Comparison of Controller-Setting Techniques as Applied to Second-Order Dead Time Processes William A. Weigandl and James E. Kegerreis School of Chemical Engineering, Purdue University, Lafayette, Ind. 47907
Several controller-setting techniques are compared for the proportional-integral control of second-order dead time plants. The measures of performance illustrated b y figures are peak height, decay ratio, integral of the absolute error, and integral of the time times the absolute error. The results show that setting techniques based on first-order dead time models are too conservative when applied to second-order dead time plants.
M a n y processes are effectively represented by a secondorder with dead time model (Latour et al., 1967). In many installations, control of these processes is achieved by using the conventional proportional and integral modes (Buckley, 1964; Gallier and Otto, 1968; Hazebraek and van der Kaerden, 1950). I n this work, several setting techniques for P-I controllers are compared with respect to the quality of control when applied to second-order dead time systems. In particular, we consider the Ziegler-Xichols (Ziegler and Sichols, 1942) technique with the ultimate period-gain rules (Coughanowr and Koppel, 1965),This technique uses the entire second-order dead time model of t'he process to arrive a t controller sett,ingsLe., we do not use the Ziegler-Nichols technique with t,he first,order dead time approximat'ion obtained from the process reaction curve. I n addition, we also consider a technique developed especially for second-order systems (Lopez et al., 1969) which, of course, also uses the entire second-order dead time model of the process. I n this work, we refer to t,his last technique as SD. Also, we consider five other techniques, but these require the process dynamics t,o be approximated by a first-order dead time model. These are the Cohen-Coon (Cohen and Coon, 1953) and 3C (Murrill and Smith, 1966) techniques, which were based on quarter decay and the minimum integral of the error, and techniques labeled IAE, ISE, and ITAE (Lopez et al., 1967; Miller et al., 1967; Smith and Murrill, 1966), which were derived on the basis of minimum integral of the absolute error, minimum integral square error, and minimum integral of the time absolute error, respectively. In t'his presentation, the setting techniques will be capitalized as above, and the measures of performance used in their derivation will always be written out. hliller et al. (1967), compare six of the seven techniques mentioned above, and they conclude that the techniques based on the minimum integral criteria are superior to the others for a unit step disturbance. They also state that, of these three, ITAE is the best. But! in so doing, they use Ziegler-Nichols with the processes reaction curve-i.e., it is used with the first-order dead time approximation. Although their conclusions are based on the transient response of a firstorder dead time process, their discussion of techniques for approximating second-order dead time processes by firstorder dead time models leaves the impression that their conclusions also hold for second-order dead time processes. 1Ic1
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To whom correspondence should be addressed. Ind. Eng. Chern. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
;Ivoy and Johnson (1967), worlcing with controller sett'ing techniques which are based on a second-order dead time model, maintain t'hat methods derived on the basis of a first-order dead time model are too coiiservat,ive when apl)lied to secondorder dead t'ime plants. They illustrate their conclusion for a P-I controller with the transient response for a single secondorder system and a single first,-order dead t,ime setting technique, namely that of Hazebraek and van der Waerdeii (1950). Some questions which then arise are: one, is the conclusion of 1lcAvoy and Johnson valid for ot,her second-order dead time processes; two, is this conclusion valid when comparisons are made with other first-order dead titne setting techniques; and three, is this conclusion valid if one compares different measures of the t,ransient response. In t'his work, we answer these questions by: one, examining this conclusioii for a wide class of second-order dead t'ime processes; two, by examining this conclusion for an increased number of first-order dead time setting techniques, namely five; and three, by examining this conclusion by comparing several quantitative measures of the transient response. The two base cases for our comparison are Ziegler-Nichols (1942) technique with the ukimate period-gain rules and the technique of Lopez et al. (1969) developed especially for second-order with dead time processes. These t'wo techniques, in contrast to the five other techniques, arrive a t the controller settings using the eiitire second order with dead time model of the process. In addition to the above questions, the results of this work yield other conclusions which concern the use of tlhe CoheiiCoon setting technique, and variat,ioii between setting techniques. Solution Technique
T o carry out the above comparisons, the closed-loop step response for a second-order dead time plant must be obtained. For this case, the closed-loop load disturbance transfer function is
K
-
Taking a comn~on deiiorniiiator in Equation 1, inserting - X ( S ) / C ( ~ S )for G,, mai~ipulatiiig, and transferring to the time domain, we obtain
(2) The input t o the process, composed of the controller output and load disturbance, will be denoted as I :
I ( t ) = u(t)
+ m(t)
(3)
Combining Equation 3 with the equation for a proportional-integral controlkr and noting that the load disturbance is a unit step, we obtain the second equation required for the closed-loop response
The values of the constants K C K Dand rl/T are now determined from the various controller tuning techniques esninined in this n.ork. The solut'ion to Equation 7 is readily obtained with the digital computer by the matrix exponential technique (Ball and *%dams,1967). The evaluation of what is the best performance for a control system is often subject to much debate. I n this paper, four common performance indices are presented: peak height, decay ratio, integral of the absolute error, and the integral of the time times the absolute value of the error. Graham and Lathrop (1953), Smith and AIurrill (1966), and Wills (1962) all consider the iiitegral of the time absolute error a n excellent performance index. Comparison of Controller-Setting Techniques
Xormalizing b y defiiiing c, = c/K,, t, = t / T , and d and differentiating Equation 4 we obtain
=
Td/T
(5)
Defining the state variables z1 = C r , x2 = d c r / d t r , and r3 = I ( t r ) . aiicl replacing the unit impulse function by an initial condition of 1.0 on I ( & ) ,Equations 5 and 6 yield the following vector-matrix differential equation : 0 -
1
1b
- KcK,
-1s noted previously, the Cohen-Coon, 3C, I.4E1 ISE, and ITAE setting techniques require that' the process to be controlled be fitted to first order with dead time. Two fitting techniques are considered in this paper. The first and more standard, used by Cohen and Cooii (1953), involves drawing a tangent at the point of inflection on the open-loop step response curve. The intersection with the abscissa is the apparent dead time, and t'he equivalent first-order time constant is the ultimate gain over the slope. This method will be called fit,ting t,echnique one. -4second fitting technique was used by Miller et al. (1967). As before, the open-loop step response curve is used. The apparelit dead time is found in the same manner, but in t'his case, the first-order time constant is taken as the time when the response reaches 63.2y0 of its ultimate value minus t'he apparent dead time. This method mill be called fit technique two. Figures 1-4 illustrate the behavior of eight PI controller setting techniques as applied to the second-order dead time system for a representative value of d . The authors have data for several other values of d-i.e., d = 0.1, 0.5, 0.7, 0.9-but the restriction to the value of d = 0.3 was made since the qualitat'ive nature of the results were the same for all d and also to conserve space. The value of d = 0.3 was also chosen because many chemical processes have values of d between 0.15 and 0.4. These figures contain measures of closed-loop performance obt'ained by setting the controller with the best
e
Figure 1 . Generalized comparison of controller-tuning techniques for PI controllers based on peak height
0.321
0.30-
bee-,
= MAJOR DEAD TIME TIME CONSTANT = 0.3
ISE I
I
I
I
I
I
I.
1
1
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
87
/*
0.6-
/*
Figure 2. Generalized comparison of controller-tuning techniques for PI controller based on decay ratio
L 0
t
I
1
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 1.0
b = RATIO OF MINOR TO MAJOR TIME CONSTANT
i
A
3c
A
A
A
.
A
AISE JIAE ITAE
$? Lz w
Figure 3. Generalized comparison of controller-tuning techniques for PI controllers based on integral of absolute error
cc-I
W I-
3
Z S cc-2 D
r - 2 - 9 - - L -
-I
0 v)
m a
_ _ _ _ _ _ -----_ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ -------
LL
0 -I
2-N
a a
W W
I-
z
0.2
-
OA
d
Oll
=
DEAD TIME = 0.3 MAJOR TIME CONSTANT
I
I
I
I
1
I
I
I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
I I .o
b = RATIO OF MINOR TO MAJOR TIME CONSTANT
settings as given by the techniques examined in this workLe., each combination of b and d represents a different secondorder dead time system which then requires a corresponding different pair of controller settings, K , and rl. The performance indices plotted on these figures are normalized with respect to the system gain and major time constant. The actual performance indices and the normalized performance indices are related b y : actual peak height = peak height X system gain; actual decay ratio = decay ratio; actual integral of the absolute error = integral of the absolute error X system gain X major time coiistant X 1.5; and actual integral of the time absolute error = integral of the time 88
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
absolute error X system gain X (major time X 1.5, The controller techiiiques evaluated were Ziegler-Sichols, with ultimate period-gam rules, the special technique of Lopez et al. (1969) referred to as SD; Cohen-Coon with fit one, Cohen-Coon with fit two, 3C, IXE, ISE,aiid IT.1E all with
fit trio. The fact, claimed Miller et a]. (1967), that these last four setting techniques give better results 1% hen used with fit two as ol)posed to fit one was verified (Kegerreis, 1970) but 15 not slion n on the.e figures. In lookiiig a t Figure 1 for peak height, me see that all of the techniques give about the qame performance. We see that
"i
12-
IO
-
8Figure 4. Generalized comparison of controller-tuning techniques for PI controllers based on integral of time absolute error
i
i
i
d =
DEAD TIME = 0.3 MAJOR TIME CONSTANT
i
A
A
A
A ISE
A
A A
IAE
A
- cc-I _ _ _ _ _ _ _ _ _ _ _ _-----_ _ _ _--Z-N --
b = RATIO OF MINOR TO MAJOR although CC-1 and ISE give the best' results, the difference between these and the average of the other five techniques is only 7-117,. We also note that the other techniques produce results n-liich T-ary by only 67, anioiig themselves. Referring t o Figure 2 , we see the rariat,ioii in decay ratio among coiitroller-setting techniques is considerably greater than for peak height. For example, for the ratio of minor to major time coiistaiit of 0.5 (b = 0.5), the largest value of deea>-rat'io is four times the smallest value. It is apparcnt from Figure 2 that SD outperforms the others, but we also note from Figure 1 that SI) also gives the largest value of peak height for inost values of b. We also see from Figure 2 that, the Z - 9 , CC-2, and ITBE techniques also give good resultsLe., decay ratios close to 0.25. Figure 3 clearly indicates that the Ziegler-Xichols t'echnique gives the best performance except for very small values of b. It is interesting to note that' the IAE technique does not yield the smallest value of the integral of the absolute error excelit for b = 0. The case of b = 0 is that of a first-order dead time system. This result indicates that, although the IAE technique gives the smallest' value of the integral of the absolute error when applied to a first-order dead t'ime system, techniques derived on the basis of a first-order dead time model do not yield the beJt results when applied t o second-order dead time plants even when a good fitting approximatio~iis used. Figure 4 yields the same conclusion for the integral of the time times the absolute error-Le., the ITAE technique is only best for b = 0. Figure 4 clearly shows t'hat the Z-N and SD techniques are the best. The advantage of Ziegler-Nichols can also be illustrated with the response time, and integral of t'he error squared (Kegerreis, 1970). It should be kept ill mind that the result's illustrated on Figures 1-4, and consequently any coiiclusions drawn from them, are for the regulator problem. However, it is wellknowi-e.g., Lopez et al. (1969)-that most controllers are used as regulators. I n fact, nine of the 11 references for this paper which present cont'roller-set'tings techniques do so for the regulator problem. The behavior for systems subject to set point cliariges will be different, and therefore, aiiy coiiclusion made here may not hold for t'lie set point problem. For examples of papers which treat' the set point problem, see Kovira et al. (1969) a i d \Tills (1962).
TIME CONSTANT
Also, it should be remembered that our results do not imply that the first-order dead t'ime techniques will do even more poorly than they do for second-order systems if applied t o higher ordered systems-Le., a third or a tenth ordered system. But what, should be kept in mind is that chemical plants are most often very high ordered systems, and i t happens t h a t the additional dynamic constant produced by the second order with dead time model yields a better approximation t o t h e process dynamics than that produced by the first order with dead time model (Lopez et al., 1969). Therefore, the question, are first order with dead time methods too conservat~ivewhen applied to second order with dead time plants, is a valid one. Conclusions
Figures 1-4 indicate that the Ziegler-Xichols (with ultimate period-gain rules) and the SD techniques give the best performances when compared with the other techniques examined in this work. Therefore, using the above two techniques which use the entire second-order dead time model as our base for comparison, we verify the opinion of McAvoy and Johnson (19671, that setting techniques derived on the basis of a first-order dead time model are generally too conservative when applied t,o second-order dead time plants. The CoheiiCoon fit two and ITAE techniques give the second best perf or manc es. A11 iiiteresting conclusion for the Cohen-Coon setting technique is the improved performance produced b y fitting technique two. Figures 1-4 clearly indicate that CC-2 is better than CC-1 for second-order dead time systems for all measures of performance except peak height', and there the difference is only 7-117,. This result is interesting in t h a t i t is fitting technique one which is normally used with the Cohen-Coon method (Cohen and Coon, 1953; Coughanowr and Koppel, 1965). The variation among setting techniques is significantly reduced when applied to first-order dead time systems. This can be seen from the variation of values of the performance measures forb = 0.0 on Figures 1-4. Nomenclature
b
c (SI
c
(0
cr
= ratio of the minor to major time = system output, Laplace domain = s j stem output, time domain = normalized system output
constants
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
89
CC-1, CC-2 = Cohen-Coon sett,iiig technique used with fit 1 and fit 2, respectively
ratio of dead time to major time constant controller transfer function minimum integral of the absolute error controller-setting technique = minimum integral of the error-squared controller-setting technique = minimum integral of the time absolute error controller-setting technique = controller gain = process gain = controller output, Laplace domain = controller output, time domain = unit step function applied a t t = 0 = controller-setting technique developed specifically for record order with dead time processes = major time coiistaiit = dead time = normalized time = load disturbance, Laplace domain = load disturbance, time domain = state variables = Ziegler-Nichols controller-setting technique based on ultimate period-gain rules = controller-setting technique based on ‘/4 decay ratio and minimum integral of the error = = =
GR1:ICK LETTERS 6(t)
=
deta function applied at t
=
0
Literature Cited
Ball, S. J., Adains, lt. K., Uak Ridge National Laboratory, Ilept. OliNI,-TAI-l933, 1967. Buckley, P. S., “Techniques of Process Control,” pp 69, 76, 90, 149, 238, Wiley, New York, N.Y., 1964. Coheii, G . B., Coori, G. A., Trans. ASJIE, 75, 827 (1953). Coughanowr, I>. ll., Koppel, I,. B., “Procesb Syitemr Analysis arid Control,” .-_- pp 241-4, 312-13, llcGraw-Hill, New York, .T
A\.
. r
Y
.)
I‘JU.).
Gallier, 1’. h-,, Otto, I{, E., Instrum. Techno/., 15, 6,; (1968). Graham, I)., Lathrop, It. C., A I E E , 72, 273 (1953). Haxebraek,. P.,, vaii der Waerden, B., Trans. ASJIE, 72, 309
(1930). Kegerreis, J. E., 1lS thesis, Purdue University, Lafayette, Ind., 1 n?o
Id,”.
Latour, P. It., Koppel, L. B., Coughanowr, L). It., I n d . Eng. Chem. Process Des. Debelop., 6 , 4.52 (1967). Lopex, A. AI., Miller, J. A4,, Smith, C. L., LIurrill, P. W., Instrum. Technol., 14, .i7 (1967). LoDez. A . 11..Smlth. C. L.. LIurrill, P. W., Brit. Chem. Enq., i 4 , i533 (1969). AIcAvoy, T. J., Johnson, E. F., Ind. Eng. Chem. Process Des. Develop., 6, 440 (1967). Miller, J. A , , Lopez, A. hl., Smith, C. L., 11urril1, P. W., Contr. Eng 1 4 , 7 2 (1967). XurriIl, P. W.,Smith, C. L., Hydrocarbon Process, 45, 105 (1966). Itovira A . A , . ~Iurrill.P. W.. Smith, C. L., Instrum Contr. S~pt.:42, 67 i1969). ’ Smith, C. L., LIurrill, P. W., I S A J . , 13, 30 (1966). Wills, D., Confr. Eng., 9 , 104 (1962). Ziegler, J., Nichols, N., Trans. A S d l E , 64, 739 (1942). RECEIVED for review February 9, 1971 ACCEPTEDOctober 12, 1971 James E. Kegerreis was supported by a fellowship provided by Union Carbide. Computer time was supplied by Purdue University. ~
Pilot-Plant Studies of Anhydrous Melt Granulation Process for Ammonium Phosphate-Based Fertilizers Robert G. lee, Robert S. Meline, and Ronald D. Young’ Tennessee Valley Authority, Muscle Shoals, Ala. 56660
An anhydrous melt process for producing ammonium phosphate-based fertilizers b y use of merchant-grade, wet-process acid has been developed by TVA on a pilot plant scale. Heat of reaction of anhydrous ammonia with the acid is used to form the melt. The melt i s produced in a tee-type reactor and granulated in a pug mill which provides the working action needed to induce crystallization of the polyphosphates. To facilitate granulation of the ammonium phosphate melt ( 1 2-57-0), the polyphosphate content is limited to a maximum of about 3070 b y controlling the temperature of the reaction. A wide range of grades, such as 28-28-0, 21 -42-0,24-24-0,19-19-19, and 17-1 7-1 7, can be produced b y using urea, ammonium nitrate, or ammonium sulfate to provide additional nitrogen and b y using potassium chloride or other sources to provide potassium. Since only small amounts of water are present in the materials fed to the granulator, no dryer is required in the process. The physical characteristics and storage properties of the products (1 5-30% of phosphates as polyphosphate) containing urea are much better than those of similar products made b y conventional processes.
T h e Tennessee Valley A\ut’Iiority has developed o n a pilot plant scale a simple melt-type granulat,ion process for the production of ammonium phosphate-based fertilizers from merchant-grade, wet-process phosphoric acid. The process is a n outgrowth of TVA’s work on t,he direct, process for production To whom correspondence should be addressed.
90
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
of ammonium polyphosphate (Meline et al., 19iO). I n practice, this process consists of operating the direct-process reaction system in the temperature range required to produce an anhydrous ammonium phosphate melt of about 12- 57- 0 grade wit,h about 20- 30% of the Projin polyphosphate form. With t,he polyphosphate cont,ent in this range, the melt can be granulated withoiit difficulty in a pug mill. The polyphosphate con-
