory described earlier. Unfortunately, this paradigm shift had its downsides as well. Rather than aiming for a good balance between mathematics and engineering, the field of systems and control became mainly mathematics driven. In particular, mathematical modeling was not given the central place in systems theory that it deserves. Robustness, i.e., the integrity of the control action against plant variations, was not given the central place in control theory that it deserved. Fortunately, this situation changed with the recent formulation and the solution of what is called the H∞ problem. The H∞ problem gives a method for designing a feedback processor as in Figure P.5 that is optimally robust in some well-defined sense. Unfortunately, the H∞ problem also falls beyond the scope of this introductory book.

A short description of the contents of this book

Both the transfer function and the state space approaches view a system as a signal processor that accepts inputs and transforms them into outputs. In the transfer function approach, this processor is described through the way in which exponential inputs are transformed into exponential outputs. In the state space approach, this processor involves the state as intermediate variable, but the ultimate aim remains to describe how inputs lead to out- puts. This input/output point of view plays an important role in this book, particularly in the later chapters. However, our starting point is different, more general, and, we claim, more adapted to modeling and more suitable for applications. As a paradigm for control, input/output or input/state/output models are often very suitable. Many control problems can be viewed in terms of plants that are driven by control inputs through actuators and feedback mecha- nisms that compute the control action on the basis of the outputs of sen- sors, as depicted in Figure P.5. However, as a tool for modeling dynamical systems, the input/output point of view is unnecessarily restrictive. Most physical systems do not have a preferred signal flow direction, and it is im- portant to let the mathematical structures reflect this. This is the approach taken in this book: we view systems as defined by any relation among dy-

xviii Preface

namic variables, and it is only when turning to control in Chapters 9 and 10, that we adopt the input/state/output point of view. The general model structures that we develop in the first half of the book are referred to as the behavioral approach. We now briefly explain the main underlying ideas.

We view a mathematical model as a subset of a universum of possibili- ties. Before we accept a mathematical model as a description of reality, all outcomes in the universum are in principle possible. After we accept the mathematical model as a convenient description of reality, we declare that only outcomes in a certain subset are possible. Thus a mathematical model is an exclusion law: it excludes all outcomes except those in a given subset. This subset is called the behavior of the mathematical model. Proceeding from this perspective, we arrive at the notion of a dynamical system as simply a subset of time-trajectories, as a family of time signals taking on values in a suitable signal space. This will be the starting point taken in this book. Thus the input/output signal flow graph emerges in general as a construct, sometimes a purely mathematical one, not necessarily implying a physical structure. We take the description of a dynamical system in terms of its behavior, thus in terms of the time trajectories that it permits, as the vantage point from which the concepts put forward in this book unfolds. We are especially interested in linear time-invariant differential systems: “linearity” means that these systems obey the superposition principle, “time-invariance” that the laws of the system do not depend explicitly on time, and “differential” that they can be described by differential equations. Specific examples of such systems abound: linear electrical circuits, linear (or linearized) me- chanical systems, linearized chemical reactions, the majority of the models used in econometrics, many examples from biology, etc.

Understanding linear time-invariant differential systems requires first of all an accurate mathematical description of the behavior, i.e., of the solution set of a system of differential equations. This issue—how one wants to define a solution of a system of differential equations—turns out to be more subtle than it may at first appear and is discussed in detail in Chapter 2. Linear time-invariant differential systems have a very nice structure. When we have a set of variables that can be described by such a system, then there is a transparent way of describing how trajectories in the behavior are generated. Some of the variables, it turns out, are free, unconstrained. They can thus be viewed as unexplained by the model and imposed on the system by the environment. These variables are called inputs. However, once these free variables are chosen, the remaining variables (called the outputs) are not yet completely specified. Indeed, the internal dynamics of the system generates many possible trajectories depending on the past history of the system, i.e., on the initial conditions inside the system. The formalization of these initial conditions is done by the concept of state. Discovering this