Chapter 9Tradeoffs and Limits ofPerformance9.1 IntroductionFundamental limits of feedback systems will be investigated in this chapter.We begin in Section 9.2 by discussing the basic feedback loop and typicalrequirements. This includes the ability to follow reference signals, effectsof load disturbances and measurement noise and the effects of process vari-ations. It turns out that these properties can be captured by a set of sixtransfer functions, called the Gang of Six. These transfer functions are in-troduced in Section 9.3. For systems where the feedback is restricted tooperate on the error signal the properties are characterized by a subset offour transfer functions, called the Gang of Four. Properties of systems witherror feedback and the more general feedback configuration with two degreesof freedom are also discussed in Section 9.3. It is shown that it is impor-tant to consider all transfer functions of the Gang of Six when evaluating acontrol system.Another interesting observation is that for systems with two degrees offreedom the problem of response to load disturbances can be treated sepa-rately. This gives a natural separation of the design problem into a designof a feedback and a feedforward system. The feedback handles process un-certainties and disturbances and the feedforward gives the desired responseto reference signals.Attenuation of disturbances are discussed in Section 9.4 where it isdemonstrated that process disturbances can be attenuated by feedback butthat feedback also feeds measurement noise into the system. It turns out143144 CHAPTER 9. TRADEOFFS AND LIMITS OF PERFORMANCEPSfrag replacementsFC P−1Σ Σ ΣreuvdxnyControllerP rocessFigure 9.1: Block diagram of a basic feedback loop.that the sensitivity function which belongs to the Gang of Four gives a nicecharacterization of disturbance attenuation. The effects of process variationsare discussed in Section 9.5. It is shown that their effects are well describedby the sensitivity function and the complementary sensitivity function. Theanalysis also gives a good explanation for the fact that control systems canbe designed based on simplified models. When discussing process varia-tions it is natural to investigate when two processes are similar from thepoint of view of control. This important nontrivial problem is discussedin Section ??. Section 9.6 is devoted to a detailed treatment of the sen-sitivity functions. This leads to a deeper understanding of attenuation ofdisturbances and effects of process variations. A fundamental result of Bodewhich gives insight into fundamental limitations of feedback is also derived.This result shows that disturbances of some frequencies can be attenuatedonly if disturbances of other frequencies are amplified. Tracking of referencesignals are investigated in Section ??. Particular emphasis is given to precisetracking of low frequency signals. Because of the richness of control systemsthe emphasis on different issues varies from field to field. This is illustratedin Section ?? where we discuss the classical problem of design of feedbackamplifiers.9.2 The Basic Feedback LoopA block diagram of a basic feedback loop is shown in Figure 9.1. The systemloop is composed of two components, the process P and the controller. Thecontroller has two blocks the feedback block C and the feedforward blockF . There are two disturbances acting on the process, the load disturbance dand the measurement noise n. The load disturbance represents disturbancesthat drive the process away from its desired behavior. The process variable x9.2. THE BASIC FEEDBACK LOOP 145is the real physical variable that we want to control. Control is based on themeasured signal y, where the measurements are corrupted by measurementnoise n. Information about the process variable x is thus distorted by themeasurement noise. The process is influenced by the controller via thecontrol variable u. The process is thus a system with three inputs andone output. The inputs are: the control variable u, the load disturbanced and the measurement noise n. The output is the measured signal. Thecontroller is a system with two inputs and one output. The inputs are themeasured signal y and the reference signal r and the output is the controlsignal u. Note that the control signal u is an input to the process and theoutput of the controller and that the measured signal is the output of theprocess and an input to the controller. In Figure 9.1 the load disturbancewas assumed to act on the process input. This is a simplification, in realitythe disturbance can enter the process in many different ways. To avoidmaking the presentation unnecessarily complicated we will use the simplerepresentation in Figure 9.1. This captures the essence and it can easily bemodified if it is known precisely how disturbances enter the system.More Abstract RepresentationsThe block diagrams themselves are substantial abstractions but higher ab-stractions are sometimes useful. The system in Figure 9.1 can be representedby only two blocks as shown in Figure 9.2. There are two types of inputs,the control u, which can be manipulated and the disturbances w = (r, d, n),which represents external influences on the closed loop systems. The out-puts are also of two types the measured signal y and other interesting signalsz = (e, v, x). The representation in Figure 9.2 allows many control vari-ables and many measured variables, but it shows less of the system structurethan Figure 9.1. This representation can be used even when there are manyinput signals and many output signals. Representation with a higher levelof abstraction are useful for the development of theory because they makeit possible to focus on fundamentals and to solve general problems with awide range of applications. Care must, however, be exercised to maintainthe coupling to the real world control problems we intend to solve.DisturbancesAttenuation of load disturbances is often a primary goal for control. Thisis particularly the case when controlling processes that run in steady state.Load disturbances are typically dominated by low frequencies. Consider146 CHAPTER 9. TRADEOFFS AND LIMITS OF PERFORMANCEPSfrag replacementsCPyuzwFigure 9.2: An abstract representation of the system in Figure 9.1. Theinput u represents the control signal and the input w represents the referencer, the load disturbance d and the measurement noise n. The output y is themeasured variables and z are internal variables that are of
View Full Document
Unlocking...