Effect of Manipulated Variables Selection on the Controllability of

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Effect of Manipulated Variables Selection on the Controllability of Chemical Processes Zhihong Yuan, Bingzhen Chen,* and Jinsong Zhao Department of Chemical Engineering, Tsinghua University, Beijing100084, China ABSTRACT: Chemical processes usually involve several alternative manipulated variables each having the potential to influence system operability. This work presents a methodology that illustrates how manipulated variables selection can relate to the openloop stability and phase behavior of chemical processes over the entire feasible operating region. Within this framework, the first step explores the steady state maps, under different manipulated variable selections. The inherent characteristics, including open-loop stability and phase behavior, are then analyzed. After that, the effect of manipulated variable selection on the static controllability can be assessed. In the third step, based on the conventional model predictive controller, closed-loop dynamic simulations, with both reference tracking and disturbance rejection, are carried out. This allows the comparison of dynamic behaviors, under different operating policies, and a validation of outcomes from the open-loop theoretical analysis. Results from both the static and the dynamic analysis reveal the influence of the manipulated variable selection on the process controllability, over the feasible operating region. These conclusions can assist in process control structure selection and process operation. The proposed method is applied to a polymerization reaction process to demonstrate its efficiency. This example emphasizes how such a methodology can help clarify and handle the causes of the complex phenomena that arise in the design, operation, and control of chemical processes.

1. INTRODUCTION The control of highly nonlinear systems is a subject of considerable importance, especially in light of recently increased standards in product quality, stricter environment regulations, and tighter safety requirements for the chemical and biochemical processes. Recent advances in computational methodologies are leading to the better control of processes and improved operations.1 However, there are many factors that can adversely affect control system design that are not easily handled or overcome. These include input/output multiplicities, open-loop unstable characteristics, and right half plane zeros/nonminimum phase behaviors determined by process design itself. It is therefore necessary to recognize these inherent characteristics and to clarify the cause of such unfavorable factors. The main objective of the traditional sequential approach to chemical plant design is to improve profits and ensure products meet the customer’s variable requirements. Operability issues, as well as switchability between operating points, are usually ignored, meaning the effects of operability on the economics of the plant can be neglected, leading to severe operating problems and financial penalties.2,3 There are many uncertainties that affect the operability of a nonlinear process. First, a process may not run at the designed operating point due to considerable uncertainties and variations. A small change in operation conditions could move the operating point from a minimum phase behavior position to the nonminimum phase behavior operating region and cause control problems.4 Second, because no model provides a perfect description of the reality, differences between the models used for developing advanced control systems and realistic chemical processes can also cause operational problems. Third, to improve the chemical processes’ efficiency, the industrial trend has been steered toward more highly integrated and complex plants, having processes with interacting process units. This causes r 2011 American Chemical Society

more interactions between processes and complicates the design of the control system. Finally, process control system design is very much an open-ended process, meaning there is not one unique correct solution.5 For a specific chemical plant, there are many possible alternative control structures. Different pairs of manipulated/controlled variables exhibit different nonlinear relationships. In fact, a control structure that is good for a specific control or economic objective may not be good for another. It is therefore important to analyze the influence of the uncertainties discussed above on the inherent characteristics of the nonlinear processes. Over the last several decades, several control methodologies have been presented to deal with the problems caused by uncertainties, such as offset free nonlinear predictive control, to track the set-point changes;6,7 constrained predictive control, to satisfy the process input/output constraints;8,9 black-box model based predictive control, to deal with highly nonlinear processes;10,11 and distributed predictive control, to tackle the problem of controlling a complete plant-wide process.12,13 Many of these controllers may be suitable for a certain operating point, or a nearby region, but may fail at a new point, which is set away from the original setting. On the other hand, parameters that are tuned for the models predictive controller are often based on trial and error do not obey a consistent rule.14 It is noteworthy that modifications to a process design can sometimes have a more profound effect on the process dynamics than on control system design.15 A thorough chemical process nonlinearity assessment, at the conceptual design stage, can therefore relieve the burden of controller design to some extent. Received: January 18, 2011 Accepted: May 4, 2011 Revised: April 20, 2011 Published: May 04, 2011 7403

dx.doi.org/10.1021/ie2001132 | Ind. Eng. Chem. Res. 2011, 50, 7403–7413

Industrial & Engineering Chemistry Research

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Figure 1. Examples for manipulated variable selection.

Motivated by the above considerations, this present study, and follow-up work, will demonstrate how uncertainties, including manipulated variable selection, catalyst deactivation/distribution, and plant-wide topology structures, can affect the inherent properties and dynamic performances of chemical processes. Skogestad16 demonstrated that the selection of manipulated variables is not a difficult task at the stage of control structure design, as these variables are generally a direct consequence of the process design itself. Inputs that have conspicuous and rapid effects on the controlled variables may be selected as the manipulated variables.17 Stephanopoulos18 and Luyben19 listed the guidelines for selecting manipulated variables to generate control structures. Recently, Seider and Lewin20 published an excellent summary of the principles for manipulated variables selection. Until now, a wide spectrum of techniques for manipulated variables selection has been proposed and most of them are relevant within the context of inputoutput loop formation. Many linear/linearization based measures, such as Relative Gain Array, Singular Values, and Condition Number, are then applied to assess the controllability of the inputoutput loop. This simplifies the development, implementation, and operation of any control strategy at the designed operating point. Most chemical processes are inherently nonlinear in nature, and those aforementioned controller design and analysis methods are based on linear process models, which may no longer be satisfactory over a wider operating range. Also, the above methods do not provide insight into characterizing the operating regimes for the stability and minimum-phase behavior of their steady states. In the extended bifurcation diagrams, the solution diagrams can be divided into four zones.21 How the inherent properties, including open-loop stability and phase behavior, vary with the values of the selected manipulated variables can be exhibited simultaneously. In terms of the relationships between manipulated variable selection and process controllability, this research work presents a systematic framework to analyze process controllability, over the feasible operating region, with different manipulated variable selections. In this framework, we demonstrate the process static controllability characteristics, with each manipulated and controlled variable coupling. First, closed-loop simulations, such as set-point tracking and disturbance rejection, are set up for each pair to test the dynamic controllability. The results are expected to give useful guidance for controlling structure selection and control algorithm design. The remainder of this article is structured as follows. Section 2 provides preliminaries for the research work related to the controllability definitions and structures for the proposed methodology. This is followed by

an illustrate example, to validate the proposed framework. In Section 3, the static controllability, under different manipulated variable selections, for the illustrated example is investigated. This section is also devoted to implementing the controller, to validate the results from the static controllability analysis, and to compare the dynamic behavior under different operating policies. All conclusions are presented in Section 4.

2. PRELIMINARY This section outlines preliminary materials that will be used in the presented research work. Challenges to be solved in the present work are listed and the basic knowledge of controllability is introduced. Based on the previously discussed problems and the listed tools, the proposed framework, as well as its detailed steps, is shown in the last subsection. 2.1. Outline of the Problem. A complex chemical process has a vast number of manipulated and controlled variables, as in the illustrated example shown in Figure 1. For a certain process, there are several pairs of manipulated and controlled variables, which lead to different maps between these two types of variable. Skogestad17 reviewed the area of plant-wide control including a control structure section. A control structure is usually selected based on the operating point, with its desirable profitability decided by the chemical engineer. As discussed previously, this design procedure often neglects the operability and can cause operation difficulties, because of the severe nonlinearity of the process.19 Analyzing the relationship between the controllability and the manipulated variable selection is a prerequisite step to designing a chemical process with optimum operability. To achieve this, three questions need to be addressed: (1) What is the steady state behavior with different types of manipulated variables? (2) Once the pairing of manipulated and controlled variables is decided, how do the inherent characteristics, such as open-loop stability and phase behavior, change with their variation over a feasible operating region? (3) Is the dynamic simulation consistent with the theoretical results obtained from the static controllability analysis? Answering the above questions may reveal the ideal control system for improving the processes safety operations and control performance. 2.2. Scope of Controllability and Phase Behavior. There are several definitions for controllability. Classically, controllability is described as “the ability of the plant to move fast and smoothly from one operating condition to another and to deal effectively with disturbances.”22 At the design stage, the term controllability is used to qualitatively compare alternative designs based on their 7404

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Figure 2. Framework for the presented methodology; Fm: manipulated variables; Yc: controlled variables.

ability to minimize process output variability, in the presence of load disturbances. Static controllability is independent of controller design. To analyze controllability, it is desirable to gain insight into what factors impose limitations and how the process behaves when subjected to changing conditions. Some process characteristics, such as right half plane zeros, time delays, and unstable steady states will limit the control performance. Therefore, a number of methods are available, mainly for linear systems, which evaluate their potential impact on closed-loop performance.23 Recently, Yuan24 presented a detailed overview of controllability and its assessment methodologies, for both linear and nonlinear processes. The techniques used for linear systems may not be suitable for highly nonlinear processes. In the present work, phase behavior is adopted as the open-loop indicator for controllability. For highly nonlinear systems, it is impossible to define a transfer function, but one can find the zero dynamics, which are the dynamics of the nonlinear systems’ inverse.25 The concept of zero dynamics is analogous to the notion of the zeros of a linear system. The term phase behavior is used in relation to the stability of the zero dynamics. A nonlinear system is termed minimum phase at a steady state point, x∼, if all eigenvalues of the zero dynamics are in the left half plane. Otherwise, it is a nonminimum phase. Besides imposing adverse effects on the control performance, nonminimum phase behavior can also present issues for the stability of the controller. The presence of such an inherent characteristic forbids the implementation of a conventional controller from the class of nonlinear inversion-based controllers, because they are unstable.26 A nonminimum phase nonlinear system can be stabilized with either longer prediction horizons or proper adjustments to the objective function, within the framework of optimal control, such as with nonlinear model predictive control.27 However, an offset-free performance cannot be achieved in the presence of a disturbance, under the conventional model predictive control framework.28 To deal with this, based on nonlinear models, Kanter proposed a new feedback control law for multivariable nonminimum-phase processes. Compared to a general model predictive controller, this framework has fewer tuning parameters to achieve closed-loop, asymptotic stability.29,30 Panjapornpon developed a nonlinear control method, which permits effective control at an unstable steady state

with NMP behavior, and ensures offset-free conditions with the uncertainties.31 This would improve the controllability of a nonlinear process. However, as the shortest prediction horizon forms the proposed control system, an analytical proof of the closed-loop system’s stability is still a problem. From a control engineer’s viewpoint, a stable and minimum phase behavior process is very desirable. However, inherent properties, such as phase behavior and open-loop stability, may change with changes in operating condition32 and variable selection, resulting in difficult control problems. 2.3. Methodology Descriptions. If a conceptually designed chemical process possesses poor controllability characteristics, one can redesign the process to improve its controllability, using the three approaches presented schematically in Figure 2: (i) a rebuilding of the process models; (ii) a reselection and pairing of Fm and Yc; or (iii) an adjustment in process topology structure. The presented work only discusses the relationship between controllability and Fm selection. For a given nonlinear system, the first and most important property is the open loop stability. To establish the stability characteristics of nonlinear systems is required to assess its controllability. Stability becomes an important consideration when feedback controllers are introduced into the system, as an unstable process would complicate its control system design. In relation to the three questions listed in Section 2.1, the framework for the proposed systematic methodology is displayed in Figure 2. In the framework, the detailed algorithm for obtaining zero dynamics from the normal form can be seen in ref 33. The framework for analyzing the influences of manipulated variable selection on controllability includes six steps, which are described as follows: Step 1. Explore the steady state behavior of the process over the feasible operating region, once the manipulated and controlled variables are selected and paired. The impact of the manipulated variables on the nonlinearity then becomes clear. Step 1.1. Compute the steady-states, via the homotopy continuation method.34 Step 1.2. Analyze the open-loop stability/instability characteristics of all the steady states. 7405



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Step 2. Extract the zero dynamics from the original chemical process, and then determine the phase behavior of the process at each steady state. Step 3. Analyze the static controllability of the process, based on the results of steps 1 and 2. Step 4. Implement the controller to validate previous theoretical predictions. Step 5. Compare the realistic dynamic performance with the safety and dynamic performance requirements. Reselect and pair the manipulated and controlled variables and repeat the above steps. Step 6. Compare the static controllability and dynamic behavior, under different manipulated variable selections, to interrogate the influences of the operating policy on the process controllability. The proposed methodology can not only show the relationship between the manipulated variable selections and process controllability, but it also reveals how the nonlinearity and other inherent characteristics vary with each selected manipulated variable, over the entire feasible operating region. The framework can also demonstrate static and dynamic controllability simultaneously. Both of the above will provide guidance for improving the processes operation performance. If the chemical process has good static and dynamic controllability, it is able to adapt to frequent changes and switches. The following section is devoted to applying the proposed technique to a classic nonlinear polymerization reaction process.

Based on the algorithm for extracting zero dynamics described in ref 4, the zero dynamics under this operating policy are formed as follows. ð0Þ

dη1 ð0Þ ð0Þ ð2Þ ¼ f1 ðη1 , η2 , ηð1Þ s , ηs Þ dt ∂f4 ð0Þ ð0Þ ð1Þ ð2Þ ðη , η2 , ηs , ηs Þ ð0Þ dη2 ∂x5 1 ð0Þ ð0Þ ð1Þ ð2Þ ¼ f2 ðη1 , η2 , ηs , ηs Þ þ ∂f4 ð0Þ ð0Þ ð1Þ ð2Þ dt ðη , η2 , ηs , ηs Þ ∂x2 1 ð0Þ ð0Þ ð2Þ Lf f4 ðη1 , η2 , ηð1Þ ð0Þ ð0Þ s , ηs Þ ð2Þ f5 ðη1 , η2 , ηð1Þ , η Þ s s ∂f4 ð0Þ ð0Þ ð1Þ ð2Þ ðη , η2 , ηs , ηs Þ ∂x2 1

ð4Þ

ð5Þ

½η1 , η2 , η3 , η4 , η5 ¼ ½Cm , CI , T, D0 , Tj

ð1Þ

In the design and operation of chemical processes, it is important to understand the behaviors and stabilities of steady states. However, the purpose of this study is to analyze the influence of manipulated variable selection on process controllability, over the whole feasible operating region. The stability of the steady states and the phase behavior are therefore investigated simultaneously. As discussed above, the phase behavior at certain operating points will be conducted by analyzing the stability of the zero dynamics at that point. The first step is to explore the steady state behavior of the process over the operating region. The extended bifurcation diagrams of the MMA polymerization reactor, depicting the temperature, T, and cooling fluid temperature, Tj, vs the initiator flow rate, F1, are in Figures 3 and 4, respectively. Figure 3 shows the relationship between T and F1, whenFI∈(0.0009,0.00321), as obtained using the homotopy continuation method. The process exhibits output multiplicities. After obtaining the steady states, how the stability of the zero dynamics varies with F1, over the whole region, can be analyzed. These two results are demonstrated by Figure 3 (S-MP denotes that the system is stable and has minimum phase behavior; S-NMP denotes that the system is stable, but has nonminimum phase behavior; US-MP denotes that the system is unstable with minimum phase behavior, across the operating region). It can be seen from Figure 3 that in the lower temperature region, the system is stable, but exhibits nonminimum phase behavior. In the middle temperature region, the system is unstable and has minimum phase behavior. In the higher temperature region, the stability and phase behavior do not change with variations in F1. It is known that when the system runs in the nonminimum phase behavior region, the system is disturbed using a step function, which may present an inverse response.37 This will adversely influence process operability and controller performance. In terms of inherent safety, especially for highly exothermic polymerization reactions, running in the nonminimum phase behavior or unstable regions may be avoided by either changing the operating/design parameters or by adjusting models at an early design stage.4 The relationship between TJ and F1 is displayed in Figure 4, and is similar to that in Figure 3. 3.2.2. Analysis of the Second Operating Policy. By following a similar procedure, the second operating policy involves selecting F and Fcw as manipulated variables to control D0 and TJ. Set

½y1 , y2 ¼ ½D0 , Tj

ð2Þ

½η1 , η2 , η3 , η4 , η5 ¼ ½Cm , CI , T, D0 , Tj

ð6Þ

½u1 , u2 ¼ ½FI , Fcw

ð3Þ

½y1 , y2 ¼ ½D0 , Tj

ð7Þ

3. CASE STUDIES In this section, a methyl methacrylate (MMA) polymerization reactor is selected as an example to demonstrate the efficiency of the proposed method, for analyzing the effect of manipulated variable selection on controllability. The reason for selecting this particular case study is that the model for its reactor is accurate and often used as a benchmark for the performance testing of various novel control algorithms. 3.1. Process Description. The polymerization system is a bulk free-radical MMA polymerization, with azo-bis-isobutyronitrile (AIBN) as the initiator and toluene as solvent. The heat of the exothermic reaction is removed using a cooling jacket. Using the standard reaction mechanism35 and the appropriate assumptions for model building,36 an original mathematical model was developed by Daoutidis.37 The dynamic model is taken from ref 35 without considering D1. In this work, the controlled variables are the cooling fluid temperature, Tj, and the molar concentration of the dead polymer chains, D0. The manipulated variables include the initiator volumetric flow rate, F1, the cooling water volumetric flow rate, Fcw, and the monomer feed stream volumetric flow rate, F. The kinetic/constant and process design/ operation parameters are the same as those listed in ref 36. 3.2. Static Controllability Analysis. In this section, the static controllability, including phase behavior and the stability of steady states, will be analyzed using two different operating policies. 3.2.1. Analysis of the First Operating Policy. In the first case study, it is assumed that D0 and Tj are maintained within a region around the desired operating state by manipulating F1 and Fcw. Set

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Figure 3. Relationship between the reactor temperature T and initiator volumetric flow rate F1.

Figure 4. Relationship between the cooling fluid temperature, TJ, and initiator volumetric flow rate, F1.

½u1 , u2 ¼ ½F, Fcw

The expression for the zero dynamics may differ under different manipulated variables. Under the second operating policy, the zero dynamics of the polymerization reactor has the following form. ð0Þ

dη1 ð0Þ ð0Þ ð0Þ ð2Þ ¼ f1 ðη1 , η2 , η3 , ηð1Þ s , ηs Þ dt ð0Þ

þ

ðCmin η1 Þ ð1Þ ηs

ð0Þ

ð0Þ

ð0Þ

ð2Þ f4 ðη1 , η2 , η3 , ηð1Þ s , ηs Þ

ð9Þ

ð0Þ

dη2 ð0Þ ð0Þ ð0Þ ð2Þ ¼ f2 ðη1 , η2 , η3 , ηð1Þ s , ηs Þ dt þ

ð0Þ

η2

ð0Þ

ð8Þ

ð0Þ ð0Þ ð0Þ ð2Þ f ðη , η2 , η3 , ηð1Þ s , ηs Þ ð1Þ 4 1 ηs

ð10Þ

dη3 ð0Þ ð0Þ ð0Þ ð2Þ ¼ f3 ðη1 , η2 , η3 , ηð1Þ s , ηs Þ dt þ

ð0Þ

ðTin η3 Þ ð1Þ ηs

ð0Þ

ð0Þ

ð0Þ

ð2Þ f4 ðη1 , η2 , η3 , ηð1Þ s , ηs Þ

ð11Þ

The extended bifurcation diagrams for the MMA polymerization reactor, depicting the temperature, T, and cooling fluid temperature, TJ, vs the monomer flow rate, F, are in Figures 5 and 6, respectively. It was found that under the second operating policy, the steady state map of the system becomes more complex. In certain regions, such as T∈(345,440) and Tj∈(330,400), the system exhibits multiple steady-states and input/output multiplicities with different stabilities and phase behavior characteristics. When F∈(0,0.6), the stabilities and the phase behaviors vary with the monomer volumetric flow rate, F, which can be seen from Figure 5. An unstable nonminimum phase behavior region also exists, which is a huge challenge for control algorithm design. In the lower monomer flow rate subregion, even with minor changes 7407



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Figure 5. Relationship between the reactor temperature T and monomer volumetric flow rate F.

Figure 6. Relationship between the cooling fluid temperature, TJ, and monomer volumetric flow rate, F.

in F, the operating point switches from a stable to an unstable region, or from a minimum phase behavior region to a nonminimum phase behavior region. Without the proper controller in the unstable subregion, a minor disturbance could cause a runaway situation and/or serious accidents. In this subregion, the system has poor controllability. All of these characteristics pose inherent limitations on the process operation. When F > 0.6, the zero dynamics are stable, at any steady state, and the system only shows minimum phase behavior in this subregion. In the subregion, where 0.6 < F < 2 and 350 < T < 420, heat generation dominates heat removal, causing open-loop instability. This will complicate the control system design. Therefore, both the openloop instability and the nonminimum phase behavior need to be identified and eliminated/avoided in the process design stage to relieve the burden on the process controller design and process operation. The relationship between reactor temperature and the manipulated variables is similar to the relationship between cooling

fluid temperature and the manipulated variables. Figures 7 and 8 present the steady state solution diagrams for the two temperatures vs cooling fluid flow rate, Fcw. From these two figures, the process exhibits multiplicity. The reactor temperature, T, exhibits output multiplicity, but the cooling fluid temperature exhibits both input and output multiplicity. In the lower reactor temperature region, the system has nonminimum phase behavior. Input multiplicity in TJ - Fw may cause a sudden destabilization of the control system. 3.2.3. Remarks. The above two subsections discuss the effects of manipulated variable selection on process static controllability, over the entire operating region. Under both operating policies, the open-loop stabilities and phase behaviors are analyzed. According to the first operating policy, both the reactor temperature and the cooling fluid temperature exhibit output multiplicity, but both temperatures would show input/output multiplicities under the second operating policy. If the initiator and monomer flow rates are kept as constant, the TJ exhibits 7408



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Figure 7. Relationship between the reactor temperature, T, and cooling fluid flow rate Fcw.

Figure 8. Relationship between the cooling fluid temperature, TJ, and cooling fluid flow rate Fcw.

input and output multiplicity, but T exhibits only input multiplicity. In addition, the input multiplicity in TJ Fw may cause a sudden destabilization of the control system. These characteristics pose inherent limitations on the control performance. The system has a poorer controllability under the second operating policy. In terms of inherent safety, initiator volumetric flow rate and cooling fluid flow rate would be the optimal manipulated variables. The system would have a better dynamic performance under operating policy I, upon comparison to operating policy II. The next section is devoted to implementing the dynamic closedloop simulations, to both validate the above theoretical analysis and to compare the dynamic behavior under the two different policies. 3.3. Dynamic Controllability Analysis. Industrial polymerization reactors are normally equipped with a control device, whose design and complexity relies on the control objectives and the manipulated variable selections. Open-loop stability and phase behavior analysis may play an important role in assessing the effect of manipulated variables on the controlled variables

and help prevent poor process controllability. Throughout the past several decades, there have been a number of articles with different control algorithms for polymerization reactors.38,39 Our aim, however, was to both validate the theoretical analysis results and to demonstrate the efficiency of the proposed methodology, for analyzing the effects of different operating policies on the controllability. A suitable operating policy for the operation and control of polymerization reactors can then be determined. To validate the results from the above theoretical analysis, several closed-loop dynamic simulations, including set-point tracking and disturbance rejection, under two operating policies were carried out to demonstrate the dynamic controllability characteristics. The system was run under conventional model predictive control to explore the dynamic behavior. 3.3.1. Set-Point Tracking. To illustrate and compare the setpoint tracking performances of the two operating policies, we selected several operating points, located in different subregions with different inherent characteristics. The transition sequences were fixed as (I) AfBfCfA; (II) A1fB1fC1fA1; (III) 7409



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A2fB2fC2fD2fA2. The characteristics of all these points are listed in Table 1. Figure 9 shows the responses of reactor and cooling fluid temperatures to set point changes, following the sequence AfBfCfA. All three points are in the minimum phase behavior subregion, so the inverse response does not occur. The process can track the set-point quickly. The simulation results show that both the reactor and cooling fluid temperatures can be stabilized in the desired reference without difficulty. Figure 9 illustrates the responses of the two temperatures to changes in set-point, under the second operating policy. As shown in Figures 5 and 6, under the second operating policy, the system exhibits a more complex nonlinear behavior. Regulating the system with operating policy II will be more difficult than with operating policy I. Figure 10 shows the responses to the set-point changes in the subregion where F > 0.6. Table 1. Characteristics of Selected Operating Points S-MP

US-MP

Operating Policy I

A

B, C

Operating Policy II

A1, C2

B1, C1

S-NMP

US-NMP

D2

A2, B2

When F < 0.6, the stability and the phase behavior vary with monomer volumetric flow rate, F. A minor change in F could change the characteristics of the operating point. The set-point tracking performance, following the sequence A2fB2f C2fD2fA2, is depicted in Figure 11. It is noteworthy that both points A2 and B2 are in the unstable nonminimum phase behavior subregion. Due to unfavorable characteristics and input multiplicity, Figure 11 shows that a sudden destabilization in reactor temperature occurs at these two points. The conventional model predictive controller cannot stabilize the reactor temperature to the desired reference. A sudden destabilization also occurs at points C2 and D2. If the process is operated under F < 0.6, an advanced and more efficient controller needs be implemented to ensure a stable and safe running system. The cooling fluid temperature can also track the set-point changes. There were significant oscillations, especially in the transient within the unstable nonminimum phase behavior subregion. Figure 11 indicates that the inverse response in reactor temperature cannot be overcome using a conventional model predictive controller, due to the unstable zero dynamics at point D2, where the system exhibits nonminimum phase behavior. Mathematical analysis predicted that, in the stable nonminimum phase behavior operating region, the system is subject to an

Figure 9. Set point temperature tracking with sequence I.

Figure 10. Set point temperature tracking under sequence II. 7410



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Figure 11. Set point temperature tracking under sequence III.

Figure 12. Temperature trajectories for a (5% disturbance in inlet temperature under operating policy I.

inverse response and the simulations are in agreement with the theoretical results. In the unstable nonminimum phase behavior subregion, an oscillation and a sudden destabilization would occur. Ma explained the relationship between the input multiplicity and the sudden destabilization during operation of the polymerization reaction process.20 The above dynamic simulations reinforce the above theoretical theory analysis. It is known that these characteristics are inherent properties of the process itself and they should be removed by modifying the process design to improve the overall control performance. 3.3.2. Disturbances Rejection. In the above subsection, the influence of manipulated variable selection on set-point tracking was studied. In this subsection, the effect of manipulated variable selection on disturbances rejection is discussed. An increase in inlet feed temperature causes an increase in cooling fluid flow rate, to maintain the processes initial condition. The reactor temperature will increase due to the increase in feed temperature, so more cooling fluid will be required.22 Hence, an increase in the inlet temperature will cause significant control problems. At points A and A1, the effects of a ( 5% disturbance in inlet feed temperature, Tin, on the dynamic behavior of the process, under the two operating policies, are illustrated as follows. Figures 12 and 13 show the simulated regulatory performances, under the two operating policies, with both positive

and negative changes in feed inlet temperature. When faced with both positive and negative steps, the cooling fluid temperatures could return to the original set points under both operating policies. However under operating policy II, the reactor temperature reached new operating points. Under operating policy I, the response could eliminate the disturbance in feed inlet temperature more efficiently than that achievable with operating policy II. In short, operating policy I can counter disturbances more efficiently than policy II. It is concluded that different operating policies impose different influences on the inherent and dynamic characteristics. These results are in agreement with those from the mathematical model analysis. 3.3.3. Remarks. The closed-loop dynamic simulations, including the reference tracking and disturbance rejections, have been established to explore the dynamic behavior of the system under two different operating policies. The results suggest that operating policy I is superior to II in terms of their ability to track and reject disturbances. At the conceptual design stage, an effort should be made to avoid any undesirable inherent properties. This can be done by adjusting the design/operating parameters or by reselecting the manipulated variables to improve both process set-point tracking and disturbance rejection performance. 7411



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Figure 13. Temperature trajectories for a (5% disturbance in inlet temperature under operating policy II.

3.4. Summary. In this section, the effect of the manipulated variable selection on process controllability, over the entire operating region, has been investigated using a polymerization reaction example process. First, the inherent characteristics, including phase behavior and open-loop stability, are analyzed using the process model. Second, theoretical predictions are validated via closed-loop dynamic simulations. The results demonstrate that the proposed method can be applied to analyze the relationships between manipulated variable selections and process controllability, over the whole operating region. The proposed method can guide process control structure selection. The case study involved a specific polymerization reaction in a CSTR, but the lessons learned can be applied to a variety of polymerization reactions in which complex nonlinearity can frequently arise, in both industrial practice40 and large-scale complex process networks.

4. CONCLUSIONS This paper is devoted to probing the influence of manipulated variable selection on process controllability, via open-loop controllability analysis and closed-loop dynamic simulation. Since both industrial processes and large process networks involve many alternative manipulated variables, understanding the relationship between these variable selections and the open-loop controllability is crucial for controller design and tuning. In summary: 1. For nonlinear processes, their nonlinearity depends on the selection and pairing of the manipulated variables and the controlled variables. Taking the MMA polymerization reactor as an example, the process exhibited complex nonlinear behavior under operating policy II, but may show less complex nonlinear behavior under operating policy I. 2. For a nonlinear process, the inherent characteristics, such as stability and phase behavior, may change with changes in the manipulated variable. With a certain manipulated variable value, the process is open-loop unstable with nonminimum phase behavior. Even minor disturbances may cause a runaway, so an efficient and advanced controller is required to keep the process running safety. It is critical to identify the inherent characteristics over whole operating region. 3. Although singular value analysis allows a systematic investigation into the effect of the manipulated variable selection on the nonminimum phase characteristics of the process,41 they will not reveal their influences on open-loop stability

over the entire feasible operating region. Using the framework presented in this paper, the effect of the manipulated variables on both phase behavior and open-loop stability can be visualized simultaneously. 4. As discussed in our previous work,32 when integrating the open-loop stability and phase behavior analysis in a process design, the whole process operating region can be divided into S-MP subregions: the S-NMP subregion, the US-MP subregion, and the US-NMP subregion. Within the open-loop stable and minimum phase behavior subregions, the system will run smoothly. In the US-NMP subregion, the system is difficult to stabilize and a runaway may occur, even with very minor disturbances. This is both dangerous and a significant challenge for control system design. The closed-loop dynamic simulations, discussed throughout this study, validate the above theoretical analysis. It is therefore important to avoid the process running in the US-NMP subregion at the conceptual design stage, by either adjusting the design/ operating parameters or reselecting the manipulated variables. 5. Input multiplicity that can cause a sudden destabilization problem in the control system should be removed at the design stage. 6. The proposed methodology was applied in an example, to demonstrate how to clarify and deal with the causes of the complex phenomena that can arise in the design, operation, and control of a chemical process. The results listed above are obtained from the current design conditions of a classical polymerization reaction process. Should these design parameters be changed, the problems must be reassessed. The presented systematic framework for analyzing the relationship between controllability and manipulated variable selection, over the whole operating region, would provide useful guidelines for designing an appropriate control system and any potential design alternatives. Future work will focus on extending this methodology to other complex reaction systems and largescale process networks.

’ AUTHOR INFORMATION Corresponding Author

*Tel: þ86 10 62781499. Fax: þ86 10 62770304. E-mail: [email protected]. 7412



’ ACKNOWLEDGMENT We gratefully acknowledge financial support from the National Natural Science Foundation of China (Grant 20776010) and the Doctoral Fund of Ministry of Education of China (Grant 20100002110021). ’ REFERENCES (1) Biegler, L. T.; Zavala, V. M. Large-scale nonlinear programming using IPOPT: An integrating framework for enterprise-wide dynamic optimization. Comput. Chem. Eng. 2009, 33, 575–582. (2) Perkins, J. D. The integration of design and control the key to future processing systems? In Proceedings of 6th World Congress of Chemical Engineering, Melbourne, 2001. (3) Seferlis, P.; Georgiadis, M. C. The Integration of Process Design and Control; Elsevier: Amsterdam, 2004. (4) Yuan, Z. H.; Chen, B. Z.; Zhao, J. S. Phase behavior analysis for industrial polymerization reactors. AIChE J. 2011In press. (5) Luyben, M. L.; Tyreus, B. D. Plantwide control design procedure. AIChE J. 1997, 43, 3161–3174. (6) Huang, R.; Biegler, L. T.; Patwardhan, S. C. Fast offset-free nonlinear model predictive control based on moving estimation. Ind. Eng. Chem. Res. 2010, 49, 7882–7890. (7) Maeder, U.; Borrelli, F.; Morari, M. Linear offset-free model predictive control. Automatica 2009, 45, 2214–2222. (8) Marusak, P. M.; Kuntanapreeda, S. Constrained model predictive force control of an electrohydraulic actuator. Control Eng. Prac. 2011, 19, 62–73. (9) Li, P.; Wendt, M.; Wozny, G. A probabilistically constrained model predictive controller. Automatica 2002, 38, 1171–1176. (10) Causa, J.; Karer, G.; Zupancic, B. Hybrid fuzzy predictive control based on genetic algorithms for the temperature control of a batch reactor. Comput. Chem. Eng. 2008, 32, 3254–3263. (11) Nascimento Lima, N. M.; Manenti, F.; Wolf Maciel, M. R. Fuzzy model-based predictive hybrid control of polymerization processes. Ind. Eng. Chem. Res. 2009, 48, 8542–8550. (12) Xu, S. C.; Bao, J. Control of chemical processes via output feedback controller networks. Ind. Eng. Chem. Res. 2010, 49, 7421–7445. (13) Lu, J. F.; Christofides, P. D. Distributed model predictive control of nonlinear systems subject to asynchronous and delayed measurements. Automatica 2010, 46, 52–61. (14) Garriga, J. L.; Soroush, M. Model Predictive Control Tuning Methods: A Review. Ind. Eng. Chem. Res. 2010, 49, 3505–3515. (15) Morari, M. Design of resilient processing plants. III. A general framework for the assessment of dynamic resilience. Chem. Eng. Sci. 1983, 38, 1881–1991. (16) Skogestad, S. Plantwide control: The search for the selfoptimizing control structure. J. Process Control 2000, 10, 487–507. (17) Skogestad, S. Control structure design for complete chemical plants. Comput. Chem. Eng. 2004, 28, 219–234. (18) Stephanopoulos, G.; Ng, C. Perspectives on the synthesis of plant-wide control structures. J. Process Control 2000, 10, 97–111. (19) Luyben, W. L.; Tyreus, B. D.; Luyben, M. L. Plantwide Process Control; McGraw-Hill: New York, 1999. (20) Seider, W. D.; Seader, J. D.; Lewin, D. R. Process Design Principles: Synthesis, Analysis, and Evaluation, 3rd ed.; John Wiley and Sons: New York, 2009. (21) Meel, A.; Seider, W. D. Game Theoretic Approach to Multiobjective Designs: Focus on Inherent Safety. AIChE J. 2006, 52, 228–246. (22) Ma, K. Process design for controllability of nonlinear systems with multiplicity. Ph.D Thesis, University College London: London, 2002. (23) Skogestad, S.; Postlethwaite, I. Multivariable Feedback Control; John Wiley & Sons: New York, 1996. (24) Yuan, Z. H.; Chen, B. Z.; Zhao, J. S. An overview on controllability analysis of chemical processes. AIChE J. 2011, 57, 1185–1201. (25) Kravaris, C.; Kantor, J. C. Geometric methods for nonlinear process control.1 Background. Ind. Eng. Chem. Res. 1990, 29, 2295–2310.

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Effect of Manipulated Variables Selection on the Controllability of

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