Nonlinear Control of Chemical Processes: A Review

Znd. Eng. Chem. Res. 1991,30,1391-1413

1391

REVIEWS

Nonlinear Control of Chemical Processes: A Review B. Wayne Bequette Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590

Recent developments in nonlinear systems theory combined with advances in control system hardware and software make the practical application of nonlinear process control strategies a reality. This review article surveys nonlinear control system techniques ranging from ad hoc or process-specific strategies to predictive control approaches based on nonlinear programming. The capabilities of these techniques to handle the common problems associated with chemical processes, such as time delays, constraints, and model uncertainty are discussed. Although the recent progress in nonlinear control is encouraging, a significant number of goals for future research in nonlinear control of chemical processes are detailed.

I. Motivation It is well-recognized that one of the characteristics of chemical processes that presents a challenging control problem is the inherent nonlinearity of the process. In spite of this knowledge, chemical processes have been traditionally controlled by using linear systems analysis and design tools. A major reason that the use of linear systems theory has been so pervasive is that there is an analytical solution, hence there are generally more rigorous stability and performance proofs. Also, the computational demands for linear system simulation (and implementation) are usually quite small when compared to a nonlinear simulation. Obviously, the use of a linear system technique is quite limiting if a chemical process is highly nonlinear. Progress in nonlinear control theory, combined with computer hardware advances, now allow advanced, nonlinear control strategies to be successfully implemented on chemical processes. In spite of the recognized importance of process nonlinearities, a number of review articles from the early 1980s mention nonlinear systems techniques only in passing. Ray (1983)surveys the field of multivariable process control, with nonlinear systems playing a small role. During the first three engineering foundation conferences on chemical process control (Foss and Denn, 1976;Seborg and Edgar, 1982; Morari and McAvoy, 1986) the only papers that focused on nonlinearities were Ray (1982), Shinnar (1986),and Morshedi (1986). During the past five years there has been a significant increase in the number of control system techniques that are based on nonlinear systems concepts. A goal of this paper is to review these new control techniques that are based explicitly on a nonlinear dynamic representation of the chemical process, as well as techniques that have existed for decades. Since this is a survey paper, most of the details of the techniques reviewed are left to the original literature sources. A broad overview of chemical process control in general is presented by Fisher (1991). A motivation of my review is to provide an entry point to the rapidly expanding field of nonlinear chemical process control. A particularly detailed tutorial of nonlinear control based on differential geometry has been provided by Kravaris and Kantor (1990a,b). McLellan et al. (1990b)

Table I. Common Process Characteristics multivariable interactions between manipulated and controlled variables unmeasured state variables unmeasured and frequent disturbances high-order and distributed processes uncertain and time-varying parameters Constraints on manipulated and state variables deadtime on inputs and measurementa

review error trajectory control methods and place a number of these in the context of differential geometric techniques. I will include differential geometric techniques in my review, but refer the reader to these articles for more detail. I will not review the literature on adaptive control (Seborg et al., 1986),since it is based on a linear system representation of the process.

11. Introduction Chemical manufacturing processes present many challenging control problems, including nonlinear dynamic behavior. Other common proceas Characteristicsthat cause control difficulty for linear and nonlinear systems alike are shown in Table I. A number of critiques from the mid1970s discussed the limitation of the so-called modern control theory for handling many of these problems (Foas, 1973;Lee and Weekman, 1976;Kestenbaum et al., 1976). Recent research efforts have concentrated on providing control system design techniques to handle many of the characteristics shown in Table I. Adaptive control (Seborg et al., 1986) was promoted as a technique to solve the nonlinear problem by 'relinearizing" the process model as the process moved into different "linear" operating regions, as well as to estimate time-varying parameters (generally linear system based). Robust control system design techniques (Doyle and Stein, 1981;Doyle, 1982)were developed to account for model uncertainty. Internal model control (IMC) (Garcia and Morari, 1982)was developed to provide a transparent framework for process control system design and to explicitly handle manipulated variable constraints. Holt and Morari (1985)analyzed the effect of process deadtime in multivariable systems. Morari (1987)reviewed the three critiques (Fose, 1973;Lee and Weekman, 1976;Kestenbaum et al., 1976)and con-

0888-5885/91/2630-139I$O2.50/0@ 1991 American Chemical Society

1392 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991

cluded that the most promising and widely open areas are the control of nonlinear systems and adaptive control. The primary emphasis in this review paper is on techniques that use an explicit, nonlinear model of the process for control system design or implementation. The capability of these techniques for dealing with the other process characteristics will play an important role as a basis for comparison between the approaches. This paper has the following structure. First, I will discuss nonlinear dynamic chemical process models in order to introduce the nomenclature and mathematics that will be used in this paper. Next, I discuss complexities that are introduced by nonlinear systems, compared with linear systems. I will then use the following categories for nonlinear control techniques: (a) special or ad hoc strategies; (b) internal model; (c) differential geometric; (d) reference system synthesis; (e) robust control system design; (f) model predictive; (g) control using open- or closed-loop oscillations. After the review of these techniques, I will conclude the paper with a personal perspective on research needs in nonlinear process control. 111. Chemical Process Modeling An important part of any control system design is the development of a mathematical representation of the process. Even if linear control system design techniques are used, verification is normally performed via simulation using a nonlinear process model. Process modeling is by no means trivial, and the model complexity should be a function of the planned use of the model. Process modeling has been discussed by Denn (1986),while computational methods are emphasized by Ramirez (1989). Shinnar (1981)discusses the relationship between modeling and design of control systems for chemical reactors. He stresses that control must become an essential part of process and reactor design, since some of the most important decisions that affect control are made during the pilot-plant operation and steady-state process design stages. Shinnar (1986)states that most of the research problems facing process control are related to model sensitivity. This paper is based on the control of chemical processes that are described by lumped parameter models, that is, systems that can be modeled by seta of fmt-order ordinary differential-algebraicequations. In addition, deadtime on process inputs or output variables is included. Deadtime can be due to transport (flow through pipes) or measurement delays (analytical instrumentation, etc.). Mathematically, the representation is dx/dt = f(~,u(t-O),p,l) dynamic modeling equations (1)

=0 algebraic equilibrium relationships, etc. (2) y = gP(z(t-4)) state-output relationships (3) where x = state variables, u = manipulated variables, p = parameters, 1 = load disturbances (measured or unmeasured), y = output or measured variables, 0 = deadtime between manipulated and state variables, and 4 = deadtime between state and measured variables. This type of formulation may also be used for distributed parameter systems that have been “lumped” by discretizing the partial differential equations and forming ordinary differential equations. Many times a model is of very high order, which could cause computational difficulties. Loffler and Marquardt (1990) have recently discussed techniques to reduce the order of nonlinear differential-algebraic process models.

€!,(X,U,P)

It should be noted that most of the control strategies reviewed in this paper are based on the control of processes governed by simple, low-order, lumped parameter models. The exceptions include several papers involving staged separation processes (higher order, lumped parameter) (e.g., Joseph et al., 1989)and a crystallization (distributed parameter) process (Eaton and Rawlings, 1990a). IV. Complexities Introduced by Nonlinear Systems A number of papers have provided insight into operational problems created by nonlinearities in chemical processes, even though control design procedures were not specifically developed to overcome these problems. Silverstein and Shinnar (1982)study the effect of process design on the stability and control of fixed bed catalytic reactors with feed/effluent heat exchange. The emphasis is on operation at the open-loop unstable operating point. They find that design specifications such as inert spheres in the reactor and heat-exchanger bypasses can have a tremendous impact on the stability of the feedback system. Small perturbations in the operating conditions for exothermic CSTRs may cause control and operability problems, as shown by Ray (1982). One of the primary limitations is the constraint on coolant temperature, which is the manipulated variable for the example systems. A complete study of the complex open-loop behavior of CSTRs is presented by Uppal et al. (1974). The classic series of papers by his and Amundson (1958)analyze the dynamics of CSTR’s with proportional feedback control. Fox et al. (1984)have shown that the optimization of chemical processes can create zeros of the transfer function on imaginary axis, if the controlled variable is the objective function in the optimization problem. This causes the process gain to change sign depending on the operating condition; rarely will an offset-free linear controller exist for this case. Koppel (1982)discusses the possibility of input multiplicities in chemical process control. Input multiplicities occur when more than one set of manipulated variables can produce a desired set of steady-state outputs. A CSTR with reversible, series reactions is presented as an example. Koppel shows that, after a square pulse disturbance in inlet concentration, the linear control system brings the output variables back to the desired steady state, yet the manipulated variables may go to a new steady-state value, depending on the length of time of the disturbance. In a practical situation, the new steady-state values of the manipulated variables may not be economically favorable. Morari (1983)has shown that systems with input multiplicities have eigenvalues of the steadystate gain matrix that cross the imaginary axis. When an odd number of these eigenvalues cross the axis simultaneously, no linear system with integral action can maintain stability. All of these results provide additional justification for nonlinear control of chemical processes. The performance limitation of conventional, linear proportional-integral-derivative(PID) controllers applied to nonlinear processes is shown by Chang and Chen (1984), who use controller gains as bifurcation parameters in their analysis. A substrate inhibition bioreactor model is used to illustrate behavior such as multiple equilibrium points, limit cycles, tori, and chaotic strange attractors in the closed-loop system. Doherty and Ottino (1988)review chaos, with a particular emphasis on chemical engineering applications. Many chemical processes are designed conservatively to avoid complex operating regimes, as discussed by Seider et al. (1990).Several proceas examples are presented where a different design was more economically favorable, but

Ind. Eng. Chem. Res., Vol. 30,No. 7,1991 1393

n-Tll-7 Coolant in

Roduct

Figure 2. Control block diagram for double cascade control of a CSTR. Coolant

Figure 1. Instrumentation diagram for double cascade control of a CSTR.

would result in a more complex operating problem. An argument is presented for the coordination of design, operation, and control optimization to reduce overdesign. Seider et al. (1991)note that recent advances in model predictive control of nonlinear systems make operation at optimum economic conditions a realistic goal. V-XIII. Control System Design Techniques V. Linear Control-PID and Cascade Although linear control system strategies will not be reviewed in any detail in this paper, the ubiquitous proportional-integral-derivative(PID) controller is worthy of discussion. PID has been the dominant control strategy for decades and will remain the method used on a large fraction of all control loops for decades to come. The PID algorithm is

e ( t ) = y,,,(t)

- y(t)

(5)

where the variables are defined in the Nomenclature section. Although the PID algorithm is intuitive and quite simple, many advanced algorithms reduce to PID algorithms for certain types of models (Rivera et al., 1986). One reason that the majority of the loops will remain PID is cascade control: virtually all advanced controllers cascade to flow controllers, which are easily handled by PID. Furthermore, many loops have double cascade control. For example, many advanced control strategies for exothermic continuous stirred tank reactors (CSTR's) assume that coolant temperature is the manipulated variable. This means that the advanced strategy is changing the setpoint to the coolant temperature controller, which in turn, is changing the setpoint of a coolant flow controller. It is assumed that the dynamics of the coolant temperature are quite a bit faster than the primary control loop and that the flow control dynamics are faster still. The instrumentation diagram for a double cascade system is shown in Figure 1, while the control block diagram is shown in Figure 2. The vast majority of CSTR control problems reviewed in this paper have double cascade control strategies. It is generally assumed that the dynamics of the inner loops are very rapid, so the feedback system can be analyzed as a standard feedback controller with coolant temperature as the manipulated variable.

The primary disadvantage to the PID controller is that it is not based on a process model (also an advantage); knowledge of process dynamics, such as deadtime or nonlinearity, is not explicitly used in the control law. The obvious advantage to the PID controller is its simplicity. VI. Special and Ad Hoc Strategies It was realized in the 1950s that one of the limitations to control system performance was due to the nonlinear dynamic behavior of the process. Initial solutions consisted primarily of hardware modifications to remove major nonlinearities. Shinskey (1962)used controller input or output function generators to correct for process gain changes due to process inputs or loads. Example processes were pH, pressure vessel, heat exchanger, and flow. Solutions to the problem typically consisted of hardware such as equal-percentage valves or square-root extractors. Jones et al. (1963)developed a procedure for optimum feedback control systems for nonlinear processes by using nonlinear control elements. The optimization criteria was a minimization of the weighted integral squared error of all of the controlled variables. A total of 13 examples was presented, with optimum linear and nonlinear controllers presented for each case. A nonlinear controller was shown to yield superior results to a linear controller, for a CSTR example analyzed in more detail than the other more generic examples. Tuning Parameters as a Function of Error. Perhaps the easiest way to create a nonlinear controller is to make slight modifications of the PID tuning parameters as a function of error. For example, the proportional gain can be varied in the following fashion (Luyben, 1989):

k, = k,*(1

+ b*le(t)l)

(6)

An advantage to this strategy is that the controller is insensitive to measurement noise when the output is close to the setpoint. The control action becomes more vigorous the further the output is from the setpoint. Marroquin and Luyben (1972)have studied four different configurations of a temperature cascade control system on a batch reactor, with (a) master controller gain varied by the error in the master loop, (b) master controller gain varied by the error in the slave loop, (c) slave controller gain varied by the error in the slave loop, and ( d ) slave controller gain varied by the error in the master loop. They found that case d, slave gain varied by master loop error, gave better control than a linear control system, particularly under widely varying load conditions. Both simulation and experimental results are presented. The primary controlled variable was reactor temperature and

1394 Ind. Eng. Chem. Res., Vol. 30,No. 7, 1991

the secondary controlled variable was jacket temperature. Split-range control of hot and cold water valve positions was used as the manipulated variable. Cheung and Luyben (1980) have studied the "widerange" controller developed by Shunta and Fehervari (1976) for control of liquid level (particularlysurge vessels). This controller is a PI controller with proportional gain and integral time that are varied as a function of the error. The nonlinear controller was more versatile than the linear controllers studied, but it was difficult to tune, and its responses were hard to predict. Also, performance with respect to large noise amplitude was not satisfactory. Because of these problems, dual-range controllers that operate as piecewise linear approximations of the nonlinear controller were developed. The controller that was highly recommended was a limited output change (LOC) controller, where the controller output rate of change was limited to a maximum response specification. Simulation results were verified experimentally in the thesis by Cheung (1978). Parameter Scheduled Controllers. Rugh (1987) presents a nonlinear PID controller design procedure. The major assumption is that the plant transfer function can be represented with coefficients that are continuous functions of the manipulated input. The nonlinear PID controller that is developed has parameters that are continuous functions of the manipulated input. Examples of a series of noninteracting tanks are used for illustration. The nonlinear PID controller maintains constant closedloop dynamic behavior, while the linear PID controller has responses varying from over-damped to under-damped, depending on the operating condition. Jutan (1989) develops a nonlinear PI controller by establishing a function that contains information of past trends in the output variable. Relatively minor improvements over PID controller performance are shown. Tsogas and McAvoy (1985) have reported results for single input-single output one-sided gain scheduling control of the overload composition in distillation columns. The controller proportional gain was varied as a function of the distillate composition but was not varied if the overhead purity was less than the desired purity. McDonald and McAvoy (1985) have characterized the gains and time constants for multivariableinput-output relationships in high-purity-distillation columns as a function of the operating point. Dynamic matrix control (DMC) (Cutler and Ramaker, 1980) was used, based on the first-order + deadtime models as a function of operating point. McDonald and McAvoy showed that the gain and time conatant scheduled controller was superior to linear DMC for moderate-purity towers. For high-purity towers only gain scheduling was used; again it was superior to linear DMC. Marini and Georgakis (1984) have developed a nonlinear reaction-rate controller for low-density-polyethylene (LDPE) reactors. They were particularly interested in operating at open-loop unstable operating points. The LPDE model was based on imperfect mixing; they were able to show that the unstable dynamics of the LPDE reactor are associated with the deviation from the steady state of the reaction rate. The ideal nonlinear controller that they developed adjusts the initiator concentration to keep the reaction rate at its steady-state value. The linearized version of this controller is fundamentally a PI controller with proportional gain and integral time that are functions of the steady-state operating conditions (i.e., a gain and integral time scheduled controller). This scheduled controller was much more effective than con-

stant-parameter controllers. Under certain conditions there is an incentive to use the full nonlinear controller over the scheduled one. The work presented by Marini and Georgakis leads to the idea of extensive variables (Georgakis, 1986). Nonlinear Feedforward Control. Haskins and Sliepcevich (1965) used the invariance principle of control to design controllers for a stirred tank heater. Functional relationships are found that change the manipulated variables so that the time derivatives of the desired output variables are zero. The resulting controllers are linear or nonlinear feedforwardlfeedback controllers. They found that the linear controllers gave satisfactory control quality, so that there was little justification for nonlinear control for disturbance rejection for this process. They noted that nonlinear control becomes more important if setpoint changes are made. The results presented are based on simulations and experiments. Luyben (1968) has developed nonlinear feedforward control strategies for a number of CSTR configurations. Nonlinear feedforward control was compared with linear feedforward and open-loop control strategies and performed well even with model uncertainty. For a number of configurations the strategy has an explicit solution, but for many configurations the nonlinear feedforward solution has an implicit solution. This is not considered a drawback since a computer is used for implementation. Luyben also developed nonlinear feedforward controllers for batch and tubular reactors. Rovaglio et al. (1990) use a complex, nonlinear distillation model on-line to determine the manipulated variable value that best rejects measured feed composition disturbances. The method implemented is in the form of cascade control since the setpoint of a temperature on a particular tray is determined so that a concentration ratio in the bottoms stream is maintained constant. They also show good results by using a simplified relationship for feedforward control of feed composition disturbances. Variable Transformation Approaches. Another approach to nonlinear control is to select some function of measured variables (a synthetic output) that creates a more linear relationship between input and output variables. Ryskamp (1982) suggested several ways to implicitly decouple Controllers in a distillation column. One method was to use logarithms of the composition variables to linearize the dynamic behavior. Koung and Harris (1987) have used logarithmic transformations of distillate and bottoms composition in distillation as output variables, obtaining better closed-loop results than with a more complex partial linearization technique (see section VIII). Logarithmic transformations were also used by Georgiou et al. (1988) to create more linear input-output relationships, which were then used in the form of DMC for multivariable control of high-purity distillation. They found that this strategy yields better results than linear DMC and is much simpler than gain and time constant scheduling DMC (McDonald and McAvoy, 1985). Skogestad and Morari (1988a) find that the use of logarithms eliminates the effect of nonlinearity at high frequency, while Skogestad and Morari (1988b) show that controllers that use logarithmic compositions can effectively operate over a wide range of operating conditions. Georgakis (1986) has presented an approach based on "extensive variables" of a process, which are generally related to the total energy or mass balance of a unit, or the deviation of the reaction rate from its steady-state behavior (Marini and Georgakis, 1984). Standard linear control system design techniques can then be used for the control

Ind. Eng. Chem. Res., Vol. 30,NO. 7, 1991 1395 d

Figure 3. Nonlinear internal model control structure.

of the transformed or extensive process variables. The overall control strategies are inherently multivariable and nonlinear and result in systems that maintain closed-loop characteristics over a larger operating region than simple linear control strategies. The term chemical reaction invariant is used for a state variable that is not affected by the chemical reactions taking place in the system. For example, with the reaction A B, CA+ C, would be a reaction invariant. Waller and Makila (1981) use reaction invariants to analyze CSTR and pH systems. They show that a control system can be developed that uses a nonlinear controller gain which is inversely proportional to the process gain between measured pH and control action; the overall loop gain is then constant. Wright and Kravaris (1991) and Wright et al. (1991)develop the notion of a strong acid equivalent which can be calculated on-line from pH measurements and a nominal titration curve of the inlet stream. The resulting relationship between the titrating stream flow rate and the strong acid equivalent output is linear, so a simple PI controller is used for control. The closed-loop pH response is nonlinear; if a linear pH response is desired, Kravaris and Wright show how to design a globally linearizing controller (GLC) (see section VIII). Summary of Special Techniques. A primary advantage to a number of these techniques is that they are direct extensions of standard PID controllers, so they are easy to understand. Accurate knowledge of the nonlinear model is not generally required, since the controller parameters are tuned on-line. A disadvantage of the transformed variable approach is that tight, decoupled control of the transformed (or extensive) variables does not necessarily assure tight, offset-free control of the physical variables.

-

VII. Internal Model Approaches Garcia and Morari (1982) have presented internal model control (IMC) as a general structure that uses a linear process model in parallel with the actual process. Although they have shown that the IMC structure can be rearranged into the traditional feedback control structure, there are a number of reasons for using the IMC design procedure. For an open-loop stable process and an open-loop stable controller, the nominal closed-loop system is guaranteed to be stable. Also, inherent limitations to closed-loop performance due to open-loop characteristics are easy to identify. Advantages to implementing the IMC structure include manipulated variable constraint handling and deadtime compensation. Economou et al. (1985, 1986) developed a nonlinear approach for IMC (NLIMC), using operator theory. The general NLIMC structure is shown in Figure 3. Fundamentally, the idea is to invert the nonlinear plant to form the controller. Newton iterations are used to find the manipulated variable action for the next time step that would move the plant to the desired output at the end of the time step. The resulting control action is equivalent to a deadbeat controller for a linear system. With a filter

in the loop, it is basically a deadbeat controller at each time step, with the filter providing a trajectory for the plant to reach. By adjusting the filter properly, one can "tune" the manipulated- or state-variable responses, however there is no guarantee that particular constraints will be satisfied. Results were presented for an adiabatic, exothermic CSTR, operating in a region where the linearized process gain changes sign. Economou et al. noted that the operator inversion method of Hirschorn (1979) failed, in all of the practical cases considered, to provide a satisfactory approximation to the exact inverse. Since the Hirschorn inverse implies the use of higher order derivatives, it is sensitive to noise and numerical errors. Also, Kravaris and Kantor (1990a) show that the Hirschorn inverse suffers from internal instability due to pole-zero cancellations at the origin. Daoutidis and Kravaris (1990a) demonstrate that the construction of internally stable non-Hirschorn inverses is possible. A disadvantage to the NLIMC control structure, as presented in the literature, is that open-loop unstable processes can not be handled since the formulation is equivalent to an open-loop observer on the state variables. For some process models, a reduced order observer may lead to a stable feedback system. Li and Biegler (1988) find that the Newton-type control law used by Economou et al. (1986) is equivalent to a quadratic programming problem without constraints. They develop a form of successive quadratic programming (SQP) to handle the constraints on manipulated and state variables. Li and Biegler also note that manipulatedvariable constraints have a similar function to the filter in the NLIMC structure, but that time domain constraints are easier to interpret than a frequency domain filter. A primary limitation to the technique is that a perfect model is assumed; they suggest that an on-line parameter estimation technique should be added to update the model. Two CSTR problems are used to illustrate the effect of state and manipulated variable constraints. They find that tighter bounds on the manipulated variables do not necessarily lead to poorer performance in all output variables. Li et al. (1990) determine a minimum sample time requirement for stability and show the effect of a relaxation factor on control stability and performance. They also formulate an integral control objective to minimize intersample oscillationsin the output variables. One CSTR example from Li and Biegler (1988) and a pIi problem are used to illustrate the approach. Parrish and Brosilow (1986,1988) developed nonlinear inferential control (NLIC) to estimate unmeasured disturbances and follow a desired setpoint trajectory. The technique as developed does not account for state variable constraints and cannot handle systems where the process gain changes sign in the operating region. There is a limitation on the sample time since a finite difference (Euler) representation for the process model is used. This technique was extended by Hidalgo and Brosilow (1990) to handle an open-loop unstable styrene polymerization reactor. The process model used is a three-state model, while the simulated plant is a four-state model. A Runge-Kutta integration technique is used, rather than Euler, so sample time is less important than in the Parrish and Brosilow case. The disturbance effect and modeling errors are lumped into a single variable that is estimated, the concentration of initiator in the reactor. The disturbance estimator determines a constant value for the initiator concentration over the last sampling interval that results in a model reactor temperature that matches the measured reactor temperature. A one-step-ahead approach is used to calculate a control action that will bring the output to

1396 Ind. Eng. Chem. Res., Vol. 30,No. 7, 1991

its desired value at the end of the time horizon. Cooling water flow is the primary manipulated variable, reactor temperature is the output variable, and a time horizon of two sample times is used. If the cooling water saturates, then monomer flow is manipulated and a time horizon of one sample time is used. The reason that this approach can be used for open-loop unstable processes is that state variable identification is implicit in the output matching technique. This technique can be considered a limiting case of the multistep nonlinear model predictive control techniques that are reviewed in section XI. Summary of Internal Model Approaches. These techniques are easy to understand and are direct extensions of the linear IMC approach. A filter factor is varied for robustness and performance tuning, and manipulated variable constraints can be explicitly handled in the Li and Biegler (1988) formulation. Model uncertainty issues have not been rigorously addressed in the NLIMC formulation. One- or two-step approaches are used, which generally results in control action similar to minimal prototype control for linear systems. Multistep approaches are reviewed in section XI. VIII. Differential Geometric Approaches In recent years differential geometry has been used as an effective tool for the analysis and design of nonlinear control systems, similar to the way that Laplace transforms and linear algebra have been used for linear control system analysis and design (Isidori, 1989). An understanding of the structural characteristics of nonlinear systems can be obtained using differential geometric concepts. An overview of geometrical methods for process control is given by Kantor (1987). Henson and Seborg review geometrical control methods (1990b) and present a general (unified) approach (1989). The reader is encouraged to read the tutorial by Kravaris and Kantor (1990a,b) for many of the details of differential geometric-based control system design. McLellan et al. (1990b) review error trajectory techniques and place them in the context of differential geometric approaches. Here, I plan to sketch the philosophy of the approaches and emphasize the contributions of various literature sources. The majority of the papers presenting differential geometry approaches are based on input-linear systems. The general form for these models is (for a single input-single output system) dx/dt = f ( x ) + g(x)u (7) Y = h(x)

(8)

where x is a vector of states, u is the manipulated input, and y is the measured output. Kravaris and Kantor (1990a,b) note that Lie algebra in nonlinear systems replaces matrix algebra in linear systems. Define the Lie derivative as

which is the directional derivative of the function h(x) in the direction of the vector f ( x ) . One may also differentiate h(x)first in the direction of f ( x ) , and then in the direction of g ( x )

Also, higher order Lie derivatives can be written as

L)h(x) = Lf[Lf%(X)]

(11)

' -?'

'''TLC"t

taw feedbafk

conmller

process

ourput

compensator

1

Figure 4. Globally linearizing control structure.

The relatiue order of a system is the least positive integer r for which L&pZ(x) # 0 (12) The relative order represents the number of times that the output y must be differentiated with respect to time to recover the input u. For linear transfer function models, this is the difference in the orders of the denominator and numerator polynomials. Three ways of designing state feedback transformations have been used for chemical process control. State equation linearization (Hunt et al., 1983) makes the closed-loop state model linear, while input/output linearization (or globally linearizing control (GLC); (Kravaris and Chung, 1987) makes the input/output closed-loop system linear. Approximate or partial linearization (Krener, 1984) finds transformations that linearize the system to order 2 or higher. Kravaris and Kantor (1990b) discuss full linearization which makes both the state equations and the input/output behavior linear; this can be used for only a restricted class of systems. In general static control-linear state feedback compensator law is written as u = p(x) + d x ) u (13) where u is a reference input. A separate control law is then developed based on the reference input. State Equation Linearization. A state feedback compensator law is found so that the relationship between the reference input, u, and the dynamic state equation is linear (Hunt et al., 1983) after a coordinate transformation, 5 = T(x)and u = Tn+l(x,u) d[/dt = A [ + bu (14)

A pole-placement type of state feedback control law is usually then developed (e.g., Hoo and Kantor, 1985). A disadvantage to this approach is that placement of the closed-loop poles does not assure that good (offset-free) control of the desired output is obtained. Another major disadvantage is that a set of partial differential equations must be solved ta determine the variable transformations; an analytical solution is only available in special cases. Also, involutivity conditions can be quite restrictive, for systems with more than two states. Kravaris and Kantor (1990b) provide a CSTR example where the activation energies of two reactions must be equal to satisfy involutivity conditions. Input / Output Lineariaation (Globally Linearizing Control). A state feedback compensator law is found so that the relationship between the reference input, u, and the output, y, is linear (Kravaris and Chung, 1987) dx/dt = [ f ( x ) + g ( x ) &)I + g(x) q ( x ) u (15) Y =W) (16) Any linear design technique can then be used for the controller. A block diagram of the globally linearizing control structure is shown in Figure 4. Kravaris and Chung (1987) show that the state feedback compensator law ( r = relative order)

Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1397

v U =

- k-0 iP&!(h) PC&fr'h(x)

(17)

yields a linear input/output behavior between the output y and the reference input v Y(S)

=

1

Po + 01s + pg2

+ ... + &s' 4 s )

(18)

Choosing a linear controller transfer function with the form

Po + 01s + P2s2 + + (cs + 1)' - 1 e..

v(s) =

(19)

results in a critically damped closed-looptransfer function

A number of important chemical processes have a relative order of 1;this approach will yield a firsborder closed-loop response for these processes. A disadvantage to the input/output linearization approach is that it can only be used for minimum-phase systems; some extensions to nonminimum-phase systems are discussed later. Approximate Linearization. Krener (1984) has developed an approach that finds a coordinate transformation and state feedback that forms an order K approximate linear system dE/dt = A t

+ bv + O(x,u)*+l

(21)

where 5 and v are the transformed states and inputs. Application Papers. Some of the earliest results using differential geometric concepts for chemical process control were presented by Hoo and Kantor (1985,1986a,b). Hoo and Kantor (1985) show linearizing transformations for a two-state CSTR model (Uppal et al., 1974) and use pole placement control on the resulting linear input state system. Hoo and Kantor (1986a) present a strategy to control an open-loop unstable bioreactor. The extended Kalman filter approach is used for state estimation, and results on the effect of plant/model mismatch are presented. Aluko (1988) uses state equation linearization on the two-state CSTR model of Uppal et al. (1974) and investigates the effect of tuning parameter selection on the steady-state offset when both composition and temperature are controlled. Ogunnaike (1986) shows how transformations can be found to transform nonlinear single input-single output (and single state) systems into linear systems. Explicit solutions for control action are obtained for systems that are linear in the manipulated variable. Several level control systems are used as examples. The obvious limitation to this work is that it only handles single-state systems. Kravaris and C h u g (1987) present an approach (globally linearizing control, GLC) based on linearizing the input-output behavior of the process. A simple PI controller is then used on the linear input-output variables. They illustrate the technique on a batch reactor with a series reaction, tracking a desired temperature setpoint trajectory. Their strategy is less sensitive to process noise than the PI strategy that is used for comparison. An open-loop state observer is used to estimate the unmeasured state variables. Kravaris and Chung show that conditions necessary to use the Hunt-Su-Meyer (1983) approach cannot be satisfied for temperature control of the chemical reactor example that they have chosen.

Alaop and Edgar (1989) use an approxhate linearization approach (Krener, 1984) to control a SISO heat exchanger. A simple two-state nonlinear model is used to control a 40-state plant; closed-loop performance is significantly better than PI control. Robustness appears to be a problem since the control quality is significantly degraded with a 10% error in the overall heat transfer coefficient. Henson and Seborg (1990a) study SISO strategies for continuous fermentors with productivity as the output variable. PI controllers cannot be successfully used over a wide range because of a gain change at the maximum productivity. Henson and Seborg find that state equation linearization can be used if the substrate feed concentration (sf)is the manipulated variable; however, oscillatory behavior is observed. They also find that state equation linearization cannot be used if dilution rate is the manipulated variable. Input/output linearization can be used for both substrate feed and dilution rate as manipulated variables. Extremely vigorous manipulated variable action results when sf is used as the manipulated variable; and ad hoc modification improves the performance of the input/output linearization controller with sf as the manipulated variable. Perfect measurements of both state variables has been assumed in this work. Alvarez-Gallegos (1988) has found the transformations for state equation linearization of a three-state CSTR Cjacket dynamics are included) where cooling jacket flow rate is the manipulated variable. An approach to examine robustness is suggested; however, no numerical results are presented. Alvarez et al. (1989) study a CSTR with jacket temperature as the manipulated variable and show that the nonlinear controller that they develop emulates the form of a PID controller with variable parameters. Alvarez-Gallegos and Alvarez-Gallegos (1988) use state equation linearization to control continuous fermentation processes. With biomass concentration as the controlled variable, only substrate feed concentration can be used as the manipulated variable. Robustness is studied by finding the limits on parameter variation for closed-loop stability. Since the substrate concentration is not measured, it is estimated by using derivatives of the biomass concentration. Robustness. Kantor and Keenan (1987) derive a sufficient condition for stability of control systems designed using state equation linearization. Bounds on a state-dependent disturbance for which the nominal state feedback control remains stable are determined. Kantor and Keenan note that the choice of controller constants can be used to improve robustness, but this cannot be done arbitrarily. The coordinate transformation may be changed to improve robustness. The series/parallel van de Vusse reaction system is wed as an application example. For this system the model uncertainty that can be tolerated is a strong function of the operating point. Doyle I11 and Morari (1990) use a conic-sector-based approach to synthesize robust (stability and performance) controllers using state equation linearization, approximate linearization, and linear (P,PI) control. The method treata unmeasured disturbances and unmodeled dynamics. Bounds are placed on the state variables to guarantee the invertibility of the linearizing transformations. They synthesize controllers for a CSTR operating at an unstable steady state. Doyle I11 and Morari conclude that the approximate linearization strategy is more practical than state equation linearization. Chen and Chang (1990) examine the effect of unknown fast dynamics in nonlinear systems with state equation linearization. Their studies indicate that dimension mis-

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match may be the most important modeling error, because it eliminates the good performance offered by high-gain feedback control. Calvet and Arkun (1990)design stabilizing P controllers for the state equation linearization strategy, subject to modeled (but unmeasured) disturbances. PI controllers are designed to eliminate offset in the output. The application example is an open-loop unstable CSTR with feed temperature and concentration disturbances. The controlled variable and manipulated variables are concentration and coolant temperature, respectively. If temperature is the controlled variable, then the manipulated variable must be reactor feed flow rate, for temperature to be linearly extracted from the states. Many of the proofs and algorithms used in this work are presented in Calvet and Arkun (1989). Kravaris and Palanki (1988a,b) analyze the robustness of the GLC strategy using a Lyapunov-based approach. Given upper bounds on the modeling error, they design a nonlinear state feedback law that guarantees stability and performance for all perturbations with the given bounds. In (1988a) two process examples used are based on an isothermal, liquid-phase, multicomponent chemical reaction in a CSTR. The model error is due to an unmodeled side reaction and an error in a component flow rate. Feedforward Control. Feedforward and feedback linearization has also been considered by Calvet and Arkun (1988a,b) using the state equation linearization approach. Calvet and Arkun (1988b) apply a linear IMC control structure to the system that results from linearizing transformations. For a CSTR example the control design is quite complex, because input and state transformations are first developed. Since the reactor is operated at an open-loop unstable operating point, the transformed system is then stabilized by proportional state feedback. An output variable that can be extracted linearly from a transformed state is then selected to implement the linear IMC structure. The manipulated variable was coolant temperature, requiring composition as the output variable, since temperature cannot be extracted linearly. If feed flow rate was the manipulated variable, then reactor temperature could be chosen as the output variable; results were not shown for this case. Daouditis and Kravaris (1989) extend GLC to incorporate feedforward/feedback (FF/FB) strategies for SISO nonlinear systems. They define a disturbance relative order ( p ) which is used to classify the type of feedforward controller needed. The relationships p > r, p = r, and p C r indicate that no feedforward, static feedforward, and dynamic feedforward control are needed, respectively. The algorithm is demonstrated by composition control of a system of three CSTR’s in series; the GLC FF/Fl3 strategy is shown to be superior to GLC alone. This approach is extended to multivariable systems by Daoutidis et al. (1990),where a disturbance relative order is calculated for each disturbance/output pair. A continuous, exothermic, methyl methacrylate polymerization reactor is used as an application example; coolant flow rate and initiator flow rate are the manipulated variables and temperature and number-average molecular weight are the controlled variables. As expected, the MIMO FF/FB strategy outperformed the MIMO GLC strategy, which outperformed two linear SISO PI loops. Kravaris et al. (1989) present a FF/FB interpretation of the GLC controller for trajectory tracking in batch processes. The result is an output feedback controller for systems with relative order 1,obtained by combining the

state feedback with a reduced-order open-loop observer. The example that they present is batch copolymerization of styrene and acrylonitrile using xylene as a solvent. Good results are obtained with manipulated variable noise and with errors in the initial conditions of the model state variables. Manipulated Variable Constraints. The procedure developed by Calvet and Arkun (1988b; discussed in the feedforward section above) implicitly handles manipulated variable constraints. An input mapping transformation is used to back calculate the transformed manipulated variable when the physical manipulated variable is constrained. The limitation to this approach is that the controller is not explicitly accounting for the effect of the manipulated variable constraint. Also, there does not appear to be a direct extension to multiple input systems. They illustrate the performance of this control system on the exothermic CSTR, by moving from an open-loop stable to unstable operating point. Processes with Deadtime and Nonminimum Phase Characteristics. A technique for deadtime compensation is developed by Kravaris and Wright (1989) for SISO systems that have deadtime on the manipulated variable. The process must be open-loop stable and the “deadtime-free” part of the process must have a stable inverse. Robust stability results are developed for (a) errors in deadtime and (b) unstructured linear multiplicative uncertainty. The simulation example of an endothermic CSTR with series reactions verifies the tuning requirements for stability. The concept of a controller that “inverts” a process (controller poles cancel the process zeros) is well established in linear control theory. On the basis of Byrnes and Isidori (1985), Kravaris (1988) shows an input/output linearization of nonlinear systems that is the analogue of pole-zero cancellation in linear systems. Analogous to linear systems, if there are right-half-planezeros, then the closed-loop system will not be internally stable if pole-zero cancellation is used. Kravaris and Daoutidis (1990)extend the linear system idea of placing controller poles at the reflection of the process right-half-plane zeros to nonlinear systems; this approach is limited to second-order systems. The performance of this approach is illustrated on an isothermal series/parallel reaction system. Wright and Kravaris (1990) develop a minimum-phase-output predictor that extracts a minimum-phase output, in a fashion similar to the way that a Smith predictor extracts a delay-free output for linear systems. A GLC strategy is then used on the minimum-phase output. Nikolaou and Manousiouthakis (1990) have used input-output analysis to show that (for open-loop stable nonlinear systems) stability of the system’s inverse is sufficient but not necessary for stability of the exact-linearizing feedback loop. This is true regardless of whether input/output or state equation linearization, static feedback, or dynamic feedback is used. They find that the exact-linearized system cannot have an arbitrary form and must contain the right-half-plane zeros of the transfer function of the linear model around the steady state considered. Multiinput-Multioutput Systems. Hoo and Kantor (198613) use state equation linearization to design an MIMO control system to handle a mixed-culture bioreactor. The manipulated variables are dilution rate and inhibitor addition rate while the transformed states are cell density of the substrate insensitive species, the log ratio of cell densities, and the net difference in specific growth rates. Convergence to a steady state from a wide variety

Ind. Eng. Chem. Res., Vol. 30, No.7, 1991 1399 of initial conditions is obtained; however, proportional feedback control of the transformed variables leads to offset in the physical variables. An extended Kalman filter is used to estimate unmeasured states when the only measurement available is the total cell mass. Simulations show that satisfactory control is obtained when significant measurement noise is present. Kravaris and Soroush (1990) develop an input/output linearization control strategy for MIMO systems (MIMO GLC). The application example is a methyl methacrylate and vinyl acetate semibatch copolymerization. They obtain good rejection of initial condition errors in the state variables and good setpoint tracking. Kravaris and Soroush find that the closed-loop system became unstable when manipulated variable constraints were encountered. Daoutidis et al. (1990) extend this approach to feedforwardlfeedback control (reviewed in the feedforward section). Alsop and Edgar (1990) use an approximate linearization approach and a low-order, nonlinear, MIMO model of a distillation column for control of a rigorous model of the column. Levine and Rouchon (1989) use nonlinear perturbation rejection techniques, based on static-state feedback, to reject feed composition disturbances in distillation columns. The model used for control purposes is based on a low-order aggregated model, which is obtained from a full-order model using singular perturbation techniques. Their approach has been successfully applied in the industrial operation of a 42-tray depropanizer. McLellan et al. (1990a) show that distillation columns satisfy requirements for “disturbance decoupling”, where the process outputs can be isolated from input and state disturbances. Castro et al. (1987) use a bilinear estimation technique for a low-order aggregated nonlinear distillation model and develop a control law based on the work by Isidori (1989) for disturbance decoupling. Castro et al. (1990) apply disturbance decoupling to a low-purity, nine-stage binary distillation column. Alvarez et al. (1990) use state equation linearization (based on a disturbance-dependent state feedback transformation) to control a free-radical polymerization CSTR. Conversion and temperature are controlled, initiator feed rate and heat removal rate are manipulated, and inlet conversion and temperature are measured disturbances. The multivariable control scheme was justified because nonlinear control of temperature alone does not guarantee closed-loop stability. Systems T h a t Are Not Input Linear. Two approaches to input/output linearization to include systems where the input occurs nonlinearly are given by Henson and Seborg (1990~).Method 1 is to define a new input which is the time derivative of the control input, resulting in a dynamic-state feedback that is more sluggish that a static-state feedback with the same relative order of the open-loop system. Method 2 requires the solution of a nonlinear algebraic equation for the manipulated input, generally resulting in a faster response than the first method. Henson and Seborg study a set of two CSTR’s in series, which have a manipulated variable (coolant flow rate) that appears nonlinearly in the modeling equations. They found that the static approach (method 1)had better disturbance rejection and was less sensitive to modeling errors than the dynamic approach (method 2). A pH control problem, where the output variable (pH) is an implicit function of the states, is presented by Henson and Seborg (1989). State-Variable Estimation. Since differential geometric techniques are based on state-variable feedback,

it is necessary that all state variables be measured or estimated. The vast majority of the papers in this area have assumed that all of the state variables are measured. The extended Kalman filter has been used by Hoo and Kantor (1986a,b) to estimate unmeasured concentrations in bioreactor systems. Alvarez-Gallegos and Alvarez-Gallegos (1988) use derivatives of the biomass concentration to estimate the substrate concentration in a fermentation process. Castro et al. (1990) use derivatives of the overhead and bottom compositions in binary distillation to estimate unmeasured state variables. When random measurement noise was included, the control action was not smooth, indicating that there may be some sensitivity to noise. Kantor (1989b) has used an approach suggested by Krener and Isidori (1983) to construct a nonlinear observer for a two-state CSTR model with a first-order reaction. Linear error dynamics are obtained and the observer converges exponentially fast in the linear coordinates. Limitations are that the feed and coolant temperature must be measured, the method is only adequate for first-order reactions, and model uncertainty issues are not addressed, Preliminary feedback results using state equation linearization are presented. This observer is used by Limqueco and Kantor (1990) to study the closed-loop behavior of a CSTR under three different operating conditions, including strong parametric sensitivity, ignition/extinction behavior, and nonlinear oscillations. Lower and upper bounds are placed on the control action; however these bounds are not explicitly included in the control law formulation. Satisfactory closed-loop results were presented for all three cases; again, no model uncertainty effects were considered. Kravaris and Chung (1987) obtain good results using an open-loop state observer on an endothermic batch CSTR with significant temperature measurement noise; a PI controller could not successfully handle the noise. Daoutidis and Kravaris (199Oa) develop output feedback control structures for minimum-phase nonlinear processes. For open-loop stable processes, an open-loop observer is implicitly incorporated in the output feedback control law (equivalent to the approach by Kravaris and Chung (1987)). For open-loop unstable processes, a reduced-order observer that makes explicit use of the output and its derivatives (note the susceptibility to measurement noise) is used to form a two-degree-of-freedomcontrol structure. If the process is relative order 1,then no output derivativea are required. This paper provides a clear differential geometric interpretation of nonlinear internal model control, with minimal and nonminimal process inverse realizations. Geometric Approach for Optimal Feedback Control. Palanki et al. (1990) develop a geometric approach to synthesize state feedback control laws for tracking singular arcs in optimal control problems. They use a styrene-AIBN-toluene batch solution polymerization example where the objective is to obtain a final product that meets average molecular weight and initiator concentration specifications. The other example involves the maximal production of ethanol in a fed batch fermentation process with time-varying parameters. Summary of Differential Geometric Approaches. The key to the differential geometric approaches is the input-linear model, which allows linearization through static-state feedback. Advantages include the ability to globally stabilize a system that satisfies certain conditions. Some initial robustness issues have also been addressed. Since these techniques are based on state and control input

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transformations, all of the states must be measured or estimated. A disadvantage to the state equation linearization approach is that performance specificationscannot be easily expressed in terms of the original process variables. In general, good control of the transformed variables does not assure that the original process variables will be adequately controlled. Also, the development of the linearizing variable transformations for state equation linearization may require the solution of PDE’s; rarely will an analytical solution exist. Input/output linearization is generally limited to minimum-phase systems. Manipulated variable constraints can not be explicitly included in the control law formulation. Also,differential geometric techniques are based on a continuous-time framework. In practice, these control strategies would be implemented using a computer; however no analysis of discretization effects has been published in the chemical process control literature. Arapostathis et al. (1989) and Grizzle and Kokotovic (1988) show that feedback linearizability can be destroyed through the introduction of the usual sample and hold devices.

IX. Reference System Synthesis Techniques Reference system synthesis (RSS) techniques use a model of the process and a desired process response to determine a feedback control law. This idea is similar to approaches that have been used for linear discrete systems. Dahlin’s controller (Seborg et al., 1989), for example, specifies a first-order + deadtime response trajectory. In RSS, a model of the process is written = f(x,u,p,l) (22) and a desired response is specified 2, = f ( ~ s p , ~ , ~ ~ , l ) (23) One example of a desired response incorporates proportional-integral action

A control function

u = k(X,,,X,P,l) (25) is found so that the model matches the desired response (2 = XJ. If the process is input linear, then an explicit solution for the manipulated variable arises; if the process is nonlinear in the input, then an implicit solution will generally be obtained. Bartusiak et al. (1989) have shown examples for RSS for both linear and nonlinear processes, with linear and nonlinear desired responses. In one example the reactor temperature of a system with a single irreversible reaction in an adiabatic reactor is controlled by manipulating the inlet temperature. A first-order response specification yielded an easy to implement, analytical control law. A more complex CSTR example included cooling dynamics and measurement lags on temperatures and concentrations. A cascade control structure was designed to implement RSS under open-loop unstable operating conditions. The RSS controller had much better responses to large temperature setpoint changes, compared to a linear controller. Bartree et al. (1989) have extended the RSS procedure to handle systems with deadtime. They show that RSS yields superior results to a Smith predictor controller, for uncertain deadtimes. Their analysis shows that the Smith predictor assumes that the present model mismatch remains the same for t d units of time into the future, while

the RSS controller uses process dynamics to translate the model mismatch into the future. The RSS controller will always be stable if the model deadtime is greater than the plant deadtime. Adebekun and Schork (1989a,b) have used the reference system synthesis approach to control a methyl methacrylate (MMA) polymerization reactor. The first paper (1989a) assumes that perfect measurements are available. The second paper (1989b) developes three different estimation techniques for the unmeasured state variables: (i) a full-order estimator, (ii) a reduced-order estimator, and (iii) a two-time-scale filter. Two types of controllers are developed. A nonsquare (least-squares) controller handles the case where more states are specified than manipulated inputs. In another case, a subset of the state variables is controlled, resulting in a square control system. Good control of an open-loop unstable process is obtained, even with significant measurement noise. Adebekun and Schork (1991a) globally stabilize a CSTR with nth-order kinetics using RSS. An adaptation mechanism is developed by Adebekun and Schork (1991b) to handle nth-order reaction systems with unknown heat-transfer coefficientsand heat of reactions. Balchen et al. (1988) apply RSS (internal decoupling) to a nonisothermal CSTR with a first-order exothermic reaction. Uncertainties in the heat capacity and activation energy are considered. One severe limitation to the internal decoupling approach is that the number of states must be equal to the number of manipulated inputs. Lee and Sullivan (1988) develop an RSS procedure that they call generic model control (GMC). On the basis of response specifications,they derive other common feedback control strategies from GMC, generally by using two tuning parameters. A rather simple, single-state nonlinear process is considered as a nonlinear control example. Henson and Seborg (1990b) have shown that GMC and internal decoupling are implicitly based on differential geometric concepts, with systems that have a relative order of 1. A complete case study of GMC of a single-effect evaporator system is presented by Lee et al. (1989) and Newell and Lee (1989). GMC is compared with DMC and MVSISO control and shows excellent dynamic behavior. Cott et al. (1989) show that nonlinear-steady-state models combined with approximate dynamics can be effectively used for nonlinear control using GMC. A distillation column example is used, where GMC outperforms both conventional and material balance control. Lee et al. (1990) develop a general approach for handling deadtime in the RSS framework and use the single-effect evaporator as an application example. Brown et al. (1990) extend GMC to handle constraints, by solving a single-step constrained optimization problem. A number of tuning parameters must be defined, because their approach combines desired constraint variable and output variable trajectories into a single objective function. A single-stage evaporator and a CSTR are used as application examples. My feeling is that a predictive horizon approach is a much more satisfactory way of handling the constrained optimization problem (see section XI). Cott and Macchietto (1989) use GMC for temperature control of an exothermic batch reactor. Filtered temperature measurements are which is then used to estimate a single parameter (QIUA), used in a discrete-form GMC law. Riggs and Rhinehart (1988) compare a GMC approach (process model based control, PMBC) and nonlinear IMC for an exothermic continuous stirred tank reactor (CSTR) and claim that PMBC is more robust. A major disadvantage of the PMBC approach is the finite difference

Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1401 approximation to the model solution, which places a maximum value on the control sample time. Riggs and Rhinehart also use PMBC for a single-tube steam heated exchanger. Williams et al. (1990) use PMBC to control wastewater pH by adjusting base flow rate at two injection points. A titration curve is generated by adjusting two fictitious acid parameters, using three measured pH/base concentration data pairs. Simulation results include the effect of measurement noise and feed wastewater concentration disturbances. An approximate model is used by Riggs (1990) in a PMBC approach to control a propylene sidestream draw column. The steady-state approximate model is converted into a dynamic model by assuming first-order relationships. Liu (1967) designs a controller such that the desired response for each output is a linear function of the error in that output. One application is the CSTR model with second-order kinetics. Concentration and temperature are controlledby manipulating the inlet and coolant flow rates. Manipulated variable constraints are handled by changing the desired state reponse dynamics when a constraint is encountered. An absorption column control problem is presented as a second example. Hutchinson and McAvoy (1973) use the approach by Liu (1967) to design a nonlinear, noninteracting control strategy for a multivariable pressured stirred-tank heater system. The nonlinear control system was compared with time-optimal control. Experimental results verified the theoretical predictions. Bhat et al. (1990) assume a first-order reference model for temperature dynamics in a CSTR and develop a control strategy that does not require knowledge of reaction kinetics, activation energies, or heats of reaction. Knowledge of heat-transfer coefficients and areas are required. Two tuning parameters are A,, the desired first-order response time and k, a proportional gain to force the error between the reference model and the process output to vanish. Temperature and the time derivative of temperature are necessary for this control strategy, so their technique is sensitive to large measurement noise. Jayadeva et al. (1990) extend this approach to handle a pH process (which has a nonlinear implicit output equation). Bequette (1989) has developed a single-step-ahead control law for a CSTR that also does not require knowledge of reaction kinetics. All unknown parameters are lumped into a single parameter which is estimated at each time step. A "filtered" nonlinear discrete deadbeat controller is then formulated. Good results are obtained for an open-loop unstable process with noisy temperature measurements. Summary of Reference System Synthesis. Many of the RSS techniques can be considered to be a t the "art" stage, since the specification of the form of a reference system affects the type of control law that results; of course, this can be an advantage to the creative individual. The desired response trajectory is not limited to a linear system; Bartusiak et al. (1989) have suggested forms of nonlinear responses. Like other model inverse approaches, RSS produces an unstable controller when the plant has right-half-plane zeros. Another disadvantage is that the time derivative of the output must be directly coupled to a manipulated variable to use this approach. This problem can be alleviated if a cascade approach is used, as shown by Bartusiak et al. (1989). The form of the RSS controller for a CSTR has been found to be equivalent to the controller obtained from an input/output linearization approach (Bartusiak et al., 1989). Indeed, Henson and Seborg (1990b) have shown that RSS methods of GMC and internal decoupling are implicitly based on differential geometric techniques, for

desired trajectory (future)

actual outputs (past)

Rcmnt time

Horizon maximum

U

-

mnimum

1 + L pastconmi moves

_*

Control Horizon

Figure 5. Predictive control approach.

systems with a relative order of 1. The RSS framework appears more intuitive to the nonspecialist than the differential-geometric-based approaches. The reviews and tutorials by McLellan et al. (1990b) and Kravaris and Kantor (1990a,b) are recommended reading, since they provide excellent introductions to differential geometric techniques. The differential geometric techniques are recommended over the reference system synthesis approaches because of their more general nature (not limited to systems of relative order 1).

X. Robust Control System Design Doyle I11 et al. (1989) extend robust control system design procedures based on structured singular value theory (Doyle, 1982) to nonlinear systems. The restrictive assumptions are that the nonlinearity lies in a conic sector and that the states are bounded. The CSTR model from Uppal et al. (1974) is studied und.er two different parameter sets. Linear controllers are designed to function over desired windows of operation in the phase plane. Schaper et al. (1990) use a statistical approach to analyze the robustness properties of closed-loop systems in the time domain. They consider concentration control of a CSTR subject to feed temperature disturbances, using coolant temperature as a manipulated variable. Khambanonda et al. (1990a) develop a modified sector bound approach to test for the stability of nonlinear closed-loop systems. They use the Krener (1984) transformation to decompose a nonlinear operator in a linear dynamic part and a nonlinear static part. The accuracy of their stability test is shown in the heat-exchanger example of Alsop and Edgar (1989). Khambanonda et al. (1990b) extend this approach to handle nonlinear controllers (designed using approximate linearization), again using the heat-exchanger example. XI. Predictive Control Approaches During the past decade there has been an increasing use of linear model predictive control (LMPC) techniques. A survey of model predictive control, including applied and theoretical papers, has been performed by Garcia et al. (1989). The most cited LMPC technique is dynamic matrix control (DMC) (Cutler and Ramaker, 1980). DMC is based on selecting a set of L future manipulated variable moves (control horizon), to minimize an objective function based on a sum of the squares of the differences between model predicted outputs and a desired output variable trajectory over a prediction horizon, R, as shown in Figure 5. Although the DMC optimization is performed for a sequence of future control moves, only the next control

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move is implemented. Model uncertainty and process disturbances are handled by calculating an additive disturbance as the difference between the process measurement and the model prediction at the current time step. It is assumed that the future disturbances are equal to the current disturbance, and a new trajectory is calculated. Notice that predictive control is really an open-loop optimal control strategy (Lapidus and Luus, 1967) with feedback provided by the disturbance estimate (which also compensates for model uncertainty). There is an analytical solution to the DMC method, since it is based on unconstrained control; constraints are handled artifically through weighting factors for the speed of response. Quadratic dynamic matric control (QDMC) (Garcia and Morshedi, 1986) was developed to explicitly incorporate process constraints, on both the manipulated variables and the process outputs. Ricker (1985) discusses an approach similar to QDMC but presents the material in a more generic fashion and develops the idea of manipulated variable blocking to reduce the size of the optimization problem. Nonlinear Programming Solution. Predictive control strategies have been well-received by industry because they are intuitive and explicitly handle constraints. One limitation to the existing methods is that they are based on linear systems theory and may not perform well on highly nonlinear systems. A direct extension of the linear model predictive control methods results when a nonlinear dynamic process model is used, rather than a linear convolution model. The objective of nonlinear predictive control (NLPC) is to select a set of future control moves (control horizon) in order to minimize a function based on a desired output trajectory over a prediction horizon, as shown is Figure 5. A general mathematical formulation is

k+R

The objective function is the sum of the squares of the residuals between the model predicted outputs and the setpoint values over the prediction horizon of R times steps (26a). The optimization decision variables are the control actions L time steps into the future; after the Lth time step it is assumed that the control action is constant (260. Notice that absolute (26d) and velocity (26e) constraints on the manipulated variables are explicitly included in this formulation. State- and output-variable constraints are included in (26g,h). Although the optimization is based on a control horizon, only the fmt control action is implemented. After the first control action is implemented, plant output measurements are obtained. Compensation of plant/model mismatch is performed, and the optimization is performed again.

There are a number of important issues that must be addressed in the solution of (26a-i). The choice of constrained optimizationtechnique ia one of the first decisions that must be made. Another major issue is how to solve the dynamic model constraints (26b). Since many of the state variables are unmeasured, a decision must be made about the proper initial conditions for the state variables at the beginning of the prediction horizon (26i). The model outputs (y,) are a function of the state variables (264; however a correction must be applied to ym to obtain a better prediction of the outputs (y d). Initial Conditions and Model &put Predictions. For open-loop stable systems, the initial conditions for the state variables (26i) can be obtained from the solution of the model over the previous time step, based on the manipulated action actually implemented on the plant (this is equivalent to using an open-loop observer). The DMC characterization of the future additive disturbances as the difference between the model output and the plant output at the current time step can be used to predict the output over the prediction horizon d ( k ) = ~ ( k-)ym(k) (27) for i = 1, R (28) ypred(k+i) = y,(k+i) + d(k) For open-loop unstable systems, a state-variable identification procedure should normally be used (either explicitly or implicitly), so that the model state variables do not significantly diverge from the actual state variables. For linear open-loop unstable systems (with an unstable model), a state-variable estimation strategy must be used to satisfy internal stability requirements (for a more complete discussion of internal stability see Morari and Zafiriou (1989)). Sistu and Bequette (1990b) and Eaton and Rawlings (1990b) have found that an open-loop observer can be used on open-loop unstable nonlinear systems, under certain conditions. Intuitively, one expects that measuring all state variables (usually impossible) would provide the best initial conditions. Sistu and Bequette (1990b) have shown that this is not the case for systems with parameter or model structure uncertainty. Poor dynamic response and steady-state offset can occur if all state variables are measured and used as initial conditions with plant/model mismatch. Solution of the Dynamic Model Equations. Several methods can be used to handle ordinary differential equation equality constraints (26b) with a constrained nonlinear optimization program: (i) sequential solution, iteratively solving the ODEs as an “inner loop” to evaluate the objective function, (ii) simultaneous solution, transforming the ODEs to algebraic equations which are solved as nonlinear equality constraints in the optimization, (iii) intermediate solution,transforming the ODEs to algebraic equations which are solved as an “inner loop” to evaluate the objective function, and (iv) linear approximation, approximating either by a single linearization over the prediction horizon or by a linearizationat a number of time steps in the prediction horizon. I feel that the approach used for the solution of the ODEs is not nearly as important as the other issues involved, such as selection of the initial conditions for the model at each time step and the adjustments of the ”tuning parameters”. Sequential Solution Using an ODE Solver. A classical optimal control formulation is to use a two-stage approach where an optimization routine serves as an outer loop to iteratively select new seta of manipulated variable moves, while an ODE solver is used to integrate the dy-

Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1403 namic equations at each iteration of the optimization. The objective function gradients can be determined by (a) finite differences based on small changes in the manipulated variables or (b) simultaneously integrating the adjoint differential equations and the model differential equations (Jang et al., 1987a,b; Chen and Joseph, 1987; Joseph et al., 1989). Using (a) requires the integration of fewer ODEs, at the expense of performing the integration more often. Joseph and co-workers integrate the model ODE'S and sensitivity equations for the optimal solution. The prediction and control horizons are the same length (R= L). They use a first-order filter for the setpoints, which eliminates the deadbeat type of control that normally results when the prediction horizon and control horizon are the same. Jang et al. (1987a) illustrate their approach to a two-CSTR system, with measurement noise and manipulated variable constraints. Jang et al. (198713) study the CSTR example from Economou et al. (1986) and compare their approach with PID and self-tuningcontrol. Modeling errors, measurement noise, and state-variable constraints are included. Morshedi (1986) presents an extension of QDMC called universal dynamic matrix control (UDMC). The modeling equations are integrated by using any nonlinear ODE solver. An efficient approach developed by Morshedi et al. (1986) is used to compute the Jacobian of the states with respect to the manipulated variables, that is, the sensitivity of the differential equations. The mean-value theorem is used to develop a linear analytical solution to the sensitivity equations, reducing the computational load significantly. Balchen et al. (1989a) apply a general nonlinear predictive control technique to an electrometallurgical process, which has a long sample time (2.7 h). Balchen et al. (198913) formulate a predictive optimization problem with constraints on the state and control variables, but indicate that the solution appears to be quite time-consuming. They recommended the use of the maximum principle and parametrize each manipulated input to calculate an optimum feedback (LQG) around the optimal trajectory. Simultaneous Solution. Another approach is to reduce the ODEs to algebraic equations by using a weighted residual technique (Finlayson, 1980). The algebraic equations are then solved as equality constraints in a nonlinear program. Cuthrell and Biegler (1987,1989) have used sequential quadratic programming (SQP) to solve optimal control problems. In this case the optimal control profile is calculated and implemented, without compensation for model uncertainty or disturbances. This solution approach results in more decision variables since the values of the state variables at each collocation point are included as decision variables. Since the SQP method does not require that constraints be satisfied at each iteration, there should be faster convergence to the optimum. A major advantage to this approach is that state-variable constraints are easily handled. A major disadvantage to an infeasible path approach, such as SQP, is that termination may occur at an infeasible point. This may be all right in off-line strategies, where a new initial guess could be made and the solution performed again. In on-line strategies, some "outer layer" type of approach must be used to detect such a failure. Rawlings and co-workers (Eaton and Rawlings, 1990a,b; Eaton et al., 1989; Patwardhan et al., 1990)have used this simultaneous approach in a predictive control formulation (which they call NMPC) by using the standard DMC additive disturbance procedure to account for plant/model mismatch. Patwardhan et al. (1990) analyze the effect of

parameter uncertainty and manipulated-variable constrainta using NMPC of an exothermic CSTR and compare their results with state equation linearization (Hoo and Kantor, 1985) and linear control. Intermediate Solution. An intermediate approach is to use the sequential optimization and simulation approach, but to solve algebraic equations resulting from a weighted residual technique rather than ODEs. The primary advantage is that the optimization problem is much smaller than in the simultaneous simulation/optimization method. This intermediate approach is used by Bequette (1990,1991) and Sistu and Bequette (1990a,b). Bequette (1990) shows the effect of manipulated-variable velocity constraints on the control of a biochemical reactor. Bequette (1991) has found that, for SISO systems, the control horizon can generally be set to one-time step with the prediction horizon varied for performance and robustness. These results are consistent with linear MPC (Maurath et al., 1988). Linearization Approaches. A number of researchers have developed MPC approaches based on a linearization of the plant model for the prediction phase. Brengel and Seider (1989) use analytical derivatives to integrate the plant equations into the future. The model is linearized several times per control interval for accuracy. They mention plant/model mismatch compensation and present some results, but fail to discuss the technique used. Li and Biegler (1989) linearize the solution along the projected trajectory in their multistep Newton approach. They develop a rigorous solution procedure; however they assume a perfect model. No techniques are presented for dealing with disturbances or model error. Garcia (1984) uses a nonlinear model to account for the effect of previous control moves plus a linearization of the model for the optimization of future control moves. He demonstrates this technique on a batch reactor control problem. A major advantage to this approach is that it allows the use of proven linear MPC software to be used on a nonlinear problem. Peterson et al. (1990) use a linear model to determine a sequence of future control moves. These future moves are implemented on a nonlinear model of the process, which will generally give different results than the prediction of the linear model. The linear model is updated by use of successive substitution, until it approximatea the solution of the nonlinear model. Once convergence is achieved, then the normal DMC procedure is used. Peterson et al. find that successive substitution can converge slowly and also diverges if the process gain changes sign (which can happen if operation is around an open-loop unstable operating point, for example),80 they recommend calculation of control actions using a secant method. The major disadvantage to the secant method is that a onestep-ahead approach is used; a first-order filter must then be added to cope with modeling errors and aggressive control action. This approach is applied to a batch polymerization model, with both SISO and MIMO control. The MIMO problem is effectively reduced to two SISO problems by sampling and controlling temperature with a sample time of 9 s, while molecular weight is sampled every 12 min. Hernandez and Arkun (1990) use a neural network to represent the nonlinear model. The extended DMC approach of Peterson et al. (1990) is then used for the predictive control implementation on a CSTFt example. Morningred et al. (1990) assume that linear dynamics can be separated from a nonlinear discrete process model and use a strategy similar to state equation linearization (Hunt et al., 1983) to provide an exact transformation of

1404 Ind. Eng. Chem. Res., Vol. 30,No. 7, 1991

the nonlinear model to a linear model. A formulation similar to dynamic matrix control is then used for nonlinear predictive control. The approach is adaptive because nonlinear parameters are estimated on-line. They show that this adaptive nonlinear predictive control approach outperforms adaptive linear predictive control, PI control, and their nonadaptive, nonlinear predictive control strategy on a CSTR example. A Nonlinear Programming Approach to Process Identification. As discussed earlier, for open-loop stable systems the model equations can be continuously integrated in time, with the final values at one time step as the initial conditions for the next time step. For open-loop unstable systems a state-variable identification method must be used. Even for open-loop stable systems there are good reasons to explicitly account for the plant/model mismatch. If unmeasured state-variable constraints are included in the optimization, then it is very important that the model reflect the actual operating conditions. The plant/model mismatch can be due to a number of factors, including: uncertain parameters, unknown state variables, unmeasured disturbances influencing the process, error in the structure of the process model (different order or type of model), and measurement noise. Nonlinear programming provides a natural formulation for process identification if nonlinear predictive control is used. The objective of the optimization-based approach is to minimize a measure of model/plant mismatch over an estimation horizon, by using process parameters, unmeasured state variables, and loads as decision variables. If actual physical disturbances can be modeled, then they can be estimated as part of a parameter estimation strategy. A typical least-squares objective function can be used, based on a data window (E) of past measurements. For a single output measurement the objective function can be stated as b

where the decision variables are the unmeasured state variables at the beginning of the estimation horizon, estimated parameters, and loads over the horizon. Joseph and co-workers (Jang et al., 1987a,b; Chen and Joseph, 1987) use an estimation phase, based on optimization (Jang et al., 1986), to update model parameters and unmeasured state variables. They continuously estimate parameters but do not pass new values to the model unless the parameters change by greater than 5 % . Sistu and Bequette (1990a,b) use a threshold approach, based on measurement statistics, to activate the estimation procedure. Their approach is similar to that used by Pavlechko et al. (1985) for linear, adaptive-predictive control. Sistu and Bequette (1990a) and Bequette (1991)have shown the advantage of using multirate sampling of temperature and composition measurements for rapid disturbance rejection in an endothermic CSTR. The approaches by Jceeph and co-workers and Sistu and Bequette (1990a,b) for updating state-variable values are also different. If there is measurement noise, Sistu and Bequette include the measured variables at the beginning of the estimation horizon as estimated variables, while Joseph et al. use the measured values at the beginning of the estimation horizon &e the initial values for integration of the model equations. Li and Biegler (1990) use the parameter estimation code GREG (Caraecoteios and Stewart, 1985) in conjunction with their multistep nonlinear approach (Li and Biegler, 1989) to control uncertain chemical processes. They illustrate

the effectiveness of this approach on a pH control example, with uncertain parameters and measurement noise. Rawlings et al. (1989) and Meadows and Rawlings (1990) estimate parameters on-line in a semibatch reactor, using sequential quadratic programming for optimization and DASSL for integration of the ODE’S. The estimation algorithm computes an estimate of the amount of component A initially in the reactor, along with the uncertainty in the estimate. This parameter uncertainty is used to estimate the uncertainty in the amount of component B required. A predictive control algorithm is used to add component B for end-point control, without adding excess B. An average of 30-40% reduction in batch time is obtained, compared to the current open-loop control by the process operator. Open-Loop Unstable Processes. One major justification for nonlinear control is that a single strategy can maintain good performance characteristics over a wide operating range. A particularly taxing example is control of systems that have multiple steady states, some that are open-loop stable and others that are open-loop unstable. Control of an open-loop unstable CSTR is considered by Patwardhan et al. (1990), who find that the prediction horizon must be equal to the control horizon (R = L). This is not an intuitive result and we have not found it necessary for R = L for the open-loop unstable processes that we have studied (Bequette, 1990; Sistu and Bequette, 1990b). Schmid and Biegler (1990) apply the multistep Newton approach of Li and Biegler (1989) to a fluid catalytic cracking unit (FCCU) under open-loop unstable operating conditions. A proportional controller is used as the inner loop of a cascade control strategy to stabilize the process (Biegler, 1990). The multistep Newton controller is found to handle 1% measurement noise but fails with 2 % noise. Brengel and Seider (1989) show satisfactory results for control of the open-loop unstable CSTR of Li and Biegler (1988). Although not stated in their paper, Brengel and Seider perform implicit state-variableestimation since both state variables are measured. Bequette (1990) uses state-variable and load disturbance estimation for tight regulatory control of a bioreactor at an open-loop unstable operating point. Biomass concentration is measured while the reactor substrate and feed substrate concentrations are estimated. Sistu and Bequette (1990b) perform a comprehensive analysis of NLPC of a CSTR, including operation at open-loop unstable operating conditions. One interesting result is that an open-loop observer can be used to control the process at the unstable point. Performance with a state and parameter estimator is much better, however. Deadtime or Distributed Parameter Systems. Bequette (1991) and Sistu and Bequette (1990a) include deadtime compensation in their NLPC strategy, and show the effect of time-delay uncertainties. Li and Biegler (1989) incorporate deadtime in their multistep approach to nonlinear control and use a CSTR example with deadtime between state variables. Eaton and Rawlings (199Oa) use NLPC on a batch crystallization (distributed parameter) process. They use orthogonal collocation and Gaussian quadrature to form ODE’S from the integrodifferential equations. The concentration profile that results from temperature control is shown to be superior to isothermal operation (a factor of 10 reduction in objective function is achieved). Robustness. Rawlinga and co-worker8 (Eaton and Rawlings, 19Wa; Eaton et al., 1989) compute the sensitivity of the optimal solution to model parameters and manip ulated variables. They show the bounds on manipulat-

Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1405 ed-variable uncertainty for a given variability in the objective function for a batch reactor control problem. Sistu and Bequette (1990a) show the effect of model structure uncertainty for an endothermic CSTR, while Sistu and Bequette (1990b) show the effect of model structure uncertainty for an exothermic CSTR. Zafiriou (1990) uses operator theory (contraction mapping principle) to study the robustness properties on linear model predictive controllers subject to input and output constraints (the “hard” constraints make the predictive controller nonlinear). Necessary and/or sufficient conditions for nominal and robust stability are derived. Several examples, including a subsystem of the Shell standard control problem (Prett et al., 1989),are used for illustration. Relationship between Nonlinear Inferential Control and NLPC. The single-stepapproach of Hidalgo and Brosilow (1990) can be considered a type of NLPC, with prediction and control horizons of one step. The approach that they use for estimation is also based on one time step. The same results would be obtained from the estimation technique of Bequette (1990,1991) if an estimation horizon of one step was used, and if the unmeasured state variables in the model were merely integrated in time rather than being estimated at each time step. Similarly, the control algorithm of Hidalgo and Brosilow can be viewed as a predictive controller with a single prediction step. The primary advantage is that a constrained nonlinear programming code does not have to be used to implement this strategy. A major disadvantage is that constraints are only handled implicitly, not explicitly (boundsare simply placed on the manipulated variables). Summary of Predictive Control Approaches. Nonlinear predictive control approaches appear to handle many more of the common process characteristics (Table I) than the other nonlinear control techniques. They are extensions of linear model predictive techniques that have been successfully applied to a number of multivariable control problems in industry during the past decade. Constraints on manipulated (as well as state and output) variables are explicitly handled. Deadtime compensation is an inherent feature of these techniques. These approaches are intuitive and easily understood by a process operator. The major disadvantage of these techniques is that there is no gumantee of convergence or rigorous proofs of robustness with respect to stability and performance. Computational time could be an issue with complex models that have relatively fast dynamics. I have found that the most important issue in implementing NLPC is determining the initial conditions for the state variables in the prediction horizon optimization. XII. Control Using Open- or Closed-Loop Oscillations Although the objective of this paper is to review nonlinear feedback control techniques, there are some nonlinear open-loop techniques worthy of mention. Cinar and co-workers have presented a series of papers dealing with open-loop vibrational control of an open-loop unstable CSTR. Simulation and experimental results have been shown for the second-order exothermic reaction between sodium thiosulfate and hydrogen peroxide. Cinar et al. (1987a,b) show theoretical and experimental results for oscillations in the input flow rate and note that a major benefit to vibrational control is that output measurements are not necessary. They note an extension that combines feedback control with vibrational control and mention that work is in progress in this area. Cinar et al. (1987c), Rigopoulos et al. (1988), and Shu et al. (1989) show that

forcing the input flow and concentration leads to improved conversion. Indeed, a higher reactor productivity can be achieved at a given average temperature. Bruns and Bailey (1975) use “push-pull” controllers to control a single-state nonlinear system near an open-loop unstable operating point. Examples of this simple system include a enzyme-catalyzed reaction with substrate inhibition kinetics and ethylene hydrogenation in an isothermal CSTR. The controllers are essentially relays with hysteresis, which are designed by use of describing function analysis and Tsypkin’s method. Bruns and Bailey (1977) extend this technique to a twestate-variable CSTR modeL The manipulated variable is coolant temperature and the controlled variable is reactor temperature. Sterman and Ydstie (1990a) analyze the feasibility of periodic operation of CSTR’s and use relay controllers with hysteresis and integral action. The effects due to measurement errors and disturbances are minimized with this strategy. Sterman and Ydstie (1990b) analyze open-loop control of a multiinput gas-liquid CSTR. The phase shift between the synchronized inputs is an important variable. Summary of Control Using Oscillations. A major advantage to these types of control strategies is that chemical reactor productivity may be improved for certain reaction systems. The open-loop methods do not require measurement feedback, which can be an advantage if mwurements are unavailable or hard to obtain. However, without measurements, the control system cannot correct for disturbances or model error. XIII. Other Techniques Cebuhar and Costanza (1984) have used optimal control theory for bilinear systems to handle a steam-jacketed exchanger, a simplified CSTR, and a two-state CSTR. Alvarez-Gallegos and Alvarez-Gallegos (1982) have applied optimal control of discrete, input-linear multivariable processes to a fermentation process. Svoronos et al. (1981) develop an approach for estimation and control of bilinear single inputaingle output systems and apply the technique to a bilinear, isothermal CSTR with success. Yeo and Williams (1987) have developed a procedure for predictive control of bilinear systems. Takamatsu et al. (1989) develop a robustness based design approach for bilinear process models. A sliding-mode technique for controller surge veasels has been developed by Kantor (1989a). The controller operates by switching between two nonlinear feedback rules. The idea is derived from an approach used by McDonald et al. (1986) to determine the outlet flow from a vessel that results in the smallest maximum rate of change in outflow, while satisfying level constraints. Suarez-Cortez et al. (1989) apply sliding-mode control to a continuous fermentation process. A major disadvantage to this technique is that significant spikes occur in the manipulated variables when there is parameter uncertainty. Fernandez and Hedrick (1987) use a slidingmode-control law on an input/output linearized (Kravaris and Chung, 1987) system, to achieve more global robustness results. The approach is applied to a MIMO CSTR example. One of the limitations to the approach is the chattering of the manipulated-variable action, which is generally unacceptable in practice. McLellan (1990) provides a complete comparison of robust GLC (Kravaris and Palanki, 1988a,b) and the sliding-mode method of Fernandez and Hedrick (1987). Lu and Holt (1990) use a min-max optimization procedure to find the optimal control for a discrete, nonlinear process. This formulation finds the best control for the worst case plant, based on parameter bounds. The con-

1406 Ind. Eng. Chem. Res., Vol. 30,No. 7, 1991

troller is implemented as a table look-up controller. A partial solution to the Shell control problem (Prett et al., 1989), a distillation column model with manipulatedvariable uncertainty and constraints, is presented. Agarwal and Seborg (1987a) have extended SISO linear self-tuning control strategies to discrete nonlinear SISO systems. They present simulation results for a CSTR model from Steadman (1978) and show superior performance over linear adaptive and other nonlinear adaptive strategies. Agarwal and Seborg (1987b) extend their previous results to the multivariable case and apply this approach to a distillation model from Wong (1985). Wong and Seborg (1986a,b)develop a low-order nonlinear model of a distillation column by finding correlating process gains and time constants as a function of the inputs and states. They develop a multivariable control law (which also handles deadtime) that can be interpreted as a variablegain, PI control system. A similar approach for developing a low-order nonlinear model has been used by Alsop (1987) in distillation column control using differential geometric techniques. The obvious advantage to this approach, is that complex, nonlinear state-space models are not needed since low-order nonlinear models are constructed from plant input-output behavior. Computational savings can be significant, and modeling time could be quite a bit shorter. The disadvantage to this approach is that the process conditions must remain in the region where the model was developed; that is, interpolation is useful but extrapolation cannot be handled. Nikolaou and Manousiouthakis (1989) develop a hybrid approach that combines input-output and state-space theories to examine the stability and performance of nonlinear feedback systems. Continuous stirred tank heaters and CSTRs are used as examples to illustrate the notions of gains and incremental gains over sets. Manousiouthakis (1990) has used a game theoretic approach to robust controller synthesis that is applicable to linear and nonlinear systems alike. The idea is to find the controller that achieves the best bound on the worst closed-loop performance than can occur over all possible plants. Manousiouthakis suggests that, for linear systems, the gap between two measures (minimax to maximin) for linear control provides a range for any improvement that nonlinear control may offer over linear control. An equivalent relationship should be true for nonlinear systems. Daoutidis and Kravaris (1990b) develop an approach to evaluate alternative control configurations for chemical processes. Relative order is used as the main analysis tool; they find that the generic calculation of relative order requires only structural information on the process. This approach is used on an evaporator, CSTR, and heat-exchanger network. The variable coupling conclusions agree with physical intuition in all of the cases.

XIV. Summary of Nonlinear Chemical Process Control Techniques for the control of nonlinear chemical processes have been reviewed, from initial strategies involving simple hardware modifications in the 1950s to recent predictive control approaches based on nonlinear programming. A concise list of the unit operations that have been controlled by use of these techniques is presented in Table 11. Nonlinear-programming-based (nonlinear predictive control) techniques appear to handle the majority of the common chemical process characteristics shown in Table I. Particularly important is the explicit constraint handling ability. The major disadvantage of these approaches is the

computational time required to perform the prediction optimization. The input/output linearization approach has the major advantage of an explicit solution for the manipulated variable, in many cases. Disadvantages include the restrictions on the class of systems that can handled (nonminimum phase; there are recent developments lifting these restrictions, however), the continuous system nature of the control law, and the inability to explicitly handle constraints. The reference system synthesis approaches are seen to be special cases of the differential-geometricbased techniques.

XV. F u t u r e Research Directions A major criticism of chemical process control research in the past was that the theoreticians were not developing solutions for the “real” control problems (Foss,1973). Rather, chemical process control researchers were directly using control techniques developed in other fields, where the control problems were significantly different than chemical process problems. This is no longer a valid criticism of chemical process control in general, since many of the most successful techniques (such as model predictive control) were developed by researchers in chemical process control. Also, these techniques were developed to specifically handle common problems such as manipulatedvariable constraints and time delays. Researchers in nonlinear chemical process control are not merely translating results from other fields to process control problems. Although many of the differential geometric control ideas have been ‘borrowed” from other fields, chemical engineers have certainly made a substantial contribution in this area; progress is being made toward lifting many of the restrictions on these techniques. The major contributions to optimization-based (nonlinear predictive) control have been made by chemical engineers. These approachers can be considered a mesh between the fields of optimization and simulation but represent a practical approach to handling the common chemical process control problems. The progress is nonlinear chemical process control during the past five years is encouraging; however there remains much research to be performed. This is not surprising since it has taken many decades of linear systems research to reach the current level of advanced linear control system technology. In this section I sketch a number of areas where I feel that significant research in nonlinear process control needs to be conducted. Experimental Verification and Control Strategy Comparison. Since few experimental applications have been reported, it is important that experimental work be conducted in nonlinear control. It is also important that a particular nonlinear control strategy be compared with other nonlinear strategies; it is even more important that nonlinear techniques be compared (fairly) with the best available linear system techniques. A t least an antireset windup PI controller should be used for systems that hit constraints. For a fair comparison the linear controllers must have access to the same measurements used by the nonlinear controllers. For example, comparing a nonlinear controller that has both temperature and composition measurements with a linear controller with only temperature measurements is hardly fair. There have been a number of papers in the literature that compare multivariable nonlinear control with decentralized linear control (SISO-PI); it appears that, in many cases, a multivariable linear strategy would have adequate performance. Model Development. Model development is obviously the most important part of a nonlinear-model-based

Ind. Eng. Chem. Res., Vol. 30, No.7, 1991 1407 Table 11. Chomiod Prwerm ControlledUiing Nonlinear Control Techniques Absomtion Biochemical Reactors Alvarez-Gallegos and Alvarez-Gallegos (1982,1988) Beauette (1990) Brengel and Seider (1989) Eaton and Rawlings (1990b) Henson and Seborg (1990a) Hoo and Kantor (1986a,b) Kravaris (1988) Lien and Wang (1990) Palanki et al. (1990) Suarez-Cortez et al. (1989) Chemical Reactors Adebekun and Schork (1989a,b, 1991a,b) Agarwal and Seobrg (1987a) Aluko (1988) Alvarez-Gallegos (1988) Alvarez et al. (1989,1990) Balchen et 81. (1989b) Bartusiak et al. (1989) Bequette (1989,1991) Bhat et al. (1990) Brengel and Seider (1989) Brown et al. (1990) Bruns and Bailey (1975, 1977) Calvet and Arkun (1988b, 1990) Cebuhar and Costanza (1984) Cinar et al. (1987a-c) Daouditis and Kravaris (1989) Daouditis et al. (1990) Doyle I11 and Morari (1990) Doyle et al. (1989) Eaton and Rawlings (1990a) Eaton et al. (1989) Economou and Morari (1985) Economou et al. (1986) Fernandez and Hedrick (1987) Garcia (1984) Georgakis (1986) Henson and Seborg (1989, 1990~) Hernandez and Arkun (1990) Hildago and Brosilow (1990) Hoo and Kantor (1985) Jang et al. (1987a,b) Jones et al. (1963) Kantor and Keenan (1987) Kravaris (1988) Kravaris and Chung (1987) Kravaris and Palanki (1988a) Kravaris and Soroush (1990) Kravaris and Wright (1989) Kravaris et al. (1989) Li and Biegler (1988) Li et al. (1990) Limqueco and Kantor (1990)

Liu (1967) M d n i and Georgakis (1984) Marroquin and Luyben (1972) Meadow and Rawlings (1990) Momingred et al. (1690) Nikolaou and Manousiouthakis (1989) Parrish and Brosilow (1986) Palanki et al. (1990) Patwardhan et al. (1990) Peterson et al. (1989) Rawlings et al. (1989) Riggs and Rhinehart (1988) Rigopoulos et al. (1988) Schaper et al. (1990) Schmid and Biegler (1990) Shu et al. (1989) Silverstein and Shinnar (1981) Sistu and Bequette (1990a,b) Steadman (1978) Sterman and Ydstie (1990a,b) Svoronos et al. (1981) Waller and Makila (1981) Wright and Kravaris (1990) Crystallization Eaton and Rawlings (1990) Distillation Agarwal and Seborg (1987b) Alsop (1987) Alsop and Edgar (1990) Castor et al. (1987, 1989) Cott et al. (1989) Georgakis (1986) Georgiou et al. (1988) Joseph et al. (1989) Koung and Harris (1987) Levine and Rouchon (1989) Lu and Holt (1990) McDonald and McAvoy (1985) Riggs (1990) Rovaglio et al. (1990) Ryskamp (1982) Skogestand and Morari (1988a,b) Tsogas and McAvoy (1985) Wong and Seborg (1986a,b) Evaporator Brown et al. (1990) Lee et al. (1989) Newell and Lee (1989) Flow Shinskey (1962) Heat Exchange Alsop and Edgar (1989) Cebuhar and Costanza (1984)

strategy. Simple models (such as ideal CSTRs) are readily available; however, it is the complex systems that will probably justify the use of nonlinear techniques. There is a great need to build a "toolbox" of nonlinear chemical process models that are more complex, or are composed of a number of simple models. The development of techniques to provide a rational basis for determining the proper complexity of models is important. These models should be developed with a consideration of control performance in mind. Systems with a large degree of parametric uncertainties, frequent disturbances, and poor measurements may not justify complex models. Reduction of the order of complex models will be an important area of research. State-Variableand Parameter Estimation. Most of the nonlinear controller techniques assume that all state variables are measured or have been estimated. In prac-

Haskins and SlieDcevich (1965) Hutchinson and McAvoy '(1973) Khambanonda et al. (1990a,b) Nikolaou and Manousiouthakis (1989) Parrish and Brosilow (1988) Riggs and Rhinehart (1988) Shinskey (1962) Level Cheung and Luyben (1980) Kantor (1989) Ogunnaike (1986) Rugh (1987) Shunta and Fehervari (1976) PH Henson and Seborg (1989) Jayadeva et al. (1990) Li and Biegler (1990) Li et al. (1990) Parrish and Brosilow (1988) Shinskey (1962) Waller and Makila (1981) Williams et al. (1990) Wright and Kravaris (1991) Wright et al. (1991) Pressure Shinskey (1962) Other Processes Balchen et al. (1989a);electrometallurgical Experimental Resulta Cheung (1978) Haskins and Sliepcevich (1965) Hutchinson and McAvoy (1973) Levine and Rouchon (1989) Marroquin and Luyben (1972) Wright et al. (1990) Systems with Deadtime Kravaris and Wright (1989) Bequette (1991) Sistu and Bequette (1990a) Lee et al. (1990) Li and Biegler (1989) Manipulated Variable Constrainta Bequette (1990b) Calvet and Arkun (1988b) Eaton and Rawlings (1990a) Eaton et al. (1989) Hidalgo and Brosilow (1990) Li and Biegler (1988) Li et al. (1990) Patwardhan et al. (1990) Sistu and Bequette (1990b)

tice, the estimation of unmeasured state variables is not a trivial problem, particularly in conjunction with the estimation of unknown parameters. There is a great need for further research in this area, or at least the incorporation of existing techniques (extended W a n filter, etc.) in the solution of these problems. Information on the uncertainty of state variables and parameters should be used to adjust the performance requirements of a feedback system. Industrial Case Studies. The need for industrial case studies has been noted in the past; however, with the exception of the Shell control problem (Prett et al., 1989) no case studies have been prepared. A number of detailed problems need to be developed that include aspects of most of the following: time-varying parameters, variable deadtime, model uncertainty (including unmodeled reactions, etc.), manipulated- and state-variable constrainta,

1408 Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991

and measured and unmeasured disturbances. Perhaps these problems should be formulated and sent to academic and industrial researchers in the same fashion that the Dow batch reactor parameter estimation problem (Biegler et al., 1986) was handled. McFarlane et al. (1990) have made a detailed presentation of a fluid catalytic cracking unit simulation model, which has great potential for case studies in advanced control. Model Structure Uncertainty. There has been some work illustrating parametric uncertainty,but little research has been performed illustrating the effect of model structure uncertainty. Any model (whether linear or nonlinear) is a simplified representation of the actual process. Invariably, there are additional transport lags, mixing effects, heat losses, and other nonlinearities that are not captured in the model; thus, there is a great need for research quantifying the effect of model structure uncertainty. Also, more rigorous analysis of parametric uncertainty effects on robust stability and performance needs to be performed for most of the control strategies. Control System Synthesis and Impact of Design on Control. There is a need for quick screening tools (similar to the RGA for linear systems) to determine if it is worth the rigor to develop a nonlinear control strategy for a particular process. Manousiouthakis and Nikolaou (1989) have made a step in the right direction with the development of a nonlinear block relative gain (NLBRG) as a measure of input-output interaction in decentralized control of nonlinear systems. Also, extensions of the game theoretic approach of Manousiouthakis (1990) should provide bounds on possible performance increase for nonlinear controllers. Daoutidis and Kravaris (1990b) have developed an initial approach for structural control system synthesis. There has been much discussion about the importance of the impact of process design on process control, but very little research has been performed in this area. Initial progress is reported by Brengel and Seider (19901, who use essentially a case study (brute force) type of approach and incorporate their nonlinear predictive control strategy (Brengel and Seider, 1989). Implementation Issues. Practical implementation of nonlinear control will probably require multilevel strategies. A supervisory layer will needed to determine if enough process information is available to verify that the nonlinear model is accurate enough for control purposes. Perhaps an expert system approach should be used to switch between linear and nonlinear algorithms or even open-loop control. The proper operator interface will become even more important than it already is as more complex controllers are used. The operator should always have an understanding of why a controller is making a manipulated variable change. Deadtime and Constraints. Some of the important linear system problems such as deadtime and manipulated-variable Constraints have been addressed in a few nonlinear control publications, but there is much work to be done. Variable deadtime or deadtime estimation techniques need to be developed. Also, the identification of time-varying manipulated-variable constraints is important. Concepts such as failure sensitivity have not been addressed for nonlinear systems. Computational Issues. Few publications have addressed computational issues; usually, it is assumed that the control law calculation can be performed instantaneously when compared with the dominant time constant of the process. It is important that studies be performed that account for the computational "lag". This may be par-

ticularly important for the techniques that have been based on a continuous time representation of the process, such as many of the differential geometric approaches. Desirability of Linear Responses. The majority of nonlinear control system techniques have a goal of obtaining a linear system. If variable transformations are used, the goal is obtaining a transformed linear system and then using linear control system design techniques. The primary advantages are that it is easy to design stable linear controllers and it is easy to understand linear closed-loopresponses. What needs to be addressed is the desirability of a linear closed-loop response. Are there many cases in which the control quality would be better (by some measure) if the closed-loop response was nonlinear? What leads us to believe that we should have the same speed of response, no matter where we are in the operating region? Process Control Publications. As I noted earlier, it is desirable to compare the various nonlinear and linear control techniques in our control publications. I believe that it is particularly important to include manipulatedvariable responses in addition to controlled-variable responses, although this is currently done in much less than half of all control publications. "Tighter" controlledvariable responses generally lead to more "vigorous" manipulated-variable responses; it is absolutely necessary to consider this in any control system comparison. Acknowledgment Financial support from a National Science Foundation research initiation award (NSF CTS-8910362) is gratefully acknowledged. Nomenclature b = tuning constant d = additive disturbance e = error E = estimation horizon f = nonlinear dynamic state function g = nonlinear dynamic function between inputs and states h = nonlinear function between states and outputs k, = proportional gain 1 = load disturbance L = control horizon p = parameter p ( x ) = linearizing state feedback function q ( x ) = linearizing state feedback function R = prediction horizon r = relative order u = manipulated input u = reference input x = state y = output Greek Letters 5 = transformed state X = closed-loop time constant p = disturbance relative order q = integral time constant TD = derivative time constant e = input-state deadtime 4 = state-output deadtime 0 = predictive control objective function II = estimation objective function Subscripts m = model prediction pred = corrected model prediction r = desired response sp = setpoint

Ind. Eng. Chem. Res., Vol. 30, No. 7, 1991 1409

Acronyms CSTR = continuous stirred tank reactor DMC = dynamic matrix control FCCU = fluid catalytic cracking unit GMC = generic model control GLC = globally linearizing control GPC = generalized predictive control GRG = generalized reduced gradient IMBC = integrated model based control IMC = internal model control LDPE = low-density polyethylene LMPC = linear model predictive control LOC = limited output change LQG = linear quadratic Gaussian LQR = linear quadratic regulator MAC = model algorithmic control MIMO = multiinput-multioutput MPC = model predictive control MVSISO = multivariable single input-single output

NLBRG = nonlinear block relative gain NLIC = nonlinear inferential control NLIMC = nonlinear internal model control NLPC = nonlinear predictive control NMPC = nonlinear model predictive control ODE = ordinary differential equation PID = proportional integral derivative PMBC = process model based control QDMC = quadratic dynamic matrix control RGA = relative gain array RSS = reference system synthesis SISO = single input-single output SQP = sequential quadratic programming UDMC = universal dynamic matrix control Literature Cited

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Received for review August 8, 1990 Revised manuscript received January 15, 1991 Accepted February 1, 1991

KINETICS AND CATALYSIS Continuous Operation of the Berty Reactor for the Solvent Methanol Process Chandrasekhar Krishnan and J. Richard Elliott, Jr.* Department of Chemical Engineering, The University of Akron, Akron, Ohio 44325-3906

Jozsef M.Berty Berty Reaction Engineers Ltd., 1806 Bent Pine Hill, Fogelsville, Pennsylvania 18051-9712

In the solvent methanol process (SMP), an inert and selective solvent removes methanol as soon as it is formed from syngm. Conversion in the conventional vapor-phase methanol synthesis is limited because of equilibrium limitations due to the reverse reaction, but data presented in this paper demonstrate that high conversions can be obtained in the SMP. Rate data have been collected for the SMP a t operating conditions typical of the vapor-phase process (7.8-10 MPa, 493-513 K). Single-pass H2 and CO conversions range from 30 to 80%. In some cases, conversions are higher than those predicted by vapor phase equilibrium calculations based on the feed composition, providing that SMP is able to overcome the equilibrium barrier. Rates are 2-3 times lower than those encountered in the vapor-phase process owing to pore diffusion limitations from the presence of the liquid. 1. Introduction Methanol is synthesized catalytically from H2,CO, and COO(synthesis gas). The following are the main reactions: CO + 2H2 * CHSOH (1) C02 + H2 + CO

+ H2O

(2)

The catalyst typically preferred is Cu/ZnO/A1203,which exhibits optimum activity in the temperature range 473-543 K (Chinchen et al., 1988). The strong exothermicity of reaction 1 (-91 kJ/mol) implies that the equilibrium CH30H content decreases with increasing temperature. In order to obtain reasonable rates and conversions, operating pressures in the range 5-10 MPa (50-100atm) are required. Single-pass conversions of CO and H2 are still only around 15-30%, resulting in the recycle of large quantities of unconverted reactants. This

* T o whom correspondence should be addressed.

leads to substantial fixed and operating costs in a CH30H plant. The initial concept of a three-phase CH30H process was put forward by Chem Systems Inc. (Sherwin and Blum, 1975). In that process, the use of a high heat capacity, inert paraffinic oil in the synthesis loop affords good temperature control of reaction 1 but product CH30H is removed from the vapor phase and single-pass conversions of H2 and CO are only slightly better than in the vapor-phase process. Recently, the use of two novel converters to improve syngas conversion to CH30H has been reported. In the gas-solid-solid trickle flow reactor (GSSTFR) (Westerterp et al., 19871, a fine adsorbent powder selectively picks up CH30H as soon as it is formed, thereby decreasing the equilibrium limitation on the forward reaction. The reactor system with interstage product removal (RSIPR) (Westerterp et al., 1988),on the other hand, achieves a high conversion per pass by using a series of Lurgi type reactors with CH30H adsorbers installed after each reactor.

0888-5885/ 91/ 2630-1413$02.50/ 0 0 1991 American Chemical Society