Nonlinear Control Elements I 143 FIG 5.18. Maladjustments in the ~3/‘-y-y- z program parameters are easy to eltoa great t,j too long q too high diagnose. Time 2. When e = ez, the on-off output’ drops to 0 percent and the time delay begins. The on-off operator remains in control and preload is sustained. 3. At the end of the delay period, transfer is made to the proportional- plus-reset-plus-derivative controller and preload is replaced with COII- troller out’put, starting reset uct’ion. By this time, the error and its derivative should both be zero, so the controller output will equal the preload setting. Transfer is therefore ‘Lbumpless.” The dual-mode system gives the best set-point response attainable. Optimal switching, by definition, is unmatched in the unsteady state, while t,he linear controller provides the regulation necessary in the steady state. But any control system is only as good as the intelligence with which it is supplied. In the event of maladjustments in the three param- etcrs el, q, and id, the track of the controlled variable will be imperfect. The value of el will vary directly with the difficulty of the process. As the process difficulty decreases, the controlled variable is less a function of load, and hence has more tolerance for inaccuracies in the control parameters. But the degree of performance improvement provided by dual-mode control also varies directly with process difficulty. The dual-mode system needs six adjustments, which fall into t\vo independent groups. Settings of proportional, reset, and derivative only pertain t’o t,he steady state, while the program settings are in effect else- where. Consequently, adjusting the dual-mode system is no more diffi- cult than adjusting two separate controllers. Rules for setting the program parameters are self-evident : 1. ,\Inladjustment of el causes overshoot or excessive damping. 2. Excess t)ime delay turns the controlled variable downward after the set point is reached. 3. An incorrect preload setting introduces a bump after the t’ime- delay interval. The effect’s of these mnlndjust~mcnts are graphically demonst’rnted in Fig. 5.18. Recall the specifications which were set, forth at the beginning of the section 011 dual-mode control. *\ Insimum speed has been provided by the on-off controller. The programmed switching crit’icnlly damps the loop as the set point is approached. Offset is climinntcd by reset iu the 144 1 Selecting the Feedback Controller linear controller. FinalIy, noise of magnitude less than el will not actuate the on-off operator and therefore will be no more of a problem than in a linear system. Although complicated and costly, dual-mode control cannot be matched for performance. NONLINEAR TWO-MODE CONTROLLERS It has been demonstrat(ed that a loop whose gain varies inversely with amplitude is prone to limit-cycle. Any controller with similar charac- teristics can promote limit cycling in an otherwise linear loop. On-off controllers are in this category. So any nonlinear device that is purposely inserted into a loop for the sake of engendering stability must have the opposite characteristic: gain increasing with amplitude. The only stabilizing nonlinear devices discussed up to this point have this property -it was manifested as a dead zone in the three-state controller and as the linear mode in the dual-mode system. It is not difficult to .visualize a desirable combination of properties for a general-purpose nonlinear controller. In fact, the characteristics out- lined for a dual-mode system apply: the controller should have high gain to large signals, low gain to small signals, and reset action. The variation of gain with error amplitude can be accomplished continuously or piecemeal. A Continuous Nonlinear Controller It is possible to create a cont.roller with a continuous nonlinear function whose gain increases with amplitude. In contrast to the three-state controller, its gain in the region of zero error would be greater than zero, with integrating action to avoid offset. But its change in gain with amplitude should be less severe than that of a dual-mode system. Thus it would be more tolerant of inaccuracy in the control parameters. The continuous nonlinear controller could be mathematically described by the expression IIZ = F flel e + $j / e dt > (5.12) In this way its gain varies with the absolute magnitude of the error. A suitable linear function can be used. flel = p + (’ ,:““’ (5.13) where p is an adjustable parameter representing linearity and e is ex- pressed in percent. If /3 = 1.0, the controller is linear. But as @ approaches zero, the control function becomes square law, taking the Nonlinear Control Elements I 145 FIG 5 19. The proportional charac- teristic of a continuous nonlinear controller displays variable damping. -o+ Deviation shape of the parabolic sections shown in Fig. 5.19. It is not desirable for fl to equal zero, since this would render the controller essentially insensitive to small signals, and offset would result. A value of p in the vicinity of 0.1 would make the minimum gain of the controller 10/P. A characteristic of this sort produces varying degrees of damping in the closed loop. If a linear controller were used to regulate a given linear process, a certain proportional gain could be found which would produce uniform oscillations. A straight line represent’ing this gain, labeled “zero damping, ” is superimposed on the curve in Fig. 5.19. If the proportional gain of t’he linear controller were halved, the closed loop would exhibit >a-amplitude damping. The controller gain representing ji-amplitude damping is also indicated. The nonlinear characteristic crosses both these contours of constant damping. Between the intersections are three distinct stability regions. In the region surrounding zero deviation, damping heavier than $a-ampli- tude persists, while adjacent to it on both sides are regions of lighter damping and consequently faster recovery. There is st,ill another region on each side where damping is less than zero-representing instability. Should a deviation arise large enough to fall into this last area, it will be amplified with each succeeding cycle. To gain a better insight into the response of this nonlinear charac- teristic in a loop with a linear process, the input-output graph of Fig. 5.20 has been constructed. Notice how heavily a small signal is damped. Damped oscillations in a linear loop theoretically go on forever. But with a nonlinear characteristic of the kind shown, damped oscillations cannot persist beyond one or two cycles. On the ot,her hand, a large signal causes more corrective action than a linear controller, appro- priately damped, could provide. A sufficiently larger deviation could promote instability, however, so the proportional band of the nonlinear cont>roller must be adjusted for the largest anticipated deviation. As with other nonlinear controllers, set-point response exceeds what is obt’ainable with linear modes. This is because set-point changes are 146 1 Selecting the Feedback Controller FIG 5.20. If the initial deviation is not extreme, it may be damped within one cycle. m normally greater and more rapid t’han load disturbances, taking advan- tage of the region of higher gain. Load disturbances make their appear- ance as a slow departure of the controlled variable from the set point. Since a linear controller has more gain in the region close about the set point, it will generally respond more effectively to small load changes. A comparison of the responses of linear and nonlinear three-mode con- trollers is shown in Fig. 5.21. A nonlinear two-mode controller seems generally to outperform a linear two-mode controller. The nonlinear function provides an extra margin of stability similar to what can be attained with derivative. In cases where so much noise is superimposed on the measurement t’hat derivative cannot be used, a nonlinear function can be quit’e valuable. Another feature of the nonlinear controller is its extreme tolerance of gain changes in the loop. Response to upsets of moderate magnitude appear virtually identical over a proportional band range of 4 : 1 or more. Consequently little care need be given to the set’tings of proportional and reset, save for the possibility of bringing the unstable region too close to the set point. Time Time FIG 5.21. A three-mode nonlinear controller exhibits better set-point response but poorer load response than its linear counterpart. Nonlinear Control Elements I 141 Linear 0’ Time, set 60 40 0 Time, set FIG 5.22. The nonlinear two-mode controller is superior in all respects on a noisy flow loop. Flow Control A flow measurement is always accompanied by noise. This noise is attenuated somcwhnt by the wide proportional band of the controller and passed on to the valve. If the noise is of any magnitude, the valve may be stroked suflicicntly to introduce ac+ual changes in flow. The nonlinear function is an efficient noise filter, in that, it rejects small-amplitude signals. The result is smoother valve motion and a more stable loop. Figure 5.22 shows comparative records for linear and nonlinear control of a noisy flow loop. The nonlinear controller has proven to be quite effective on pulsating flows too, where the disturbance is periodic rather than random. Level Control J,evel measurements are often noisy because of splashing and turbu- lence. In addition, the surface of a liquid tends to resonate hydrau- lically, producing a periodic signal superimposed on the average level. Since the liquid-level process cannot respond fast enough for a change in valve position to dampen these fluctuations, they ought to be disregarded by the controller. A nonlinear controller does just this, sending a smooth signal to the valve. It was pointed out in Chap. 3 that many tanks with level controls are intended as surge vessels. In these applications, tight control is inadmis- sible because it frustrates the purpose of the vessel. A wide proportional band with reset was suggested for control. But the nonlinear controller is, in fact, ideal for this application for two reasons: 1. 1Iinor fluctuations in liquid level will not be passed on to the valve, providing smooth delivery of flow. 2. lllajor upsets will be met by vigorous corrective action, ensuring that the upper and lower limits of the vessel will not be violated. This application is often referred to as LLaveraging level control,” because it is desired that the manipulated flow follow the avcragc level in 148 1 Selecting the Feedback Controller -20 _ _ ~ 100 Ls 4 lL 0 Time, hr FIG 5.23. The nonlinear two-mode con- troller prevents minor fluctuations in level from affecting delivery of flow. the tank. Averaging is really a dynamic process and can be accomplished with a suit’able lag. But adding a lag would only serve to reduce the speed of response. The nonlinear function, however, provides filt,ering without sacrificing speed. A typical record of level in a surge vessel and the corresponding output of its nonlinear controller are presented in Fig. 5.23. pH Control The neutralization process has been described as unusually diffkult to control because of the extreme nonlinearitjy of the pH curve. Limit cycling (*an he encountered when a linear controller is used, because loop gain varies inversely with deviation. This, t,hen, is a natural application for t,he nonlincnr (*ontroller whose gain varies directly with deviation. In fact, any process prone to limit cycbling can benefit by its USC. The nonlinear function in the controller need not be a perfect complement FIG 5.24. A nonlinear controller can give uniform damping to a pH loop. Nonlinear Control Elements I 149 m FIG 5.25. A discontinuous nonlinear controller employs a high-low limiter. to the process curve, because any contribution it can make will be an improvement over a linear function. And if the linearity, p, is adjustable, a reasonable fit can be made. The input-output graph of Fig. 5.24 shows how a constant loop gain is achieved. A Discontinuous Nonlinear Controller The nonlinear function shown in Fig. 5.19 can be approximated by three straight lines. The center is essentially a dead zone where little or no control action takes place. This function is not difficult to intro- duce into a linear controller; it involves sending the controlled variable to the set-point input through high and low limits. Within the limits, there is no error signal; elsewhere an error is developed as the differenw between the measurement and the neartir limit. ITigure 5.25 describes the arrangement of the instrument and its proportional function. Pro- portional, rcsct, high, and low limits are adjustable. This nonlirwnr coiltrollcr is often used in nvernging lcvcl npplic*ations. Its dead zone is also a vnluablr feature in the pH-caontrol system dcwribcd in Chap. 10. PROBLEMS 5.1 .I lincnr proress is found to be undmn~wd unrltr proportional c*ontrol with a hand of 20 Iwrwrlt. I\-hat will hal)lwn if the band is reduced to 10 Iwrccnt ; to 5 percent? 5.2 .I thernml ~~roccss with IO-SW dcstl tinw rind n Gnin lag is to be cooled with refrigerant sul)l)licd front a solrnoiti val\-c. If the ~nl\-c is left on, the tcm- ptrnturc falls to O’F; \vhtln it, is off, thr trnllwrnture riws to GOOF. bktimatc the Iwriod nnrl nliil)litudc of the limit q.c*lc if thcx on-off controllfr wcrc Iwrfcct. 5.3 ‘I’hc on-off cwntrollcr used for the, ~~roc’css in Prob. 5.2 wtunlly hns :L difkwntial gal) of 2°F. lCstinmt(~ the lwriotl and :m~plitudr of the limit ryclr, taking the difftrcntial gal) into account. 150 1 Selecting the Feedback Controller 5.4 A lever is driven by a bidirectional constant-speed motor to a position determined by a three-state controller. The motor has a speed of 10 percent of full stroke per second, and an inertial time constant of 1.0 sec. Differential gap in the controller is 2 percent of full stroke. How wide does the dead zone have to be to prevent limit cycling? What would be the period of the cycling? 5.5 A batch chemical reactor is to be brought up to operating temperature with a dual-mode system. Full controller output supplies heat through a hot- water valve, while zero output opens a cold-mater valve fully; at 50 percent out- put, both valves are closed. While full heating is applied, the temperature of the batch rises at, l”F/min; the time constant of the jacket is estimated at 3 min, and the total dead time of the system is 2 min. The normal load is equivalent to 30 percent, of controller output. Estimate the required values for the three adjustments in the optimal switching program. 5.6 h given linear process is undamped with a proportional band setting of 50 percent for a linear controller. If a continuous nonlinear controller is used with a linearity setting of /3 = 0.2, how narrow can the proportional band be set and still tolerate an error of 20 percent? [...]... variable Secondary Controller Primary controller FIG 6. 1 Cascade control resolves the process into two parts, each within a closed loop Improved Control through Multiple Loops FIG 6. 2 The primary controller sees a closed loop as a part of the process It must be recognized, however, that cascade control cannot be employed unless a suitable intermediate variable can be measured Many processes are so arranged... instrumentation 4 Nonlinear control functions As an example of how equipment might be protected by a selective control system, consider a compressor whose discharge is ordinarily on FIG 6. 15 Motor speed is manipulated by whichever controller has the lower ouput Time, hr 167 1 6 8 1 Multiple-loop Systems FIG 6. 16 A high selector is used to permit control of the peak reactor temperature flow control, except that... two variables as the input to a controller is not recommended Improved Control through Multiple Loops FIG 6. 6 In the recommended sgstern, the ratio calculation is outside the loop I 161 m Rotio station circuit, making I’ = KY if X is controlled, or r = X/K if Y is controlled Figure 6. 6 shows the set-point calculation In this configuration, one of the variables becomes controlled and the other serves... RATIO CONTROL SYSTEMS In a ratio control system, the true controlled variable is the ratio K of two measured variables X and Y: K=; (6. 3) Control is usually effected by manipulating a valve influencing one of the variables, while the other is uncontrolled or “wild.” The obvious way to implement the ratio control function is by computing X/Y, as shown in Fig 6. 5 But this is not the best way Figure 6. 5... generate a given controller output is E = 1 e dt = Rim Demand flow t” e FIG 6. 12 An integral control sgstern is subject to volume offset d L Time 165 1 6 6 1 Multiple-loop Systems The actual volumetric offset is the percent integrated error E times the maximum flow rate: EF = mV (6. 8) Each valve position, hence each flow rate, has a related volume offset On the face of each digital control station are... FIG 6. 7 A single master station can set all the plant streams through ratio stations To flow controllers station Individual rotio stations 169 Secondary 1 Multiple-loop Systems I FIG 6. 8 Several secondary loops can be set in cascade from one primary controller c always equal 100 percent Then the master station or primary controller sets the true total flow Since the prime manipulated variable in Fig 6. 8... for the response of CP with respect to 1’2 Refer to the block diagram in Fig 6. 3 Let g, and g, be vectors representing gain and phase of the process and the controller, respectively Then c2 = mg, c2 = (r2 - c2)gcgp c2u + g&J = rzgcg, Process I I FIG 6. 3 The input to the secondary loop is r2, its output is 0 1 56 1Multiple-loop Systems The vector gain of the closed secondary loop will be designated g02:... upon the other Each controller will have its own measurement input, but only the primary controller can have an independent set point and only the secondary controller has an output to the process The manipulated variable, the secondary controller, and its measurement constitute a closed loop within the primary loop Figure 6. 1 shows the configuration The principal advantages of cascade control are these:... of automatically modifying a controller to satisfy a combination of func153 0 1 5 4 1 Multiple-loop Systems tions of a controlled variable The common denominator in all these situations is the manipulation of a single final element through more than one control loop CASCADE CONTROL The output of one controller may be used to manipulate the set point of another The two controllers are then said to be... them to respond like proportional controllers The controller arrangement is shown in Fig 6. 19 for the pressure-flow system that appeared in Fig 6. 15 Automatic transfer from one controller to another takes place at the instant when the outputs are equal This fact, coupled with the common reset signal, means that transfer is bumpless 1 7 0 1 Multiple-loop Systems FIG 6. 19 The output of the selector is . variable. Secondary Controller Primary controller FIG 6. 1. Cascade control resolves the process into two parts, each within a closed loop. Improved Control through Multiple Loops FIG 6. 2. The primary controller sees. constant gain for the primary loop. RATIO CONTROL SYSTEMS In a ratio control system, the true controlled variable is the ratio K of two measured variables X and Y: K=; (6. 3) Control is usually effected by. the ratio of two vari- ables as the input to a controller is e - Controller c not recommended. Improved Control through Multiple Loops I 161 FIG 6. 6. tern, the the loop. In the recommended sgs- ratio