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Other IFAC Publications AUTOMATIC A the journal of IFAC, the International Federation of Automatic Control Editor-in-Chief: G. S. Axelby, 211 Coronet Drive, North Linthicum, Maryland 21090, USA Published bi-monthly IFAC PROCEEDINGS SERIES General Editor: Janos Gertler, Department of Electrical and Computer Engineering, George Mason University, Fairfax, Virginia, USA NOTICE TO READERS If your library is not already a standing/continuation order customer or subscriber to these publications, may we recommend that you place a standing/continuation or subscription order to receive immediately upon publication all new volumes. Should you find that these volumes no longer serve your needs your order can be cancelled at any time without notice. A fully descriptive catalogue will be gladly sent on request. ROBERT MAXWELL Publisher MODEL BASED PROCESS CONTROL Proceedings of the IF AC Worhhop Atlanta, Georgia, USA, 13-14 June, 1988 Edited by T. J. McAVOY Department of Chemical and Nuclear Engineering University of Maryland, USA Y. ARKUN School of Chemical Engineering, Georgia Institute of Technology, USA and E. ZAFIRIOU Department of Chemical and Nuclear Engineering and Systems Research Center, University of Maryland, USA Published for the INTERNATIONAL FEDERATION OF AUTOMATIC CONTROL by PERGAMON PRESS OXFORD · NEW YORK · BEIJING · FRANKFURT SÄO PAULO · SYDNEY · TOKYO · TORONTO U.K. Pergamon Press pic, Headington Hill Hall, Oxford OX3 OBW, England U.S.A. Pergamon Press, Inc., Maxwell House, Fairview Park, Elmsford, New York 10523, U.S.A. PEOPLE'S REPUBLIC Pergamon Press, Room 4037, Qianmen Hotel, Beijing, People's Republic of China OF CHINA FEDERAL REPUBLIC Pergamon Press GmbH, Hammerweg 6, D-6242 Kronberg, Federal Republic of Germany OF GERMANY BRAZIL Pergamon Editora Ltda, Rua Eça de Queiros, 346, CEP 04011, Paraiso, Sâo Paulo, Brazil AUSTRALIA Pergamon Press Australia Pty Ltd., P.O. Box 544, Potts Point, N.S.W. 2011, Australia JAPAN Pergamon Press, 5th Floor, Matsuoka Central Building, 1-7-1 Nishishinjuku, Shinjuku-ku, Tokyo 160, Japan CANADA Pergamon Press Canada Ltd., Suite No. 271, 253 College Street, Toronto, Ontario, Canada M5T 1R5 Copyright © 1989 IFAC All Rights Reserved. No part of this publication may be reproduced, stored in a retneval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or other­ wise, without permission in writing from the copynght holders. First edition 1989 British Library Cataloguing in Publication Data Model based process control: proceedings of the IFAC workshop, Atlanta Georgia, USA, 13-14 June 1988. 1. Process control I. McAvoy, Thomas J. (Thomas John), 1940— II. Arkun, Y III. Zafiriou, E. IV. International Federation of Automatic Control 670.42'7 ISBN 0-08-035735-0 These proceedings were reproduced by means of the photo-offset process using the manuscripts supplied by the authors of the different papers. The manuscripts have been typed using different typewriters and typefaces. The fay-out, figures and tables of some papers did not agree completely with the standard requirements: consequently the reproduction does not display complete uniformity. To ensure rapid publication this discrepancy could not be changed: nor could the English be checked completely. Therefore, the readers are asked to excuse any deficiencies of this publication which may be due to the above mentioned reasons. The Editors Printed in Great Britain by A. Wheaton £sf Co. Ltd., Exeter IFAC WORKSHOP ON MODEL BASED PROCESS CONTROL Sponsored by (IFAC) — International Federation of Automatic Control Working Group on Process Control Organized by American Automatic Control Council International Programme Committee (IPC) T. McAvoy (Chairman), USA J. Balchen, The Netherlands D. Bonvin, Switzerland D. W. Clarke, UK T. Edgar, USA E. D. Gilles, FRG H. Koivo, Finland M. Kümmel, Denmark M. Morari, USA A. J. Morris, Canada G. Schmidt, FRG D. Seborg, USA T. Takamatsu, Japan National Organizing Committee (NOC) Y. Arkun (Chairman) A. Palazoglu J. Schork E. Zafiriou FOREWORD This volume contains the proceedings of the IFAC sponsored Workshop on Model Based Process Control. The Workshop was also sponsored by the American Automatic Control Council and IFAC/s Working Group on Process Control. The Workshop was held in Atlanta, Georgia on June 13 and 14, 1988. A total of 99 people from 18 countries attended the Workshop. Of thé attendees 39 came from industry. This large industrial attendance indicates the strong interest of practicioners in model based control techniques. The Workshop format consisted of an invited tutorial and four invited case studies. All of the case studies deal with actual industrial applications. A key message that comes through from these papers is that industry has bought into the model based technology. Multivariable, constrained model based control is being applied effectively today. The case studies were followed by 15 papers that were selected from 31 contributed papers. The authors of the selected papers came from 8 different countries. Thus, the Workshop provided an international forum to discuss the area of model based process control. Several important research topics and needs emerged from the Workshop. These included process identification, state space approaches to the prediction of the effect of disturbances, control of nonsquare plants, real-time detection of and recovery from abnormal operating conditions due to ill-conditioning and feasibility with respect to constraints, the effect of hard constraints on the stability of the model predictive methods, model predictive control of distributed parameter systems, nonlinear model predictive control and robustness considerations. Academic and industrial case studies demonstrated the need for further research in these areas. T. J. McAvoy Y. Arkun E. Zafiriou vu Copyright© IF AC Model Based Process TUTORIAL Control, Georgia, USA 1988 MODEL PREDICTIVE CONTROL: THEORY AND PRACTICE Manfred Morari*, Carlos E. Garcia** and David M. Prett** *Chemical Engineering, 206—41, California Institute of Technology, Pasadena, California 91125, USA **Shell Development Company, P.O. Box 1380, Houston, Texas 77001, USA ABSTRACT We refer to Model Predictive Control (MPC) as that family of controllers in which there is a direct use of an explicit and separately identifiable model. Control design methods based on the MPC concept have found wide acceptance in industrial applications and have been studied by academia. The reason for such popularity is the ability of MPC designs to yield high performance control systems capable of operating without expert intervention for long periods of time. In this paper the issues of importance that any control system should address are stated. MPC techniques are then reviewed in the light of these issues in order to point out their advantages in design and implementation. A number of design techniques emanating from MPC, namely Dynamic Matrix Control, Model Algorithmic Control, Inferential Control and Internal Model Control, are put in perspective with respect to each other and the relation to more traditional methods like Linear Quadratic Control is examined. The flexible constraint handling capabilities of MPC are shown to be a significant advantage in the context of the overall operating objectives of the process industries and the 1—,2—, and oo norm formulations of the performance objective are discussed. The application of MPC to nonlinear systems is not covered for brevity. Finally, it is explained that though MPC is not inherently more or less robust than classical feedback, it can be adjusted more easily for robustness. INTRODUCTION The petro-chemical industry is characterized as having Each one of these automation layers plays a unique and very dynamic and unpredictable marketplace conditions. complementary role in allowing a company to react rapidly For instance, in the course of the last 15 years we have wit­ to changes. Therefore, one layer cannot be effective with­ nessed an enormous variation in crude and product prices. out the others. In addition, the effectiveness of the whole It is generally accepted that the most effective way to gen­ approach is only possible when all manufacturing plants erate the most profit out of our plants while responding to are integrated into the system. marketplace variations with minimal capital investment is Although maintaining a stable operation of the process was provided by the integration of all aspects of automation of possibly the only objective of control systems in the past, the decision making process. These are: this integration imposes more demanding requirements. In • Measurements: The gathering and monitoring of pro­ the petro-chemical industries control systems need to sat­ cess measurements via instrumentation. isfy one or more of the following practical performance criteria: • Control: The manipulation of process degrees of free­ dom for the satisfaction of operating criteria. This • Economic: These can be associated with either main­ typically involves two layers of implementation: the taining process variables at the targets dictated by single loop control which is performed via analog con­ the optimization phase or dynamically minimizing an trollers or rapid sampling digital controllers; and the operating cost function. control performed using realtime computers with rel­ • Safety and Environmental: Some process variables atively large CPU capabilities. must not violate specified bounds for reasons of per­ • Optimization: The manipulation of process degrees of sonnel or equipment safety, or because of environmen­ freedom for the satisfaction of plant economic objec­ tal regulations. tives. It is usually implemented at a rate such that the controlled plant is assumed to be at steady-state. • Equipment: The control system must not drive the Therefore, the distinction between control and opti­ process outside the physical limitations of the equip­ ment. mization is primarily a difference in implementation frequencies. • Product Quality: Consumer specifications on products • Logistics: The allocation of raw materials and schedul­ must be satisfied. ing of operating plants for the maximization of profits • Human Preference: There exist excessive levels of and the realization of the company's program. variable oscillations or jaggedness that the operator will not tolerate. There can also be preferred modes of operation. 2 Manfred Morari, Carlos E. Garcia and David M. Prett In addition, the implementation of such integrated systems since it is only in this form that it is possible to compare it is forcing our processes to operate over an ever wider range with other schemes. Then the several existing forms of con­ of conditions. As a result, we can state the control problem strained MPC are reviewed, concluding with the nonlinear that any control system must solve as follows: MPC approaches. Although the issue of model uncertain­ ties in MPC techniques is not dealt with in this paper, On-line update the manipulated variables to satisfy multi­ some comments on robustness of MPC are included. ple, changing performance criteria in the face of changing plant characteristics. HISTORICAL BACKGROUND The whole spectrum of process control methodologies in The current interest of the processing industry in Model use today is faced with the solution of this problem. The Predictive Control (MPC) can be traced back to a set of difference between these methodologies lies in the particu­ papers which appeared in the late 1970's. In 1978 Richalet lar assumptions and compromises made in the mathemati­ et al. described successful applications of "Model Predic­ cal formulation of performance criteria and in the selection tive Heuristic Control" and in 1979 engineers from Shell of a process representation. These are made primarily to (Cutler & Ramaker, 1979; Prett & Gillette, 1979) outlined simplify the mathematical problem so that its solution fits "Dynamic Matrix Control" (DMC) and report applications the existing hardware capabilities. The natural mathe­ to a fluid catalytic cracker. In both algorithms an explicit matical representation of many of these criteria is in the dynamic model of the plant is used to predict the effect of form of dynamic objective functions to be minimized and of future actions of the manipulated variables on the output. dynamic inequality constraints. The usual mathematical (Thus the name "Model Predictive Control"). The future representation for the process is a dynamic model with its moves of the manipulated variables are determined by op­ associated uncertainties. The importance of uncertainties timization with the objective of minimizing the predicted is increasingly being recognized by control theoreticians error subject to operating constraints. The optimization and thus are being included explicitly in the formulation of is repeated at each sampling time based on updated infor­ controllers. However, one of the most crucial compromises mation (measurements) from the plant. made in control is to ignore constraints in the formulation Thus, in the context of MPC the control problem includ­ of the problem. As we explain below these simplifications ing the relative importance of the different objectives, the can deny the control system of its achievable performance. constraints, etc. is formulated as a dynamic optimization It is a fact that in practice the operating point of a plant problem. While this by itself is hardly a new idea, it con­ that satisfies the overall economic goals of the process will stitutes one of the first examples of large scale dynamic lie at the intersection of constraints (Arkun, 1978; Prett optimization applied routinely in real time in the process and Gillette, 1979). Therefore, in order to be successful, industries. any control system must anticipate constraint violations and correct for them in a systematic way: violations must The MPC concept has a long history. The connections not be allowed while keeping the operation close to these between the closely related minimum time optimal control constraints. The usual practice in process control is to problem and Linear Programming were recognized first by ignore the constraint issue at the design stage and then Zadeh & Whalen (1962). Propoi (1963) proposed the mov­ "handle" it in an ad-hoc way during the implementation. ing horizon approach which is at the core of all MPC al­ Since each petro-chemical process (or unit) is unique we gorithms. It became known as "Open Loop Optimal Feed­ cannot exploit the population factor as is done in other back" . The extensive work on this problem during the sev­ industries (e.g., aerospace). That is, we cannot afford ex­ enties was reviewed in the thesis by Gutman (1982). The treme expenses in designing an ad-hoc control system that connection between this work and MPC was discovered by we know will not work in another process and therefore Chang & Seborg (1983). its cost cannot be spread over a large number of applica­ Since the rediscovery of MPC in 1978 and 1979, its pop­ tions. Due to the increase in the number of applications ularity in the Chemical Process Industries has increased of this type (resulting from the need to achieve integra­ steadily. Mehra et al. (1982) review a number of applica­ tion) , this implies an enormous burden both in the design tions including a superheater, a steam generator, a wind and maintenance costs of these loops. In our experience, these costs more than offset the profitability of any ad-hoc tunnel, a utility boiler connected to a distillation column control system. and a glass furnace. Shell has applied MPC to many sys­ tems, among them a fluid catalytic cracking unit (Prett and Gillette, 1979) and a highly nonlinear batch reactor In conclusion, economics demand that control systems (Garcia, 1984). Matsko (1985) summarizes several suc­ must be designed with no ad-hoc fixups and transparent cessful implementations in the pulp and paper industries. specification of performance criteria such as constraints. Our experience has demonstrated that Model Predictive Several companies (Bailey, DMC, Profimatics, Setpoint) Control (MPC) techniques provide the only methodology offer MPC software. Cutler & Hawkins (1987) report a to handle constraints in a systematic way during the de­ complex industrial application to a hydrocracker reactor sign and implementation of the controller. Moreover, in involving seven independent variables (five manipulated, its most general form MPC is not restricted in terms of two disturbance) and four dependent (controlled) variables the model, objective function and/or constraint function­ including a number of constraints. Martin et al. (1986) ality. For these reasons, it is the only methodology that cites seven completed applications and ten under design. currently can reflect most directly the many performance They include: fluid catalytic cracker - including regen­ criteria of relevance to the process industries and is capa­ erator loading, reactor severity and differential pressure ble of utilizing any available process model. This is the controls; hydrocracker (or hydrotreater) bed outlet tem­ primary reason for the success of these techniques in nu­ perature control and weight average bed temperature pro­ merous applications in the chemical process industries. file control; hydrocracker recycle surge drum level control; reformer weight average inlet temperature profile control; In this paper the MPC methodology is reviewed and com­ analyzer loop control. The latter has been described in pared with other seemingly identical techniques. We par­ more detail by Caldwell & Martin (1987). Setpoint (Gros- ticularly emphasize the unconstrained version of MPC didier, 1987) has applied the MPC technology to: fixed Model Predictive Control 3 and ebulating bed hydrocrackers; fluid catalytic crackers; nitude vanishes as i —> oo. Thus, in the time domain we distillation columns; absorber/stripper bottom C<z compo­ have the truncated impulse response model sition control and other chemical and petroleum refining n operations. y(k) = J2H (k-i) (7) iU In academia MPC has been applied under controlled condi­ i=l tions to a simple mixing tank and a heat exchanger (Arkun and with the definitions et al., 1986) as well as a coupled distillation column sys­ tem for the separation of a ternary mixture (Levien, 1985; H{ = Hi- Hi., (8) Levien and Morari, 1986). Parish and Brosilow (1985) compare MPC with conventional control schemes on a i heat-exchanger and an industrial autoclave. H = ^H (9) i j Most applications reported above are multivariable and 3 = 1 involve constraints. It is exactly these types of prob­ the truncated step response model lems which motivated the development of the MPC con­ trol techniques. Largely independently a second branch of n MPC emerged, whose main objective is adaptive control. y(k) = Y^HiAu(k-l) (10) Peterka's predictive controller (1984), Ydstie's extended- horizon design (1984) and EPSAC developed by DeKeyser where et al. (1982, 1985) are in this category as well as Clarke's generalized predictive control algorithm (1987a,b). These Au{k) = u{k) - u{k - 1) (11) developments are essentially limited to SISO systems with extension to the MIMO case conceptually straightforward and Hi are the step response coefficients. Depending on but very involved when the details are considered. The the time delay structure of the system the leading step constrained case is not considered in any detail in these pa­ response coefficient matrices may be zero or have zero el­ pers. Because of the different underlying philosophy these ements. algorithms are outside the focus of this paper. Nevertheless some cross references will be useful at times because these MPC ALGORITHM FORMULATIONS algorithms were largely developed for the nonadaptive case The name "Model Predictive Control" arises from the with the adaptation added in an ad hoc manner based on manner in which the control law is computed (Fig. 1). At recursive least squares (or similar) parameter estimates. the present time k the behavior of the process over a hori­ The stability and robustness of the adaptive scheme was zon p is considered. Using a model the process response generally not analyzed. to changes in the manipulated variable is predicted. The moves of the manipulated variables are selected such that MODELS the predicted response has certain desirable characteris­ All derivations in this paper will be carried out for general tics. Only the first computed change in the manipulated MIMO systems. Occasionally, in the interest of providing variable is implemented. At time k + 1 the computation is special insight SISO systems are going to be discussed sep­ repeated with the horizon moved by one time interval. arately. The idea of MPC is not limited to a particular sys­ We will demonstrate how Dynamic Matrix Control (DMC) tem description, but the computation and implementation and Model Algorithmic Control (MAC) are derived. All depend on the model representation. Depending on the other MPC algorithms which have been proposed are very context we will readily switch between state space, trans­ similar. fer matrix and convolution type models. We will assume the system to be described in state space by Dynamic Matrix Control x(k) = Ax{k - 1) + Bu(k - 1) (l) The manipulated variables are selected to minimize a y(k) = Cx(k) (2) quadratic objective. P min S2\\y(k + i\k)-r(k + i)\\l L For zero-initial conditions the equivalent transfer matrix representation is + ||Att(* + *-l)||! (12) y(z) = P{z)u{z) (3) 4 where P{z) = C{zI-A)-1B (4) y{k + l\k) = Y^HiAu(k + i-i)+ Σ Hi&u{k + t-i) Because most chemical engineering processes are open-loop stable our discussion will be limited to stable systems. The extension of the presented results to unstable systems is de­ +d(k + i\k) (13) scribed elsewhere (for example, Morari et al., 1988). When A is stable the inverse in (4) can be expanded into a Neu- man series n d(k + t\k) = d{k\k) = y (k) - Σ HiAu{k - i) (14) P(z) = f2CAiBz-'-i (5) m *=o -.: P Σ CiMk + l\k) + CtAk + ^ - 1) + cJ' < 0; j = 1, n t=l c where Hi are the impulse response coefficients, whose mag­ 4 Manfred Morari, Carlos E. Garcia and David M. Prett y{k + l\k) predicted value of y at time 2. The number of input moves m is not used for tuning k + I based on information (m = p). available at time k 3. The disturbance estimate (14) is filtered. Let y(k) d{k + t\k) predicted value of additive be the measurement and y(k) the model prediction. disturbances at process output Then the disturbance estimate is defined recursively at time k + i based on information available at time k d{k+l\k) = ad{k+i-l\k) + {l-a){y (k)-j){k)) (16) m ym{k) = measurement of y at time k Au(k + l) = u(k + l) -u{k + t-l) with d(k\k) = 0,0 < a < 1. Equation (16) adds a Hi,i = 1, n = model step response matrix first order exponential filter with adjustable parame­ coefficient ter a in the feedback path (Fig. 2A). For r = 0 this is n = truncation order equivalent to augmenting Q. a is a much more direct nc = number of constraints and convenient tuning parameter than the weights, P horizon length (in general = horizon length, etc. in the general MPC formulation. p» n) a is directly related to closed loop speed of response, m number of manipulated bandwidth and robustness, but does not affect nom­ variable moves in the future inal (P = P(z)) stability. Garcia & Morari (1982, {Au{k + I) = 0 1985a,b) have analyzed the effect of this filter in de­ W > m; m < p) tail. xTQx weighting matrices ANALYSIS C3' C3 c3 constant matrices The MPC formulation (5)-(8) looks reasonable and attrac­ tive, and has been used extensively in industrial applica­ The prediction of the output (13) involves three terms on tions. However, a complete and general analysis of its the RHS: The first term includes the present and all fu­ properties (stability, robustness and performance) is not ture moves of the manipulated variables which are to be possible with the currently available tools. In general, the determined as to solve (12). The second term includes resulting control law is time varying and cannot be ex­ only past values of the manipulated variables and is com­ pressed in closed form. We would like to compare MPC pletely known at time k. The third term is the predicted with other design techniques and discuss alternate formula­ disturbance d which is obtained from (7). d(k + i\k) is tions and extensions. For this purpose we will concentrate assumed constant for all future times (i > 0). At time first on the unconstrained case because only here a rigor­ k it is estimated as the difference between the measured ous analysis is possible. The we discuss the different ways output y(k) and the output predicted from the model. In by which constraints can be handled and their implica­ block diagram notation (14) corresponds to a model P in tions. Finally we will review some extensions to nonlinear parallel with the plant P (Fig. 2A) with the resulting feed­ systems. back signal equal to d(k\k). Equations (12)-(15) define a Quadratic Program which is solved on-line at every time UNCONSTRAINED MPC step. This "controller" is represented by block Q in Fig. Without constraints (12)-(14) is a standard linear least 2A. squares problem which can be solved explicitly quite eas­ Though computationally more involved than standard lin­ ily. With the moving horizon assumption a linear time ear time invariant algorithms, the flexible constraint han­ invariant controller is found. Garcia and Morari (1985b) dling capabilities of MPC are very attractive for practical have shown how to obtain the controller transfer function applications: A stuck valve can be simply specified by the from the linear least squares solution. operator on the console as an additional constraint for the Structure optimization program. The algorithm will automatically adjust the actions of all the other manipulated variables to Garcia and Morari (1982) were the first to show that the compensate for this failure situation as well as possible. In structure depicted in Figs. 2A and C is inherent in all an unexpected emergency which a traditional fixed- logic MPC and other control schemes. It will be referred to scheme might find difficult to cope with, MPC will keep as Internal Model Control (IMC) structure in this paper. the process operating safely away from all constraints or Here P is the plant, P a model of the plant and Q (Qi allow the operator to shut it down in a smooth manner. and Q2) the controller (s). y is the measured output, r the reference signal (setpoint), u the manipulated variable and Model Algorithmic Control d the effect of the unmeasured disturbances on the output. The total MPC system which has to be implemented con­ MAC is distinctive from DMC in three aspects. sists of the model P and the controller (s) Q (Qi and Q2) and is indicated by the shaded box in Fig. 2. In this sec­ 1. Instead of the step response model involving Au, an tion and throughout most of the paper we will assume that impulse response model involving u is employed. If the model P is a perfect description of the plant (P = P). the input u is penalized in the quadratic objective, We will retain the super tilde (~), however, to emphasize then the controller does not remove offset. This can the distinction between the real plant P and the model be corrected by a static offset compensator (Gar­ cia and Morari, 1982). If the input u is not penal­ P which is a part of the control system. The IMC struc­ ized then extremely awkward procedures are neces­ tures in Figs. 2A and C have largely the same characteris­ sary to treat nonminimum phase systems (Mehra and tics. Initially we will concentrate our analysis on Fig. 2A. Rouhani, 1980). Subsequently the advantages of employing two controller blocks Qi and Q as in Fig. 2C will be addressed. 2 Model Predictive Control 5 The following three facts are among the reasons why MPC e = (I-PQ1)d-(I-PQ2)r (21) is attractive. Here in the absence of model error the two controller blocks Fact 1: The IMC structure in Fig. 2A and the classic make it possible to design independently for good distur­ control structure in Fig. 2B are equivalent in the sense bance response and setpoint following. The equivalent that any pair of external inputs {r, d} will give rise to the classic feedback controller is described by same internal signals {u,y} if and only if Q and C are related by u = Q(I - Pgi)-1Qr1Q2r - Qi(I - PQi^y (22) Q = C(I + PC)~l (17) ± C = Q(I-PQ)-1 (18) = Cr - Cy x 2 Fact 2: If P = P then the relation between any input An excellent historical review of the origins of the structure and output in Fig. 2A is affine* in the controller Q. In in Fig. 2A which has as its special characteristic a model particular in parallel with the plant is provided by Frank (1974). It appears to have been discovered by several people simul­ = PQ(r -d) + d (19) taneously in the late fifties. Newton, Gould and Kaiser y (1957) use the structure to transform the closed loop sys­ tem into an open loop one so that the results of Wiener e = y-r = (I-PQ)(d-r) (20) can be applied to find the i?2-optimal controller Q. When Fact 8: If P is stable, then the MPC system in Fig. 2A the Smith Predictor (Smith, 1957) is written in the form shown in Fig. 3 where P* is the SISO process model with­ is internally stable if and only if the classic control system out time delay it can be noticed that its structure also with C defined by (18) is internally stable. In particular contains a process model in parallel with the plant. In­ when P = P the MPC system is stable if and only if Q is dependently Zirwas (1958) and Giloi (1959) suggested the stable. predictor structure for the control of systems with time These facts have the following important implications: delay. Horowitz (1963) introduced a similar structure and called it "model feedback". Because of Fact 1 the performance of unconstrained MPC is not inherently better than that of classic control as one Frank (1974) first realized the general power of this struc­ might be led to believe from the literature. Indeed, for any ture, fully exploits it and extends the work by Newton MPC there is an equivalent classic controller with identical et al. (1957) to handle persistent disturbances and set- performance. points. Youla et al. (1976) extend the convenient UQ- parametrization" of the controller C to handle unstable Q can be considered an alternate parametrization of the plants. In 1981 Zames ushers in the era of #«,-control classic feedback controller C, albeit one with very attrac­ utilizing for his developments the Q-parametrization. At tive properties: The set of all controllers C which gives present it is used in all robust controller design method­ rise to closed loop stable systems is essentially impossible ologies. to characterize. On the contrary the set of all controllers Q with the same property is simply the set of all stable Unaware of all these developments the process industries Q's (Fact S). Furthermore all important transfer func­ both in France (Richalet et al., 1978) and in the U.S. (Cut­ tions (e.g., (19) and (20)) are affine in Q but nonlinear ler and Ramaker, 1979; Prett and Gillette, 1979) exploit functions of C (Fact 2). From a mathematical point of the advantages of the parallel model/plant arrangement, view it is much simpler to optimize an affine function of Q Brosilow (1979) utilizes the Smith Predictor parametriza­ by searching over all stable Q's than it is to optimize a non­ tion to develop a robust design procedure and Garcia and linear function of C subject to the complicated constraint Morari (1982, 1985 a,b) unify all these concepts and re­ of closed loop stability. From an engineering viewpoint it fer to the structure in Fig. 2A as Internal Model Control is attractive to adjust a controller Q which is directly re­ (IMC) because the process model is explicitly an internal lated to a setpoint and disturbance response (19) and (20) part of the controller. and where (in the absence of model uncertainty) closed The two-degree-of-freedom structure is usually attributed loop stability is automatically guaranteed as long as Q is to Horowitz (1963). It has been analyzed by many people stable. On the other hand even when C is a simple PID since (e.g., Vidyasagar, 1985). controller it is usually not at all obvious how closed loop performance is affected by the three adjustable parameters Tuning Guidelines and for what parameter values the closed loop system is stable. As apparent from (19) Q plays the role of a feed­ As we will analyze in more detail below the problem forward controller. The design of feedforward controllers is (12)-(14) is very closely related to the standard Linear generally much simpler than that of feedback controllers. Quadratic Optimal Control problem, for which a wealth of powerful theoretical results is available. In particular, ex­ The main limitation of MPC in Fig. 2A is apparent act conditions on the tuning parameters are known which from (20): Both the disturbances d and the reference sig­ yield a stabilizing feedback control law. Because of the fi­ nals r affect the error e through the same transfer matrix nite horizon in (12) most conditions which guarantee that (/ — PQ). If r and d have different dynamic characteris­ (12)-(14) will lead to a stabilizing controller are only suffi­ tics it is clearly impossible to select Q simultaneously for cient. Thus, at this time the tuning of MPC has to proceed good setpoint tracking and disturbance rejection. For the largely by trial and error with these sufficient conditions "Two-Degree-of-Freedom- Structure" in Fig. 2C and the as guidelines. equivalent classic structure in Fig. 2D (for P = P) we find For simplicity, the theorems below (Garcia & Morari, 1982) * The relation between x and y is called affine when are formulated for SISO systems without delays. Equiv­ y = A + Bx. alent results for MIMO systems have been derived (Gar-

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