ANNUAL REVIEW
MORTON M. DENN
CHEMICAL REACTION ENGINEERING Optimization, Control, and Stability
A focus upon those subjects which are of particular pertinence in reaction engineering forms this p a r t of the review he areas of optimization, control, and stability are I t is our intention in this review to focus upon those subjects which are of particular pertinence in reaction engineering. Several general reviews through 1966 are available (ZA, 4A-6A), and a fairly accurate overview of current control-related research activity is to be found in the scope of papers for the annual Joint Automatic Control Conference (7A). The recent text of Lapidus and LUUS(3.4) is a comprehensive treatment of the optimal control and certain aspects of the stability of engineering processes described by lumped-parameter models, with particular emphasis on some reactor examples. Theory continues to far outpace implementation, and one of our primary goals in this review is to point to those areas of research which have direct bearing on the design, operation, and control of chemical reactors. Some helpful comments are to be found in a recent paper of Williams ( 7 A ) . I t is perhaps worth noting parenthetically at this point that in areas which are mathematical in nature and cross traditional disciplinary boundaries there has been an unfortunate tendency among some chemical engineering authors to duplicate research results from other disciplines with “chemical engineering examples.’’
Tbroad in their scope.
M . Denn is Associate Professor of Chemical Engineerzng, Dejartment o j Chemical Engineering, Uniuersity of Delaware, h’ewark, Del. 19777. Professor Denn is also author of a book, “Ofitimization by Variational Methods,” McGraw-HzII, to be published earli, summer 1969. AUTHOR Morton
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I N D U S T R I A L ANL, i N G l N E E R l N G C H E M I S T R Y
This is an important contribution in a text or tutorial paper, but the availability of journals in other disciplines is such that one cannotjustify the use of space in chemical engineering research journals for the application of established techniques to problems of no inherent interest. The year 1967 was better than most in this respect in stability, control, and optimization, but the exceptions remain. Computation of Optima
The determination of an optimum by means of a mathematical model usually reduces to the maximization or minimization of an algebraic function subject to algebraic constraints, or a problem in “mathematical programming,” and several new works in this area have appeared (7B, 3B, 4B, 8B, Z1B). The text by Wilde and Beightler (21B), in particular, is an attempt to unify from a single point of view the numerous approaches to function minimization and, while it is disappointingly sparse in the number of practical examples and less broad in scope than the title “Foundations of Optimization” would suggest, it is nonetheless a major contribution toward consolidating an expanding literature and a valuable source for computational procedures. For processes whose profit function and constraints are expressible as “generalized polynomials,” sums of products of powers of the process variables, as is often the case in the use of engineering correlations, an application of the theory of inequalities termed “geometric programming” can reduce the computational effort. Developed in a series of papers over the past several years, the originators of the basic theory have now written a text (4B),and a lengthy chapter in Wilde and Beightler (27B) contains further applications and extensions. In steep descent algorithms of function minimization, convergence of the iterations in practical situations often
requires the development of a positive definite “weighting” matrix which rotates the gradient vector to define a search direction in the space of process parameters. Such methods of “conjugate directions” have been the object of considerable study and are reviewed in Wilde and Beightler (27B). Among the several new contributions (2B, 7B, 78B, 22B) there is particular significance in that of Zangwill (ZZB), who has demonstrated a failure to establish a linearly independent set of conjugate directions which occurs in some algorithms and may result in nonconvergence. The major computational significance of conjugate direction methods is that they do not require the evaluation of second derivatives of the function being minimized, will converge from poor starting approximations, and converge near the optimum with the same speed as the second-order Newton-Raphson method (27B). Some authors have discussed a method of constructing a diagonal weighting matrix which has been termed the “two-derivative method’’ (73B, 77B, 79B). The method is not new and is computationally inconvenient in requiring second derivatives. Processes whose defining variables evolve in space or time, such as tubular, batch, or unsteady-state wellmixed reactors, are modeled by differential or difference equations, and maximization or minimization problems for such processes are generally referred to as “variational problems.” The text of Wilde and Beightler (27B) has a very limited treatment of computational algorithms for variational problems, but the book of Lapidus and Luus ( 3 A ) devotes substantial space to this subject and serves as a good guide to the extensive literature on the subject. Several papers appeared in 1967 in the traditional chemical engineering literature (5B, 6B, 9B-72B, 76B) but they do not substantially extend the coverage in the text (3A). When process structure needs to be considered, several approaches are possible. Some of these are considered in Wilde and Beightler (27B) and several new papers (74B, 75B, 20B) in which references to other approaches may be found.
reactor performance. Others studied include the distribution of cold feed along a reactor to maximize conversion ( I C , 5C) and design of a polymerization reactor to maximize conversion to a specified molecular weight distribution (7C). Dyson and coworkers (2C) have solved several model reactor problems analytically, while Pacey and Rustin (6C) have utilized plant data and regression analysis for an optimization. Transient Operation
When a linear process is forced with a periodic signal the response is periodic with a time-averaged response equal to the output that would result from forcing at the mean input. This is not true for a nonlinear process, so that the same average expenditure of effort varied over time might result in better average performance than a constant input. Horn and Lin (5D) have shown how to use only steady-state information to determine if the performance at an optimal steady state can be improved upon by periodic forcing. Their approach is based upon the observation that a n optimal steady state cannot be improved upon by transient operation only if it satisfies all conditions for a transient optimum. Douglas ( 2 0 , 3 0 ) and Gore ( 4 0 ) have also considered periodic reactor operation. Douglas and Gaitonde ( 3 0 )have studied the response of a reaction system in an unforced limit cycle about an unstable steady state and have concluded that the potential improvement resulting from the nonlinear averaging might sometimes call for the use of positive feedback control to enhance, rather than damp, output fluctuations. Chou et al. ( 7 0 ) and Jackson ( 6 0 ) have studied another transient reactor optimization problem, the time variation required to optimize performance with a decaying catalyst. I n the simplest cases of a single irreversible reaction or a decay rate which depends only upon temperature, the policy is to adjust the temperature to maintain a constant conversion. Chou credits an independent derivation to Szepe and Levenspiel. I n an unpublished paper Pollock ( 7 0 ) has obtained similar results for a model of a catalytic reformer.
Optimal Design
Only a small number of applications of optimization principles to improved reactor design were published in 1967 (7C-8C), but several of these are striking in the demonstration of the high level of information which can be extracted with nominal effort. Gunn (3C) and Thomas and Wood (8C) have considered the distribution of a bifunctional catalyst to optimize selectivity and have found that in some simple situations the solution is to compartmentalize the reactor with constant distributions in each section. Horn and Tsai (4C) have nicely demonstrated how the linear adjoint equations can be utilized as sensitivity equations to determine the effect of local and global (recycle, bypass) mixing on
Optimal Control
I n the specification of an optimum control system, results are meaningful for practical implementation only when expressible as a feedback-feedforward system, where a measurement of the process state, entering disturbances, and time left to run are sufficient to determine the control settings. A complete theoretical treatment reducible to this form is available in general only for linear and certain two-variable nonlinear systems. For linear systems the optimal feedback control is often a relay (two-position) system with a nonlinear switching function, though when the performance criterion is an integral of quadratic terms in deviations VOL. 6 1
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from steady state of both state and control variables, the optimum is a multivariable linear feedback-feedforward control. T h e quadratic term in control deviations is sometimes called an “energy constraint.” The most comprehensive treatment is in the text by Athans and Falb (ZE). There is a very good treatment of optimal linear feedback control in Lapidus and Luus (3A), but the chapter on the optimal control of nonlinear systems is not concerned with reduction to feedback-feedforward form. A new book by Bellman (3E) is concerned entirely with the linear-quadratic problem. The nonlinearity of the controller in even the simplest linear situations and the computational hurdles in more complex problems have prompted interest in the quantitative relationship between easily implementable control laws and optimal controls. Simple feedback laws can be derived from stability considerations, and this work continues ( 3 4 72E). Denn (4E) has demonstrated that such a control, which is a relay system with linear switching for a linear process, is in fact the optimum for a performance index which is not significantly different from an integral of a quadratic in the state deviations. Such “inverse problems,” in which the control is specified and the criterion for which it is optimal is then determined, were first studied in a control context by Kalman ( 9 E ) . Other extensions of less importance in process work have appeared ( I E , 78E). The significance of the inverse problem is that for many control situations an economic performance index is not available, and “keeping fluctuations small” is the only requirement. The demonstration that a control law is the best for a reasonable criterion of “smallness” is assurance that it will perform well over the entire control period. Fuller (5E) and Millman and Katz (75E)have considered similar problems. I n each case the performance of linear feedback controllers, relay in the first study and three-mode in the second, was quantitatively compared to optimum controllers for appropriate performance indices. Only negligible differences in performance were observed in each case for good choices of parameters in the linear control laws. The latter study is of particular interest in that it is concerned with maximizing conversion in batch reactors for series and parallel reactions. The performance index used by Fuller to evaluate the linear switching laws was an integral of a quadratic in state deviations, relating the results closely to the work of Denn (4E). A number of other pertinent studies of approximations to optimal controls have also appeared (7E, 73E, 74E, 16E). Attention has also been paid to the question of choice of an objective function from some other viewpoints. Waltz (79E) has performed some simple but interesting calculations on the incorporation of secondary optimization criteria by allowing slight variations from the opti48
INDUSTRIAL A N D ENGINEERING CHEMISTRY
mum in the primary criterion, and he has shown the possibility of large improvements in total performance. Johnson (8E) has studied the Chebyshev “minimum-ofmaximum deviation” criterion, a meaningful but little used measure of performance. Fuller (6E) has critically examined the use of the energy constraint as a penalty function for incorporating saturation limits on the control settiny into the analysis of linear processes and has found that the increase in the value of the quadratic objective may be as high as 5OyG,a potentially large trade-off for the convenience of using multivariable proportional feedback in place of a linearly switching relay controller. The greatest nuniber of other papers on optimal control of particular interest in chemical reactor analysis in 1967 were concerned with optimal control of distributed systems (IOE, 17E, 77E, ZOE, 27E). There is a growing literature in this area, with many fundamental contributions from chemical engineers, but the papers presented at the Joint Automatic Control conference (7A) indicate unfortunate duplication of effort and lack of general awareness of the chemical engineering contributions. The references cited can serve as a guide to the earlier basic studies. Identification and Adaptation
Process identification, the determination of a working process model from on-line measurements of input and output data, is a subject of obvious importance in reactor analysis which has generally received only superficial treatment from chemical engineers. A symposium on “Identification in Automatic Control Systems’’ was sponsored by the International Federation of Automatic Control, and two reviews prepared for that symposium have recently been published (3F, 6 F ) . The major effort has been devoted to the identification of linear systems, and the work of Rogers and Steiglitz (79F) is notable here. These authors have investigated the maximum likelihood estimation of process parameters in the presence of process noise and have shown how the problem reduces to one in mathematical programming with multiple local solutions. There have been numerous other studies ( I F , 4F, 5F,9F,72F, 18F, 22F, 23F). The paper of Hutchinson and Shelton (7OF) is a good demonstration of the practical application of spectral analysis ( 7 IF). A pattern recognition algorithm has also been used (74F). Pattern recognition would appear to have applications in reaction engineering beyond this one case, and a new review (75F) is worth noting. The identification of nonlinear processes is a more difficult task, and most approaches are based on the construction of hereditary models which predict response based on previous observations. Harris and Lapidus (8F) have written a good tutorial on the W’iener-Bose
method and applied it to reactor situations there and in a more technical presentation (7F). Several others also contributed in this area (73F, 20F). Closely related to the identification problem is the area of adaptive control. Several publications of interest in reaction engineering appeared in 1967 (ZF, 76F, 77F, 27F). Control Applications
Disappointingly little use has been made of the results noted in the preceding two sections in applications. T h e work of Astrom (7G), though not concerned with a reactor, can be cited as a notable exception, and Jeffers et al. (8G) report the use of an optimal relay controller on a nuclear reactor. Shah (75G, 16G; see also 4G) has presented two detailed accounts of simulation studies for purposes of control and optimization in ammonia and ethylene production, and several useful studies have been concerned with settings for simple feedback controllers ( 7 0 6 7 2 6 ) and other conventional means of control (72G, 74G). Interest in the manipulation of system structure to simplify control analysis, usually to produce noninteracting control, continues (2G, 3G, 6G, 7G, QG). T h e amount of information required for such manipulations generally implies a model of sufficient accuracy that, with the formulation of an objective, a truly optimal control can be determined with no more effort. This will, in general, contain interaction, but further output constraints can be imposed. I n the same vein, the modal control analysis of Gould and Schlaepfer (5G) can be criticized on the grounds that at the same level of model sophistication, the problem is susceptible to attack using optimal control theory. Stability
Most stability studies of interest in reaction engineering in 1967 were concerned with the uniqueness and stability of steady states in packed and cmpty tubular reactors ( I H , ZH, 5H-QH, 77H-73H). Padberg and Wicke ( 7 7H) showed experimentally the existence of multiple steady states in a packed system. Eight papers by Amundson and coworkers (ZH, 5H-QH), taken as a whole, provide a nice demonstration of the variety of methods available for the study of stability problems, as well as presenting results of intrinsic physical interest. I n other work, Luecke and McGuire (4H) have investigated the use of Krasovskii’s theorem to generate stability regions with quadratic Liapunov functions. Their best results remain conservative. Matsuura and Kat0 (70H) have shown how concentration equations alone in an isothermal reactor can lead to instabilities for reasonable rate expressions. New results in controller stability, such as those of Popov and Kalman
(3H), have not yet been exploited in reactor analysis, though such studies are under way. REFERENCES General (1A) Joint Automatic Control Conference Preprints (Philadelphia, June 28-30, 1967), Lewis Winner for ISA, New York, 1967. (2A) Lapidus, L., “Control, Stability, and Filtering,” IND.ENC. CHEM.,59, No. 4, 28-38 (1967). (3A) ,Lapidus, L., and Luus, R., “Optimal Control of Engineering Processes,” Blaisdell, Waltham, Mass., 1967. (4A) Riinsdorp, J. E., “Chemical Process Systems and Automatic Control,” Chem. Eng. Prog., 63, No. 7, 97-116 (1967). (5A), Sargent, R. W. H., “Integrated Design and Optimization of Processes,” tbzd.. NO. 9. 71-78 (1967). . , (6A) Williams, T. J., “Annual Review: Computers and Process Control,” IND. ENC.CHEM.,59, No. 12, 53-68 (1967). (7A) Williams, T. J., “University Research Results for Industrial Process Control,” Instr. Tech., 14, No. 11, 49-56 (1967). Computation of Optima (1B) Abadie, J. (ed.), “Nonlinear Programming,” John Wiley, New York (1967). (2B) Daniel, J. W., “Convergence of the Conjugate Gradient Method with Computationally Convenient Modifications,” Num. Math., 10, 125-31 (1967). (3B) Dudnikov, E. E., and Rybashov M . V., “Methods for the Solution of Mathematical Programming Problems 0; General Purpose Analog Computers (Survey),” Automation Remote Control, 28, 788-826 (1967). (4B) Duffin, R. J., Peterson, E. L., and Zener, C., “Geometric Programming: John Wiley, New York, 1967. (5B) Fine, F. A., and Bankoff, S. G., “Control Vector Iteration in Chemical Plant Optimization: Accessory Minimization Problem of Jacobi,” IND.ENG. CHEM. FUNDAM., 6 , 288-93, (1967). (6B) Fine, F. A., and Bankoff, S. G., “Second-Variational Methods for Optimization of Discrete Systems,” ibid., pp 293-99. (7B) Goldstein, A. A., and Price, J. F.,“An EfficientAlgorithm for Minimization,” Num Math., IO, 184-89 (1967). (8B) Jacobs, 0. L. R., “An Introduction to Dynamic Proqramming, the Theory of Multistage Processes,” Chapman and Hall, London, 1967. (9B) L y i d u s , L.,“The Control ofNonlinear Systems via Second Order Approximations, Chem. Eng. Prog., 63, No. 12, 64-72 (1967). (10B) Lapidus, L., and Luus, R., “ T h e Control of Nonlinear Systems. I : Direct Search on the Performance Index,” A.I.Ch.E. Journal, 13, 101-08 (1967). (1 1B) Lee, E. S., “ Quasilinearization in Optimization: A Numerical Study,” ibid., pp 1043-51. (12B) Luus, R., and Lapidus, L., “ T h e Control of Nonlinear Systems. 11: Convergence by Combined First and Second Variations,” ibid., pp 108-13. (13B) Merthot, J. C., and Cholette, A,, Letter to the Editor, Can.J . Chem. En
