CHA PTER 10
CHAPTER1 0Cascade ControlObjectives of the Chapter To review classical cascade control. To present an alternate way of thinking about cascade control that leads to improvedperformance. To introduce controller design methods that accommodate process uncertainty.Prerequisite ReadingChapter 3, “One-Degree of Freedom Internal Model Control”Chapter 4, “Two-Degree of Freedom Internal Model Control”Chapter 5, “MSF Implementations of IMC Systems”Chapter 6, “PI and PID Controller Parameters from IMC Design”Chapter 7, “Tuning and Synthesis of 1DF IMC Controllers for Uncertain Processes”Chapter 8, “Tuning and Synthesis of 2DF Control Systems”241
242Cascade Control Chapter 1010.1 INTRODUCTIONCascade control can improve control system performance over single-loop control whenevereither: (1) Disturbances affect a measurable intermediate or secondary process output thatdirectly affects the primary process output that we wish to control; or (2) the gain of thesecondary process, including the actuator, is nonlinear. In the first case, a cascade controlsystem can limit the effect of the disturbances entering the secondary variable on theprimary output. In the second case, a cascade control system can limit the effect of actuatoror secondary process gain variations on the control system performance. Such gainvariations usually arise from changes in operating point due to setpoint changes or sustaineddisturbances.A typical candidate for cascade control is the shell and tube heat exchanger ofFigure 10.1.SteamFRPRTRFeedEffluentCondensateFigure 10.1 A shell and tube heat exchanger.The primary process output is the temperature of the tube side effluent stream. Thereare two possible secondary variables, the flow rate of steam into the exchanger and thesteam pressure in the exchanger. The steam flow rate affects the effluent temperaturethrough its effect on the steam pressure in the exchanger. The steam pressure in theexchanger affects the effluent temperature by its effect on the condensation temperature ofthe steam. Therefore, either the steam flow rate or the steam pressure in the exchanger canbe used as the secondary output in a cascade control system. The choice of which to usedepends on the disturbances that affect the effluent temperature.If the main disturbance is variations in the steam supply pressure, due possibly tovariable steam demands of other process units, then controlling the steam flow with thecontrol valve is most likely to be the best choice. Such a controller can greatly diminish theeffect of steam supply pressure variations on the effluent temperature. However, it is still
10.1 Introduction243necessary to have positive control of the effluent temperature to be able to track effluenttemperature setpoint changes and to reject changes in effluent temperature due to feedtemperature and flow variation. Since there is only one control effort, the steam valve stemposition, traditional cascade control uses the effluent temperature controller to adjust thesetpoint of the steam flow controller, as shown in Figure 10.2.SteamFlowFRC teFigure 10.2 Cascade control of effluent temperature via steam flow control.If feed flow and temperature variations are significant, then these disturbances can beat least partially compensated by using the exchanger pressure rather than the steam flow asthe secondary variable in a cascade loop, as shown in Figure tFeedEffluentCondensateFigure 10.3 Cascade control of effluent temperature via shell side pressure control.
244Cascade Control Chapter 10The trade-off in using the configuration of Figure 10.3 rather than that of Figure 10.2is that the inner control loop from the steam pressure to the valve stem position may notsuppress variations in valve gain as well as with an inner loop that uses the valve to controlthe steam flow rate. This consideration relates to using a cascade control system to suppressthe effect of process uncertainty, in this case the valve gain, on the control of the primaryprocess variable, the effluent temperature. We will have a lot more to say about usingcascade control systems to suppress process uncertainty in the following sections.To repeat, cascade control has two objectives. The first is to suppress the effect ofdisturbances on the primary process output via the action of a secondary, or inner controlloop around a secondary process measurement. The second is to reduce the sensitivity of theprimary process variable to gain variations of the part of the process in the inner controlloop.As we shall demonstrate, cascade control can be usefully applied to any process wherea measurable secondary variable directly influences the primary controlled variable throughsome dynamics. We will also demonstrate that despite frequent literature statements to thecontrary, inner loop dynamics do not have to be faster than the outer loop dynamics.However, the traditional cascade structure and tuning methods must be modified in order forcascade control to achieve its objectives when the inner loop process has dynamics that areon the order of, or slower than, the primary process dynamics.10.2 CASCADE STRUCTURES AND CONTROLLER DESIGNSFigure 10.4 shows the traditional PID cascade control system block diagram (Seborg et al.,1989). This is the cascade structure associated with figures 10.2 and 10.3. For Figure 10.2,the secondary process variable y2 is the steam flow rate, while for Figure 10.3, it is the shellside steam pressure. In both cases, the primary variable y1 is the effluent tionp2y2p1ProcessInner loopOuter loopFigure 10.4 Traditional cascade block diagram.y1
10.2 Cascade Structures and Controller Designs245One of the objectives of this section is to present methods for obtaining the parametersof the PID controllers of Figure 10.4 from a well-designed and well-tuned IMC cascadecontrol system, just as we did for single-loop control systems in Chapter 6.Figure 10.5 shows an IMC cascade block diagram that accomplishes the sameobjectives as Figure 10.4. There are other equivalent IMC cascade structures to that givenby Figure 10.5 (Morari and Zafiriou, 1989). However, the configuration of Figure 10.5 isconvenient because it suggests that controller q2 should be designed and tuned solely tosuppress the effect of the disturbance d2 on the primary output y1, and also convenientbecause both controller outputs u1 and u2 enter directly into the actuator. As we shall seelater, this last point facilitates dealing with control effort saturation. However, for theremainder of this section we shall ignore the saturation block in order to study the designand tuning of IMC controllers. These IMC controllers will then be used to obtain the PIDcontroller parameters in Figure 10.4, as was done in Chapter 6 for single-loop controlsystems.d1d2r q1 ( s )u1usuy2p2 ( s ) p2 ( s )u2 y1p1 ( s ) p1 ( s ) d2q2 ( s ) d1Figure 10.5 IMC cascade structure.From Figure 10.5, the transfer functions between the inputs to the inner loop, u1 andd2, and the secondary process output y2 arep ( s )u1 ( s ) (1 p 2 ( s )q 2 ( s ))d 2 ( s ).(10.1)y2 (s) 2(1 ( p ( s ) p ( s ))q ( s )222The transfer functions between the setpoint and disturbances and the primary process outputy1 arep p q r ( s ) (1 p 2 q 2 ) p1d 2 ( s ) (1 p1 p 2 q1 ( p2 p 2 ) q 2 ) d1 ( s ).(10.2)y1 ( s ) 1 2 1 (1 ( p p ) p q ( p p )q )112 1222
246Cascade Control Chapter 10In Eq. (10.2) we have suppressed the dependency of all transfer functions on the Laplacevariable s to keep the equation on one line. Based on equations (10.1) and (10.2) we observethe following:(1) If the lag time constants of the primary process p1(s) are large relative to those ofthe secondary process p2(s) then the inner loop controller q2(s) should be chosen so that thezeros of (1 p2 ( s )q2 ( s )) cancel the small poles (i.e., large time constants) of p1 ( s ) as outlined in Chapter 4. Otherwise, q2(s) should simply invert a portion of p2 ( s ) as describedin chapters 3 and 7.(2) The outer loop controller q1(s) should approximately invert the entire processmodel p1 p2 ( s ) , as described in chapters 3 and 7.(3) The IMCTUNE software can be used to design and tune both q1(s) and q2(s).We recommend tuning q2(s) with the outer loop open, and then tuning q1(s) with theinner loop closed. That is, first find the filter time constant ε2 for q2(s), and then find ε1 forq1(s). According to the denominator of Eq. (10.2), the tunings for q1(s) and q2(s) interact.Therefore, some adjustment of ε2 may be necessary after obtaining ε1.Having obtained the IMC controllers for Figure 10.5, we would like to use thesecontrollers to obtain the PID controllers in Figure 10.4 in a manner similar to that for singleloop controllers described in Chapter 6. Unfortunately, however, we can do so only veryapproximately. Figure 10.5 can be rearranged, ignoring the saturation block, as given byFigure 10.6.d1d2rq1q 2 1 ( s )q2 ( s )(1 p2 q2 ( s ))p2 ( s )y2p1 ( s )y1 p1 ( s ) d1Figure 10.6 IMC cascade control with a simple feedback inner loop.The controller given by q2 ( s ) /(1 p2 ( s )q2 ( s )) can often be well approximated by aPID controller, as described in Chapter 6. Again, IMCTUNE can be used to obtain thiscontroller. However, obtaining PID1 in Figure 10.4 is not so straightforward. Collapsing thefeedback loop through p1 ( s ) , while leaving the inner loop alone, yields Figure 10.7.
10.2 Cascade Structures and Controller Designsd2rC1 ( s )q 2 1 ( s )247d1(1 ( p2 ( s) p2 ( s)) q2 ( s))(1 p1 ( s ) p2 ( s)q1 ( s) ( p2 ( s ) p2 ( s )) q2 ( s ))q2 ( s )(1 p2 q2 ( s ))p2 ( s )y2y1p1 ( s )C1 ( s ) q1 ( s)(1 ( p2 ( s ) p2 ( s)) q2 ( s )) /(1 ( p2 ( s ) p2 ( s)) q2 ( s ) p1 ( s ) p2 ( s )q1 ( s ))Figure 10.7 Standard feedback form of Figure 10.6.The controller C1(s) in Figure 10.7 cannot be realized because it contains the processtransfer function p2(s), which is uncertain and cannot be made part of the controller. We canhowever approximate p2(s) with its model p2 ( s ) . In this case C1(s) becomesC1 ( s ) q1 ( s ) /(1 p1 ( s ) p2 ( s )q1 ( s )).(10.3)Another difference between figures 10.6 and 10.7 is that even if the model p1 ( s ) is aperfect representation of the process, the pulse created by the inner loop response to thedisturbance d2(s) (i.e., d 2 ( s ) /(1 p1 ( s ) p 2 ( s )q 2 ( s )) for a perfect model p2 ( s ) ) feeds backaround the outer loop of Figure 10.7. Since the primary controller cannot suppress thispulse, it continues around the loop until it dies out.Even using the approximation given by Eq. (10.3) to obtain a PID controller does notreduce Figure 10.7 to the standard PID cascade diagram of Figure 10.4 because C1(s) inFigure 10.7 is multiplied by q2 1 ( s ) . If q2 1 ( s ) is a lead (which will generally occur only ifthe process description is quite uncertain), then q2 1 ( s ) can be approximated by apolynomial via a Taylor’s series. This polynomial can be multiplied into the PID controllerobtained from C1(s) to obtain a new PID controller after dropping higher order terms. Evenif q2 1 ( s ) is a lag, it may still be possible to approximate the term C1(s) q2 1 ( s ) by a PIDcontroller. However, the necessary approximations will have to be carried out by hand,following procedures in Chapter 6, as the current version of IMCTUNE does not carry outthe necessary manipulations.Two rather long examples of cascade control of uncertain processes follow. Theindividual processes in both examples are first-order lags plus dead time and havesignificant process uncertainty. In the first example, the secondary process output dynamicsare significantly faster than the primary process dynamics. In the second example, theprimary and secondary process dynamics have similar dynamic behavior.
248Cascade Control Chapter 10Example 10.1 Secondary Process has Faster Dynamicsthan the Primary Processp1 ( s ) K 1e T s; 0.8 K1 1.2, 17.5 T1 22.5, 14 τ 1 16τ1 s 1(10.4a)p2 ( s) K 2 e T s; 0.6 K 2 1.8,τ2s 1(10.4b)122 T2 4, 1 τ 2 3(a) IMC System DesignFollowing the suggestions in chapters 7 and 8, we use the upper-bound gains and dead timesand the lower-bound time constants for the process models. 22 .5s1.2e p1 ( s ) 14 s 1 4s1.8e p2 ( s ) s 1(10.4c)(10.4d)Computing the 2DF feedback controller for the inner loop (see Figure 10.5), usingIMCTUNE with the outer loop, open givesq2 ( s) ( s 1)(9.05s 1).1.8(4.4s 1) 2(10.5a)Figure 10.8 shows the tuning curves, while Figure 10.9 shows typical time responses to astep disturbance in the inner loop. Data for both figures was obtained from IMCTUNE.
10.2 Cascade Structures and Controller Designs249 Partial Sensitivity Function 101100Upper-bound10 1Lower-bound10 210 310 210 1100101Frequency (rad/unit time)Figure 10.8 Cascade inner loop tuning using controller given by Eq. (10.5a).0.50.4K1 τ1 T1 K2 τ2 T2 1.2 14 17.5 .6 1 2Output0.3K1 τ1 T1 K2 τ2 T2 .8 16 17.5 .6 1 20.20.10 0.1K1 τ1 T1 K2 τ2 T2 1.2 14 17.5 1.8 3 4050100150200250300TimeFigure 10.9 Responses to a step inner loop disturbance (d2) with the outer loop open.Having obtained the inner loop controller, the outer loop controller can be obtained from thecascade facility of IMCTUNE, and isq1 ( s ) (15s 1).2.16(16.87 s 1)(10.5b)
250Cascade Control Chapter 10The tuning curves for the outer loop of the cascade, using Eq. (10.5b), are shown inFigure 10.10. Also in this figure are the closed-loop upper-bound and lower-bound curvesfor a single-loop controller for a model and controller of 26 .5s2.16 e p(s) 15s 1q( s) (15s 1).2.16 (14.3s 1)(10.6)Recall from equations (10.4a) and (10.4b) that the overall process isKe T s( τ1 s 1)( τ 2 s 1)Upper and Lower Bounds of the Magnitudeof the Complementary Sensitivity Functionp( s) 0.48 K 2.16, 19.5 T 26.5, 14 τ 1 16 1 τ 2 3.10110010 1Cascade Control10 210 3Single-Loop Control10 210 1100101Frequency (rad/unit time)Figure 10.10 Comparison of closed-loop setpoint to output responses.Based on the closed-loop frequency responses we can conclude that the fastestresponses of the single-loop system are slightly faster than those of the cascade system, butmore importantly, the slowest responses are significantly slower. Figures 10.11 and 10.12support these conclusions. Note the different time axes in figures 10.11 and 10.12.
10.2 Cascade Structures and Controller Designs2511.41.2Output10.8K1 τ1 T1 K2 τ2 T20.61.2 16 22.5 1.8 3 40.41.2 14 17.5 1.8 3 40.20.8 14 17.5 0.6 1 20050100150200250TimeFigure 10.11 Step setpoint responses for the cascade control system of Figure 10.5.1.41.2Output10.8K τ1 τ2 T0.62.16 16 3 26.50.42.16 14 1 19.50.2000.48 14 1 19.5100200300400Time500600700Figure 10.12 Step setpoint responses for the single-loop control system, using Eq. (10.6).
252Cascade Control Chapter 10The reason for the improved setpoint response of the cascade system is that the innerloop of the cascade reduces the effect of gain uncertainty in the inner loop process. To showthat this is so, Figure 10.13 compares the closed-loop frequency responses of the cascadesystem with that of a single-loop controller. The process is the same as that given byequations (10.4a) and 10.4b), except that instead of a lower-bound of 0.48, the lowerbounds (lb) are 1.1 and 1.44. That is, the single-loop process isp( s) Ke T slb K 2.16, 19.5 T 26.5, 14 τ 1 16 1 τ 2 3. (10.7)( τ1 s 1)( τ 2 s 1)The model and controller for the process of Eq. (10.7) are the same as given in Eq. (10.6)and are repeated for convenience: 26 .5s2.16 e p(s) 15s 1q( s) (15s 1)2.16 (14.3s 1)Upper and Lower Bounds of the Magnitudeof the Complementary Sensitivity FunctionA lower-bound gain of 1.44 corresponds to a secondary process (i.e., p2 ( s ) ) with again of 1.8 and no gain uncertainty. A lower-bound gain of 1.1 corresponds to a secondaryprocess whose gain varies between 1.375 and 1.8. In other words, the effect on the outerloop of the ratio of the maximum to minimum gain variation of the secondary process hasbeen reduced from a ratio of 3 to a ratio of 1.3. The slowest time responses are compared inFigure 10.14.101Upper Bounds10010 1Lower BoundsCascade Control10 2Single-Loop Control Lower-Bound Gain 1.44Single-Loop Control Lower-Bound Gain 1.110 310 310 210 1Frequency (rad/unit time)Figure 10.13 Comparison of cascade and single-loop control systems.100
10.2 Cascade Structures and Controller Designs2531.4Single-Loop Control Lower-Bound Gain 1.441.2Single-Loop Control Lower-Bound Gain 1.1Output1.0Cascade Control0.80.6Process parametersK1 τ1 T1 K2 τ2 T2 .8 14 17.5 .6 1 20.40.20050100150200250TimeFigure 10.14 Comparison of slowest responses to a step setpoint change.We now return to the cascade control system responses to a step disturbance to theinner loop, but this time with the outer loop closed. The time responses for the sameprocesses as in Figure 10.9 are shown in Figure 10.15. From this figure we conclude thatthere is no need to retune the inner loop.Figure 10.16 shows the effect of using the single-degree of freedom IMC controllergiven by Eq. (10.8) on the response to a step disturbance in the inner loop. These responsesshould be compared with those of Figure 10.15.q2 ( s) ( s 1).1.8(4.18s 1)(10.8)The filter time constant of 4.18 in Eq. (10.8) yields an Mp of 1.05. That is, thecontroller is tuned so that the worst-case overshoot of the inner loop output y2 to a stepsetpoint change to the inner loop is about 10% with the controller q2 in the forward path.This controller is then used in the feedback path of the inner loop in Figure 10.5.
254Cascade Control Chapter 100.50.4K1 τ1 T1 K2 τ2 T2 1.2 14 17.5 .6 1 2Output0.30.2K1 τ1 T1 K2 τ2 T2 .8 16 17.5 .6 1 20.10K1 τ1 T1 K2 τ2 T2 1.2 14 17.5 1.8 3 4 0.1 0.2050100150200250300TimeFigure 10.15 Responses to a step inner loop disturbance (d2) with the outer loop closed.0.60.5K1 τ1 T1 K2 τ2 T2 1.2 14 17.5 .6 1 2Output0.40.30.2K1 τ1 T1 K2 τ2 T2 .8 16 17.5 .6 1 20.10 0.1K1 τ1 T1 K2 τ2 T2 1.2 14 17.5 1.8 3 4050100150Time200250300Figure 10.16 Responses to a step inner loop disturbance using the controller given byEq. (10.8).
10.2 Cascade Structures and Controller Designs255While the inner loop disturbance responses using the single-degree of freedomcontroller Eq. (10.8) are significantly slower than the 2DF controller given by Eq. (10.5a),the responses of the output y1(t) to setpoint changes to the outer loop are only slightlyslower than those given in Figure 10.11.(b) PID Cascade Controller DesignsSection 10.2 discusses methods for approximating the IMC cascade control system with thetraditional cascade system of Figure 10.4. Figure 10.7 shows the IMC equivalentconfiguration. For convenience, this figure is repeated in Figure 10.17.d2rC1 ( s )q 2 1 ( s )d1(1 ( p2 ( s ) p2 ( s ))q2 ( s ))(1 p1 ( s ) p2 ( s )q1 ( s ) ( p2 ( s ) p2 ( s ))q2 ( s ))q2 ( s )(1 p2 q2 ( s ))p2 ( s )y2y1p1 ( s )C1 ( s ) q1 ( s )(1 ( p2 ( s ) p2 ( s )) q2 ( s )) /(1 ( p2 ( s ) p2 ( s )) q2 ( s ) p1 ( s ) p2 ( s )q1 ( s ))Figure 10.17 Standard feedback form of an IMC cascade control system.Recall that the controller C1(s) in Figure 10.7 is not realizable because it containsterms involving the inner loop process p2(s), which varies within the uncertainty set andcannot be part of the controller. Therefore we suggested replacing p2(s) with itsmodel, p 2 ( s ) . This givesC1 ( s ) q1 ( s ) /(1 p1 ( s ) p2 ( s )q1 ( s )).(10.9a)IMCTUNE provides the following PID controllers from the IMC controllers obtainedpreviously:q2 ( s) PID2 1.79(1 1 /(12.05s ) 1.68s ) /(14.7 s 1).(10.9b)Inner loop:p 2 q 2 ( s ))(1 Outer loop:C1 ( s ) PID1 .234(1 1 /( 23.77 s ) 5.35s /(.29 s
Chapter 6, “PI and PID Controller Parameters from IMC Design” Chapter 7, “Tuning and Synthesis of 1DF IMC Controllers for Uncertain Processes” Chapter 8, “Tuning and Synthesis of 2DF Control
Chuo Kikuu cha Morogoro cha Kilimo kuwa Chuo Kikuu cha Sokoine cha Kilimo. Seneti ya Chuo Kikuu Cha Sokoine Cha Kilimo (SUA) mwaka 1991 ilipitisha uamuzi kwamba kila mwaka Chuo kitakuwa kinamuenzi Hayati Edward Moringe Sokoine kwa kuadhimisha kumbukizi ya kifo chake kilichotokea tarehe 12 Aprili, 1984.
Wananchi wengi wanavifahamu vyama vikuu vitatu nchini yaani Chama cha Mapinduzi (CCM), Chama cha Demokrasia na Maendeleo (CHADEMA) na Chama cha Wananchi (CUF) Wananchi wengi wangewachagua wagombea wa Chama cha Mapinduzi (CCM) kama uchaguzi ungefanyika wakati wa mahojiano. Idadi ya wananchi ambao hawako karibu na chama chochote cha .
Kiswahili Kidato cha 4 Kitabu cha Mwanafunzi iii DIBAJI Mwanafunzi Mpendwa Bodi ya Elimu Rwanda inayo heshima kwa kutoa kitabu hiki cha Somo la Kiswahili Michepuo mingine kidato cha nne kitabu cha mwanafunzi .Kitabu hiki kitakuwa
CHARACTER SHEET Level Other skills: Craft - INT Perform - CHA Knowledge - INT Profession - WIS Ranks Racial, Feats, Synergy Skill Bonus Misc Armour Check Penalty Skill Ranks Level Adjustment Hit Die Untrained Acrobatics DEX-Appraise INT Bluff CHA Climb STR-Diplomacy CHA Disable Device DEX Disguise CHA Escape Artist DEX Fly DEX---Handle Animal CHA
Lakshmi Gayathri Mantra Om Mahaa Devyai Cha Vidhmahe Vishnupatnyai Cha Dheemahi Thanno Lakshmi Prachodhayaath Saraswathi Gayathri Mantra Om Vaak Devyai Cha Vidhmahe Virinchi Pathniyai Cha Dheemahi Thanno Vaani Prachodhayaath Durga Gayathri Mantra Om Kaatyayanaya Vidhmahe Kanya Kumari Dhimahi Tanno Durgih Prachodhayath . Shiva Gayathri Mantra
Cha 408-02 The use of a mobile jack implies the necessity of using protection routes. It absolutely forbidden the vehicle lifting using the front suspension arms or the rear axle as supporting points. The following type of mobile jack is using the sockets, the support Cha 408 – 01 or Cha 408 – 02 in order to place the route Cha 280 – 02.
Jim Geenen Down Home NC Member / CHA partner Fall/winter 2018-19 www.downhomenc.org Ann Geers HCHHSA Board Board member/ CHA partner Fall 2018 Aleasa Glance Haywood County Schools Student Services Director/ CHA partner Fall-winter 2018-19 www.haywood.k12.nc.us Vicki Gribble Mountain Projects Health Care Navigator / CHA partner Winter
Misericordes sicut pater. Hãy biết xót thương như Cha trên trời. Vị chủ sự xướng: Lạy Cha, chúng con tôn vinh Cha là Đấng thứ tha các lỗi lầm và chữa lành những yếu đuối của chúng con Cộng đoàn đáp : Lòng thương xót của Cha tồn tại đến muôn đời !
