Title PageContents1. Introduction1.1 Introduction1.2 A Brief HistoryThe Centrifugal GovernorThe Emergence of ControlThe Second Wave1.3 Process Control1.4 Manufacturing1.5 Robotics1.6 Power1.7 Aeronautics1.8 Electronics and Communications1.9 Automotive1.10 Computing1.11 Mathematics1.12 Physics1.13 Biology1.14 Summary2. Feedback2.1 Introduction2.2 Simple Forms of Feedback2.3 Representations of Feedback Systems2.4 Properties of Feedback2.5 Stability2.6 Open and Closed Loop Systems2.7 Feedforward2.8 Summary3. Dynamics3.1 Introduction3.2 Two views on dynamics3.3 Ordinary Differential Equations3.4 Laplace Transforms3.5 Frequency Response3.6 State Models3.7 Linear Time-Invariant Systems3.8 Summary4. Simple Control Systems5. Feedback Fundamentals6. PID Control7. Specifications8. Feedforward Design9. State FeedbackIndex7Specifications7.1 IntroductionIn this chapter we will discuss how the properties of a control system canbe specified. This is important for control design because it gives the goals.It is also important for users of control so that they know how to specify,evaluate and test a system so that they know it will have the desiredproperties. Specifications on a control systems typically include: stabilityof the closed loop system, robustness to model uncertainty, attenuation ofmeasurement noise, injection of measurement noise, and ability to followreference signals. From the results of Chapter 5 it follows that theseproperties are captured by six transfer functions called the Gang of Six.The specifications can be expressed in terms of these transfer functions.Essential features of the transfer functions can be expressed in terms oftheir poles and zeros or features of time and frequency responses.7.2 Stability and Robustness to Process VariationsStability and robustness to process uncertainties can be expressed interms of the loop transfer function L= PC, the sensitivity function andthe complementary sensitivity functionS=11 + PC=11 + L, T =PC1 + PC=L1 + L.Since both S and T are functions of the loop transfer function specifica-tions on the sensitivities can also be expressed in terms of specifications onthe loop transfer function L. Many of the criteria are based on Nyquist’s252From _Control System Design_by Karl Johan Åström, 2002Copyright 2002, Karl Johan Åström. All rights reserved.Do not duplicate or redistribute.7.2 Stability and Robustness to Process VariationsFigure 7.1 Nyquist curve of the loop transfer function L with indication of gain,phase and stability margins.stability criterion, see Figure 7.1. Common criteria are the maximum val-ues of the sensitivity functions, i.e.Ms= maxωhS(iω)h, Mt= maxωhT(iω)hRecall that the number 1/Msis the shortest distance of the Nyquist curveof the loop transfer function to the critical point, see Figure 7.1. Also recallthat the closed loop system will remain stable for process perturbations∆P provided thath∆P(iω)hhP(iω)h≤1hT (iω)h,see Section 5.5. The largest value Mtof the complementary sensitivityfunction T is therefore a simple measure of robustness to process varia-tions.Typical values of the maximum sensitivities are in the range of 1 to 2.Values close to one are more conservative and values close to 2 correspondto more aggressive controllers.Gain and Phase MarginsThe gain margin gmand the phase marginϕmare classical stability cri-teria. Although they can be replaced by the maximum sensitivities it isuseful to know about them because they are still often used practically.253Chapter 7. SpecificationsThe gain margin tells how much the gain has to be increased before theclosed loop system becomes unstable and the phase margin tells how muchthe phase lag has to be increased to make the closed loop system unstable.The gain margin can be defined as follows. Letω180be the lowestfrequency where the phase lag of the loop transfer function L(s) is 180○.The gain margin is thengm=1hL(iω180)h(7.1)The phase margin can be defined as follows. Letωgcdenote gaincrossover frequency, i.e. the lowest frequency where the loop transferfunction L(s) has unit magnitude. The phase margin is then given byϕm=π+ arg L(iωgc)(7.2)The margins have simple geometric interpretations in the Nyquist dia-gram of the loop transfer function as is shown in Figure 7.1. Notice thatan increase of controller gain simply expands the Nyquist curve radially.An increase of the phase of the controller twists the Nyquist curve clock-wise, see Figure 7.1.Reasonable values of the margins are phase marginϕm= 30○− 60○,gain margingm= 2 − 5. Since it is necessary to specify both margins tohave a guarantee of a reasonable robustness the marginsgmandϕmcanbe replaced by a single stability margin, defined as the shortest distanceof the Nyquist curve to the critical point−1, this distance is the inverseof the maximum sensitivity Ms. It follows from Figure 7.1 that both thegain margin and the phase margin must be specified in order to ensurethat the Nyquist curve is far from the critical point. It is possible tohave a system with a good gain margin and a poor phase margin andvice versa. It is also possible to have a system with good gain and phasemargins which has a poor stability margin. The Nyquist curve of the looptransfer function of such a system is shown in Figure 7.2. This system hasinfinite gain margin, a phase margin of 70○which looks very reassuring,but the maximum sensitivity is Ms= 3.7 which is much too high. Sinceit is necessary to specify both the gain margin and the phase margin toendure robustness of a system it is advantageous to replace them by asingle number. A simple analysis of the Nyquist curve shows that thefollowing inequalities hold.gm≥MsMs− 1φm≥ 2 arcsin12Ms(7.3)2547.2 Stability and Robustness to Process VariationsFigure 7.2 Nyquist curve of the loop transfer function for a system with good gainand phase margins but with high sensitivity and poor robustness. The loop transferfunction is L(s)=0.38(s2+0.1s+0.55)s(s+1)(s2+0.06s+0.5.A controller with Ms= 2 thus has a gain margin of at least 2 and a phasemargin of at least 30○. With Ms= 1.4 the margins are gm≥ 3.5andφm≥45○.Delay MarginThe gain and phase margins were originally developed for the case whenthe Nyquist curve only intersects the unit circle and the negative real axisonce. For more complicated systems there may be many intersections andit is more complicated to find suitable concepts that capture the idea of astability margin. One illustration
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