Chapter 9Robustness and PerformanceQuotationAuthors, citation.This chapter treats robustness and performance. It begins with analysisof a simple controller. It is shown that seemingly reasonable design choicesgives a closed loop system that is extremely sensitivity to parameter varia-tions. New concepts, which give intuition and make it possible to character-ize robustness and performance quantitatively, are introduced. Propertiesthat fundamentally limits achievable performance are also discussed.9.1 IntroductionFundamental properties of feedback systems will be explored in this chap-ter. There are many requirements on a control system the ability to followreference signals, to suppress external disturbances and effects if measure-ment noise and process variations. It is important to understand all theseissues both to be able to analyze, and specify a system. Trade-offs betweenrobustness and performance is a key issue for design.Robustness is the ability of the closed loop system to be insensitive tocomponent variations. It is one of the most useful properties of feedback.Robustness is also what make it possible to design feedback system basedon strongly simplified models. It is therefore essential to have a good un-derstanding of robustness and to have ways of expressing it quantitatively.One of the key questions is to describe variations in system dynamics. Usingstate space concepts this can be done by varying the parameters of a system.There are however many variations that are not captured by such an ap-proach. For example, some dynamic phenomena many have been neglected.223224 CHAPTER 9. ROBUSTNESS AND PERFORMANCEIf they were included the model would have had more states. There may alsobe small time delays that have been neglected in a model. For linear modelvariations are well captured by transfer functions. Using transfer functionsit is possible make perturbations that correspond to additional dynamics aswell as time delays.It is also necessary to have quantitative ways to express how well afeedback system performs. Measures of performance and robustness areclosely related. The sensitivity function introduced in Section ?? expressesboth how well disturbances are affected by feedback and how sensitive theclosed loop system is to small perturbations of the process dynamics. Thefact that similar concepts are used makes it easy to make trade-offs betweenrobustness and performance.The structure of a feedback controller is another fundamental issue.There is a sharp distinction between two classes of systems. In systemswith error feedback only the error signal is accessible through the sensors. Atypical example is track following in a CD player where only the deviationfrom the track is measured. In other systems both the reference and theprocess output are available for measurement. It is then possible to com-pletely separate command signal following from robustness and disturbanceattenuation by using a controller with two degrees of freedom. The feedbackis designed to deal with disturbances and robustness and feedforward is usedto obtain the desired response to command signals. It is possible to have acomplete decoupling of command signal following from disturbance atten-uation and robustness. For systems with error feedback all issues must bedeal with using feedback.When specifying the performance of a control system it is common prac-tice to relate it to how well the system is able to follow command signals.This is not sufficient. For linear systems where the controller has error feed-back it is necessary to consider four transfer functions, the Gang of Four, tohave a complete understanding of the behavior of the system. For a systemwith a controller having two degrees of freedom it is necessary to considersix transfer functions, the Gang of Six, to completely specify the behaviorof the system. The specifications should also reflect this. It is interestingthat in spite of all complications specifications for many problems can becaptured by a few parameters.Some systems are intrinsically difficult to control, typical examples areunstable systems with time delay. It is important to understand the un-derlying reasons and to have some feel for how they relate to fundamentalsystem properties and to sensing and actuation. If the difficulties can bespotted at an early stage of the design they can be remedied by moving or9.2. AN EXAMPLE 225adding sensors and actuators or by redesigning the system. This is one ofthe strong reasons for investigating dynamics and control at an early stageof a design. It is also a reason why design engineers should be aware ofdynamics and control.9.2 An ExampleWe will begin by a simple applications of controller design using state feed-back and observers. Consider the systemdxdt= Ax + Bu =·−1 01 0¸x +·a − 11¸uy = Cx =£0 1¤y.(9.1)The system has the transfer functionGP(s) = C[sI − A]−1B =s + as(s + 1)(9.2)State FeedbackWe will begin by designing a state feedback assuming all states can bemeasured. The reachability matrixWr=·a − 1 −a + 11 a − 1¸has the determinant det Wr= a(a − 1). The system is thus reachable if ais neither 0 nor 1. A state feedback will first be designed assuming that allstates are measured. The feedbacku = −l1x1− l2x2= −Lx (9.3)gives the closed loop systemdxdt= (A − BL)xwhereA − BL =·−1 01 0¸−·a − 11¸£l1l2¤=·−1 − (a −1)l1−(a − 1)l21 − l1−l2¸.226 CHAPTER 9. ROBUSTNESS AND PERFORMANCEThis matrix has the characteristic polynomialdet·s + 1 + (a −1)l1(a − 1)l2l1− 1 s + l2¸= s2+ (1 + (a −1)l1+ l2)s + al2.Assume that a closed loop system with the characteristic polynomials2+ 2ζcωcs + ω2cis desired. Equating coefficients of equal power of s gives1 + (a −1)l1= 2ζcωcal2= ω2c.Solving this equation gives the following feedback gainsl1=2aζcωc− a − ω2ca(a − 1)l2=ω2ca(9.4)Notice that the gains become infinite for a = 0 and a = 1 when the systemslooses reachability.An ObserverNext we will design an observer for the system. The observability matrix isWo=·0 11 0¸This matrix has full rank and there are thus no restrictions in designing theobserver. The observer is given bydˆxdt= Ax + Bu + K(y −Cx) =·−1 01 0¸ˆx +·a − 11¸u +·k1k2¸(y −£0 1¤ˆx)(9.5)It follows from Equations (9.1) and (9.5) that the observer error is given byd˜xdt= (A − KC)˜x =·−1 01 0¸˜x −·k1k2¸£0 1¤˜x =·−1 −k11 −k2¸˜xThis system has the characteristic
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