Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 1132-1 136
1132
thermal measurements using a reduced time scale show that the pyrolysis mechanism is temperature dependent below 500 "C. Application of a variety of commonly used solid-state reaction expressions indicates that as the temperature increases the mechanism progresses from a diffusion-controlled reaction, through a phase-boundary process, to a reaction governed by nucleation and growth. Removal of the minerals does not change the mechanism; however, the half-life for the reaction is considerably reduced below 500 O C . Analysis of nonisothermal pyrolysis data does not require the use of a temperature-dependent rate law, and first-order kinetics with a single activation energy w ill fit the data. The apparent discrepancy between these two approaches is most likely related to the neglect of the temperature effect on the mechanism when transposing equations to enable integration to be carried out.
Literature Cited Allred, V. D. Colo Sch Mines 0.1964, 5 9 , 47. Allred, V. D. Chem. Eng. Prog. Symp. Ser. 1966, 62, 5 5 . Anthony, D. A,; Howard, J. B.; Hottel, H. C.; Meissner, H. P. "Fifteenth SymDosium on Combustion"; The Combustion Institute: Pittsburgh, PA: 1975, p 1303. Avrami, M. J. Chem. Phys. 1941, 9 , 177. Campbell, J. H.;Koskinas, G. H.; Gallegos, G.; Greg, M. Fuel 1980, 59, 718. Campbell, J. H.; Koskinas. G. H.; Stout, N. D. Fuel 1978. 5 7 , 372. Charlesworth, J. M. Ind. f n g . Chem. Process Des. Dev. 1985, preceding paper in this issue. Coats. A. W.; Redfern, J. P. Nature (London) 1964, 2 0 1 , 66. de Bruijn. T. J. W.; de Yong, W. A.: van den Berg, P. J. Thermochim. Acta 1981, 4 5 , 315. Fabuss, B. M.; Smith, J. 0.; Satterfield, C. N. I n "Advances in Petroleum Chemistry and Refining"; McKetta, J. J., Ed.: Interscience: New York. 1964.
Haddadin. R. A.; Mizyed, F. A. Ind. Eng. Chem. Process Des. Dev. 1974. 13, 332. Hancock, J. D.: Sharp, J. H. J. Am. Ceratn. SOC. 1972, 55, 74. Hubbard, A. B.; Robinson, W. E. U.S. Bur. Mines Rep. Invest. 1950, 4744. Jacobs, P. W. M.; Tompkins, F. C. Proc. R. SOC.London A 1952, 215, 265. Jacobs, P. W. M.; Tompkins, F. C. I n "Chemistry of the SolM State"; Garner, W. E., Ed.; Butterworths: London, 1955; Chapter 7. Lynch, L. J.; Webster, D. S.;Parks, T. "Proceedlngs, First Australian Workshop on Oil Shale", Lucas Heights, May 1983, p 139. Mampel, A. 2.Phys. Chem. Abt. A 1940, 187. 43, 235. Neilsen, L. E. J. Macromol. Scl. Rev. Macromol. Chem. 1969, C3(1 ) . 69. Nobie, R. D.; Harris, H. G.; Tucker, W. F. Fuel 1981, 6 0 , 573. Nuttali, H. E.; Guo, T. M.; Schrader, S.; Thakur, D. S. I n "Geochemistry and Chemistry of Oil Shales"; Miknis, F. P.; McKay, J. F., Ed.; ACS Symposium Series No. 230; American Chemical Society: Washington, DC, 1983. O'Neal, H. E.; Benson, S.W. I n "Free Radicals-Volume 2"; Kochi. J. K., Ed.; Wiley: New York, 1973; Chapter 17. Prout, E. G.:Tompklns. F. C. Trans. Faradsy SOC. 1944, 4 0 , 488. Rajeshwar, K.; Dubow, J. Thermochim. Acta 1982, 5 4 , 71. Rajeshwar, K.; Nottenburg, N.; Dubow, J. J. Mater. Sci. 1979, 14, 2025. Schnackenberg, W. D.: Prien, C. H. Ind. f n g . Chem. 1953, 4 5 , 313. Sharp, J. H.; Brindley, G. W.; Achar, B. N. N. J. Am. Ceram. SOC. 1966, 49, 379. Shewmon, P. G. "Diffusion in Solids";McGraw-Hili: New York, 1963. Shih, S. M.; Sohn, H. Y. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 420. Tang, T. 8.; Chaudhri, M. M. J. Therm. Anal. 1980, 18, 247. Thien, S. S.;Carpenter, H. C.; Sohns, H. W. NBS Spec Publ. 1968, 302, 529. Valensi, G. Compt. Rend. 1935 201, 602. Wall, L. A.; Flynn, J. H.; Strauss, S. J. Phys. Chem. 1970, 7 4 , 3237. Wallman, P. H.; Tamm, P. W.; Spars, B. G. I n "Oil Shale, Tar Sands and Related Materials"; Stauffer, H. C., Ed.; ACS Symposium Series No. 163; American Chemical Society: Washington DC, 1981. Wischin, A. Proc. R . SOC.LondonA 1939, 172, 314. Woinsky, S.G. Ind. Eng. Chem. ProcessDes. Dev. 1968, 7 , 529. Young, D. A. "Decomposition of Solids"; Pergamon: London, 1966.
Received for review March 12, 1984 Accepted October 23, 1984
A Computer Algorithm for Optimized Control Patrlcla A. S. Ralston, Kelth R. Watson,+ Ashutosh A. Patwardhat!,$ and Pradeep B. Deshpande" University of Louisville, Louisville, Kentucky 40292
There is a continuing need for simple and efficient methods for finding optimized process and controller parameters. I n this paper, a computer algorithm is described which may be used to determine optimized process constants of an overdamped secondarder-plus dead-time model and optimum tuning constants of digital PID controllers. The algorithm should find use in on-demand identification and tuning applications.
Process identification and tuning methods are important in computer control applications. The identification procedure determines the process parameters which frequently change owing to changes in operating conditions, equipment fouling, etc. The tuning procedure finds updated tuning constants in the event that the process parameters have changed. The identification procedure of concern in this paper is based on experimental input-output data from the process. A suitable disturbance is introduced into the manipulated variable while the process is operating at steady state in manual, and the process output data are 'Currently at E. I. du Pont de Nemours & Co., Midlothian, VA. *Currently a t Colorado State University, Fort Collins, CO. 0 196-4305/85/1124-1132$01.50/0
recorded. The experimental output is compared with the predicted output of an assumed process model, and the process identification procedure is applied to determine the set of process parameters which minimize the error between observed and predicted outputs. Once new process constants are found, there may be a need to determine new controller constants. In this instance, the purpose of the optimization procedure is to determine the set of controller tuning constants which minimize a user-specified measure of error under the closed-loop response curve. Several methods are available for finding optimum tuning constants of digital controllers. Among them are the method of Gallier and Otto (1969) and Fertik (1975). These investigators have developed graphs which give optimum tuning constants as functions of process parameters. The effect of sampling is included in the form of 0 1985 American Chemical
Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 1133
additional dead time of half the sampling period. In this paper, a simple optimization procedure is described which can be used for both process identification and optimal tuning. The technique can be used in real-time digital control applications. The method utilizes the LJ optimization procedure (Luus and Jaakola, 1973) and the modifications suggested by Gaddy and co-workers (Heuckroth et al., 1976; Doering and Gaddy, 1980; Martin and Gaddy, 1982). The need to investigate a new optimization procedure to process identification arose due to the fact that the earlier method developed by Brantley et al. (1982) was found to be sensitive to initial trial parameters. For a review of early work on process identification, the reader is referred to Brantley et al. (1982). The LJ optimization method has been applied for solving a variety of nonlinear programming problems in chemical engineering including one involving optimization of a sulfuric acid process (Doering and Gaddy, 1980) and another involving model reduction of systems in the z-transform domain (Jaakola and Luus, 1974; Luus, 1980; Luus and Hsu, 1981). For a discussion of more advanced methods of system identification, the reader is referred to Eykhoff (1981). We address the identification problem first. The treatment is brief, but the interested reader may refer to Stark et al. (1982) for more detailed information. Following identification, the tuning problem is addressed. For the latter, the modified LJ procedure has been applied to find the optimal tuning constants of digitial PID controllers. Met hod The original LJ optimization procedure utilizes the following 11 steps: 1. Assume initial values for the variables to be optimized and denote them as Xl(0), X,(O), X,(O), and X4(0). Note: In identification, there are four variables to be optimized ( K , ea, T ~and , T , ) , and in tuning, up to three variables are to Le optimized (Kc,q,and T ~ ) . 2. Assign initial range for each variable and denote them as rl(0), r2(0),?,(Oh and r4(0). 3. Set iteration index j to 1. 4. Read a sufficient number of random numbers between -0.5 and +0.5. Denote them as Y(k,i). Luus and Jaakola suggest 2000 random numbers. 5. Take four P random numbers from step 4 and assign them to X1, X,,X,,and X4 so that there are P sets of values which are calculated according to
p7SG-l I N I T I A L I Z E VALUES AND RANGE
A i- i + i
1
MINIMIZES PERFOMANCE
1 AND PARAMETERS T H A T ARE OPTIMUM AS OF THIS S T E P
Figure 1. Flow chart of LJ procedure for identification and tuning.
al., 1976; Doering and Gaddy, 1980; Martin and Gaddy, 1982; Goulcher and Long, 1978). These modifications are summarized by Reklaitis et al. (1983). One of these involves the modification of eq 1 according to
Xj = X i + Y(k,i)'"'ri-'
(la)
P = 100 has been suggested by Luus and Jaakola. (For optimal tuning, three P random numbers will be required.) 6. Test constraints and determine the output for each admissible set. 7. Calculate the performance index to be optimized for each admissible set. 8. Find that set which minimizes the objective function. Write its value and the associated values of XiG). Increment j by 1to j + 1. 9. If the number of iterations has reached the maximum allowed (200 iteration are suggested), end the problem. Otherwise, go to the next step. 10. Reduce the range by an amount riW = (1 - t)rjCi-l) (2)
where X iis the current best solution. When M equals 1, as in the original LJ algorithm, the effect is that Xj becomes a random variable uniformly distributed over the interval specified in the range. With M = 3, 5, 7, etc., the distribution of X i becomes more concentrated around X i . Thus, the distribution coefficient, M , can regulate the contraction or expansion of the search region. Another enhancement proposed in the literature has to do with the range reduction factor E , Instead of holding constant as in the original LJ algorithm, it has been proposed to adjust as and when warranted so to maintain a satisfactory ratio of the number of improved search points to the number of function evaluations. Martin and Gaddy (1982) report that the adaptive random procedure resulted in up to 80% reduction in the number of function evaluations to converge to the optimum with a reliability of over 90%. In this work, the effect of varying M and t has been investigated. A flow chart of the computer program which can be used to implement the original LJ procedure is shown in Figure
usually t = 0.05. 11. Go to step 5 and continue. Several modifications have been proposed to improve the efficiency of the original LJ algorithm (Heuckroth et
If the program is used for identification, the required input data are (1)disturbance input vs. time, (2) output vs. time, (3) initial trial values of the process parameters, and (4) sampling period.
Xiv) = Xjo'-l) + Y(k,i)~ ~ v - 1 )
(1)
1.
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
1134
Table I. Comparison of Two Process Identification Techniques for Two Sets of Initial Values optimized optimized init guess parameters parameters K , = 1.0 1.008" 1.008c Bd = 1.0 1.814" 0.8392' 71 = 6.0 3.947" 4.063c 7 2 = 2.0 3.423" 3.282' I = 0.00109" I = 0.00112' K , = 1.0 1.045b 1.014' s, = 1.0 3.443' 0.9609' 7 1 = 2.0 6.544' 4.821' 7 2 = 0.5 0.02' 2.523' I = O.llb I = 0.00168
i ..._.. GALLIER
AND O T T O
FERTlK
__
THIS
WORK
"Brantley et al. (1982), 212 cycles. 'Brantley e t al. (1982), 37 cycles. 'This work (200 iterations). 5
0
15
10
20
T I ME
Figure 2. Set point response of process with optimized tuning constants.
For identification of an overdamped second-order-plus dead-time process model, the output equation for use in step 6 is (Stark et al., 1982) X,
=
alXt-l
-
u~x,-Z
+ blU,-N-l + b2UL-N-2 + b 3 ~ , - N - 3
n
c bte-
,=a
XJ2
(4)
where x,, represents the experimental output, and x, is the output predicted by eq 3 in response to the same input u which is used to obtain the experimental output, xLe. The number of output samples is n,and the duration between samples is T. Minimization of the performance index I is synonymous with finding the optimum values of K,, 6d, T ~ and , r2. The flow chart shown in Figure 1 can also be used to find the optimal tuning constants of digital PID controllers. In this instance, there are up to three variables (Kc,71, and rD) to be optimized instead of four as in process identification, and, therefore, some of the steps 1-11 must be suitably modified as indicated in the following paragraphs. In this paper, the tuning constants are optimized for changes in set point, although the same procedure can be used to find optimum tuning constants for changes in the load. The output equation for use in step 6 for optimal tuning is based on the closed-loop pulse-transfer function of the process
where C = process output, R = set point, Gho= zero-order hold, G, = process-transfer function, G, = control algorithm, and z = z-transform operator. If the tuning constants are to be optimized for say a unit step change in set point, then the procedure is to take the z transform of the variou terms in eq 5 and then invert the resulting expression into the time domain to obtain the output a t the sampling instants. The process-transfer function is assumed to be known from a previous process identification study.
1 (E(dt t
integral of absolute error, IAE =
(3)
where x, and u, are the output and the input, respectively, at the ith sampling instant, and N represents the integral number of sampling periods in 19d. The constants al, a2, bl, b,, and b3 are functions of K,, od, rl,and r2. They are defined in the Nomenclature section. The performance index to be minimized is
I =
A typical set-point response is shown in Figure 2. The objective of optimization is to find the set of tuning constants which minimizes a measure of error the closed-loop response curve. Several functions of error can be written down for use in the minimization process of which two commonly encountered expressions are 0
(6)
where
E = error, ( R - C) integral of time times absolute error, ITAE
=st t
0
IEl dt ( 7 )
Equation 6 minimizes the area under the response curve in Figure 2. Equation 7 penalizes the errors which occur late in time but is forgiving of errors which occur early in time. The choice of which equation to use is made by the designer depending on the process objectives. A suitable expression such as eq 6 or 7 is used as the performance index in step 7 of the optimization procedure. Finding the minimum value of IAE or ITAE is equivalent to finding the optimal tuning constants of the digital PID controller. Results and Discussion The work reported in this paper has been carried out on the DEC 10/90computer. The identification algorithm has been tested on numerous problems. Among them is a fourth-order system having the transfer function 1 (8) Gp(s) = (0.5s + 1)(2s + l)(s + 1)2 The objective is to fit this fourth-order transfer function to the best second-order-plus dead-time process model. A unit step change in input has been used as the disturbance input. The step response of the process determined from eq 8 by Laplace transforms has served as the experimental output. The optimum process parameters obtained from the execution of the algorithm by the method of Brantley et al. (1982)and by the original LJ procedure are shown in Table I. The results for two sets of initial guesses are shown to assess sensitivities of the procedures to initial values of the parameters. For the first set of initial guesses, both programs give good results, but when the starting values are deliberately underestimated in the second set, the method of Brantley et al. (1982)converges to a first-order-plusdead-time model having a higher error as compared with that given by the original LJ procedure. The latter method converges to the correct model even with the underestimated parameters. The results of both tests for the LJ procedure when expressed in terms of T and { yielded the same second-order
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 Table 11. Ootimum Parameters in AdaDtive Random Search Procedure for Process Identification CPU time, iterat error KP ed ‘1 ‘2 min:s (A) Initial Guesses, K, = 1, e d = 1, r1 = 6, r2 = 2 200 50 50 50 50 50 50 50
0.00112 0.0012 0.00107 0.001067 0.001061 0.00119 0.00177 0.00224
1.008 1.006 1.008 1.008 1.008 1.005 1.015 1.018
200 50 50 50 50 50 50 50 50 50 50 50
0.00168 0.00269 0.00222 0.00340 0.00288 0.00393 0.00402 0.00432 0.00210 0.00627 0.20041 0.69775
1.014 1.032 1.015 1.025 1.022 1.027 1.028 1.030 1.017 1.035 .9122 .8310
-
0.8392 0.8845 0.8181 0.8049 0.8166 0.9206 0.9627 1.066
4.063 3.787 3.868 3.562 3.739 3.763 4.916 5.169
(B)Initial Guesses, K ,
-
Table 111. Optimal Tuning Constants method K” TI this work init guess
7n
(10.0, 10.0, 10.0) (5.0, 5.0, 5.0)
Gallier and Otto Fertik
1.0132 0.9983 1.0 1.1
7.2217 7.2441 6.7 5.7
IAE ~
1.5868 1.6463 1.5 0.4
= 1, e d = 1, T~ = 2, 4.821 5.390 5.084 4.449 5.441 5.740 5.769 5.843 5.083 6.142 0.9122 2.355
0.9609 1.116 0.8528 1.224 1.152 1.277 1.285 1.315 1.045 1.528 1.423 1.054
~
3.282 3.472 3.489 3.813 3.617 3.459 2.455 2.172
1.6493 1.6490 1.6038 4.1145
model having a value of T = 3.6 and { = 1.02. Note that the L J procedure requires the calculation of the performance index I based o p n samples, whereas the value of I in a given iteration may exceed the previous best in smaller than n samples. A saving in CPU time can be achieved if the computation of I for the iteration is discontinued (Luus, 1982). This feature has been incorporated in the computer program. To assess the benefits of the adaptive random search procedures, the process identification program was executed for numerous values of E and M. The results are
T~
-
1.6493 1.6545 1.6526 1.6584 1.8921 1.6524 2.0521
1.0132 1.0092 1.0014 1.0139 1.1971 1.0086 1.3519
200 50 50 50 50 50 50 50 50 50 50 50 50 50 50
1.6490 1.6537 1.6514 1.6514 1.6515 1.6502 1.6501 1.6506 1.6500 1.6494 1.6502 1.6509 6.3123 11.3651 11.7433
0.9983 0.9909 1.0013 1.0023 1.0269 1.0206 1.0194 1.0094 0.9981 0.9996 1.0067 1.0030 0.3736 0.0095 0.0010
0.125 0.15 0.175 0.20 0.175 0.175
3 5
1:11.89
1.113 3.934 0.5071
1 1
0.050 0.075 0.025 0.100 0.125 0.15 0.175 0.200 0.05 0.05 0.05 0.05
14.44 12.56 15.95 17.18 18.27 18.81 19.30 13.21 18.45 21.89 22.29
1 1 1 1 1 1 1 3 5 7
5.543e-0.5* Gp(s) = (5.111s 1)(2.222s + 1)
(9)
+
The sampling period is 0.25 time units. The optimal tuning constants of a digital PID controller resulting from the execution of the original LJ procedure are shown in Table I11 along with the values obtained by the method
(B)Initial Guesses, K, = 5.0,
-
0.10
shown in Table 11. In this instance, by increasing the range reduction factor from E = 0.05-0.175, the CPU time could be reduced from 1 min 26 s to just under 17 s, without sacrificing the accuracy. It is possible to automatically adjust within the program as and when necessary (Martin and Gaddy, 1982) but this has not been done in this work. The results in Table I1 also show that varying M did not help. The optimal tuning program has been tested on numerous models (Watson, 1982). The results presented in this paper are for digital control of a process having the transfer function
7.2217 7.2398 7.3080 7.5516 10.870 7.2755 3.0916 7.2411 7.2232 7.2765 7.1727 7.2003 7.2934 7.2609 7.2762 7.2076 7.1991 7.3092 7.2958 2.5677 0.3721 0.0408
1 1 1 1 1 1
0.05
= 0.5
2.523 1.967 2.421 1.733 1.900 1.588 1.565 1.492 2.244
Table IV. Optimum Parameters in Adaptive Random Search Procedure for Controller Tuning CPU time, iterat IAE KC ‘D min:s ‘I (A) Initial Guesses, K , = 10.0, = 10.0, T D = 10.0 200 50 50 50 50 25 50
M
t
1:08.64 13.25 14.99 16.30 16.84 16.92 20.30 21.80
1135
1.5868 1.6340 1.6348 1.5902 1.540 1.6176 1.6707
= 5.0, 1.6463 1.6709 1.6254 1.6144 1.5840 1.5801 1.5776 1.615 1.6497 1.6370 1.6308 1.6482 0.6900 TI
0.0000 0.0177
TD
t
M
2:39.86 20.84 29.61 35.92 40.0 20.46 47.44
0.05 0.075 0.10 0.125 0.15 0.10 0.10
1
2:52.97 28.66 35.73 39.45 42.63 44.56 46.09 47.31 48.19 48.89 49.72 50.25 52.36 32.67 50.96
0.050 0.075
1 1 1 1 1 1 1 1 1 1 1 1
= 5.0
3
0.100 0.125 0.15 0.175 0.200 0.225 0.250 0.275 0.300 0.325 0.275 0.275 0.275
3
5 7
1136
Ind. Eng. Chem.
Process Des. Dev., Vol. 24,
No. 4, 1985
of Gallier and Otto (1969) and Fertik (1975). The optimization was based on the IAE criteria and a unit step change in set point. The minimum values of IAE associated with the optimum set are also shown in Table 111. The closed-loop responses determined by the z-transform method are shown in Figure 2. The results in Table I11 show that the LJ procedure is insensitive to initial guesses. As in the case of process identification, some saving of CPU time has been achieved by terminating the calculation of IAE for a given iteration whenever its value exceeded the previous best. The application of the adaptive random search procedure to the tuning problem yielded the results shown in Table IV. For this case, the optimum range reduction factor turned out to be 0.125. The CPU time could be reduced from 2 min 42 s for t = 0.0.5 to about 36 s for e = 0.125, without sacrificing accuracy. The results with t = 0.125 and M = 1 were achieved in 50 iterations. The same error could be achieved in only 20 iterations when M was equal to 3, although at a slightly higher CPU time of about 48 s. Additional runs suggested that M = 3 was optimum for this example. Here too the adjustment of M and e can be automated as suggested by Martin and Gaddy (1982), but this has not been done in this work. It must be emphasized that the optimum values of M and t are specific to the problem considered in this paper. Some comments concerning the gathering of experimental data for process identification are in order. The example presented in this paper utilized step test data, but in real-life applications, pulse inputs may be more appropriate; pulse functions have a higher frequency content and the plant resumes its original steady-state operation a t the conclusion of the test. The visual examination of the experimental data gives a crude estimate of process dead time whose use as the initial guess for dd can improve the performance of the identificiation algorithm. In some cases, noise may be a significant problem. If so, one must apply the signal conditioning techniques, including analog and digital filtering, and then use the data in the identification algorithm (Deshpande and Ash, 1981). The execution times reported in this paper are quite acceptable for o n - d e m a n d identification a n d t u n i n g in digital control applications. The two time-constants-plus dead-time process model adequately represents a wide variety of processes (Hougen, 1979; Deshpande and Ash, 1981) in the chemical industry, and the random search procedure employed in this study has been shown to be effective. The experience of other investigators reported in the literature suggests that random search procedures are effective for solving low dimensional problems, such as the ones considered in this paper, but direct searches are preferred for more complex, high dimensional problems (Reklaitis et al., 1983).
Conclusions The modified Luus-Jaakola optimization procedure has been applied to determine the dynamic parameters of an over-damped second-order-plus dead-time process model. The procedure has also been applied to determine the optimal tuning constants of digital PID control algorithms. The results indicate that the procedure is not sensitive to initial trial values of the parameters. Programming skill and effort required to implement the algorithms on the computer has been found to be quite minimal, and the execution times are reasonable. The combined identification and tuning program can be used in direct digital
control applications where the identification program can be called to identify process parameters, and in the event that the process parameters have changed, the optimal tuning program can be called to retune the digital controller. Nomenclature a , = e-T/rl + e-T/r2 a2 = e-T/rle-T/Tz bl = K , [1 (72e-mT/Tz - 71e-mT/r1)/(7Z - 7 )] b, = K , [-al (71e-mT/T1 (1 e-T/r2)- 72e-m+/Tp (1 + e - T / T 1 ) / ( $
+ +
+
-
(71e-mT/7~e-T/r2 - 7ze-mT/r2e-T/Tl)/(71- .,)I C = process output in controller tuning E = error (set point - measurement) G , = controller algorithm Gho = zero-order hold G , = process-transfer function I = performance index j = iteration index K, = proportional gain K = process gain parameter in eq IA m = modified z-transform operator (1 - 8 / T ) N = integral number of sampling periods in 6d n = number of samples r = range for variable values s = Laplace transform operator T = sampling period t = time u = input variable X = variable to be optimized x = output variable in process identification x,, = actual or experimental value of output x , = predicted value of output z = z-transform operator
b3 = K , [a, -
d'=
Greek Letters t
= range reduction factor
= dead time T = time constant = damping factor 7 d = derivative time 71 = integral time 8 = fractional dead time (8, - NT) Literature Cited Bd
Brantley, R. 0.; Schaefer, R. A,. Deshpande, P. B. Ind. Eng. Chem. process Des. Dev. 1982,2 1 , 297. Deshpande, P. B.; Ash, R. H. "Elements of Computer Process Control With Advanced Control Applications"; Instrument Society of America: Research Triangle Park, NC, 1981. Doering, F. J.; Gaddy, J. L. Comput. Chem. Eng. 1980,4 , 113. Eykhoff, P. "Trends and Progress in System Identification";Pergamn Press: Elmsford, NY, 1981. Fertik, H. A. ISA Trans. 1975, 1 4 , 292. Gailier, P. W.; Otto, R. E. Inst. Techno/. 1969, 16, 65. Goulcher, R.; Long, J. J. C. Comput. Chem. Eng. 1978,2 , 33. Heuckroth, M . L.; Gaddy, J. L.; Gaines, L. D. AIChE J . 1978. 22, 744. Hougen, J. 0. "Measurements and Control Applications"; Instrument Society of America: Research Triangle Park, NC, 1979. Jaakola, T. H. I.; Luus, R . Oper. Res. 1974,2 2 , 415. Luus, R.; Hsu, J. T. C. Paper Presented at the 2nd World Congress of Chemical Engineering, Montreal, Canada, Oct 5-8, 1981, Luus, R.; Jaakola, T. H. 1. AIChE J . 1973, 19, 760. LUUS,R. rnt. J . control 1980,32, 741. Luus, R. Private Communication, Chemical Engineering Department, University of Toronto, June 1982. Martin, D. L.; Gaddy, J. L. In AIChE Symp. Ser. 1982. 2 4 , 99. Reklaitis, G. V.; Ravindran, A,; Ragsdell. K. M. "Engineering Optimization"; Wiley: New York, 1983; p 278. Stark, P. A.; Ralston, D. L.; Deshpande, P. B. Paper presented at the American Control Conference, Arlington, VA. June 1982. Watson, K . R . M. Eng. Thesis, University of Louisville, Louisville, KY, 1982.
Received for review May 15, 1984 Revised manuscript received January 24, 1985 Accepted April 1, 1985
