Ind. Eng. Chem. Res. 1996, 35, 485-487
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Approximate Dynamic Models for Recycle Systems Alan J. Hugo,*,† Paul A. Taylor, and Joseph D. Wright‡ Department of Chemical Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada
There has been recent interest in the dynamics and control of plants with recycle, due particularly to the complex dynamics that these plants can exhibit. While the transfer functions for the recycle system may be constructed from the individual linear transfer functions for each of the subunits comprising the plant, the resultant overall plant transfer function will often contain denominator deadtime terms. These denominator deadtime terms preclude the use of timedomain simulations and most of the common advanced controller design algorithms as the mathematical methods employed by these techniques require transfer functions with rational denominators. It is therefore necessary to simplify these models before they can be used, and the purpose of this paper is to present a technique suitable for these model simplifications which retains the original system stability and has a strong intuitive appeal. Introduction Systems that contain recycle are fairly common in industry, and their complex dynamics have been explored in several recent papers (Luyben, 1993a,b; Papadourakis et al., 1989). A common recycle example is a process where unreacted reactant is separated from product and returned to the inlet of the reactor (Figure 1). While individual transfer functions may be available for each component in the recycle system, it is often desired to determine the overall transfer function response so that the unit may be simulated or a multivariable controller may be applied to the entire unit. Consider the unit represented by the block diagram shown as Figure 2. Simple block diagram manipulations give the following open-loop transfer function for the entire plant:
Gp(s) )
KFe-φFs(τFs + 1) (τFs + 1)(τRs + 1) - KFKRe-(φF+φR)s
Figure 1. Reactor/separator plant. Here the separator recycles reactant back to the reactor feed. Another common recycle process is the fluidized cat cracker unit for petroleum refining.
(1)
Due to the deadtime term in the denominator, it is difficult to use this form for analysis or controller design, as most analytical techniques require a rational denominator. Indeed, it is even difficult to determine the poles of the above system, unless the exponential term is expressed as its series expansion, in which case there are an infinite number of poles. It is possible to analyze the above system in the frequency domain, as was done by several researchers (Denn and Lavie, 1982; Ohbayashi et al., 1989), in order to assess stability. Other researchers (Luyben, 1993a; Silverstein and Shinnar, 1982) simplified the problem by analyzing systems without deadtime. However, many chemical engineering systems contain deadtime, and since controller performance is significantly affected by deadtime, its presence should not be neglected. Furthermore, most advanced controller design techniques require either the time-domain step weights (Cutler and Ramaker, 1980) or discrete transfer functions (Harris and MacGregor, 1987), and relying on frequency domain analysis alone is insufficient to design these controllers. † Present address: Control Arts Inc., 616 Bobbie Drive, Danville, CA 94526. ‡ Present address: PAPRICAN, 570 St John’s Blvd., Pt. Claire, Quebec H9R 3J9, Canada.
0888-5885/96/2635-0485$12.00/0
Figure 2. Block diagram of a recycle plant. The individual transfer functions could be of any order but would generally be of first or second order plus deadtime.
There is, unfortunately, no direct way to remove the exponential term from the denominator, and approximate transformations must be used. What will be shown in this paper is a method for obtaining an approximation to eq 1 where the denominator is a rational polynomial in s. Previous Work There are a variety of techniques for obtaining simplified models of transfer functions for functions that contain exponential deadtime terms, such as moment matching and Pade approximations (Gibilaro and Lees, 1969), but these techniques have the serious disadvantage of occasionally producing unstable approximations to stable systems. Other techniques, such as dominant eigenvalue retention (Kim and Friedly, 1975), can only be applied to rational transfer functions. The problem here is not one of simply reducing the order of the polynomial (computer aided design packages do not generally have trouble with high-order systems), but © 1996 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 35, No. 2, 1996
obtaining a transfer function that is amenable to current mathematical design and analysis techniques. Perhaps the only researchers to consider this problem explicitly were Papadourakis et al. (1989). Their methodology consisted of using the method of moments on the transfer function numerator and denominator separately to obtain separate approximations to the numerator and denominator and then combining these two approximations into the final approximate transfer function. While the technique did not have the serious flaws of former techniques, it does have the following shortcomings: 1. The algorithm may result in unstable approximations to stable processes. If this occurs, the authors recommend choosing a higher order approximation and testing this approximation for stability. The problem with this is that the designer is not always sure whether the original system is stable, and thus obtaining an unstable approximation may accurately reflect the true process. 2. The simplified transfer function deadtime is the approximate numerator deadtime minus the approximate denominator deadtime. If this number is less than zero (i.e., a prediction in the transfer function), the authors recommend setting the approximation deadtime to zero. This could lead to approximations with deadtimes that are significantly different from the actual system. Equation 1 indicates that no response is possible before φF, so the approximation should also have a deadtime of at least this long (controller robustness is generally sensitive to deadtime errors). 3. It is not apparent how accurate the approximation is, and while higher order approximations should give better accuracy (unless they result in unstable approximations to stable systems), there is little to recommend to the designer when a higher order approximation is necessary. 4. Any physical meaning of the original system parameters (gains, time constants, deadtimes) is lost in the approximation. The above are shortcomings, and do not preclude use of this algorithm. Shown in the next section is a method that is computationally simple, is intuitively attractive, and does not contain any of the above limitations. Taylor Series Approximation A third-order Taylor series expansion of eq 1 around the term e-φRs ) 0 gives
KF2KRe-(2φF+φR)s KFe-φFs + Gp ≈ + τFs + 1 (τ s + 1)2(τ s + 1) F R KF3KR2e-(3φF+2φR)s (τFs + 1)3(τRs + 1)2
+ O((KFKR)3e-3φRs) (2)
The above approximation is a series expansion that shifts the denominator deadtime terms to the numerator, thus allowing standard analysis techniques to be used. Each term in the Taylor series has a physical meaning: the first term is the transfer function for an input going once through the forward pass without recycle; the second term is an input going through the forward pass, going back through the recycle pass, and then through the process once more. The result of this recycle is therefore a third-order system (that is, three first-order systems in series), with the gain and deadtime corresponding to going through the forward pass twice and the recycle pass once. The next term repre-
Figure 3. Block diagram for example system.
Figure 4. Input step change for Taylor series approximation to eq 3.
sents the input looping twice, i.e., going through the forward pass three times and the recycle pass twice. Note that the original deadtime for the system is retained; no response may occur in either the approximation or the original system until after the deadtime period of the forward transfer function φF. The order of the expansion could be increased if more accuracy is required. The steady-state error in this approximation (if the system is stable) is simply (KFKR)n/ (1 - KFKR) where n is the order of the approximation. Alternatively, the user could enforce the steady-state gain of the approximation to be the same as the steadystate gain of the original system by dividing the above equation by 1 - (KFKR)n/(1 - KFKR). A stability analysis of the approximation and of the original system may easily be obtained. Clearly the system is stable if and only if the forward and recycle transfer functions are stable, and if |KFKR| < 1.0. This later requirement means simply that the system must attenuate the affect of an input change as it travels around the recycle path. Although the functions shown here contain a single denominator deadtime term, multivariable plants will in general result in transfer functions with multiple different denominator deadtime terms. Extension of this method to the multivariable case is straightforward and shown in Hugo (1991). While the above approximation may be of high order, there is no theoretical limitation to the order of the system for discrete transfer function based controller design (Harris and MacGregor, 1987), and inverting the transfer function to the time domain for use in dynamic matrix control (Cutler and Ramaker, 1979) is straightforward. Example This example is taken from Papadourakis et al. (1989), and represents the transfer function for a reactor/separator system similar to that shown in Figure 2. A block diagram of the system relating the separator overhead composition to the inlet feed composition is shown as Figure 3. This block diagram may be ex-
Ind. Eng. Chem. Res., Vol. 35, No. 2, 1996 487
pressed as:
G1p ) kDkR1(τ0Ds + 1)e-dDs (τRs + 1)(τ1s + 1)(τ2s + 1) - kR2kD(τ0Ds + 1)e-dDs
(3)
A third-order Taylor series expansion of the above around e-dDs gives the following:
in this paper leads to a physically intuitive approximation which is simple to program, retains the stability of the original system, and gives a direct indication of the accuracy of the approximation. While the approximation results in a high-order model, this is a minor limitation for a computer aided design package. If a lower-order model is desired, standard simplification techniques may be applied to each element of the approximation. Nomenclature
G1p ≈
kDkR1(τ0Ds + 1)e-dDs (τRs + 1)(τ1s + 1)(τ2s + 1)
+
kD2kR1kR2(τ0Ds + 1)2 e-2dDs ((τRs + 1)(τ1s + 1)(τ2s + 1))2 kD3kR1kR22(τ0Ds + 1)2 e-3dDs ((τRs + 1)(τ1s + 1)(τ2s + 1))3
+
+ O(e-4dDs) (4)
Clearly, the above expansion converges if the term kDkR2 is less than 1.0, and all denominator time constants are positive. For the numeric values given in Papadourakis et al. (1989), kDkR2 equals 0.4073, the time constants are greater than zero, and therefore the transfer function is stable. Note this fact is difficult to determine from eq 3. The steady state error for the approximation above is 0.40734/(1 - 0.4073) ) 0.0464, or about 5% at steady state. If more accuracy is required, the order of the Taylor series may easily be increased, or the above may be divided by (1 - 0.0464). Step responses for a first-, second-, third-, and fourthorder approximation are given in Figure 4. Note that the plots all have the same initial response, as the higher-order series are better approximations at later times. While these approximations all appear similar to second-order systems, recycle process in general can show very high-order behavior, particularly if there is long deadtime between the processing units (Hugo, 1991). The above expansions are very easily performed using a symbolic manipulation software package such as MAPLE (Char et al., 1990). Once eq 3 is expressed in proper form, the Taylor series expansion is determined using the following single line of MAPLE code:
G expansion :) taylor (G plant, exp term, n order); where the denominator deadtime in G plant is expressed as exp term, and n order is the desired order of the Taylor series expansion. Conclusions Block diagram manipulations for typical recycle processes result in transfer functions that contain denominator deadtime terms. While it is possible to determine the frequency response of these systems, these transfer functions must be simplified if they are to be used for controller design and simulation. However, most published transfer function approximations are not suitable for systems with deadtime, particularly when there are denominator deadtime terms. The algorithm proposed by Papadourakis et al. (1989) is able to produce rational transfer functions for irrational recycle systems, but requires considerable user input with little guidance as to the accuracy or required stability of the approximation. The algorithm proposed
dD ) example deadtime kD ) example forward transfer function gain KF ) forward transfer function gain KR ) recycle transfer function gain kR1 ) example forward transfer function gain KR2 ) example recycle transfer function gain φF ) forward transfer function deadtime φR ) recycle transfer function deadtime τF ) forward transfer function time constant τR ) recycle transfer function time constant τ0D ) example forward transfer function numerator time constant τ1 ) example forward transfer function denominator time constant τ2 ) example forward transfer function denominator time constant τR ) example recycle transfer function denominator time constant
Literature Cited Char, B. W.; Geddes, K. O.; Gonnet, G. H.; Monagan, M. B.; Stephen, M. W. MAPLE. First Leaves, 3rd ed.; Watcom Publications: Waterloo, Ontario, 1990. Cutler, C. R.; Ramaker, B. L. Dynamic Matrix ControlsA Computer Control Algorithm. AIChE 86th National Meeting, Houston, TX, April 1979. Denn, M. M.; Lavie, R. Dynamics of Plants with Recycle. Chem. Eng. J. 1982, 24, 55. Gibliaro, L. G.; Lees, F. P. The Reduction of Complex Transfer Function Models to Simple Models Using the Method of Moments. Chem. Eng. Sci. 1962, 24, 85. Harris, T. J.; MacGregor, J. F. Design of Multivariable Linearquadratic Controllers Using Transfer Functions. AIChE J. 1987, 33, 1481. Hugo, A. J. Dynamics and Control of Recycle Plants. Ph.D. Dissertation, McMaster University, Hamilton, Ontario, 1991. Kim, C.; Friedly, J. C. Approximate Dynamic Models for Chemical Process Systems. Ind. Eng. Chem. Process Des. Dev. 1974, 13, 177. Luyben, W. L. Dynamics and Control of Recycle Systems. 1. Simple Open-Loop and Closed-Loop Systems. Ind. Eng. Chem. Res. 1993a, 32, 466. Luyben, W. L. Dynamics and Control of Recycle Systems. 2. Comparison of Alternative Process Designs. Ind. Eng. Chem. Res., 1993b, 32, 476. Ohbayashi, S; Shimizu, K; Matsubara, M. Dynamics of the Flash Fermentor System with Recycle. Ind. Eng. Chem. Res. 1989, 28, 1202. Papadourakis, A.; Doherty, M. F.; Douglas, J. M. Approximate Dynamic Models for Chemical Process Synthesis. Ind. Eng. Chem. Res. 1989, 28, 546. Silverstein, J. L.; Shinnar, R. Effect of Design on the Stability and Control of Fixed Bed Catalytic Reactors with Heat Feedback. 1. Concepts. Ind. Eng. Chem. Proc. Des. Dev. 1982, 21, 241.
Received for review March 23, 1995 Revised manuscript received August 18, 1995 Accepted September 25, 1995X IE950198+ X Abstract published in Advance ACS Abstracts, December 1, 1995.
