Additional Closed-Loop Frequency Response Material (Second edition, Chapter 14)

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1 Appendix J Additional Closed-Loop Frequency Response Material (Second edition, Chapter 4) APPENDIX CONTENTS J. Closed-Loop Behavior J.2 Bode Stability Criterion J.3 Nyquist Stability Criterion J.4 Gain and Phase Margins J.5 Closed-Loop Frequency Response and Sensitivity Functions J.5. Sensitivity Functions J.5.2 Bandwidth J.5.3 Closed-Loop Performance Criteria J.5.4 Nichols Chart J.6 Robustness Analysis J.6. Sensitivity Analysis J.6.2 Effect of Feedback Control on Sensitivity J.6.3 Robust Stability Summary Frequency response concepts and techniques play an important role in control system design and analysis. In icular, they are very useful for stability analysis, control system design, and robustness analysis. Historically, frequency response techniques provided the conceptual framework for early control theory and important applications in the field of communications (MacFarlane, 979). This chapter begins by presenting useful background information in Section J.. The Bode and Nyquist stability criteria in Sections J.2 and J.3 are generally applicable and, unlike the Routh stability criterion of Chapter, provide exact results for systems with time delays. These stability criteria also provide measures of relative stability, how close a system is to becoming unstable. Two useful metrics for relative stability, the gain and phase margins, are introduced in Section J.4. The frequency response of closed-loop systems is considered in Section J.5, followed by an introduction to robustness analysis in Section J.6. This last topic addresses the important question of the sensitivity of a control system to process variations and to uncertainty in the process model used to design the control system. J. CLOSED-LOOP BEHAVIOR In Chapters 2 and 3 we observed that control system design involves tradeoffs between conflicting objectives such as performance and robustness. Now we consider J

2 J2 Appendix J Additional Closed-Loop Fequency Response Material (Second edition, Chapter 4) these design objectives in more detail and introduce another issue, the effect of measurement noise. In general, a feedback control system should provide the following desirable characteristics (see Chapter 2):. Closed-loop stability 2. Good disturbance rejection (without excessive control action) 3. Fast set-point tracking (without excessive control action) 4. A satisfactory degree of robustness to process variations and model uncertainty 5. Low sensitivity to measurement noise The block diagram of a general feedback control system is shown in Fig. J.. It contains three external input signals: set point Y sp, disturbance D, and additive measurement noise, N. The noisy output Y n is the sum of the noise N and the noise-free output Y. The following analysis illustrates the fundamental limitations and engineering tradeoffs that are inherent in achieving these characteristics. The dynamic behavior of the closed-loop system in Fig. J. can be described by the following set of equations where G is defined as G! G v G p G m : Y = G d + G c G D - G c G + G c G N + K mg c G v G p + G c G Y sp E = - G dg m + G c G D - G m + G c G N + K m + G c G Y sp (J-) (J-2) U = - G dg m G c G v + G c G D - G mg c G v + G c G N + K mg c G v + G c G Y sp (J-3) These equations can be derived easily using the block diagram algebra of Chapter. They illustrate how the three external inputs (D, N, and Y sp ) affect three output variables: the actual output Y, the error E, and the controller output U. The nine transfer functions in (J-) to (J-3) completely characterize the closed-loop performance of D the control system. Because each transfer function has the same denominator, there is a single characteristic equation, G c G. Consequently, all nine closed-loop transfer functions have identical stability characteristics. We consider two examples to illustrate the insight provided by these equations. Example J. demonstrates that the control system design should never include the cancellation of a pole by a zero when both are located in the unstable region (that is, to the right of the imaginary axis). This undesirable situation is referred to as an unstable pole-zero cancellation. EXAMPLE J. Consider the feedback system in Fig. J. and the following transfer functions: Suppose that controller G c is designed to cancel the unstable pole in G p, as in the IMC design method: Evaluate closed-loop stability and characterize the output response for a sustained disturbance. The characteristic equation, G c G, becomes: or In view of the single root at s 2.5, it appears that the closed-loop system is stable. However, if we consider Eq. J- for N Y sp, G d Y = G p = G d = -.5-2s, G v = G m = + G c = 3( - 2s) s + - 3( - 2s) s + s =.5-2s = G d + G c G D = -.5(s + ) ( - 2s)(s + 2.5) D Y d Y sp K m Y sp + E P U Y Y G c G v G u + p + Y m G m Y n + + N Figure J. Block diagram with a disturbance D and measurement noise N.

3 J.2 Bode Stability Criterion J3 This transfer function has an unstable pole at s.5. Thus, the output response to a disturbance is unstable. Furthermore, other transfer functions in (J-) to (J-3) also have unstable poles. This apparent contradiction occurs because the characteristic equation does not include all of the information, namely, the unstable pole-zero cancellation. This example has demonstrated that even an exact cancellation of an unstable pole leads to instability. Consequently, an unstable pole should never be canceled with a right-half plane zero. In contrast, open-loop unstable systems can be stabilized with feedback control, as was demonstrated in Example.8. The potential problem of instability arising from an unstable pole-zero cancellation motivates the concept of internal stability. A closed-loop system is said to be internally stable if all of the closed-loop transfer functions in (J-) to (J-3) are stable (Goodwin et al., 2). This definition is equivalent to requiring that all signals in the feedback loop are bounded (Morari and Zafiriou, 989). For the rest of this book, we assume that no unstable pole-zero cancellations occur in the formation of G OL. The next example demonstrates that design tradeoffs inherent in specifying G c can be inferred from Eqs. J- to J-3. EXAMPLE J.2 Suppose that G d G p, G m K m and that G c is designed so that the closed-loop system is stable and GG c over the frequency range of interest. Evaluate this control system design strategy for set-point changes, disturbances, and measurement noise. Also consider the behavior of the manipulated variable, U. Because GG c, The first expression and (J-) suggest that the output response to disturbances will be very good, because Y/D L. Next, we consider set-point responses. From Eq. J-, Because G m K m, G G v G p K m and the above equation can be written as for GG c, + G c G L Y = K mg c G v G p Y sp + G c G Y Y sp = G c G + G c G Y Y sp L Thus, ideal (instantaneous) set-point tracking would occur. Choosing G c so that GG c also has an undesirable consequence. The output Y becomes sensitive to noise, because Y L N (see the noise term in Eq. J-). Thus, a design tradeoff is required between the set-point responses and sensitivity to noise. In the next section, we consider one of the most important and useful frequency response results, the Bode stability criterion. J.2 BODE STABILITY CRITERION The Bode stability criterion has two important advantages in comparison with the Routh stability criterion of Chapter :. It provides exact results for processes with time delays, while the Routh stability criterion provides only approximate results due to the polynomial approximation that must be substituted for the time delay. 2. The Bode stability criterion provides a measure of the relative stability rather than merely a yes or no answer to the question, Is the closed-loop system stable? Before considering the basis for the Bode stability criterion, it is useful to review the General Stability Criterion of Section.: A feedback control system is stable if and only if all roots of the characteristic equation lie to the left of the imaginary axis in the complex plane. Thus, the imaginary axis divides the complex plane into stable and unstable regions. Recall that the characteristic equation was defined in Chapter as + G OL (s) = (J-4) where the open-loop transfer function is G OL (s) G c (s)g v (s)g p (s)g m (s). The root locus diagrams of Fig. J.2 and Section.5 show how the roots of the characteristic equation change as controller gain K c changes. By definition, the roots of the characteristic equation are the numerical values of the complex variable s that satisfy Eq. J-4. Thus, each point on the root locus satisfies (J-5), which is a rearrangement of (J-4): G OL (s) = - The corresponding magnitude and argument are ƒg OL ( j)ƒ = and G OL ( j) = -8 (J-5) (J-6) In general, the ith root of the characteristic equation can be expressed as a complex number, r i a i b i j. Note that complex roots occur as complex conjugate

4 J4 Appendix J Additional Closed-Loop Fequency Response Material (Second edition, Chapter 4) K c = 2 Imaginary K c = 4 AR OL. K c = K c = 4 K c = 2 K c = K c = Real OL (deg) K c = 2 pairs. When a pair is located on the imaginary axis, the real is zero (a i ) and the closed-loop system is at the stability limit. As indicated in Chapter, this condition is referred to as marginal stability or conditional stability. When the closed-loop system is marginally stable and b i, the closed-loop response exhibits a sustained oscillation after a set-point change or a disturbance. Thus, the amplitude neither increases nor decreases. However, if K c is increased slightly, the closed-loop system becomes unstable, because the complex roots on the imaginary axis move into the unstable region. For a marginally stable system with b i, the frequency of the sustained oscillation, c, is given by c b i. This oscillatory behavior is caused by the pair of roots on the imaginary axis at s c j (see Chapter 3). Substituting this expression for s into Eq. J-6 gives the following expressions for a conditionally stable system: AR OL ( c ) = G OL ( j c )ƒ = K c = 4 Figure J.2 Root locus diagram for a process with three poles and no zeroes. OL ( c ) = G OL ( j c ) = -8 (J-7) (J-8) for some icular value of c. Equations (J-7) and (J-8) provide the basis for both the Bode stability criterion and the Nyquist stability criterion of Section J.3. Before stating the Bode stability criterion, we need to introduce two important definitions:. A critical frequency c is defined to be a value of for which OL () 8. This frequency is also referred to as a phase crossover frequency. 2. A gain crossover frequency g is defined to be a value of for which AR OL (). For K c, instability occurs if K c becomes more negative, that is, if K c becomes larger. In the subsequent analysis, assume that K c is positive, but the results are also valid for K c if K c is replaced by K c (radians/time) Figure J.3 Bode plot exhibiting multiple critical frequencies. For a marginally stable system, c g. For many control problems, there is only a single c and a single g. But multiple values can occur, as shown in Fig. J.3 for c. In this somewhat unusual situation, the closed-loop system is stable for two different ranges of the controller gain (Luyben and Luyben, 997). Consequently, increasing the absolute value of K c can actually improve the stability of the closed-loop system for certain ranges of K c. Next we state one of the most important results of frequency response analysis, the Bode stability criterion. It allows the stability of a closed-loop system to be determined from the open-loop transfer function. Bode Stability Criterion. Consider an open-loop transfer function G OL G c G v G p G m that is strictly proper (more poles than zeros) and has no poles located on or to the right of the imaginary axis, with the possible exception of a single pole at the origin. Assume that the open-loop frequency response has only a single critical frequency c and a single gain crossover frequency g. Then the closed-loop system is stable if AR OL ( c ). Otherwise, it is unstable. Some of the important properties of the Bode stability criterion are that. It provides a necessary and sufficient condition for closed-loop stability based on the properties of the open-loop transfer function. 2. Unlike the Routh stability criterion of Chapter, the Bode stability criterion is applicable to systems that contain time delays. 3. The Bode stability criterion is very useful for a wide variety of process control problems. However, for any G OL (s) that does not satisfy the required conditions, the Nyquist stability criterion of Section J.3 can be applied.

5 J.2 Bode Stability Criterion J5 D = G d Y d Y sp K m Y sp + E P U Y u + Y G c G v G p + Y m G m Figure J.4 Sustained oscillation in a feedback control system. 4. For systems with multiple c or g, the Bode stability criterion has been modified by Hahn et al. (2) to provide a sufficient condition for stability. In order to gain physical insight into why a sustained oscillation occurs at the stability limit, consider the analogy of an adult pushing a child on a swing. The child swings in the same arc so long as the adult pushes at the right time and with the right amount of force. Thus, the desired sustained oscillation places requirements on both timing (that is, phase) and applied force (that is, amplitude). By contrast, if either the force or the timing is not correct, the desired swinging motion ceases, as the child will quickly exclaim. A similar requirement occurs when a person bounces a ball. To further illustrate why feedback control can produce sustained oscillations, consider the following thought experiment for the feedback control system in Fig. J.4. Assume that the open-loop system is stable and that no disturbances occur (D ). Suppose that the set point is varied sinusoidally at the critical frequency, y sp (t) A sin( c t), for a long period, of time. Assume that during this period the measured output, y m, is disconnected so that the feedback loop is broken before the comparator. After the initial transient dies out, y m will oscillate at the excitation frequency c because the response of a linear system to a sinusoidal input is a sinusoidal output at the same frequency (see Section 4.2). Suppose that two events occur simultaneously: (i) the set point is set to zero, and (ii) y m is reconnected. If the feedback control system is marginally stable, the controlled variable y will then exhibit a sustained sinusoidal oscillation with amplitude A and frequency c. To analyze why this special type of oscillation occurs only when c, note that the sinusoidal signal E in Fig. J.4 passes through transfer functions G c, G v, G p, and G m before returning to the comparator. In order to have a sustained oscillation after the feedback loop is reconnected, signal Y m must have the same amplitude as E and 8 phase shift relative to E. Note that the comparator also provides 8 phase shift due to its negative sign. Consequently, after Y m passes through the comparator, it is in phase with E and has the same amplitude, A. Thus, the closed-loop system oscillates indefinitely after the feedback loop is closed because the conditions in Eqs. J-7 and J-8 are satisfied. But what happens if K c is increased by a small amount? Then, AR OL ( c ) is greater than one, the oscillations grow, and the closed-loop system becomes unstable. In contrast, if K c is reduced by a small amount, the oscillation is damped and eventually dies out. EXAMPLE J.3 A process has the third-order transfer function (time constant in minutes) 2 G p (s) = (.5s + ) 3 Also, G v. and G m. For a proportional controller, evaluate the stability of the closed-loop control system using the Bode stability criterion and three values of K c :, 4, and 2. For this example, G OL = G c G v G p G m = (K c )(.) 2 (.5s + ) 3 () = 2K c (.5s + ) 3 Figure J.5 shows a Bode plot of G OL for three values of K c. Note that all three cases have the same phase angle plot, because the phase lag of a proportional controller is zero for K c. From the phase angle plot, we observe that c 3.46 rad/min. This is the frequency of the sustained oscillation that occurs at the stability limit, as discussed above. Next, we consider the amplitude ratio AR OL for each value of K c. Based on Fig. J.5, we make the following classifications: K c AR OL (for c ) Classification.25 Stable 4 Marginally stable 2 5 Unstable

6 J6 Appendix J Additional Closed-Loop Fequency Response Material (Second edition, Chapter 4) AR OL OL (deg) In Section 2.5. the concept of the ultimate gain was introduced. For proportional-only control, the ultimate gain K cu was defined to be the largest value of K c that results in a stable closed-loop system. The value of K cu can be determined graphically from a Bode plot for transfer function G G v G p G m. For proportional-only control, G OL K c G. Because a proportional controller has zero phase lag, c is determined solely by G. Also, (J-9) where AR G denotes the amplitude ratio of G. At the stability limit, c, AR OL ( c ), and K c K cu. Substituting these expressions into (J-9) and solving for K cu gives an important result: K cu = (J-) AR G ( c ) The stability limit for K c can also be calculated for PI and PID controllers, as demonstrated by Example J.4. EXAMPLE J.4 Consider PI control of an overdamped second-order process (time constants in minutes), 5 G p (s) = (s + )(.5s + ) Determine the value of K cu. Use a Bode plot to show that controller settings of K c.4 and I.2 min produce an unstable closed-loop system. Find K cm, the maximum value of K c that can be used with I.2 min and still have closed-loop stability. Show that I min results in a stable closed-loop system for all positive values of K c. K c = AR OL () = K c AR G () c (rad/min) Figure J.5 Bode plots for G OL 2K c /(.5s ) 3. G m = G v = K c = 2 K c = 4 In order to determine K cu, we let G c K c. The open-loop transfer function is G OL K c G where G G v G p G m. Because a proportional controller does not introduce any phase lag, G and G OL have identical phase angles. Consequently, the critical frequency can be determined graphi- AR (deg) Figure J.6 Bode plots for Example J.4: Curve A: G P (s) cally from the phase angle plot for G. However, curve A in Fig. J.6 indicates that c does not exist, because OL is always greater than 8. As a result, K cu does not exist, and thus K c does not have a stability limit. Conversely, the addition of integral control action can produce closed-loop instability. Curve B in Fig. J.6 indicates that an unstable closed-loop system occurs for G c (s).4 ( /.2s), because AR OL when OL 8. To find K cm for I.2 min, we note that c depends on I, but not on K c, because K c has no effect on OL. For curve B in Fig. J.6, c 2.2 rad/min and the corresponding amplitude ratio is AR OL.38. To find K cm, multiply the current value of K c by a factor, /.38. Thus, K cm.4/ When I is increased to min, curve C in Fig. J.6 results. Because curve C does not have a critical frequency, the closed-loop system is stable for all positive values of K c. EXAMPLE J.5 Find the critical frequency for the following process and PID controller, assuming G v G m : G p (s) = Curve B: G OL (s); G c (s).4a +.2s b Curve C: G OL (s); G c (s).4a + s b e -.3s (9s + )(s + ) c (rad/min) G c (s) = 2a + 2.5s + sb Figure J.3 shows the open-loop amplitude ratio and phase angle plots for G OL. Note that the phase angle crosses 8 at three points. Because there is more than one value of c, the Bode stability criterion cannot be applied. However, the Nyquist stability criterion presented in Section J.3 can be used to determine stability. c a b a c b

7 J.3 Nyquist Stability Criterion J7 J.3 NYQUIST STABILITY CRITERION The Nyquist stability criterion is similar to the Bode criterion in that it determines closed-loop stability from the open-loop frequency response characteristics. Both criteria provide convenient measures of relative stability, the gain and phase margins, which will be introduced in Section J.4. As the name implies, the Nyquist stability criterion is based on the Nyquist plot for G OL (s), a polar plot of its frequency response characteristics (see Chapter 4). The Nyquist stability criterion does not have the same restrictions as the Bode stability criterion, because it is applicable to open-loop unstable systems and to systems with multiple values of c or g. The Nyquist stability criterion is the most powerful stability test that is available for linear systems described by transfer function models. The Nyquist stability criterion is based on two concepts from complex variable theory, contour mapping and the Principle of the Argument. We briefly review these concepts in Appendix K. More detailed descriptions are available elsewhere (Brown and Churchill, 28; Franklin et al., 22). We now present one of the most important results of frequency domain analysis: Nyquist Stability Criterion. Consider an open-loop transfer function G OL (s) that is proper and has no unstable pole-zero cancellations. Let N be the number of times that the Nyquist plot for G OL (s) encircles the (, ) point in the clockwise direction. Also let P denote the number of poles of G OL (s) that lie to the right of the imaginary axis. Then, Z N P where Z is the number of roots (or zeros) of the characteristic equation that lie to the right of the imaginary axis. The closed-loop system is stable if and only if Z. Some important properties of the Nyquist stability criterion are. It provides a necessary and sufficient condition for closed-loop stability based on the open-loop transfer function. 2. The reason that the (, ) point is so important can be deduced from the characteristic equation, G OL (s). This equation can also be written as G OL (s), which implies that AR OL and OL 8, as noted earlier. This point is referred to as the critical point. 3. Most process control problems are open-loop stable. For these situations, P, and thus Z N. Consequently, the closed-loop system is unstable if the Nyquist plot for G OL (s) encircles the critical point, one or more times. 4. A negative value of N indicates that the critical point is encircled in the opposite direction (counterclockwise). This situation implies that each countercurrent encirclement can stabilize one unstable pole of the open-loop system. 5. Unlike the Bode stability criterion, the Nyquist stability criterion is applicable to open-loop unstable processes. 6. Unlike the Bode stability criterion, the Nyquist stability criterion can be applied when multiple values of c or g occur (cf. Fig. J.3). Control system design based on Nyquist plots is described elsewhere (Golnaraghi and Kuo, 29; Luyben and Luyben, 997). Example J.6 illustrates the application of the Bode and Nyquist stability criteria. EXAMPLE J.6 Evaluate the stability of the closed-loop system in Fig. J. for The time constant and time delay have units of minutes, and Obtain c and K cu from a Bode plot. Let K c.5k cu and draw the Nyquist plot for the resulting open-loop system. G p (s) = 4e-s 5s + G v = 2, G m =.25, G c = K c The Bode plot for G OL and K c is shown in Fig. J.7. For c.69 rad/min, OL 8, and AR OL.235. For K c, AR OL AR G and K cu can be calculated from Eq. J-. Thus, K cu / Setting K c.5k cu gives K c The Nyquist plot for K c 6.38 is shown in Fig. J.8 (lowfrequency data for.4 have been omitted). Note that the point is encircled once. Applying the Nyquist stability criterion gives N, P, and Z. Thus, the larger value of K c causes the closed-loop system to become unstable. Only values of K c less than K cu result in a stable closed-loop system. AR OL OL (deg) c =.69 rad/min (rad/min) Figure J.7 Bode plots for Example J.6, K c.

8 J8 Appendix J Additional Closed-Loop Fequency Response Material (Second edition, Chapter 4) Imaginary 2 AR OL AR c = GM 4 2 (, ) 2 Real g c 2 4 OL (deg) g 8 Phase margin At this point, it is appropriate to summarize the relative advantages and disadvantages of the Bode and Nyquist plots. The Bode plot provides more information than the Nyquist plot, because the frequency is shown explicitly. In addition, it facilitates analysis over a wide range of frequencies due to its logarithmic frequency scale. Another advantage of the Bode plot is that it allows the open-loop frequency response characteristic to be graphically constructed from the characteristics for the individual transfer functions, G c, G v, G p, and G m, as shown in Chapter 4. The chief advantage of the Nyquist plot is that the Nyquist stability criterion is more widely applicable than the Bode stability criterion. J.4 GAIN AND PHASE MARGINS 6 Figure J.8 Nyquist plots for Example J.6: K c.5k cu If the process changes or the controller is poorly tuned, the closed-loop system can become unstable. Thus, it is useful to have quantitative measures of relative stability that indicate how close the system is to becoming unstable. The concepts of gain margin (GM) and phase margin (PM) provide useful metrics for relative stability. Let AR c be the value of the open-loop amplitude ratio at the critical frequency c. Gain margin GM is defined as GM! AR c (J-) According to the Bode stability criterion, AR c must be less than one for closed-loop stability. An equivalent stability requirement is that GM. The gain margin provides a measure of relative stability, because it indicates how much any gain in the feedback loop can increase before instability occurs. For example, if GM 2., either process gain K p or controller gain K c could be doubled and the closed-loop system would still be stable, though it probably would be very oscillatory. Next, we consider the phase margin. In Fig. J.9, g denotes the phase angle at the gain-crossover fre- Figure J.9 Gain and phase margins on a Bode plot. quency g where AR OL. Phase margin PM is defined as (J-2) The phase margin also provides a measure of relative stability. In icular, it indicates how much additional time delay can be included in the feedback loop before instability will occur. Denote the additional time delay as max. For a time delay of max, the phase angle is max (see Section 4.3.5). Thus, max can be calculated from the expression or PM! 8 + g PM = max g a 8 b max = a PM g ba 8 b (J-3) (J-4) where the (/8) factor converts PM from degrees to radians. Graphical representations of the gain and phase margins in Bode and Nyquist plots are shown in Figs. J.9 and J.. PM Imaginary GM Figure J. Gain and phase margins on a Nyquist plot. g c Real G OL ( j)

9 J.4 Gain and Phase Margins J9 The specification of phase and gain margins requires a compromise between performance and robustness. In general, large values of GM and PM correspond to sluggish closed-loop responses, while smaller values result in less sluggish, more oscillatory responses. The choices for GM and PM should also reflect model accuracy and the expected process variability. Guideline. In general, a well-tuned controller should have a gain margin between.7 and 4. and a phase margin between 3 and 45. Recognize that these ranges are approximate, and that it may not be possible to choose PI or PID controller settings that result in specified GM and PM values. Tan et al. (999) have developed graphical procedures for designing PI and PID controllers that satisfy GM and PM specifications. The GM and PM concepts are easily evaluated when the open-loop system does not have multiple values of c or g. However, for systems with multiple g, gain margins can be determined from Nyquist plots (Doyle et al., 992). AR (deg) Ziegler-Nichols Tyreus-Luyben 2 (rad/min) EXAMPLE J.7 For the FOPTD model of Example J.6, calculate PID controller settings for the two tuning relations in Table 2.6: (a) Ziegler-Nichols (b) Tyreus-Luyben Assume that the two PID controllers are implemented in the parallel form with a derivative filter (.) in Table 8.. Plot the open-loop Bode diagram and determine the gain and phase margins for each controller. For the Tyreus-Luyben settings, determine the maximum increase in the time delay max that can occur while still maintaining closed-loop stability. From Example J.6, the ultimate gain is K cu 4.25 and the ultimate period is P u 2/ min. Therefore, the PID controllers have the following settings: Controller I D Settings K c (min) (min) Ziegler-Nichols Tyreus-Luyben The open-loop transfer function is G OL = G c G v G p G m = G c 2e-s 5s + Figure J. shows the frequency response of G OL for the two controllers. The gain and phase margins can be determined by inspection of the Bode diagram or by using the MATLAB command margin (rad/min) Figure J. Comparison of G OL Bode plots for Example J.7. Controller GM PM c (rad/min) Ziegler-Nichols Tyreus-Luyben The Tyreus-Luyben controller settings are more conservative due to the larger gain and phase margins. The value of max is calculated from Eq. J-4 and the information in the above table: (76 )( rad) max = =.7 min (.79 rad/min)(8 ) Thus, time delay can increase by as much as 7% and still maintain closed-loop stability. Although the gain and phase margins provide useful metrics for robustness, they can give misleading results for unusual situations. For example, the Nyquist plot for a stable open-loop process in Fig. J.2 exhibits large GM and PM values, but the Nyquist curve passes very close to the critical point. Thus, the closed-loop system is not very robust and a small process perturbation could cause instability. This potential shortcoming of the gain and phase margins can be avoided by considering

10 J Appendix J Additional Closed-Loop Fequency Response Material (Second edition, Chapter 4) PM Imaginary GM the sensitivity functions that are introduced in the next section. The open-loop transfer function and its frequency response characteristics play a key role in control system design and analysis. In the loop-shaping approach, the controller is designed so that the open-loop transfer function has the desired characteristics. These techniques are described in books on advanced control theory (Doyle et al., 992; Skogestad and Postlethwaite, 25). J.5 CLOSED-LOOP FREQUENCY RESPONSE AND SENSITIVITY FUNCTIONS Real G OL ( j) Figure J.2 Nyquist plot where the gain and phase margins are misleading. The previous sections have demonstrated that openloop frequency response characteristics play a key role in the analysis of closed-loop stability and robustness. However, the closed-loop frequency response characteristics also provide important information, as discussed in this section. while T is the closed-loop transfer function for set-point changes (Y/Y sp ). It is easy to show that (J-6) As will be shown in Section J.6, S and T provide measures of how sensitive the closed-loop system is to changes in the process. Let S(j and T(j denote the amplitude ratios of S and T, respectively. The maximum values of the amplitude ratios provide useful measures of robustness. They also serve as control system design criteria, as discussed below. Define M S to be the maximum value of S(j for all frequencies: (J-7) The maximum value M S also has a geometrical interpretation (Åström and Hägglund, 995). Let G OL (s) G c (s)g(s). Then M S is the inverse of the shortest distance from the Nyquist plot for G OL (s) to the critical point. Thus, as M S decreases, the closed-loop system becomes more robust. The second robustness measure is M T, the maximum value of T(j): (J-8) M T is also referred to as the resonant peak. Typical amplitude ratio plots for S and T are shown in Fig. J.3. If the feedback controller includes integral action, offset is eliminated for set-point changes or sustained disturbances. Thus, at low frequencies, T(j) n and S(j) n, as shown in Fig. J.3. It is easy to prove that M S and M T are related to the gain and phase margins of Section J.4 (Skogestad and Postlethwaite, 25): GM Ú GM Ú + S + T = M S! max S( j)ƒ M T! max ƒt( j)ƒ M S M S -, PM Ú 2 sin- a 2M S b PM Ú 2 sin - a b M T 2M T (J-9) (J-2) J.5. Sensitivity Functions The following analysis is based on the block diagram in Fig. J.. Define G as G! G v G p G m and assume that G m K m and G d. Two important concepts are now defined: S! sensitivity function (J-5a) + G c G T! G cg complementary sensitivity function + G c G (J-5b) Comparing Fig. J. and Eq. J-5 indicates that S is the closed-loop transfer function for disturbances (Y/D), AR M T M S / 2 S( j) BW T( j) Figure J.3 Typical S and T magnitude plots. (Modified from Maciejowski (989).

11 J.5 Closed-Loop Frequency Response and Sensitivity Functions J.4.2 Closed-loop amplitude ratio M T =.25 2 Set-point step response T( j)..8.6 y Frequency p BW Time Figure J.4 Typical closed-loop amplitude ratio T(j) and set-point response. Designing a controller to have a specified value of M T or M S establishes lower bounds for GM and PM. For example, when M T.25, (J-2) indicates that GM.8 and PM 47, numerical values that satisfy the guidelines given in Section J.4. Equations J-9 and J-2 imply that better performance (corresponding to a larger value of M T ) is achieved at the expense of robustness, because the lower bounds for GM and PM in Eqs. J-9 and J-2 become smaller. In general, a satisfactory tradeoff between performance and robustness occurs for the following range of values (Åström et al., 998): Guidelines. For a satisfactory control system, M T should be in the range of..5, and M S should be in the range of J.5.2 Bandwidth In this section we introduce an important concept, the bandwidth. A typical amplitude ratio plot for T and the corresponding set-point response are shown in Fig. J.4. The bandwidth BW is defined as the frequency at which T(j) > The bandwidth indicates the frequency range for which satisfactory set-point tracking occurs. In icular, BW is the maximum frequency for a sinusoidal set point to be attenuated by no more than a factor of.77. The bandwidth is also related to speed of response. In general, the bandwidth is (approximately) inversely proportional to the closed-loop settling time. An alternative definition of the bandwidth is sometimes used. The bandwidth can also be defined as the frequency for which S(j).77 when S(j) crosses.77 from below, as shown in Fig. J.3 (Skogestad and Postlethwaite, 25). Fortunately, the two bandwidth definitions typically result in similar numerical values. Furthermore, the values of BW and the gain crossover frequency g are usually very close. The curves in Fig. J.4 are similar to the corresponding plots for an underdamped second-order system (see Sections 5.4 and 4.3.3). For Fig. 4.3 a peak amplitude ratio of.25 corresponds to a damping coefficient of.5 for an underdamped second-order system. This analogy provides support for the previous guideline that M T should have a value between. and.5. J.5.3 Closed-Loop Performance Criteria Ideally, a feedback controller should satisfy the following criteria:. In order to eliminate offset, T(j) S as S. 2. T(j) should be maintained at unity up to as high as frequency as possible. This condition ensures a rapid approach to the new steady state during a set-point change. 3. As indicated in the guidelines, M T should be selected so that. M T The bandwidth BW and the frequency T for M T should be as large as possible. Large values result in fast closed-loop responses. Satisfying these criteria typically requires a compromise. For example, the requirement that M T.5 means that the controller gain K c cannot be too large. However, smaller values of K c result in smaller values of BW and T. Next, we consider the desired closed-loop frequency response characteristics for disturbances. Ideally, we would like to have the closed-loop amplitude ratio S(j) be zero for all frequencies. However, this ideal situation is physically impossible for feedback control, and thus a more realistic goal is to minimize S(j) over as wide a frequency range as possible. According to the guideline, the controller should be designed so that.2 M S 2.. J.5.4 Nichols Chart The closed-loop frequency response can be calculated analytically from the open-loop frequency response. Again, consider Fig. J., and assume that the sensor dynamics are negligible so that G m (s) K m. From Fig. J.

12 J2 Appendix J Additional Closed-Loop Fequency Response Material (Second edition, Chapter 4) Open-loop amplitude ratio, AR OL 2 AR CL CL AR CL CL Open-loop phase angle, OL 9 8 Figure J.5 A Nichols chart. [The closed-loop amplitude ratio AR CL ( ) and phase angle CL ( ) are shown as families of curves.] and Eq. J-5b, it follows that the closed-transfer function for set-point changes Y/Y sp is a function of G OL, G OL Y = T = (J-2) Y sp + G OL where G OL! G c G v G p K m. Denote the open-loop frequency response characteristics by AR OL and OL, and the corresponding closed-loop quantities, AR CL and CL. Equation J-2 and the analytical techniques of Chapter 4 can be employed to derive analytical expressions for AR CL and CL as functions of AR OL and OL (Luyben and Luyben, 997). The Nichols chart in Fig. J.5 provides a graphical display of the closed-loop frequency response characteristics for set-point changes when G m (s) K m. Contours of constant AR CL and CL are shown on a plot of AR OL vs. OL. In a typical Nichols chart application, AR OL and OL are calculated from G OL (s) and plotted on the Nichols chart as a series of points. Then AR CL and CL are obtained by interpolation. For example, if AR OL and OL at a certain frequency, then interpolation of Fig. J.5 gives AR CL.76 and CL 5 for the same frequency. The Nichols chart can be generated in the MATLAB Control Toolbox by a single command, nichols. The Nichols chart served as a useful graphical technique for designing control systems prior to 96, but it has become less important now that control software is widely available. Control system designs based on Nichols charts, including applications where G m (s) K m, are described elsewhere (Golnaraghi and Kuo, 29; Franklin et al., 22). The following example illustrates PID controller design for an M S robustness constraint. EXAMPLE J.8 Consider a fourth-order process with a wide range of time constants that have units in min (Åström et al., 998): G = G v G p G m = (s + )(.2s + )(.4s + )(.8s + ) (J-22) Calculate PID controller settings based on following tuning relations in Chapter 2: (a) Ziegler-Nichols tuning (Table 2.4) (b) Tyreus-Luyben tuning (Table 2.4) (c) IMC tuning with c.25 min (Table 2.) (d) Simplified IMC (SIMC) tuning with c.25 min (Skogestad, 23) For s (c) and (d) use a second-order plus time-delay model derived from Skogestad s model approximation method (Section 6.3.). Determine sensitivity peaks M S and M T for each controller. Compare the closed-loop responses to step changes in the set-point and the disturbance using the parallel form of the PID controller without a derivative filter: G ' c (s) = K c c + (J-23) I s + Ds d Assume that G d (s) G(s). The Bode plot for G(s) is not shown but indicates that c.8 rad/min, P u 2/ c.562 min, and K cu /AR c 3.2. Applying Skogestad s model approximation procedure gives: G ' (s) = e -.28s (s + )(.22s + ) (J-24)

13 J.6 Robustness Analysis J3 Table J. Controller Settings and Peak Sensitivities for Example J.8 Controller K c I (min) D (min) M S M T Ziegler-Nichols Tyreus-Luyben IMC Simplified IMC y Time (min) Z-N The SIMC controller in Table J. was calculated using the tuning relations in Table 2.5 for I 8. The PID controller settings are compared in Table J.. The M S and M T values determined from the sensitivity plots are also summarized in Table J.. The controller settings and sensitivity values in Table J. indicate that the Z-N controller settings are the most aggressive and the IMC settings are the most conservative. In fact, the sensitivity values for Z-N controller are unacceptably large, according to the guidelines. The closed-loop responses for a unit set-point change at t and a step disturbance (d 5) at t 4 min are shown in Fig. J.6. The ZN controller provides the worst setpoint response but the best disturbance response. The IMC controller is quite sluggish, because the approximate SOPTD model has a relatively small time delay. Similar results were obtained for the FOPTD model in Example 2.4. The SIMC and T-L controllers provide the best overall performance of these four controllers. However, a PID controller with improved performance for this example (not shown) can be obtained using a Direct Synthesis approach based on disturbance rejection (Chen and Seborg, 22). T-L IMC SIMC Figure J.6 Closed-loop responses for Example 4.8. A set-point change occurs at t and a step disturbance at t 4 min. Example J.8 has demonstrated that a variety of PID controllers can have approximately the same M S value but different performance characteristics. Thus, although the guidelines for M S and M T provide useful limits concerning controller robustness, controller performance should also be considered. Robustness metrics such as M S, M T, GM, and PM should be evaluated in conjunction with controller design methods, especially the model-based techniques of Chapter 2. J.6 ROBUSTNESS ANALYSIS In order for a control system to function properly, it should not be unduly sensitive to small changes in the process or to inaccuracies in the process model, if a model is used to design the control system. A control system that satisfies this requirement is said to be robust or insensitive. It is very important to consider robustness as well as performance in control system design. First, we explain why the S and T transfer functions in Eq. J-5 are referred to as sensitivity functions. J.6. Sensitivity Analysis In general, the term sensitivity refers to the effect that a change in one transfer function (or variable) has on another transfer function (or variable). Suppose that G changes from a nominal value G to an arbitrary new value, G dg. This differential change dg causes T to change from its nominal value T to a new value, T dt. Thus, we are interested in the ratio of these changes, dt/dg, as well as in the ratio of the relative changes: dt/t dg/g! sensitivity (J-25) The sensitivity is of icular interest because it is dimensionless and independent of the units of G and T. It is evaluated for a specific condition such as G and T. We can write the sensitivity in an equivalent form: dt/t dg/g = a dt dg b G T (J-26) The derivative in (J-26) can be evaluated after substituting the definition of T in (J-5b): dt dg = G cs 2 (J-27)

14 J4 Appendix J Additional Closed-Loop Fequency Response Material (Second edition, Chapter 4) Substitute (J-27) into (J-26). Then substituting the definition of S in (J-5a) and rearranging gives the desired result: (J-28) Equation J-28 indicates that the sensitivity is equal to S. For this reason, S is referred to as the sensitivity function. In view of the important relationship in (J-6), T is called the complementary sensitivity function. J.6.2 Effect of Feedback Control on Sensitivity Next, we show that feedback reduces sensitivity by comparing the relative sensitivities for open-loop control and closed-loop control. By definition, openloop control occurs when the feedback control loop in Fig. J. is disconnected from the comparator. For this condition: (J-29) Substituting T OL for T in Eq. J-25 and noting that dt OL /dg G c gives: dt OL /T OL dg/g dt/t dg/g = + G c G = S a Y Y sp b OL = T OL! G c G = adt OL dg b G G = G T c OL G c G = (J-3) Thus, the sensitivity is unity for open-loop control and is equal to S for closed-loop control, as indicated by (J-28). Equation J-5a indicates that S if G c G p, which usually occurs over the frequency range of interest. Thus, we have identified one of the most important properties of feedback control: Feedback control makes process performance less sensitive to changes in the process. As indicated in the previous section, it would be desirable to make S very small at all frequencies. But this is not possible owing to the following integral constraint: The Bode Sensitivity Integral (Goodwin et al., 2). Consider a stable feedback control system with an open-loop transfer function, G OL (s) G c (s)g(s)e s. Assume that G(s) has no poles to the right of the imaginary axis. Then, L q ln S( j)ƒ d = (J-3) Thus, if S is small for a icular frequency range, it has to be large for other frequencies. This phenomenon is sometimes referred to as the waterbed effect, because pushing down on one of a waterbed causes another to rise. Similar integral constraints are available for specific situations, including systems with right-half plane poles and zeros (Skogestad and Postlethwaite, 25). J.6.3 Robust Stability The robustness of a control system can be analyzed theoretically if the degree of process variability (or model uncertainty) can be characterized. The uncertainty can be structured (for example, parameter variations) or unstructured (for example, variations in the transfer function or frequency response). In this section, we consider the important concept of robust stability. The related concept of robust performance is described elsewhere (Morari and Zafiriou, 989; Skogestad and Postlethwaite, 25). In robustness analysis, the unstructured uncertainty is often expressed as bounds on a nominal process model. For example, suppose that G ' is the process model used for control system design, and that G is the transfer function for the actual (but unknown) process. They can be related by an unstructured model uncertainty, (s): G(s) = [ + (s)]g ' (s) (J-32) Rearranging Eq. J-32 indicates that (s) can also be interpreted as the relative model error: (s) = G(s) - G' (s) G ' (s) (J-33) Robustness analysis is typically based on a magnitude bound for the uncertainty ( j)ƒ () (J-34) where () is a known function of frequency. The multiplicative uncertainty description in (J-32) provides the basis for an important robust stability result. The following theorem guarantees closed-loop stability for a specified degree of model uncertainty. Robust Stability Theorem. Consider the feedback control system in Fig. J. and a specific controller G c. Assume that (i) The closed-loop system is stable for the nominal closed-loop system that consists of G c and the nominal process model, G '. (ii) The magnitude of the unstructured uncertainty (s) in (J-33) is bounded by () in (J-34). Also assume that each G(s) in (J-32) that is generated by this uncertainty structure has the same number of right-half-plane poles. Then the closed-loop system is robustly stable for all G that satisfy (J-33) and (J-34), if and only if

15 J.6 Robustness Analysis J5 T ' ( j)ƒ 6 or, equivalently, () for all 7 also (J-35) allows determination of how much modeling error can be tolerated for a icular controller design, as is demonstrated by the following example. (J-36) where denotes the nominal complementary sensitivity function, T ' G c G ' /( + G c G ' ). The proof for this theorem is available in textbooks on advanced control such as Bélanger (995). Some important consequences of this theorem are (Skogestad and Postlethwaite, 25) that. The theorem has a graphical interpretation, as shown in Fig. J.7. Robust stability is guaranteed if the uncertainty region at each frequency, a circle with radius G OL (j)(), does not include the critical point at (, ). Figure J.7 shows the uncertainty region for one frequency; the circles for the other frequencies have been omitted. In many practical problems, not all of the perturbations allowed by Eq. J-34 are possible. For example, upper and lower limits on model parameters such as a gain or time constant produce uncertainty regions that are not circular, as they appear in Fig. J.7. For these situations, the robust stability theorem provides sufficient (but not necessary) conditions. Consequently, the results tend to be conservative, as will be demonstrated in Example J In general, when the uncertainty bound () is large, the controller design must be more conservative so that T ' ( j)ƒ will be small enough to satisfy Eqs. J-35 and J-36. In summary, if the conditions of this theorem are satisfied, robust stability is guaranteed for the assumed process uncertainty description in Eqs. J-32 to J-34. The theorem T ' T ' ( j)ƒ () 6 for all 7 PM Imaginary GM Real G OL ( j) Figure J.7 Graphical interpretation of the Robust Stability Theorem. EXAMPLE J.9 Consider the nominal process model: G ' (s) = The PID controller settings for the IMC method and c.25 are: K c.22, I 6 min, D.5 min and derivative filter =.. This value of c was chosen to provide a peak sensitivity value of.96 (Chen and Seborg, 22). (a) Suppose that a small measurement time constant m was neglected in developing the nominal model. Thus, the actual process transfer function G is given by G(s) = -.6( -.5s) s(3s + ) G ' (s) m s + Use the robust stability theorem to determine the largest value of m for which robust stability can be guaranteed. (b) Repeat (a) using an exact stability analysis. (c) Do the answers of (a) and (b) agree? If not, explain which estimate is more accurate and why the discrepancy occurs. (a) The complementary sensitivity function for the nominal model and the PID controller is T ' G c G ' = + G c G ' -.6( -.5s) a b(-.22)a + s(3s + ) 6s +.5s (.).5s + b = -.6( -.5s) + a b(-.22)a + s(3s + ) 6s +.5s (.).5s + b Simplifying gives T ' -9.66s s 2 +.s +.95 = 2.75s s s 2 +.s +.95 The model uncertainty expression is given by (J-33): (s) = G(s) - G' (s) - m s = (J-37) G ' (s) m s + The bound () in (J-34) can be set equal to the magnitude of (s): m () = ( j)ƒ = (J-38) 2 2 m 2 + The robust stability theorem can be used to determine the maximum value of m for which robust stability can be guaranteed. For example, a plot of T ' ( j)ƒ () vs. can be

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