Genetically Tuned Decentralized Proportional-Integral Controllers for

Ind. Eng. Chem. Res. 2010, 49, 1297–1311

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Genetically Tuned Decentralized Proportional-Integral Controllers for Composition Control of Reactive Distillation C. Sumana and Ch. Venkateswarlu* Process Dynamics and Control Group, Chemical Engineering Sciences, Indian Institute of Chemical Technology, Hyderabad-500 007, India

This paper presents a genetic algorithm (GA) based autotuning method to design a decentralized proportionalintegral (PI) control system for composition control of a highly interactive and nonlinear reactive distillation column. The control relevant characteristics such as nonlinearities, interactions, and stability are analyzed for assessing the complexity of the process. The objective of GA tuning is to account the multivariable interactions and nonlinear dynamics of the process to find a unique set of parameters for the control system that is robust to all kinds of disturbances. The performance function in GA is formulated by incorporating the dynamic state information of the process derived from its model for various closed-loop disturbance conditions. The controller tuning problem of this multivariable process is resolved as an optimization problem and multiloop PI controllers are designed by exploiting the powerful global search features of GA. An estimator is designed to provide the compositions which serve as inferential measurements to the controllers. The performance of the proposed GA-tuned decentralized control scheme is evaluated by applying it to a metathesis reactive distillation column, and the results are compared with conventionally tuned PI controllers. The results demonstrate the better regulatory and servo performance of the GA-tuned PI controllers for composition control of reactive distillation column. 1. Introduction Reactive distillation, the process of performing chemical reaction and multistage distillation simultaneously has emerged as a favorable alternative to conventional reactor-separator configurations. This combined operation is especially useful for the chemical reactions in which chemical equilibrium limits the conversion and can be advantageously used for those reactions that occur at temperatures and pressures suitable to the distillation of components. Reactive distillation significantly enhances the overall conversion of certain equilibrium reactions. These advantages of reactive distillation over conventional configurations of reactors followed by separators have motivated a renewed interest in the use of reactive distillation technology for the production of important chemicals. However, in reactive distillation, the interaction between the simultaneous reaction and distillation introduces a much more complex behavior compared to conventional processes and leads to challenging problems in design, optimization, and control. In distillation and reactive distillation, temperature control is widely used in place of composition control because temperatures can be readily measured and directly incorporated into the control scheme. Temperature controllers can satisfactorily maintain the product compositions in binary columns as well as certain multicomponent columns where nonlinearities and interactions are not severe. This is not always satisfactory for many multicomponent distillation and reactive distillation processes, as stage temperatures do not correspond exactly to the product compositions. Reactive distillation poses additional difficulties owing to the dynamic interaction of reaction kinetics, mass transfer, and thermodynamics, and these interactions often cause counteracting influences on the process. Such an interactive and nonlinear reactive separation process can be better controlled by using composition controllers. However, composition control based on direct composition measurement is difficult * To whom correspondence should be addressed. Tel.: +91-4027193121. Fax: +91-40-27193626. E-mail: [email protected].

because online composition analyzers suffer from measurement delays as well as high investment and maintenance costs. Thus, inferential estimation of compositions as well as estimator based control has become highly relevant for reactive distillation. The proportional-integral (PI) and proportional-integralderivative (PID) controllers are extensively used in many industrial control systems despite significant advancement in modern control theory. Decentralized (multiloop) PI/PID controllers are used in many multi-input multi-output (MIMO) processes, because their structure is simple, the design is fast, and the principle is easier to understand. These PI/PID controllers are found to provide satisfactory performance for many MIMO systems.1–5 Multiloop PI/PID controllers with different tuning techniques have also been reported for composition as well as temperature control of distillation.6–8 The tuning techniques employed with these controllers include the BLT tuning,9,10 the relay feedback tuning,11–13 the independent IMC tuning,14 and the iterative continuous cyclic tuning.15 Most of these tuning techniques rely on transfer function models to obtain the response parameters which are then substituted in the Ziegler-Nichols-type tuning rules to determine the controller parameters. However, when the complexity of a multivariable process increases, its nonlinear dynamics changes to a great extent, thus making the conventional controller tuning extremely difficult. The true dynamics of the processes that exhibit severe nonlinearities and loop interactions cannot be realized in terms of transfer function models, and the tuning procedures that rely on such models can lead to improper design of the controller. The detuning of controllers while taking into account the nonlinearities and interactions for all disturbance conditions in complex processes like reactive distillation is not easy to achieve. Conventional PI/PID controllers have also been reported for reactive distillation processes for direct composition control16–18 as well as estimator-based composition control.19 However much emphasis is not laid on tuning studies of these controllers. Recently, PI controller tuning methods based on relay feedback and step tests have been studied for reactive

10.1021/ie9008474  2010 American Chemical Society Published on Web 12/14/2009

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Ind. Eng. Chem. Res., Vol. 49, No. 3, 2010

distillation,20 and it was found that relay feedback tests provide better results. However, these methods consider single loop tuning rules and require the process to exhibit sustained oscillations to determine the ultimate gain and period. In these methods, the detuning of controllers is performed either to preserve the loop stability or to meet some performance specifications. Therefore, despite the abundance of PI and PID tuning techniques, the development of new methods continues unabated. The design of stable and robust PI/PID controllers for composition control of highly nonlinear and interactive processes like reactive distillation that operate in a wide range of operating conditions is a major concern. In recent years, the stochastic search methods such as genetic algorithm (GA) have been receiving considerable atttention because of their ability to achieve high efficiency and to locate a global optimum in problem space. Because of its high potential for global optimization, GA is recognized as a powerful tool in many control-oriented applications such as parameter identification and control system design.21,22 As far as distillation is concerned, GA is used to derive a feedback feed-forward control system23 for a distillation column, where linearized transfer function model of the column has been used to design a temperature control system. The objective of this study is to present a simple and effective composition control scheme for highly interactive and nonlinear processes like reactive distillation. Since controller tuning plays a significant role in improving the performance of such processes, an efficient tuning method based on GA is presented to design a decentralized PI control system for a reactive distillation process. The control relevant characteristics such as nonlinearities, interactions, and stability are analyzed for assessing the complexity of the process. The controllers tuning is performed off-line based on the model of the process. The objective of GA tuning is to account the multivariable interactions and nonlinear dynamics of the process to find a unique set of parameters for the control system that is robust to all kinds of disturbances. The performance function in GA is formulated by incorporating the dynamic state information of the process derived from its model for various closed-loop disturbance conditions. The controller tuning problem is resolved as an optimization problem and multiloop PI controllers are designed by exploiting the powerful global search features of GA. An estimator is designed to provide the compositions which serve as inferential measurements to the controllers. The performance of the proposed GA-tuned decentralized control scheme is evaluated by applying it for the composition control of a metathesis reactive distillation column, and the results are compared with conventionally tuned PI controllers. 2. The Process and Its Characteristics The process considered is an olefin metathesis which finds wide applications in petrochemical industry. In this process, the olefin is converted into lower and higher molecular weight olefin products. When a reactive distillation column is designed for simultaneous reaction and separation of top and bottom products with high purities, the column typically exhibits very strong interactions and nonlinear behavior. Such high purity reactive separations impose constraints on the column operation which makes the controller design crucial. Therefore, before considering the design of an inferential control scheme, it is important to explore the characteristics of the process operation in terms

Figure 1. Inferential control scheme for reactive distillation.

of its nonlinearities, interactions, and stability. The schematic of the estimator supported composition control scheme is shown Figure 1. 2.1. Process Description. In olefin metathesis reaction, 2 mol of 2-pentene reacts to form 1 mol each of 2-butene and 3-hexene, 2C5H10 S C4H8 + C6H12 pentene butene hexene

(1)

The normal boiling points of components in this reaction allow an easy separation between the reactant 2-pentene (310 K), top product 2-butene (277 K), and bottom product 3-hexene (340 K). The steady state design aspects of this metathesis reactive distillation has been studied by Okasinski and Doherty.24 The reaction kinetics for metathesis reaction are given by the following equations:

(

kf ) 3553.6 exp Keq ) 0.25

)

6.6 (kcal/(g · mol)) min-1 RT

(2)

R3 ) 0.5kf[x12 - (x2x3 /Keq)] where, R3 is the rate of formation of hexene, x1, x2, and x3 are the mole fractions of pentene, butene, and hexene, respectively. The dynamic model representing the process involves mass and component balance equations with reaction terms, and algebraic energy equations supported by vapor-liquid equilibrium and physical properties. The assumptions made in the formulation of the model include adiabatic column operation, negligible heat of reaction, negligible vapor holdup, liquid phase reaction, and physical equilibrium in streams leaving each stage. The assumption of negligible energy accumulation on the plates changes differential energy balance equations to the algebraic equations. More details of the model can be referred from Alejski and Duprat.25 These model equations are also given in appendix A. Antoine coefficients for all the components of the system are obtained from Boublik and Elesevier.26 Tray hydraulic computations involve Francis weir formula. Enthalpy calculations involve empirical relations for heat capacities, which are the functions of temperature and pressure. The coefficients of heat capacity equations for gases are referred from Perry27 and for liquids are taken from Yaws.28 The column specifications and steady state details are given in Table 1. The rigorous mathematical model of the reactive distillation is solved by using Euler’s method with a step size of 0.001 h.

Ind. Eng. Chem. Res., Vol. 49, No. 3, 2010 Table 1. Column Details and Steady State Operating Conditions operating pressure operating temperature (°K) reflux drum reboiler feed total trays feed tray condenser reboiler distillate composition pentene butene hexene bottom product composition pentene butene hexene feed composition flow rates (Kmol/h) feed feed bottom product reflux vapor boil-up hold ups (Kmol) reboiler reflux drum reboiler heat duty (kcal/hour)

1 atm 281.1 333.5 308.2 12 7 total condenser partial reboiler (mole fraction) 0.064854 0.934597 0.000583 (Mole fraction) 0.063969 0.000484 0.935512 liquid pentene with unit mole fraction 100 50 50 200 225

to be quite different and also exhibit large variations for different step changes in input conditions. The gains have shown an increasing trend for the increase of step disturbance in positive direction and a decreasing trend for the increase of step disturbance in negative direction. A large variation in gains is observed with respect to the magnitude and direction of disturbance and the variation appeared to be more in the bottom loop. The higher magnitude of gains for positive step disturbances compared to those of negative disturbances indicates the column sensitivity toward increase in input conditions. These studies thus signify the presence of strong nonlinearities in the metathesis reactive distillation column. 2.3. Interaction and Stability Aspects. To design multiloop controllers for a multivariable nonlinear reactive distillation, it is important to examine the interaction effects in the column. Relative gain analysis29 (RGA) provides a measure for process interactions and suggests the most effective pairing of controlled and manipulated variables. The steady state gains of the top and bottom responses are used to evaluate the RGA of the column. For a system with two loops, the steady state gain matrix, K is given by

24.8 3.1 6450000

2.2. Nonlinearity Analysis. To design an efficient inferential control scheme for an interactive and nonlinear reactive distillation column, it is important to understand the dynamic characteristics of the process. The total degrees of freedom available with the column are reboiler heat load, reflux flow rate, and top and bottom product flow rates. The top and bottom product flow rates are manipulated to maintain the levels in reflux drum and reboiler. The reflux flow rate and reboiler heat load can be used to control the top and bottom product compositions. The multifunctional nature of reactive distillation causes strong nonlinearities, which must be assessed and analyzed with respect to changes in the operating conditions. The nonlinearity of the metathesis reactive distillation can be evaluated by examining the open loop responses of the top and bottom product compositions toward the changes in varying magnitudes of input conditions. Figure 2a and Figure 2b show the response behavior of compositions for input disturbances of different step magnitudes in both positive and negative directions. These results show that increase of reflux flow rate increases the response of top product composition, but decreases the bottom product composition response. Similarly the increase of reboiler heat load increases the bottom product composition response, but decreases the response of top product composition. The wide disproportionalilty in the steady state deviations of the responses in both directions for the proportional changes in input conditions indicate the violation of the principle of superposition thus depicting the nonlinear behavior of the system. From these results, it is observed that the magnitude of steady state response deviation from 1% to 3% in R are found to be considerably different to that of 3% to 5%. Similar trend is observed for the case of changes in reboiler heat load. The violation of the principle of superposition and the highly unsymmetrical behavior of the system as exhibited in Figure 2a and Figure 2b reveals the nonlinearity of the metathesis reactive distillation. The steady state gains of top and bottom product responses toward positive and negative step changes in inputs are also plotted as shown in Figure 2c and Figure 2d. These steady state gains of top and bottom product responses for the same magnitude of positive and negative step disturbances are found

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K)

[

K11 K12 K21 K22

]

(3)

where, K11 )

( ) ∂xD ∂R

Qr

;

K12 )

( )

( )

∂xD ∂xB ; K21 ) ∂Qr R ∂R

Qr

;

K22 )

( ) ∂xB ∂Qr

R

The RGA is defined as

[

λ 1-λ 1-λ λ

Λ)

]

(4)

where λ)

1 K12K21 1K21K22

In this work, the top product composition is controlled by manipulating reflux flow rate and the bottom product composition is controlled by reboiler heat load. The RGA results of metathesis reactive distillation column evaluated for (5% step changes in Qr and R are given by Λ)

[

7.089 -6.089 -6.089 7.089

]

(5)

The higher the value of λ from unity is, the more severe are the loop interactions. The RGA matrix evaluated above signifies that the reactive distillation column is highly interactive. The stability of the control structure is evaluated by using the Niederlinski Index29 (NI), which is given by NI )

|K|

(6)

N

∏K

jj

j)1

where, N is the number of control loops in the system, and |K| is the determinant of steady state gain matrix K and Kjj is its diagonal element. For a control structure to be stable, it is necessary that the index should be positive. For the 2 × 2 control of metathesis reactive distillation column, the NI is found to be

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Figure 2. Nonlinearity analysis: dynamic responses (a,b); steady state gains (c,d).

1.8589. This fulfills the stability condition for the control structure of metathesis reactive distillation column. 3. Controller Design Using Genetic Algorithms The aim is to present a simple and effective composition control scheme for a nonlinear reactive distillation column involved in the production of high purity components. Since controller tuning has significant influence in specifying the performance of such an interactive and multivariable process, in this work, an automatic and robust tuning method based on genetic algorithms is presented for designing the PI controllers. To find a unique set of parameters that satisfy the performance of the controllers for various disturbance conditions, the tuning problem is resolved as an optimization problem by formulating an objective function and evaluating the parameters of the controllers by minimizing the function through GA search procedure. The highly nonlinear and multimodal nature of the objective function of the process with lack of derivative motivates the use of GA to solve this optimization problem. 3.1. Genetic Algorithms. Genetic algorithms (GAs) are stochastic search algorithms that are based on the mechanics of natural selection and genetics.31 GAs can effectively handle nonconvex and nonlinear optimization problems and have high potential for finding the global optimum. A GA aims at evolving an optimal solution by a randomized information exchange through a sequence of probabilistic transformations governed by a selection scheme biased toward high quality solutions. Since GAs use a population of potential solutions and the probabilistic transition rules to create new solutions, they usually have higher probability of finding the global optimum.30 Many

applications of GAs have been reported to solve different optimization problems.31,32 GAs encode the candidate solutions of an optimization algorithm as a string of characters which are usually binary digits. The string is called a chromosome and the variables that are coded into a chromosome form substrings. The length of the string is usually determined according to the solution accuracy. The genetic algorithm considers a number of random strings for the variables, which together form a population. GA modifies and updates the population iteratively, searching for good solutions of the optimization problem. Each iteration step is called a generation. GA begins with a population of random strings representing the variables and then evaluates the strings fitness by using a fitness function, F(x) defined by F(x) )

1 1 + f(x)

(7)

where f(x) is an objective function. The specification of the fitness function is an important aspect of GA search, because the solution of the optimization problem and the performance of the algorithm depend on this function. Inspired by the “survival of the fittest” idea, the GA maximizes/minimizes the fitness value. The fitness function evaluates the fitness of every individual in the population until the end condition such as the generation number or degree of convergence is satisfied. If the end condition is not satisfied, the next step will be population evolution using genetic operators. The algorithm starts with the generation of an initial population. This population contains individuals which represent initial estimates for the optimization problem. The fitness of every individual in the population is obtained through a cost function


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Figure 5. Performance evaluation for GA tuning for load disturbance.

Figure 3. Flowchart for GA based multiloop PI controller tuning.

multiple copies in the mating pool in a probabilistic manner. Crossover is applied after selection to produce new individuals by merging the chromosomes of two individuals (parents) to obtain two other individuals (children). Parent pairs are randomly chosen from the selected population and a crossover operator is used for merging. Mutation is the last operation in which a mutation operator is applied to perform bit-wise mutation with a specified mutation probability. In mutation, better strings are created by carrying out local search around the current solution and diversity in population is maintained. Selection, crossover, and mutation are repeated for a fixed number of generations. 3.2. Controller Tuning. The PI controllers for top and bottom product compositions of reactive distillation column are defined by R(t) ) R0 + k1D(xD(t) - xDset) + k2D

∫ (x (t) - x set) dτ ∫ (x (t) - x set) dτ

Qr(t) ) Qr0 + k1B(xB(t) - xBset) + k2B

τ

D

0

D

τ

0

B

B

(8)

Figure 4. Performance evaluation for GA tuning for xD set-point change.

evaluation and new individuals are generated for the next generation. The population evolution for the next generation is performed using genetic operators such as selection, crossover, and mutation. Selection chooses individuals from the previous population for reproduction according to their fitness values. Individuals with better fitness values survive to form a mating pool. The essential idea of these operators is to pickup the above average strings from the current population and insert their

where R(t) and Qr(t) are the manipulated reflux flow rate and reboiler heat loads with R0 and Qr0 as their initial steady state values. The tuning parameters k1D, k2D are the proportional and integral constants of the top loop, and k1B, k2B are the corresponding parameters of the bottom loop. Each candidate in the GA population represents a set of these four parameters, which are referred as genes. These genes form a chromosome that will self-evolve by reproduction, cross over and mutation operations sequentially to generate a new population with improved objective. The basic aim of GA is to identify a single set of optimal or near optimal controller settings that can yield good controller performance for any disturbance condition. 3.2.1. Formulation of Objective Function. The objective function for GA search is formulated such that it quantifies the

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controller performance by taking into account the nonlinear dynamics of the process operating under different disturbance modes. For PI controllers tuning problem, the x in the objective function f(x) in eq 7 represents the proportional and integral parameters of both top and bottom controllers. Thus, the f(x) in eq 7 is denoted as Jo, which is defined by Jo ) f(k1D, k2D, k1B, k2B)

(9)

For n number of process disturbance modes, Jo is evaluated as n

Jo )

∑J

(10)

i

i)1

where Ji is the performance criterion corresponding to the ith disturbance mode, which is the weighted sum of the individual loop performances as given by Ji ) wDJDi + wBJBi

(11)

Here JDi , JBi are the top and bottom loops objectives, and wD, wB are the weightages assigned to them. GA searches for the candidate solutions by exploring the search space stochastically at several points and evaluating Jo at each of these points. Each set of updated candidate solutions of GA population in the search process represents the parameters of the top and bottom controllers, which are implemented on the model of the process to evaluate Jo. The function Jo quantifies the deviation of the actual response with the desired response and is minimized in order to establish the optimal values of the controller parameters. The GA tuning thus accounts for the information concerning all kinds of process disturbances and leads to the design of a robust decentralized PI control scheme for the multivariable process. 3.2.2. Desired Response Specifications for GA Tuning. The desired response for each disturbance condition is specified on the basis of certain characteristics such as peak overshoot, undershoot, settling time, response time, and offset. The time bound limits for the desired response trajectories are set such that any response falling within these bounds can be considered as the desired response. The objective function in GA controller tuning problem can be explicitly expressed by means of the response deviations from the prespecified time-bound limits for each disturbance condition. The individual objectives in eq 11 are evaluated as tlim it

JDi ) f(k1D, k2D) )

∑ (max[(LL (t) - x (t)), 0] + D i

D

t)0

max[(xD(t) - ULDi (t)), 0])∆t tlim it

JBi ) f(k1B, k2B) )

∑ (max[(LL (t) - x (t)), 0] + B i

B

t)0

max[(xB(t) - ULBi (t)), 0])∆t LLDi ,

ULDi ,

LLBi ,

ULBi

(12)

The and are the user defined continuous functions specifying the lower and upper bounds for the top and bottom product compositions for ith disturbance case and their corresponding time limits are t1, t2, and t3. These limits are specified on the basis of studying the response characteristics of the respective disturbance condition. Because of the interactive nature of the column, any process disturbance will result in an opposite effect in both the responses. On the basis of the analysis of the responses resulting from the model of process, the time bound limits that specify the lower and upper boundaries for the responses of individual disturbance cases can be set as

UL(t) ) xset + l1 LL(t) ) xset - l2 ;t < t1 UL(t) ) xset + l3 ;t < t < t2 LL(t) ) xset - l4 1 UL(t) ) xset + l5 ;t < t < t3 LL(t) ) xset - l6 2 UL(t) ) xset + l7 ;t > t3 LL(t) ) xset - l7

(13)

The above equation gives the general representation of the time bound upper and lower limits for top and bottom loop responses. The magnitudes of l1-l7 and the time limits t1-t3 are chosen from the shape of the desired response curve. More specifically, l1-l4 depends on the allowable peak overshoot and undershoots, l5 and l6 signify the decay ratio, and l7 stands for the allowable offset value. The time limits t1, t2, and t3 depend on the rise time, response time, and settling time of the desired response. These limits are specific to individual response and vary for each disturbance case. The GA procedure for computing the optimal tuning parameters of PI controllers is illustrated as a flowchart in Figure 3. The tuning procedure is carried out off-line using the process model. This procedure takes into account of the dynamic state information of the process for different disturbance conditions and incorporates this information in the performance function, J0 for evaluating the tuning parameters. The dotted block in the flowchart of Figure 3 describes the objective function evaluation in GA search procedure. For each set of updated controller parameters, J0 evaluation requires the simulation of the closed loop system for all disturbance conditions. For instance, in run i corresponding to the ith disturbance case, the top and bottom responses are generated and the individual objectives Ji are evaluated. J0 is then obtained from the weighted summation of these individual objectives. For illustration, consider the response curves and time bound limits in Figures 4 and 5 that represent the cases of set point change in top product composition and load disturbance, respectively. The subfigures a and b in Figure 4 illustrate the set-point tracking of top loop response and the coupling rejection of bottom loop response, respectively. The shaded area formed by parts of the response curve that lie outside the specified bounds contributes to the objective function which has to be minimized by GA search procedure to determine the optimal parameters of the controllers. The shaded area shown in subfigures a and b of Figures 4 and 5 represents the top and bottom loop objectives JDi and JBi for the respective disturbance condition. These objectives provide the quantification for how far is the actual response from the desired response over the specified period of time. The parameters of the controllers are to be determined such that for any disturbance condition, the process output response should lie with in the bounds specified for desired response. Since the controller design accounts for the nonlinear dynamics of the process to establish a unique set of controller parameters under different disturbance conditions, the controller parameters are not specific to a particular disturbance and are valid for all types of disturbances. 4. Composition Estimation The purpose of online state estimation in reactive distillation is to obtain reliable estimates of compositions defining the column operation using the process knowledge including a dynamic model and readily available data from process mea-


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Figure 6. GA tuning results.

Figure 7. Comparison of actual and estimator results for open loop disturbances.

surement sensors, and to incorporate the estimated compositions in a control scheme designed for the process. In this study, an extended Kalman filter (EKF) is designed to obtain the instantaneous composition information from the temperature measurements of the reactive distillation column. The time varying model of the nonlinear process is represented by x˙(t) ) f(x(t), t) + w(t),

x(0) ) x0

(14)

where x(t) is n dimensional state vector, f is a nonlinear function of state x(t) and w(t) is an additive Gaussian noise with zero mean. The nonlinear measurement model with observation noise can be expressed as y(tk) ) h(x(tk)) + ν(tk)

(15)

where h is a nonlinear function of state x(tk). The expected values of noise covariance matrices for the initial state x(0), process

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x ) [xb,1 xb,2 x1,1 x1,2 · · · xNt,1 xNt,2 xd,1 xd,2]T

(17)

Composition estimator design based on a rigorous mathematical model of multicomponent reactive distillation is not suitable for realistic situations, since it is difficult to obtain the measured values of liquid and vapor flow rates and tray holdups with time. Moreover, implementation of a rigorous model-based composition estimator for high dimensional reactive distillation also requires more computational effort. Therefore, a simplified model that assumes constant tray and reflux drum holdups, and constant vapor and liquid flow rates is considered for designing the composition estimator. The elements fi.j of F matrix in eq B3 of the estimation algorithm are obtained by taking the partial derivatives of the component balance equations with respect to the state vector (Appendix C). The temperature model of a measurement tray is given by the equation

( ) bj

Tm ) aj - log

sat Pm,j Pm

- cj

(18)

Nc

∑P

sat m,jxm,j

1

where a, b, and c are coefficients in vapor liquid equilibrium relation. The elements of H matrix in eq B6 of the estimation algorithm are obtained by taking the partial derivatives of eq 16 with respect to the state vector (Appendix C). The initial state covariance matrix Po, Process noise covariance matrix Q, and measurement noise covariance matrix R are the design parameters in the estimation algorithm, which need to be selected appropriately. More details concerning the algorithm can be can be referred elsewhere.33 5. Analysis of Results

Figure 8. Closed loop responses of actual and estimated compositions for simultaneous set-point changes in top and bottom product compositions. Table 2. Controller Settings step test parameters Ku Pu KZN TZN detuning factor proportional integral

relay feedback test

top loop

bottom loop

top loop

bottom loop

1850 0.3833 832.50 0.3194 1.4 594.64 1329.79

1.0 × 10 0.1825 45.0 × 109 0.1521 300 150000000 3277778

1845.3 0.3881 830.37 0.3221 1.1 754.88 2113.88

5.47 × 109 0.1610 2.46 × 109 0.1336 20 123185000 46092500

11

noise w(t), and observation noise V(tk) are given by the following relations, P0 ) E[(x0 - x(0))(x0 - x(0))T] Q(t) ) E[w(t) wT(t)] R(tk) ) E[V(tk) VT(tk)]

(16)

where P0 is initial state covariance matrix, Q(t) is process noise covariance matrix and R(tk) is observation noise covariance matrix. The EKF estimation algorithm is given in Appendix B. The state vector for the metathesis reactive distillation column is given by

The design and implementation results of the proposed strategy are discussed as follows. 5.1. GA Tuning. The GA search procedure described in the earlier section is employed to optimize the tuning parameters of the top and bottom controllers, k1D, k2D, k1B, and k2B. The optimization involves binary coding for the controller tuning parameter vector with a string length of 48 bits comprising 4 substrings each of length 12 bits. The initial population size is kept as 40 and the generations considered are 200. On the basisof the steady state gains of the process, it has been observed that the proportional and integral parameters for the top loop are in the order of 102 and 103, respectively, and both these parameters for the bottom loop are in the order of 109. These parameters are appropriately scaled in the range of [0-10] in order to facilitate a uniform search space for GA implementation. The cross over and mutation probabilities are selected as 0.85 and 0.005, respectively. For every GA evaluation, the population string parameters are decoded and multiplied by their respective scaling factors and implemented on the process model to compute the objective function. The disturbance cases considered for objective function evaluation include steady state operation, set point changes of top and bottom loops, load disturbances such as increase and decrease in feed flow rate. The time bound limits for the desired top and bottom responses are realized by analyzing the response characteristics for each disturbance condition. On considering equal importance for both top and bottom products of metathesis reactive distillation column, the weightages wB and wD are assigned to be 0.5. Figure 6 depicts the convergence of the fitness function, objective function, and the scaled GA controller parameters with respect to the number of generations. The results of Figure 6 show that


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Figure 9. Comparison of controllers for top loop set-point change (a,b) and bottom loop set point change (c,d).

Figure 10. Comparison of controllers for load disturbances: 3% increase in FL (a,b); 3% decrease in FL (c,d); 5% of each xF2 and xF3 as impurities in feed (e,f).

the integral parameters of top and bottom loops have converged in less than 100 generations, whereas the convergence of the proportional parameters took more than 120 generations. The GA evaluated proportional and integral parameters of the top loop are 487.67 and 5431.01, and those of the bottom loop are 23833940 and 81587300. 5.2. Composition Estimation. The composition vector to be estimated in metathesis reactive distillation is of dimension [28 × 1]. The state estimator that is supported by the simplified

dynamic model of reactive distillation considers the temperature measurements as its inputs and provides the estimates of compositions at every time instant. The temperature measurements of reactive distillation column are obtained through bubble-point calculation procedure by solving the rigorous model of the process. To reflect a real situation, the temperature data of every sampling instant is corrupted with a random Gaussian noise of zero mean and a standard deviation of 0.2 °C. For the metathesis reactive distillation column considered

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Table 3. Comparison of Controllers for Various Disturbance Cases IAE s. no.

disturbance case

step test

relay feedback test

autotuned GA

1 2 3 4 5 6 7

2% increase in xD set point 2% increase in xB set point 3% decrease in FL 3% increase in FL 5% butene impurities in feed 5% hexene impurities in feed 5% butene + 5% hexene impurities in feed overall performance

27.38 58.80 17.78 31.38 17.53 13.69 31.60 198.17

17.46 46.53 11.05 19.89 11.05 9.01 19.90 134.9

10.32 18.08 6.66 11.06 6.23 6.05 11.37 69.78

in this work, the optimal temperature measurements that serve as inputs to the estimator are established by using an empirical observability gramian-based methodology that has been reported in recent literature by the authors of this study.34 On the basis of this methodology, the temperatures of trays 3 and 12 are found to be the best measurements, and these are used for composition estimation in metathesis reactive distillation column. The elements of the matrices Po, Q, and R involved in the estimator are heuristically selected, and these matrices with their respective diagonal elements as 0.0005, 0.0005, and 5.0 are found to provide effective estimator performance. The estimator is studied toward the effect of different disturbances, and the results in Figure 7 illustrate the state tracking efficiency

of the estimator for multiple step changes in R and Qr under open-loop condition. 5.3. Controller Results. The estimated compositions are used as inferential measurements to the GA tuned PI controllers. The composition of the top product, butene is controlled by manipulating the reflux flow rate, and the bottom product, hexene is controlled by manipulating the reboiler heat load. The liquid levels in reboiler and reflux drum are maintained by adjusting the top and bottom product flow rates using conventional PI controllers. The proportional and integral terms for both the level controllers are selected as -1250 and -10, respectively. Figure 8 compares the inferential measurement

Figure 11. Comparison of controllers for change in thermodynamic model parameters.


1307

Figure 12. Comparison of controllers for change in kinetic model parameters.

tracking efficiency of the GA tuned PI controllers with the actual compositions for simultaneous set point changes in both top and bottom product compositions. The composition control results of the GA tuned PI controllers are compared to those of the conventionally tuned controllers. The conventional tuning methods considered are based on step test and relay feedback test.20 In step test method, the ultimate gain (Ku) and the ultimate period (Pu) for each loop are obtained from the respective open-loop responses for step change in feed flow rate, where as, in relay feedback method, the values of Ku and Pu for each loop are obtained from the sustained oscillations caused by open-loop relay input disturbance. These values of Ku and Pu are then used to evaluate the Ziegler Nichol’s settings (Kzn, Tzn) for each control loop. These settings are further detuned by trial and error by analyzing the process responses in order to eliminate the loop interactions. The values of Ku, Pu, Kzn, and Tzn for both these methods along with the detuning factors and controllers parameters are given in Table 2. Figure 9 compares the servo performance of the GA tuned controllers with the conventionally tuned controllers for both top and bottom loops. The subplots a and b corresponding to xD setpoint change in Figure 9 show that the GA tuned PI controllers have exhibited faster convergence than the conventionally tuned controllers. The subplots c and d corresponding to xB set point change in Figure 9 indicate the inferior performance of the conventional controllers in tackling the multivariable nonlinear interactions, whereas the GA tuned controller has exhibited much better performance. Figure 10 compares the regulatory performance of the GA tuned and conventionally tuned controllers for both the top and bottom loops for different disturbance conditions such as step increase and step decrease in feed flow rate, and step

Table 4. Comparison of Controllers for Changes in Thermodynamic and Kinetic Model Parameters IAE uncertainty in parameter 1% increase in a 1% decrease in a 1% increase in b 1% decrease in b 1% increase in c 1% decrease in c +0.5% ramp change in activation energy -5% ramp change in frequency factor

step test

relay feedback test

autotuned GA

55.10 40.80 35.14 41.84 35.23 30.39 59.71

35.28 24.76 21.01 26.30 22.04 18.22 41.68

17.62 12.58 10.28 12.82 10.79 8.97 20.58

39.1

26.17

12.80

decrease in feed composition. In all these disturbance cases, the top loop results show the better performance of the GA tuned PI controller by means of less overshoot/undershoot and faster convergence compared to the conventional controllers. The bottom loop responses show that even though GA tuned controller has resulted small overshoot/ undershoot, it provides much faster convergence compared to the conventional controllers. From these results, it is observed that the controller tuned by relay feedback test has shown very slow convergence, whereas the controller tuned by step test resulted in a large offset. Table 3 provides the quantitative performance of the controllers expressed in terms of IAE for different disturbance conditions introduced at time instant 2 h up to a period of 10 h. These IAE results for different disturbance cases indicate the better performance of the GAtuned PI controllers. The robustness of the controllers is evaluated by considering uncertainties in kinetic and thermodynamic model parameters. The uncertainties considered in thermodynamic

1308


parameters include step increase and decrease of the Antoine model coefficients, whereas ramp changes are considered as uncertainties in kinetic model parameters. The responses shown in Figure 11 correspond to the changes in thermodynamic parameters. In Figure 11, the subfigures a and b representing the top and bottom product responses correspond to 1% increase in the Antoine coefficients, ai, the subfigures c and d correspond to 1% decrease in the coefficients, ai, the subfigures e and f correspond to 1% increase in the coefficients, bi, and the subfigures g and h correspond to 1% decrease in the coefficients, bi. Figure 12 shows the responses corresponding to the changes in kinetic parameters. In Figure 12, the subfigures a and b representing the top and bottom product responses correspond to the ramp change in activation energy shown in subfigure c, and the subfigures d and e correspond to the ramp change in frequency factor shown in subfigure f. In all these cases, GA tuned controller has exhibited better performance in terms of faster convergence with minimum overshoot/undershoot than the conventionally tuned controllers. The bottom loop responses of these subfigures show that the controller tuned by relay feedback test has resulted slow convergence, where as the controller tuned by step test has exhibited more offset. The quantification results given in Table 4 also indicate the better performance of the GA tuned controller in the presence of uncertainties in thermodynamic and kinetic model parameters. These results thus show the robustness of the GA tuned PI controller in the presence of uncertainties in model parameters. The reason for the inferior performance of the conventionally tuned controllers is justified by the fact that they are designed based on either a single step response data or a relay feedback test data, which may not account the entire region of nonlinear process operation. Moreover, the detuning factors involved with the conventional tuning methods may not address the severe process interactions among the control loops. The better performance of the GA-tuned PI controllers can be attributed to the reasoning that these controllers are designed by taking into account the dynamic process information concerning the nonlinearities and process interactions of both the loops for different disturbances. 6. Conclusions The controller tuning methods based on relay or step tests require that the process should exhibit sustained oscillations and involve tedious detuning procedures. The controller settings that are evaluated for a specific step or relay disturbance may not be valid for the entire region of nonlinear process operation. The GA tuning method eliminates these problems by taking into account the nonlinear dynamics and multivariable interactions of the process and provides a unique set of parameters for the multiloop PI control system that is robust to all kinds of disturbances. The results on application to a metathesis reactive distillation column demonstrate the better servo and regulatory performance of the GA tuned PI controllers over the conventionally tuned controllers. This strategy can be applied to other highly nonlinear and interactive processes. Appendix A: Mathematical Model of Reactive Distillation Column The model of the process is described by the following equations.

Total Mass Balance. Total condenser dMD ) VNt - (D + R) + ∆RD dt

(A1)

Feed plate nf dMnf ) FL + Vnf-1 + Lnf+1 - (Vnf + Lnf) + ∆Rnf dt

(A2) Other plate m dMm ) Vm-1 + Lm+1 - (Vm + Lm) + ∆Rm dt Reboiler dMB ) L1 - (VB + B) + ∆RB dt Component Balance. Total condenser d(MDxD,j) ) VNtxNt,j - (DxD,j + RxD,j) + ∆RD,j dt

(A3)

(A4)

(A5)

Feed plate nf d(Mnfxnf,j) ) FLxf,j + Vnf-1ynf-1,j + Lnf+1xnf+1,j dt (Vnfynf,j + Lnfxnf,j) + ∆Rnf,j (A6) Other plate m d(Mmxm,j) ) Vm-1ym-1,j + Lm+1xm+1,j - (Vmym,j + Lmxm,j) + dt ∆Rm,j (A7) Reboiler d(MBxB) ) L1x1,j - (VByB,j + BxB,j) + ∆RB,j dt Energy Balance. Condenser dED ) VNtHVNt - (DHlD + RHlD) + ∆HrD dt

(A8)

(A9)

Feed plate nf dEnf ) FLHl + Vnf-1HVnf-1 + Lnf+1Hlnf+1 dt (VnfHVnf + LnfHlnf) + ∆Hrnf (A10) Other plate m dEm ) Vm-1HVm-1 + Lm+1Hlm+1 - (VmHVm + LmHlm) + dt ∆Hrm (A11) Reboiler dEB ) L1Hl1 - (VBHVB + BHlB) + ∆HrB dt Tray hydraulics:

(A12)

Francis weir formula hl ) (hw + how) ) (MVm /Am)

(A13)

Flow of liquid over weir MVm ) ((hl - hw)/1.33)3/2lw

(A14)

Mole fraction normalization Nc

∑ 1

Nc

xi )

∑y

i

1

)1

(A15)


Antoine Equations.

1309

Appendix C: Elements of F and H Matrices

sat ) aj log Pm,j

bj cj + Tm

(A16)

sat ym,jPm ) xm,jPm,j

(A17)

Physical Properties. Hlm ) f(Pm, Tm, xm.j)

(A18)

HVm ) f(Pm, Tm, ym,j)

(A19)

Fliq ) f(P, T, yj)

(A20)

The temperature measurement model used for state estimation is given by the equation:

( ) bj

Tm ) aj - log

sat Pm,j Pm

- cj

(A21)

Nc

∑P

sat m,jxm,j

1

Appendix B: State Estimation Algorithm The extended Kalman filter (EKF) is computed in two steps. The first is a prediction step, which is used to extrapolate the previous best estimates, and the second is a correction step by which the updated estimates are formed. Since prediction is based on process model, continuous prediction and discrete correction is employed in the estimation scheme. Prediction Equations. By starting with an initial estimate xo and its covariance Po at time zero and no measurements are taken between tk-1 and tk, the propagating expression for the state estimate and its covariance from tk-1 to tk are xˆ˙(t/tk-1) ) f(xˆ(t/tk-1), t)

(B1)

P˙(t/tk-1) ) F(xˆ(t/tk-1), t)P(t/tk-1) + P(t/tk-1)F (xˆ(t/tk-1), t) + Q(t) (B2) T

where F(xˆ(t/tk-1),t) is the state transition matrix whose i,jth element is given by F(xˆ(t/tk-1), t) )

∂fi(x(t), t) ∂xj(t)

|

(B3) x(t))xˆ(t/tk-1)

The solution of the propagated estimate xˆ(t/tk-1) and its covariance P(t/tk-1) at time tk are denoted by xˆ(tk/tk-1) and P(tk/tk-1). By using measurements at time tk, the update estimate xˆ(tk/tk) and its covariance P(tk/tk) are computed. Correction Equations. The equations to obtain corrected estimates are xˆ(tk /tk) ) xˆ(tk /tk-1) + K(tk)[y(tk) - h(xˆ(tk /tk-1))]

(B4)

P(tk /tk) ) (I - K(tk) H(x(tk)))P(tk /tk-1)

(B5)

K(tk) ) P(tk /tk-1) HT(x(tk))(H(x(tk)) P(tk /tk-1) HT(x(tk)) + R)-1

(B6)

where ∂hi(x(tk)) H(x(tk)) ) ∂x(tk)

|

x(tk))xˆ(tk/tk-1)

The recursive initial conditions for state and covariance are xˆ(t/tk-1) ) xˆ(tk /tk) P(t/tk-1) ) P(tk /tk)

(B7)

The elements of F matrix in the estimation algorithm are obtained by taking the partial derivatives of the simplified process model equations with respect to the state vector. These elements are given by f1,1 f1,2 f1,3 f2,1 f2,2 f2,4

) ) ) ) ) )

(-VbEb,11 (-VbEb,12 (L1)/Mb (-VbEb,21 (-VbEb,22 (L1)/Mb

f3,1 f3,2 f3,3 f3,4 f3,5

) ) ) ) )

(VbEb,11)/M1 (VbEb,12)/M1 (V1E1,11 - L1 + R1,11)/M1 (V1E1,12 + R1,12)/M1 (L2)/M1

f4,1 f4,2 f4,3 f4,4 f4,6

) ) ) ) )

- B + Rb,11)/Mb + Rb,12)/Mb + Rb,21)/Mb - B + Rb,22)/Mb

(VbEb,21)/M1 (VbEb,22)/M1 (-V1E1,21 + R1,21)/M1 (-V1E1,22 - L1 + R1,22)/M1 (L2)/M1

(C1)

Similarly for i ) 5, 7, 9, ... 23 and j ) i - 2 the following set of equations holds good. fi,j ) (V(i-4)/2E(i-4)/2,11)/M(i-2)/2 fi,j+1 ) (V(i-4)/2E(i-4)/2,12)/M(i-2)/2 fi,j+2 ) (-V(i-2)/2E(i-2)/2,11 - L(i-2)/2 + R(i-2)/2,11)/M(i-2)/2 fi,j+3 ) (-V(i-2)/2E(i-2)/2,12 + R(i-2)/2,12)/M(i-2)/2 fi,j+4 ) (Li/2)/M(i-2)/2 (C2) And for i ) 6, 8, 10, ..., 24 and j ) i - 2, the following set of equations holds good. fi+1,j ) (V(i-4)/2E(i-4)/2,21)/M(i-2)/2 fi+1,j+1 ) (V(i-4)/2E(i-4)/2,22)/M(i-2)/2 fi+1,j+2 ) (-V(i-2)/2En,21 + R(i-2)/2,21)/M(i-2)/2 fi+1,j+3 ) (-V(i-2)/2E(i-2)/2,22 - L(i-2)/2 + R(i-2)/2,22)/M(i-2)/2 fi+1,j+5 ) (Li/2)/M(i-2)/2 f25,23 f25,24 f25,25 f25,26 f25,27

) ) ) ) )

(VNt-1ENt-1,11)/MNt (VNt-1ENt-1,12)/MNt (-VNtENt,11 - LNt + RNt,11)/MNt (-VNtENt,12 + RNt,12)/Mnt (R)/MNt

(C3)

f26,23 f26,24 f26,25 f26,26 f26,28

) ) ) ) )

(VNt-1ENt-1,21)/MNt (VNt-1ENt-1,22)/MNt (-VNtENt,21 + RNt,21)/MNt (-VNtENt,22 - LNt + RNt,22)/MNt (R)/MNt

(C4)

f27,25 f27,26 f27,27 f27,28

) ) ) )

(VNtENt,11)/Md (VNtENt,12)/Md (-D - R + Rd,11)/Md (Rd,12)/Md

(C5)

f28,25 f28,26 f28,27 f28,28

) ) ) )

(VNtENt,21)/Md (VNtENt,22)/Md (Rd,21)/Md (-R - D + Rd,22)/Md

(C6)

1310


The values of En,11, En,12, En,21, En,22 in the evaluation of the elements of Jacobian matrix F are given by the following equations. En,11 )

Nc ∂yn,1 sat sat sat sat ) (( Psat n,j xn,j)Pn,1 - Pn,1xn,1(Pn,1 - Pn,3))/ ∂xn,1 j)1

∑

Nc

∑P

((

sat 2 n,j xn,j) )

(C7)

j)1

En,12

sat sat ∂yn,1 -Psat n,1xn,1(Pn,2 - Pn,3) ) ) Nc ∂xn,2 2 ( Psat n,j xn,j)

En,21

sat sat ∂yn,2 -Psat n,2xn,2(Pn,1 - Pn,3) ) ) Nc ∂xn,1 2 ( Psat n,j xn,j)

∑ n)1

∑ j)1

Nc

En,22 )

∂yn,2 sat sat sat sat ) (( Psat n,j xn,j)Pn,2 - Pn,2xn,2(Pn,2 - Pn,3))/ ∂xn,2 j)1

∑

Nc

((

∑P

sat 2 n,j xn,j) )

(C8)

j)1

where yn,j )

Psat n,j xn,j

∑

(

and log Psat n.j ) aj -

Nc

Psat n,j xn,j

bj cj + T n

)

j)1

(C9) The reaction terms Rn,11, Rn,12, Rn,21, and Rn,22 are given by the following equations: Rn,11 ) -4(kfnMnxn,1);Rn,12 ) 8(kfnMnxn,3); Rn,11 Rn,12 Rn,21 ) ; Rn,22 ) (C10) 2 2 where kfn is given in eq 7. The elements of H in the estimation algorithm are obtained by taking the partial derivatives of the temperature measurement expression, eq 16 with respect to state vector. These elements are given by

( )

( )

-bj(Psi,1 - Psi,3)

hi,1 )

Nc

(

∑ Ps

{ ( )} { ( )}

i,jxi,j)

j)1

aj - log

Psi,jPi

2

;

Nc

∑ Ps

i,jxi,j

j)1

-bj(Psi,2 - Psi,3)

hi,2 )

Nc

(

∑ Ps

i,jxi,j)

aj - log

j)1

Psi,jPi

Nc

∑ Ps

i,jxi,j

j)1

Appendix Notation Am ) tray area m B ) bottom flow rate, mol h-1 D ) distillate flow rate, mol h-1 F ) linearized process matrix FL ) feed flow rate Kmoles h-1 fi,j ) elements of matrix F H ) linearized process output matrix

2

(C11)

Hl ) enthalpy of liquid, kcal Kmole-1 Hv ) enthalpy of vapor, kcal Kmole-1 ∆Hrm ) heat generated on tray m due to reaction, kcal Kmole-1 hl ) height of liquid on the tray hw ) weir height how ) height of liquid over weir hi,j ) elements of matrix H Kf ) forward reaction rate constant Keq ) equilibrium reaction rate constant K ) Kalman filter gain matrix Ku ) ultimate gain: Kmol h-1(top loop), kcal h-1(bottom loop) KZN ) controller gain: Kmol h-1(top loop), kcal h-1(bottom loop) K11 ) proportional constant of top loop, Kmol h-1 K12 ) integral constant of top loop, Kmol h-2 K21 ) proportional constant of bottom loop, kcal h-1 K22) integral constant of bottom loop, kcal h-2 lw ) length of weir MB ) holdup in the reboiler, mol MD ) reflux drum holdup, mol Mm ) molar liquid hold up on tray m, Kmol h-1 MVm ) volumetric flow rate of liquid over weir from tray m Nt ) total number of trays Pn ) total pressure on tray n sat Pm,j ) saturated vapor pressure of the component j on tray m Pu ) ultimate period, h Qr ) reboiler heat load, kcal h-1 R ) reflux flow rate, Kmol h-1 ∆Rm,j ) rate of change of component j due to reaction on tray m, Kmol h-1 Tm ) temperature on tray m, °K TZN ) time period, h V ) vapor boilup rate, Kmol h-1 x ) state vector xˆ ) estimated state xm,j ) mole fraction of liquid component j on tray m xf,j ) mole fraction of liquid component j on feed tray ym,j ) mole fraction of vapor component j on tray m

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ReceiVed for reView December 29, 2008 ReVised manuscript receiVed October 30, 2009 Accepted November 24, 2009 IE9008474

Genetically Tuned Decentralized Proportional-Integral Controllers for

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