《控制理论》课程教学资源（参考书籍）Feedback Control Theory_Chapter 01 Introduction.pdf_大学文库

Preface Striking developments have taken place since 1980 in feedback control theory.The subject has become both more rigorous and more applicable.The rigor is not for its own sake,but rat her that even in an engineering discipline rigor can lead to clarity and to methodical solutions to problems. The applicability is a consequence both of new problem formulations and new mathematical so- lutions to these problems.Moreover,computers and software have changed the way engineering design is done.These developments suggest a fresh presentat ion of the subject,one that exploits these new developments while emphasizing their connection with classical control. Control systems are designed so that certain designated signals,such as tracking errors and act uator inputs,do not exceed pre-specified levels.Hindering the achievement of this goal are uncertainty about the plant to be controlled (the mat hematical models that we use in represent ing real physical systems are idealizat ions)and errors in measuring signals(sensors can measure signals only to a certain accuracy).Despite the seemingly obv ious requirement of bringing plant uncertainty explicitly into control problems,it was only inthe early 1980s that control researchers re-established the link to the classical work of Bode and ot hers by formulating a tractable mat hematical notion of uncertainty in an input-output framework and developing rigorous mat hematical techniques to cope with it.This book formulates a precise problem,called the robust performance problem,with the goal of achieving specified signal levels in the face of plant uncertainty. The book is addressed to students in engineering who have had an undergraduate course in signals and systems,including an introduct ion to frequency-domain met hods of analyzing feedback control systems,namely,Bode plots and the Ny quist criterion.A prior course on state-space theory would be advantageous for some optional sections,but is not necessary.To keep the development elementary,the systems are single-input/single-output and linear,operating in continuous time. Chapters 1 to 7 are intended as the core for a one-semester senior course;they would need supplementing with additional examples.These chapters constit ute a basic treat ment of feedback design,containing a detailed formulation of the control design problem,the fundamental issue of performance/stability robust ness tradeoff,and the graphical design technique of loopshaping, suitable for benign plants (stable,minimum phase).Chapters 8 to 12 are more advanced and are intended for a first graduate course.Chapter 8 is a bridge to the latter half of the book, extending the loopshaping technique and connecting it with notions of optimality.Chapters 9 to 12 treat controller design via optimizat ion.The approach in these latter chapters is mat hematical rather than graphical,using elementary tools involving interpolat ion by analytic functions.This mat hematical approach is most useful for mult ivariable systems,where graphical techniques usually break down.Nevert heless,we believe the setting of single-input/single-output systems is where this new approach should be learned. There are many people to whom we are grateful for their help in this book:Dale Enns for sharing his expertise in loopshaping;Raymond Kwong and Boyd Pearson for class testing the book;and Munther Dahleh,Ciprian Foias,and Karen Rudie for reading earlier drafts.Numerous 逝

Preface Striking developments have taken place since in feedback control theory The sub ject has become both more rigorous and more applicable The rigor is not for its own sake but rather that even in an engineering discipline rigor can lead to clarity and to methodical solutions to problems The applicability is a consequence both of new problem formulations and new mathematical so lutions to these problems Moreover computers and software have changed the way engineering design is done These developments suggest a fresh presentation of the sub ject one that exploits these new developments while emphasizing their connection with classical control Control systems are designed so that certain designated signals such as tracking errors and actuator inputs do not exceed prespecied levels Hindering the achievement of this goal are uncertainty about the plant to be controlled the mathematical models that we use in representing real physical systems are idealizations and errors in measuring signals sensors can measure signals only to a certain accuracy Despite the seemingly obvious requirement of bringing plant uncertainty explicitly into control problems it was only in the early s that control researchers reestablished the link to the classical work of Bode and others by formulating a tractable mathematical notion of uncertainty in an inputoutput framework and developing rigorous mathematical techniques to cope with it This book formulates a precise problem called the robust performance problem with the goal of achieving specied signal levels in the face of plant uncertainty The book is addressed to students in engineering who have had an undergraduate course in signals and systems including an introduction to frequencydomain methods of analyzing feedback control systems namely Bode plots and the Nyquist criterion A prior course on statespace theory would be advantageous for some optional sections but is not necessary To keep the development elementary the systems are singleinputsingleoutput and linear operating in continuous time Chapters to are intended as the core for a onesemester senior course they would need supplementing with additional examples These chapters constitute a basic treatment of feedback design containing a detailed formulation of the control design problem the fundamental issue of performancestability robustness tradeo and the graphical design technique of loopshaping suitable for benign plants stable minimum phase Chapters to are more advanced and are intended for a rst graduate course Chapter is a bridge to the latter half of the book extending the loopshaping technique and connecting it with notions of optimality Chapters to treat controller design via optimization The approach in these latter chapters is mathematical rather than graphical using elementary tools involving interpolation by analytic functions This mathematical approach is most useful for multivariable systems where graphical techniques usually break down Nevertheless we believe the setting of singleinputsingleoutput systems is where this new approach should be learned There are many people to whom we are grateful for their help in this book Dale Enns for sharing his expertise in loopshaping Raymond Kwong and Boyd Pearson for class testing the book and Munther Dahleh Ciprian Foias and Karen Rudie for reading earlier drafts Numerous iii

CHAPTER INTRODUCTION It must be kept in mind that a control engineers role is not merely one of designing control systems for xed plants of simply wrapping a little feedback around an already xed physical system It also involves assisting in the choice and conguration of hardware by taking a system wide view of performance For this reason it is important that a theory of feedback not only lead to good designs when these are possible but also indicate directly and unambiguously when the performance ob jectives cannot be met It is also important to realize at the outset that practical problems have uncertain non minimumphase plants nonminimumphase means the existence of right halfplane zeros so the inverse is unstable that there are inevitably unmodeled dynamics that produce substantial un certainty usually at high frequency and that sensor noise and input signal level constraints limit the achievable benets of feedback A theory that excludes some of these practical issues can still be useful in limited application domains For example many process control problems are so dominated by plant uncertainty and right halfplane zeros that sensor noise and input signal level constraints can be neglected Some spacecraft problems on the other hand are so dominated by tradeos between sensor noise disturbance rejection and input signal level eg fuel consumption that plant uncertainty and nonminimumphase eects are negligible Nevertheless any general the ory should be able to treat all these issues explicitly and give quantitative and qualitative results about their impact on system performance In the present section we look at two issues involved in the design process deciding on perfor mance specications and modeling We begin with an example to illustrate these two issues Example A very interesting engineering system is the Keck astronomical telescope currently under construction on Mauna Kea in Hawaii When completed it will be the worlds largest The basic ob jective of the telescope is to collect and focus starlight using a large concave mirror The shape of the mirror determines the quality of the observed image The larger the mirror the more light that can be collected and hence the dimmer the star that can be observed The diameter of the mirror on the Keck telescope will be m To make such a large highprecision mirror out of a single piece of glass would be very dicult and costly Instead the mirror on the Keck telescope will be a mosaic of hexagonal small mirrors These segments must then be aligned so that the composite mirror has the desired shape The control system to do this is illustrated in Figure As shown the mirror segments are sub ject to two types of forces disturbance forces described below and forces from actuators Behind each segment are three pistontype actuators applying forces at three points on the segment to eect its orientation In controlling the mirrors shape it suces to control the misalignment between adjacent mirror segments In the gap between every two adjacent segments are capacitor type sensors measuring local displacements between the two segments These local displacements are stacked into the vector labeled y this is what is to be controlled For the mirror to have the ideal shape these displacements should have certain ideal values that can be precomputed these are the components of the vector r The controller must be designed so that in the closedloop system y is held close to r despite the disturbance forces Notice that the signals are vector valued Such a system is multivariable Our uncertainty about the plant arises from disturbance sources As the telescope turns to track a star the direction of the force of gravity on the mirror changes During the night when astronomical observations are made the ambient temperature changes The telescope is susceptible to wind gusts

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CHAPTER INTRODUCTION Models Before discussing the issue of modeling a physical system it is important to distinguish among four dierent ob jects Real physical system the one out there Ideal physical model obtained by schematically decomposing the real physical system into ideal building blocks composed of resistors masses beams kilns isotropic media Newtonian uids electrons and so on Ideal mathematical model obtained by applying natural laws to the ideal physical model composed of nonlinear partial dierential equations and so on Reduced mathematical model obtained from the ideal mathematical model by linearization lumping and so on usually a rational transfer function Sometimes language makes a fuzzy distinction between the real physical system and the ideal physical model For example the word resistor applies to both the actual piece of ceramic and metal and the ideal ob ject satisfying Ohms law Of course the adjectives real and ideal could be used to disambiguate No mathematical system can precisely model a real physical system there is always uncertainty Uncertainty means that we cannot predict exactly what the output of a real physical system will be even if we know the input so we are uncertain about the system Uncertainty arises from two sources unknown or unpredictable inputs disturbance noise etc and unpredictable dynamics What should a model provide It should predict the inputoutput response in suchaway that we can use it to design a control system and then be condent that the resulting design will work on the real physical system Of course this is not possible A leap of faith will always be required on the part of the engineer This cannot be eliminated but it can be made more manageable with the use of eective modeling analysis and design techniques Mathematical Models in This Book The models in this book are nitedimensional linear and timeinvariant The main reason for this is that they are the simplest models for treating the fundamental issues in control system design The resulting design techniques work remarkably well for a large class of engineering problems partly because most systems are built to be as close to linear timeinvariant as possible so that they are more easily controlled Also a good controller will keep the system in its linear regime The uncertainty description is as simple as possible as well The basic form of the plant model in this book is y P u n Here y is the output u the input and P the nominal plant transfer function The model uncertainty comes in two forms n unknown noise or disturbance unknown plant perturbation Both n and will be assumed to belong to sets that is some a priori information is assumed about n and Then every input u is capable of producing a set of outputs namely the set of all outputs P u n as n and range over their sets Models capable of producing sets of outputs for a single input are said to be nondeterministic There are two main ways of obtaining models as described next

1.1.ISSUES IN CONTROL SYSTEM DESIGN 5 Models from Science The usual way of getting a model is by applying the laws of physics,chemistry,and so on.Consider the Keck telescope example.One can write down differential equations based on physical principles (e.g.,Newton's laws)and making idealizing assumptions (e.g.,the mirror segments are rigid). The coefficients in the differential equations will depend on physical constants,such as masses and physical dimensions.These can be measured.This met hod of applying phy sical laws and taking measurements is most successful in electromechanical systems,such as aerospace vehicles and robots.Some systems are difficult to model in this way,either because they are too complex or because their governing laws are unknown. Models from Experimental Data The second way of getting a model is by doing experiments on the physical system.Let's start with a simple thought experiment,one that captures many essential aspects of the relationships between physical systems and their models and the issues in obtaining models from experimental data.Consider a real physical system-the plant to be controlled-with one input,u,and one output,y.To design a control system for this plant,we must understand how u affects y. The experiment runs like this.Suppose that the real phy sical system is in a rest state before an input u is applied (ie.,u=y=0).Now apply some input signal u,resulting in some output signal y.Observe the pair (u,y).Repeat this experiment several times.Pretend that these data pairs are all we know about the real phy sical system.(This is the black box scenario.Usually,we know somet hing about the internal work ings of the sy stem.) After doing this experiment we will notice several things.First,the same input signal at different times produces different output signals.Second,if we hold u=0,y will fluctuate in an unpredictable manner.Thus the real physical system produces just one output for any given input, so it itself is deterministic.However,we observers are uncertain because we cannot predict what that output will be. Ideally,the model should cover the data in the sense that it should be capable of producing every experimentally observed input-output pair.(Of course,it would be better to cover not just the data observed in a finite number of experiments,but anything that can be produced by the real physical system.Obviously,this is impossible.)If nondeterminism that reasonably covers the range of expected data is not built into the model,we will not trust that designs based on such models will work on the real system. In summary,for a useful theory of control design,plant models must be nondeterministic,having uncertainty built in explicitly. Synthesis Problem A synt hesis problem is a theoretical problem,precise and unambiguous.Its purpose is primarily pedagogical:It gives us something clear to focus on for the purpose of study.The hope is that the principles learned from study ing a formal synt hesis problem will be useful when it comes to designing a real control system. The most general block diagram of a control system is shown in Figure 1.2.The generalized plant consists of everything that is fixed at the start of the control design exercise:the plant, actuators that generate inputs to the plant,sensors measuring certain signals,analog-to-digital and digital-to-analog converters,and so on.The controller consists of the designable part:it may be an electric circuit,a programmable logic controller,a general-purpose computer,or some other

ISSUES IN CONTROL SYSTEM DESIGN Models from Science The usual way of getting a model is by applying the laws of physics chemistry and so on Consider the Keck telescope example One can write down dierential equations based on physical principles eg Newtons laws and making idealizing assumptions eg the mirror segments are rigid The coecients in the dierential equations will depend on physical constants such as masses and physical dimensions These can be measured This method of applying physical laws and taking measurements is most successful in electromechanical systems such as aerospace vehicles and robots Some systems are dicult to model in this way either because they are too complex or because their governing laws are unknown Models from Experimental Data The second way of getting a model is by doing experiments on the physical system Lets start with a simple thought experiment one that captures many essential aspects of the relationships between physical systems and their models and the issues in obtaining models from experimental data Consider a real physical systemthe plant to be controlledwith one input u and one output y To design a control system for this plant we must understand how u aects y The experiment runs like this Suppose that the real physical system is in a rest state before an input u is applied ie u y Now apply some input signal u resulting in some output signal y Observe the pair u y Repeat this experiment several times Pretend that these data pairs are all we know about the real physical system This is the black box scenario Usually we know something about the internal workings of the system After doing this experiment we will notice several things First the same input signal at dierent times produces dierent output signals Second if we hold u y will uctuate in an unpredictable manner Thus the real physical system produces just one output for any given input so it itself is deterministic However we observers are uncertain because we cannot predict what that output will be Ideally the model should cover the data in the sense that it should be capable of producing every experimentally observed inputoutput pair Of course it would be better to cover not just the data observed in a nite number of experiments but anything that can be produced by the real physical system Obviously this is impossible If nondeterminism that reasonably covers the range of expected data is not built into the model we will not trust that designs based on such models will work on the real system In summary for a useful theory of control design plant models must be nondeterministic having uncertainty built in explicitly Synthesis Problem A synthesis problem is a theoretical problem precise and unambiguous Its purpose is primarily pedagogical It gives us something clear to focus on for the purpose of study The hope is that the principles learned from studying a formal synthesis problem will be useful when it comes to designing a real control system The most general block diagram of a control system is shown in Figure The generalized plant consists of everything that is xed at the start of the control design exercise the plant actuators that generate inputs to the plant sensors measuring certain signals analogtodigital and digitaltoanalog converters and so on The controller consists of the designable part it may be an electric circuit a programmable logic controller a generalpurpose computer or some other