Table V.
k9/k6 24 (670” C.) 22 (650” C.)
Rates of Free Radical Attack on Methane and Formaldehyde h/kI
Basis
33 (500’ C . ) 260 (500’ C.) 21 i 4 (670 ’ C.)
Semi-empirical estimation (Semenov, 1959) Initiated by thermal decomposition of H202 to O H . radicals (Hoare et af., 1959, 1966) Slow oxidation of methane in hydrofluoric acid-washed vessels and in boric acid-washed vessels (Blundell et af., 1965) Present work, initiated by N O in presence of HF-washed silica gel
km/h
300 (670’ C.)
Further corroborative evidence can be derived from activation energy considerations. By combining the values for kII/kI of 33 at 500’ C. (Blundell et al., 1965) and 21 at 650’ C. (this work), a value of 4 kcal. per mole is obtained for EIEII. From the above references (Semenov, 1959; Hoare, 1966), values of Ee-Eg are 6 and 4 kcal. per mole. The sequence of reactions involving the disappearance of HOZ.radicals
use of suitable reacting and quenching surfaces, but such high conversions would require chain lengths of several hundred. However, it is concluded from the results presented above that it is inconceivable on theoretical grounds, because of the more rapid simultaneous rate of radical attack on formaldehyde, to employ free radicals to provide a high conversion to formaldehyde while further oxidation is inhibited. Acknowledgment
HzOz
+
20H.
(1 3)
means that a branched-chain reaction is involved. This complicates the kinetic picture, and the normal decrease of rate a t longer ,residence times arising from the simple kinetic second-order dependence on methane and oxygen concentration will not be observed. Thus Figure 3 gives an apparently linear dependence of methane converted on residence time, due to compensation by the branching chain reactions a t longer residence times. Thus there appears to be general agreement regarding the relative rates of attack of OH. (and HOs , ) radicals on methane and formaldehyde. Conditions favoring O H . radicals, as in this work, provide optimum concentrations of formaldehyde. Throughout, the methane oxidation reaction is clearly of branching chain type, the chain length with respect to initiators being, from the present results, about 10. I n recent patents (Huttenwerke Oberhausen, 1961,1962,1963) exceptionally high methane conversions to formaldehyde have been claimed by
T h e assistance of V. L. Paul in computer programming is gratefully recorded. literature Cited
Blundell, R. V . , Cook, I\’. G. A., Hoare, D. E., Milne, G. S., Tenth Symposium (International) on Combustion, p. 445, Combustion Institute, Pittsburgh, 1965. Bombaugh, K. J., Bull, W. C., Anal. Chem. 34,1237 (1962). Egerton, A. C., Minkoff, G. J., Salooja, K. C., Combust. Flame 1, 25 (1957). Hoare, D. E., Proc. Roy. SOL.A291, 73 (1966). Hoare, D. E., Peacock, G. B., Proc. Roy. SOC.A291, 85 (1966). Hoare, D. E., Protheroe, J. B., jt‘alsh, A . D., Trans. Faraday SOL.55, 548 (1959). Huttenwerke Oberhausen A.G. (formerly Bergbau A.G.), Brit. Patents 880,873 (1961); 913,581 (1962); 926,889 (1963). Ingold, K. U., Bryce, I V . A , , J . Chem. Phyf. 24, 360 (1956). 79, 4838 (1957). McMillan, i V . G., J . A m . Chem. SOC. Semenov, N. N., “Some Problems of Chemical Kinetics and Reactivitv.” Pewamon Press. Oxford. 1959. Shtern, V. Ya.,- “The Gas Phase O’xidation of Hydrocarbons,” Pcrgamon Press, Oxford, 1964. Warren, D. R., Proc. Roy. Soc. AZ11, 22, 245 (1952). RECEIVED for review January 30, 1967 ACCEPTEDJune 29, 1967
QUALITY OF CONTROL PROBLEM FOR DEADTIME PLANTS T. J .
McAVOY’ AND E. F. JOHNSON
Department of Chemical Engineering, Princeton University, Princeton, N .J .
wo questions have to be considered in designing a control Tsystem: whether or not the control system is stable, and whether or not the quality of the control attained is good. Quality of control involves the ability of the control system to damp out quickly the effect of a disturbance on the plant. Unlike stability, quality is not a well defined concept and many different criteria have been suggested for it. There is no single answer to what constitutes good quality of control. This paper is concerned with the quality of control problem Present address, Department of Chemical Engineering, University of Massachusetts, Amherst, Mass. 440
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
associated with the selection of regulators-i.e., controllers which hold certain variables a t steady-state values-for plants with dead-time lags. The following synonyms for the term “dead-time lag” have appeared in the literature: delay time, time lag, transportation lag, transport lag, and distance velocity lag (a special case occurring in flow systems). The incorporation of a dead time into a control loop makes the over-all control system more unstable than the corresponding loop without the dead time. I n general, a poorer quality of control is attained on a plant which contains a dead time than on one which does not (Eckman, 1946).
Control quality for process plants which can be represented by a second-order transfer stage plus a dead time i s examined. A simple feedback regulator in which the controller possesses combinations of proportional, integral, and derivative modes of operation is considered. Design charts based on a minimum integral squared criterion are presented, which together with the Ziegler and Nichols rules may be used to achieve any desired quality of control.
I n this paper a simple feedback control system, shown in Figure 1, is examined. The controller is a commercially available type which possesses combinations of proportional, integral, and derivative modes of operation. This type of control system has been considered by several authors, each of whom postulated a criterion for optimum quality of control and presented design procedures based on this criterion. I n testing their proposed control systems the authors generally subject the plant to step changes in the upset variable, 6, in Figure 1 . T h e step forcing imposes a severe disturbance on the plant and its control system. Although such a change rarely occurs in practic'c, it is felt that if the control system can function well when subjected to this type of upset, it will be more than adequate in handling a less severe disturbance. Step-force testing is use'd in this paper. One criterion for optimum quality of control that has been widely used in the literature (Callander et al.? 1937; Ziegler and Nichols, 1942, 1913; Oldenbourg and Sartorius, 1948; Wolfe, 1951) is the determination of controller settings which cause dead-beat return after step forcing. Dead-beat return, often called critical damping, is the fastest possible response of the controlled variable which involves no undershoot of the steady-state value. Chien e t al. (1952) considered this criterion along with one which required 2070 undershoot. Cohen and Coon (1953) and Ream (1954) determined controller settings by specifying the subsidence ratio of the fundamen-tal component in the closed-loop transient response. Hazebsroek and van der Waerden (1950) and Wescott (1954) postulated that the integral of the square of the controlled variable: from zero to infinity should be minimized as a function of the controller parameters. Wills (1962) postulated that the integral of either the absolute value of the controlled variable or the absolute value of the controlled variable multiplied by time should be minimized as a function of the controller parameters. All of the integral criteria are applicable only to cases where integral control is used, since without this mode the resultant integrals diverge. In all of the aforementioned studies, with the exception of that of Wills, the plant step response was simulated by either a delayed ramp function or the response of a first-order transfer stage plus a dead time. I n this paper the plant is approximated by a second-order stage plus a dead time. This model, discussed by Farrington (1951), seems to be more realistic than the other two, sirice it accounts for the inertia present in physical systems and it allows a more flexible matching of the plant's characteristics. I t is assumed that the constants in the model are known-e.g., from either the pulse (Hougen and Walsh, 1961; Law and Bailey, 1963) or the step (Oldenbourg and Sartorius, 1948) forcing response of the plant or from its frequency response (Evans, 1954). A new design proc.edure is presented in this paper, by means of which one can attain any desired quality of control. This technique is based on design charts obtained by using the minimum integral square criterion on the second-order dead-time plant. A comparison with previously reported results is presented.
Plant and Controller Models
T h e object of this paper is the investigation of the control loop shown in Figure 1 . iYhen the loop is used as a regulator, the value of the reference variable, 6,) is set equal to the desired value of the controlled variable, OC. T h e plant, G, is represented by a second-order linear differential-difference equation and its transfer function is given by G(s) = exp ( - r s ) / ( w 2 s z
+ 2(ws + 1)
(1 1
where s is the Laplace variable and w, (, and T ? the dead time, are constants. Equation 1 may be normalized by defining a new Laplace variable, 7, 7
=
(2)
ST
to give
G(r)
=
exp(-r)/[(w/~)*r~4-2{(w/r)r
+ 11
(3)
the normalized plant model. Multiplying or dividing s by a constant results in a multiplication or division of time by the inverse of the same constant. All plants which have identical w to T ratios and identical values of { will have the same closedloop dynamic behavior if a normalized time (4)
t, = t / r
is used in plotting the response of the system. Thus, the variable change given by Equation 2 reduces the number of plant parameters which have to be considered to two, U / T and
i. T h e transfer function representing the controller is Gc(s) = Kp(1
+ Tt/J)(l
Td)/(l
+ CTdS)
(5)
where K , is the proportional gain, Ti is the integrating action, T d is the derivative action, and c is a constant which ranges between 0.05 and 0.10 in commercial instruments (Johnson). T,/s) in Equation 5 is sometimes written as [The term (1 (1 /3s)//3s where /3 = 1 / T i . At high values of s the gain approaches K,/c and a t low values of s the gain approaches K,//3s. With no integral action /3 -+ m . For pneumatic controllers this means an infinite resistance; for electronic controllers, a n infinite RC value in the circuit.] There are several facts which should be noted about Equation 5 . First, by setting either Ti or Td equal to zero one obtains either proportional plus derivative control or proportional plus integral control. If both parameters equal zero, simple proportional control results, and thus Equation 5 gives a general expression for all types of three-mode controllers.
+
+
Figure 1.
Simple feedback control system
VOL. 6
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OCTOBER 1 9 6 7
441
Second, the controller model has a pole-Le., the denominator equals zero-at s = -1/cTd. T h e function of this pole is to compensate the derivative operation of the controller in order to avoid excessive sensitivity. In addition, a n actual controller would have a t least one more pole due to physical limitations. However, since this pole would occur a t a large negative value of s, it would have a negligible effect on the controller dynamics and can be ignored. Third, the expression for G,(s) accounts for the interaction of the derivative and integral modes of operation which may be present in commercially available controllers (Young, 1955). By substituting Equation 2 into Equation 5 the following normalized controller model is obtained:
The substitution of r for s yields normalized controller constants. In order to determine the actual values of Ti and Td it is necessary to switch back to s. T h e charts presented in this paper show Ti X r and Td/T.
s Figure 2.
.6
Quality of Control Criferion and Design Charts
T h e criterion for quality of control which was chosen for this paper is the minimum of the integral of e,@) squared from zero to infinity as a function of the controller parameters, when the upset variable, e,, is step-forced. Hazebroek and van der Waerden (1950) used the same criterion and discussed its practicality. I n order for the criterion to be applicable, it is necessary that
e&)
+0
as
t+
m
(7)
otherwise the resulting integrals will diverge. Equation 7 is satisfied only if integral control is present. Thus, the integral squared criterion was used only to determine proportional plus integral and three-mode controller settings. The results obtained in these cases are used to suggest appropriate settings for proportional and proportional plus derivative control. T h e closed-loop transient response was calculated on a digital computer by using the Cauchy residue theorem (Nehari, 1 9 6 1 ) and a trial and error search procedure to locate the poles of the closed-loop transfer function. T h e integral of O,(t) squared was then determined from a n analytical expression. A grid of values was set u p for the controller parameters and the integral was calculated a t each point in it. The combination of controller settings which gave a minimum was then singled out and a new smaller grid set up around it. This procedure was continued until the absolute minimum was obtained. In many of the cases which were considered this minimum was fairly flat. Hazebroek and van der Waerden observed the same effect. For integral plus proportional control a two-dimensional T , and K,-had to be searched while three-mode grid-i.e., control required a three-dimensional search-i.e., Ti, K,, and Td. I n the latter case the incorporation of the compensating pole a t s = - l / c T , limits the maximum amount of derivative action which can be employed. The minimum integral occurred when the maximum value of Td was used. This effectively reduced the dimension of the grid for three-mode control to two. T h e results of the calculations are presented in Figures 2 through 9, and they have also been tabulated (McAvoy, 1964). (Results for W / T = 0.75, 1.5, and 3.0 are also tabulated.) Figures 2 and 3 apply to integral plus proportional control, while Figures 4 through 6 give three-mode controller settings for G = 0.05 in Equation 5. Figures 7 through 9 also give three-mode settings but with G = 0.10. 442
K, for proportional plus integral control
l & E C PROCESS D E S I G N A N D DEVELOPMENT
.5 .4
c. .3 I=
.2
.I
' 0
I
2
3
4
5
6
7
8
5 Figure 3.
T i for proportional plus integral control
5 Figure 4.
K, for three-mode control with c = 0.05
Figure 10 shows a typical comparison between the closedloop transient response which is obtained by designing a controller with the Ziegler and Nichols rules and the minimum integral charts. As can be seen, the latter technique gives rise to a more oscillatory response. I n general, the minimum integral square designs will yield a faster closed-loop system
1.2
1.2
I .o
I.o
.8
.8
b
6-
.6
.6 k-
.4
.4
.2
'0 Figure 6.
.2
I
2
3
4
5
5
T, for three-mode control with
6
7
8
c = 0.05
than that obtained with the dead-beat rules, but its transient response will take longer to die out. Designing for Quality of Control
Integral Plus Proportional and Three-Mode Control. T h e results obtained by using the dead-beat rules, such as those
0
0
1
Figure 9.
2
Ti for
3
4
5
5
6
7
8
three-mode control with c = 0.10
given by Ziegler and Nichols, and the minimum integral charts will bound all designs which possess a n acceptable quality of control. A designer ordinarily would not want a system whose response was slower than critical damping. Similarly, a controller that gave a faster response than the minimum integral design would be too close to instability to be useful. With this in mind it is possible to suggest a procedure whereby one can design for quality of control. The simplest method for getting good quality of control is to use either the Ziegler and Nichols rules or the minimum integral charts. A slightly more difficult procedure is to design controllers with both of these techniques and then interpolate between the results. In all three cases one is guaranteed a priorz that the resulting closed-loop system will possess a good quality of control. O n some plants the Ziegler and Nichols rules will prove to be somewhat conservative (Evans, 1954), but in the great majority of practical problems this is not true. The minimum integral charts will always give the same type of response regardless of the plant on which they are used. Thus, if the designer chooses, he can stop a t this point with the assurance that his control system will possess an acceptable quality of control. The procedure can be carried one step further by using zforms (McAvoy and Johnson, 1966) to determine the approximate closed-loop transient response. This method of obtaining the transient response is fast, accurate, and easy to use. VOL. 6
NO.
4 OCTOBER 1 9 6 7
443
.25 ZIEGLER AND NICHOLS' DESIGN .20
MINIMUM INTEGRAL SQUARE DESIGN
.I5 c
rf;
.IO
fSu .05
0. -,05
I
1
I
I
I
I
I
2
4
6
8 t
IO
I2
14
Figure 10. Typical comparison between the response of a Ziegler and Nichols control system and of a control system designed with the minimum integral charts for proportional plus integral and three-mode control
On:e the closed-loop transient response is available, the designer can judge the quality of control which has been attained and decide if his design is adequate or some modification is necessary. T o illustrate the preceding principles, suppose that one wished to design a n integral plus proportional controller for the plant
G(s) = exp(-2s)/(4s2
+ 4s + 1)
= 1.22(1
+ 0.125,'s)
(9)
T h e approximate closed-loop transient response of the resulting control system was calculated by using the z-forms and is shown in Figure 11. I t can be seen that the Ziegler and Nichols rules were too conservative in this case. The plant given by Equation 8 has a relatively small capacitance and a relatively large dead time. This combination of effects does not occur very often in practical problems. I n order to get critical damping, the designer can now use Figures 2 and 3, which give the following controller: G,(s)
= 1.64(1
+ 0.152/s)
(10 )
The approximate transient response for this control system is also given in Figure 11. T h e value of the response a t the first minimum for both systems is min = +0.090
(Ziegler and Nichols rules)
(11)
OJt) min = -0.119
(minimum integral charts)
(12)
O,(t)
A linear interpolation can be performed in order to determine a controller which will give O , ( t ) = 0 at the first minimum. The resulting controller is G,(s)
= 1.40 (1
+ 0.137,'s)
(13)
The actual transient response of the interpolated control system is shown in Figure 11. As can be seen, the procedure outlined above has allowed the designer to zero in on the control system which gives critical damping. The same approach can be used to get 5% undershoot a t the minimum or 10% or any desired value. 444
Figure 11. Example of technique for designing for specific quality of control
(8)
such that the closed-loop system which it gave was critically damped. By using the Ziegler and Nichols rules the following controller is obtained: G,(s)
t
l & E C PROCESS DESIGN A N D DEVELOPMENT
One point should be noted about the use of the minimum integral charts in the cas ; of three-mode control. If the value of c-Le., see Equation 5-for the controller which is used is not 0.05 or 0.1 but somewhere in between, it is necessary to perform a n interpolation between the charts in order to determine the minimum integral desi: n Proportional a n d Proportional Plus Derivative Control. In evaluating the quality of control which can be gotten with proportional and proportional plus derivative control, it is necessary to take into account the steady-state offset which these modes introduce. With these modes e,(t) + 1/(1
+ K D )as t
+ a
(14)
Equation 14 shows that a n increase in the proportional gain will decrease the steady-state error. However this increase will also increase the tendency toward instability. Thus, for proportional and proportional plus derivative control the designer is faced with another factor, the offset, which complicates the quality of control question. Since O J t ) remains finite as t approaches infinity, the integral square criterion cannot be used for proportional and proportional plus derivative control. However, the results which were obtained for proportional plus integral and threemode control can be used for these modes. It is suggested that Figure 2 be used for proportional settings and Figures 4, 5, 7, and 8 be used for proportional plus derivative settings. I n the study which they made, Ziegler and Nichols proposed that the value of K , which is used for proportional control be 10/9 times that which is used for integral plus proportional control. They also suggested that the same values of K , and T , be used for proportional plus derivative and three-mode control. Since their rules have proved to be successful, it is felt that the suggestions given above will also result in designs which possess a good quality of control.
Figure 12 shows a typical comparison between the results which the Ziegler and Nichols rules and Figure 2 give for proportional control. Figure 2 will produce a faster system, but again its oscillations \vi11 take longer to die out. The same conclusions will hold in a comparison of the two design procedures for proportional plus derivative control. I n this case the designer may have to interpolate between the two values of c-Le., 0.05 and 0.10-in order to arrive a t a final design. I t is again felt that the control system which is obtained by using the suggestion:j given in this section will be a limiting one. By using this control system in place of the minimum integral design, one c a i follow the procedure outlined above to obtain the quality of icontrol that is desired.
Values of R,/K, T iP,, and T J P , were found to vary somewhat as a function of a/+ and {. For the majority of plants considered in this paper the variation of these quantities is shown in Table 11. Since these parameters were found to vary in this study, it can be assumed that they would also vary in the case where a minimum integral of absolute value criterion was used.
.25 1 .20
-
.I5
-
I
ZIEGLER AND NICHOLS' DESIGN
Comparison with Previous Results
Hazebroek and van der LVaerden (1950) have given minimum integral squared design charts for proportional plus integral control. They used a first-order transfer stage and a dead time to represent the plant. Because a first-order stage does not account for inertia, its response to forcing will be faster than that of the second-order stage with similar time constants and it will yield a less stable closed-loop system. Stability is more of a limiting factor for a first-order plant than for a secondorder one. T h e results which Hazebroek and van der Waerden obtained point out this fact, since a controller which is designed with their rules will have a smaller proportional gain and a smaller integral action than that obtained from Figures 2 and 3. As a result, their design rules will give a slower closed-loop system than the minimum integral squared charts presented here. For small W / T ratios the results of the t\vo design techniques approach one another. As an example of the conservative nature of the Hazebroek and van der Waerden designs, consider the following plant :
-
.=.IO
eo .05
-
.25
HAZEBROEK AND van der WAERDEN
.20 MINIMUM INTEGRAL SQUARE
-S
15
The Hazebroek and van der Waerden controller for this plant is G,(s) = 5.24 (1
+ 0.0765/s)
(Bo (16)
Figures 2 and 3 give the following controller: G,(s) = 6.76 (1
+ 0.1751s)
-
t .IO
.O 5 0.
(1 7 )
The closed-loop transient response for both systems is shown in Figure 13. As can be seen, the Hazebroek and van der Waerden design is too conservative in this case. Because the plant model which was used in this study is more realistic than that of Hazehroek and van der LVaerden, the results prcsented here should prove to be better for practical design problems. \trills (1 962) determined optimum three-mode regulator settings for one specific second-order dead-time plant. T h e criterion which she used was that of minimizing the integral of the absolute value of i?,(t) from zero to infinity. A comparison of Wills' results and those presented in this paper is shown in Table I. K, is the ultimate proportional gain of the process and P, is the ultimate period of the process (Ziegler and Nichols, 1942). As can be seen. the minimum integral of the absolute value criterion gives rise to a smaller K , but a larger T , and T , than the minimum integral squared criterion. In general, a minimum integral of absolute value design will result in a slower closed-loop system but one in which disturbances die out faster than a minimiim integral squared design.
-.05
0
1
t
I
I
I
t
2
4
6
8 t
IO
I2
I
14
16
Figure 13. Comparison between response of a Hazebroek and van der W a e r d e n control system and of a minimum integral squared control system
Table I.
Optimum Three-Mode Controller Settings W / T = 3.16 1 = 1.74
K, / Kpmax
Ti P u
Td/Pu
c
0.423 0.686 0.690
3.10 2.98 2.67
0.290
0 0.05 0.10
Wills Figures 4 to 6 Figures 7 to 9
Table II. Variation of K,/K,,,,,
K,/K, 0.50-0.90 0.50-0.90
0.223 0.201
T,P,,, and T d / P u
Ti Pu
Td/P,
C
2 ,75-3 , 2 5 2.30-2.80 __
0.20-0,23 0,20-0.23
0.05 0.10
~
VOL. 4
~
NO. 4
OCTOBER 1 9 6 7
445
Thus, one should be cautious in extending Wills’ results to plants with different values of W / T and b. Also, it is possible for three-mode control to have a value of K,/K, max which is greater than 1.0 and still have a stable control system. Thus, the ratio, K,/K, max, is not necessarily a good measure of stability in the case of three-mode control. Wills also presented optimum three-mode controller settings for a step change in &-i.e., the reference variable in Figure 1. In this case, K,/K, max = 0.540, T I P , = 1.0, and T J P , = 0.200. Since the servomechanism problem yielded much different controller settings in Wills’ study than the regulator problem, it can be assumed that the results presented in this paper would not be directly applicable to the servomechanism problem. Conclusions
If a plant contains a dead-time lag, the ability of a designer to obtain good quality of control by means of a simple feedback system is thereby diminished. However, design charts, presented in this paper, can be used together with the Ziegler and Nichols rules to obtain any desired quality of control within the limitations imposed by the nature of the plant-i.e., second order with a dead time-and by the structure of the control system-Le., simple feedback. I n order to use the charts one has to determine appropriate values for w , {, and T in the plant model. Even if a designer would like a very specific quality of control, the amount of work involved is not very great and the calculations can be carried out by hand. A comparison between the results presented in this paper and previous work shows that minimum integral square designs obtained for a first-order plant with a dead time are too conservative when applied to a second-order dead-time plant. I n addition, previous results indicate that the optimum regulator controller settings presented in this paper are not necessarily the optimum servomechanism controller settings.
Acknowledgment
One of the authors held a National Science Foundation Fellowship during the course of this work. The numerical calculations were carried out a t the Princeton University Computation Center, which is supported in part by the National Science Foundation. The assistance of Vincent Petriano in drawing and lettering the graphs is gratefully acknowledged.
446
I & E C PROCESS D E S I G N A N D DEVELOPMENT
Nomenclature C
= constant
G
= =
G,
K, K, P, r
=
S
= = = =
Ti Td
=
t t,
max
= = =
plant transfer function controller transfer function proportional gain ultimate proportional gain ultimate period normalized Laplace variable Laplace variable integral constant derivative constant time normalized time
GREEKLETTERS 0 = 1/Ti ec = controlled variable 6, = reference variable ou = upset variable el = controller output variable ( = plant parameter 7 = dead time w = plant parameter literature Cited
Callander, A., Hartree, D., Porter, A,, Stevenson, A , , Proc. Roy. Soc. London 161, Ser. A, 460 (1937). Chien, K., Hrones, J., Reswick, J., Trans. ASME 74, 175 (1952). Cohen, G., Coon, G., Trans. A S M E 75, 827 (1953). Eckman, D., Trans. ASME 68,707 (1946). Evans, N., “Control System Dynamics,” p. 82, McGraw-Hill, New York. 1954. Farrington, G., “Fundamentals of Automatic Control,” p. 196, Chapman and Hall, London, 1951. Hazebroek, P., van der Waerden, B., Trans. A S M E 72, 309-22 (1950). Hougen, J., Walsh, R., Chem. Eng. Progr. 57, 69 (1961). Johnson, E., “Automatic Process Control Principles,” Chap. 6, McGraw-Hill, New York, 1967. Law, V., Bailey, R., Chem. Eng. Sci. 18, 189 (1963). McAvoy, T., Ph. D. dissertation, Princeton University, Princeton, N. J., 1964. McAvoy, T., Johnson, E., IND. ENG. CHEM.PROCESS DESIGN DEVELOP. 5,440 (1966). Nehari, Z., “Introduction to Complex Analysis,” pp. 82-90, Allyn and Bacon, Boston, 1961. Oldenbourg, R., Sartorius, H., “The Dynamics of Automatic Control,” pp. 65-77, Am. SOC.Mech Engrs., New York, 1948. Ream, N., Trans. Soc. Instr. Tech. 6,No. 1, 19 (1954). Wescott, J., Trans. ASME 76, 1253 (1954). Wills, D., Control Eng. 9, No. 4, 104; No. 8, 93 (1962). Wolfe, W., Trans. A S M E 73, 413 (1951). Young, A., “Introduction to Process Control System Design,” p. 37, Instruments Publishing Co., Pittsburgh, 1955. Ziegler, J., Nichols, N., Trans. A S M E 64, 759 (1942); 65, 433 (1943). RECEIVED for review October 10, 1966 ACCEPTEDMarch 20, 1967
