Nonlinear model-based control of a binary distillation column

Computers them. Engng Vol. 19, Suppl., pp. S519-S524.1995 Copyright @ 1995 Elsevier Science Ltd 009th1354(95)00068-2 Printed in Great Britain. All rights reserved 0098-1354/95 $9.50 + 0.00

Pergamon

NONLINEAR MODEL-BASED CONTROL OF A BINARY DISTILLATION COLUMN

Antonio Trotta and Massimiliano Barolo Istituto di Impianti Chimici - Universita di Padova via Marzolo, 9 - I-35 131 Padova PD ITALY E-mail: [email protected]

ABSTRACT This paper concerns the control of both product purities of a binary distillation column. Refiux and boilup flow rates are chosen as manipulated variables, and the corresponding control laws are derived within the Globally Linearizing Control framework. Reduced order models of the column are developed that allow the controller synthesis to be carried out easily. Simulation results indicate that the proposed control strategy outperforms a conventional PI control strategy by a significant amount. KEYWORDS Distillation; nonlinear control; model-based control; globally linearizing control INTRODUCTION Control of the top and bottom compositions of a distillation column can be a difficult task due to the presence of control-loop interactions and nonlinearities. Consequently, there is an increasing demand in the area of control strategies that will give better control performance than can be obtained through the use of conventional PI or PID controllers. However, it is to be noted that industry is mainly interested in easy-to-handle supports for improving process operation. Nonlinear model-based controllers have been successfully applied for the dual composition control of distillation columns (Cott et nl., 1989; Pandit et al., 1990). In these works, a steady state process model was coupled with simplified first-order dynamics by using the Generic Model Control (Lee and Sullivan, 1988) framework. However, steady state models do not model the dynamic effects of the disturbance variables. Dynamic filters are often used in order to overcome this difficulty. These filters need to be tuned on line. Barolo et al. (1994) have used a simplified dynamic model in order to derive an on-line control law for the startup and the single composition control of a binary distillation column. They used the Globally Linearizing Control (GLC; Kravaris and Chung, 1987) structure. The present work is aimed to extend the strategy of Barolo et nl. (1994) to the multi-input/multi-output (MIMO) case and to compare the performance of the GLC controller to that of a conventional PI controller. A lumped parameter dymamic process model is selected for its balance of flexibility and implementation ease. The controlled variables are the inventories of the light and heavy components in the enriching and stripping sections of the column, respectively. The manipulated streams are the reflux and the boilup rates. Results are shown through simulation examples.

CONTROLLER SYNTHESIS AND IMPLEMENTATION Kravaris and Soroush (1990) have shown that, given a nonlinear MIMO system with equal number of inputs and outputs of the form i=f(x)+&j(x)ldj

,

y, =h,(x),

i=l,..., M

j=l s519

,

(1)

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the static state-feedback

u=

(*I

provides a decoupled linear closed-loop response. In the above eqns (1) and (2), u ES”‘, y E’P, x E!R”, mSn, f(x) is a smooth vector field on 91”, g,(x),...,g,,,(x) are smooth vector fields on %“, h,(x),...,&,(x) are smooth scalar fields on 9X”, and L:hi(x) is the k-th order Lie derivative of the function h(x) with respect to the vector function p. Details are given in the work of Kravaris and Soroush (1990), and will not be presented here. Reduced-order model of a distillation column Let us consider the binary distillation column represented in Fig. 1. With a lumped parameter model, the light component balance from the top tray (J/7’)to the NP-th tray (envelope I) gives

where M is the liquid hold-up on the tray, L and V are the liquid and vapor flow rates, and x and y are the mole fractions of the light component in the liquid and in the vapor phase, respectively. This model has been shown to result in a satisfactory control performance for the single-input/single-output (SISO) case (Barolo ef al., 1994).

Figure 1. Distillation column. The heavy component balance from the bottom of the column (tray I) up to the NPE-th tray (envelope II) gives (superscript ’ indicates the heavy component mole fractions) (4) According to the models of eqns (3) and (4) the controlled outputs are ci (MY), and ci(n?ix’), , while the manipulated inputs are Lo and V,. It is easy to see that the relative orders of each output is 1. Note that feed acts like an unmeasured disturbance in models (3) and (4).

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321

Develooment of the control laws By applying eqn (2), the required static state feedback results ll=

%= [I[

l/(P,,XD) 0

u2

v, -p,o~i(Jw,

I[

0

. v2 -pzoC,(MX’)i

l/@,,Y,‘)

-P,,[w9,-, -vY)AT -G&J=1 -p2,[(Lx’)WE+,

-(vY’)NP~

1

-(Lx’)Il *(‘I

If two external PI controllers of the form v, =

&(&I ++j+,de>,

v, =

&2k2

+-,j;$E24

Cj(MX)i,d-C,tMx)i

(6)

E2 = Ci(MX’)iSd -Cj(Mx’)j

(7)

El =

are used around the input/output linearized system (subscript “d” stands for “desired”), the following

control laws can be finally obtained L, = (V)ivr +(W,

-(vy)hW

+ K,,(s, +$jis,d’)-p,,C(Mx)i P IlXD

XD

v

0

+(W, - W),,, (vy’)NPE

=

+

&2(&Z +~~~E2d’)-P2oC(MX’)i

YB'

(9)

PZlYf3’

DISTILLATION COLUMN MODEL A binary distillation example is used to illustrate the performance of the proposed control law. The physical parameters, initial conditions and design parameter are given in Tab. 1. The control objective is to maintain the top and bottom compositions by changing the reflux and boilup rates. In the simulation, the following assumptions are made: equimolal overflows, negligible vapor hold-ups, saturated liquid feed and reflux, constant relative volatility, total condenser and partial reboiler, perfect top and bottom level control. Tray hydraulics is accounted for by the Francis weir formula. Table 1. Design parameters of the distillation column. total no. of trays, NT feed tray no. , NA feed composition, zF distillate comp’n (base case), XD bottom comp’n (base case), xl8 relative volatility reflux flow rate (base case), L/F

40 20 0.5 0.9975 0.9975 1.8 1.5

bottom loop: two first order lags

0.5-1.5 min

vapor boilup (base case), V/F tray hold-up (base case), MJF reflux drum hold-up, &IF bottom hold-up, M,jF tray hydraulic time constant tray eficiency

2 0.1 mid 0.4 min-* 1 min-’ 0.1 min 100%

A4ecmrernent logs

top loop: one first order lag

1.5 min

Note that measurement lags have been introduced in both top and bottom control loops. A Runge-KuttaFehlberg algorithm with automatic step size adjustment and numerical error control has been used to integrate the set of ordinary differential equations. State Estimation The practical implementation of control laws (8) and (9) requires the knowledge of unmeasurable states. Distillation columns are usually equipped with thermocouples, which allow the temperature profile along the column to be known at any instant of time; moreover, external streams are measured on-line, as well as bottom pressure, condenser duty, and top and bottom hold-ups. Based on these measurements, a procedure for estimating on-line the unmeasurable states (internal flow rates and

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compositions) has been given by Barolo et al. (1993), and it has been experimentally shown to lead to good results in the SISO case (Barolo et al., 1994). State estimation may result inaccurate due both to simplifications in the state observers, incorrect estimation of parameters, and measurements errors. However, any inaccuracies in the state estimation will eventually result in errors in the computation of the model-terms of eqns (8) and (9), thus introducing some degrees of process/model mismatch. It should be emphasized that uncertainty description is not required to be an accurate representation of the true uncertainty (which is impossible to be known exactly), but it only needs to be a useful proxy for this, In this work, in order to examine the robustness of the model-based control laws with respect to process/model mismatch, a series of tests was run assuming a global model mismatch in the range from 0% (nominal case) up to GO%. Moreover, in the controller models tray hold-up has always been assumed constant, and equal to its nominal steady state value. Finally, no way of including the top and bottom measurement lags in the controller models has been attempted, and this fact results in a major source of process/model mismatch. Selectine the Sensor Location for Dual-Ended Control Selecting the sensor locations to regulate both ends of the column requires balancing the problem of sensor interaction and loop sensitivity. The singular-value decomposition analysis described by Moore (1992) has been used, by setting the composition on the selected tray as the controlled variable for the PI algorithm. This analysis indicates to locate the sensor pairs for the PI controllers at trays 11 and 30 (numbering from the bottom) for the bottom and top composition control loop, respectively. These locations are also used for the GLC control algorithms, i.e. NPE = 1I and NP = 30. Controller Tuning A number of procedures for tuning multiloop SISO controllers in a multivariable environment have been reported in the literature. Nevertheless these methods satisfy the main objective of arriving at reasonable controller settings with only a moderate amount of engineering and computational effort. It is never claimed that one method will produce the best results or that some other tuning structure will not give superior performance. Therefore, in order to present a meaningful comparison between the performance of the PI controllers and the proposed GLC controllers, in this work an optimization procedure has been adopted to get the values of controller tuning parameters. In the case of the GLC controllers, according to the tuning guidelines outlined by Soroush and Kravaris (1992), it has been chosen pi0 = 1, and pi, = rli (i = 1,2). Thus only two parameters need to be tuned for each GLC control loop. The sum of time-weighted absolute errors (ITAE) of both controlled variables was assumed as the objective function. A constrained Rosembrock hillclimb procedure was used to minimize the ITAE sum as a function of the tuning parameters. The ITAE values for both the GLC and the conventional PI controllers were calculated for the first 80 min after a -10% feed composition load was applied. For the PI controller the following values were found: K,, = 50 kmollmin, rID = 5 min, KcB = 400 kmollmin, and rID = 6.8 min. Perhaps surprisingly, these values are close to those obtained with the method of Shen and Yu (1994), that is 56, 16,400, and 9, respectively.

RESULTS AND DISCUSSION The performance of the proposed control laws was evaluated by extensive numerical simulations. A series of runs was performed in order to test the GLC controllers and to compare them with the conventional PI controllers, in terms of rejection of disturbances, tracking of set-points, modelling errors. In all cases, GLC performed significantly better than PI. Detailed description of the results is omitted for brevity. Figures 2a and 26 show the performance of the GLC and the PI controllers for a -10% feed composition disturbance (no process/model mismatch). The GLC controllers clearly outperform the conventional PI controllers; in fact the PI responses show increased oscillations and a slow approach to the steady state.

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0.84

10.44 11 0

10

20

30

I

I



0

40

S523

I

10

time (min)

I

I

20

30

40

time (min)

Figure 2. Control responses of a) GLC and 6) PI control laws for a -10% disturbance on feed composition.

It is worth noting that the real control objectives, that are the top and bottom compositions, are better achieved by the GLC controllers (Fig.3); the GLC responses exhibit significantly smaller deviations from the nominal values of top and bottom compositions, and the amplitude of the oscillations is significantly smaller for the GLC controllers than for the PI controllers. 0.999

0.999

b) ;

0.998

f

0.997

8 E 0.996 0 f: 2 0.995

-

~~

x’(l)

GLC PI

2 (D 0.994 3 ”

0.993 0

10

20

30

time (min)

40

0

10

20

30

40

time (min)

Figure 3. Deviations of terminal compositions from the nominal values for two symmetrical disturbances on feed composition: a) -2O%,b) +20%.

Figure 4 presents typical trends of the normalized control responses for a step change in the top controlled variable, corresponding to a step change in the desired distillate composition from 0.9975 to 0.9993. The GLC controller does not exhibit decoupled first order response, as theoretically expected, even though no process/model mismatch is assumed. This is mainly due to the fact that the measurement lags are not taken into account by the controller models. Though this fact has a detrimental effect on performance, it is clear that the GLC controller provides a better response than the PI controller. Finally, the GLC controller performance was examined when significantly less accurate models were used in the control laws. It has been found that the differences from the nominal case are negligible until the global errors are within the range of HO%. When the process/model mismatch is large (Fig.5; +30% error in the model term of eqn (S), and -30% error in the model term of eqn (9)), the performance of the GLC controllers is somewhat degraded as compared to that shown in Fig.20. Nevertheless it is still better than that of the PI controllers. In these tests the following optimal tuning parameters were used for the GLC controllers: Kc, = 25, Kc, / T,, = 5 min-’ , Kc2 = 50, and Kc2 I T,~ = 3.4 min-’ .

CONCLUSIONS

A model-based approach to the dual-composition control of a binary distillation column has been presented. The inventories of the light and heavy components in the enriching and stripping sections of

S524

European Symposium on Computer Aided Process Engineering-5 1.20

10.56

1.00

s = = 10.52 9, s ‘c g

g g

0.60

!! =

0.60

2E

0.40

c

0.20

-

GLC PI

40

50

B 10.46 = e E 8 10.44

0.00 0

10

20

30

60

10

Figure 4. Normalized responses of the controlled variables for a step increase in the top controlled variable set-points.

20

30

40

time (min)

time (min)

Figure 5. GLC control responses for -10% disturbance on feed composition under modelling errors.

the column have been chosen as the controlled variables, and a (L;v) control configuration has been used. The synthesis of model-based controllers has been carried out by means of the GLC framework. Simulation results indicate that the proposed control structure outperforms a conventional PI control strategy by a significant amount. Though modelling inaccuracies may degrade the GLC control performance, the results are still satisfactory when compared to those obtained with the PI controllers. ACKNOWLEDGEhlENT -

Financial support granted to this work by the Italian National Research COUnCil

(CNR,

Progetto Strategic0 Tecnologie Chimiche Innovative) is gratefully acknowledged.

REFERENCES Barolo, M., G. B. Guarise, S. Rienzi and A. Trotta (1993). On-line Startup of a Distillation Column Using Generic Model Control, Comput. them. Engng, 17(S), 349-354. Barolo, M., G. B. Guarise, S. Rienzi and A. Trotta (1994). Nonlinear Model-Based Startup and Operation Control of a Distillation Column: An Experimental Study, Ind. Eng. Chem. Rex, 33, 3160-3167. Cott, B. J., R. G. Durham, P. L. Lee and G. R. Sullivan (1989). Process Model-Based Engineering, Comput. them. Engng, 13,973-984.

Kravaris, C. and C. Chung (1987). Nonlinear State Feedback Synthesis by Global Input/Output Linearization, AIChE Jl, 33,592-603. Kravaris, C. and M. Soroush (1990). Synthesis of Multivariable Nonlinear Controllers by Input/Output Linearization, AZChE Jl, 36,249-264. Lee, P. L. and G. R. Sullivan (1988). Generic Model Control (GMC), Comput. them. Engng, 12,573580.

Moore, C. F. (1992). Selection of Controlled and Manipulated Variables, In: Practical Distillation Control (W. L. Luyben, Ed.), Van Nostrand Reinhold, New York. Pandit, H. G., R. R. Rhinehart and J. B. Riggs (1990). Experimental Demonstration of Nonlinear Model-Based Control of a Nonideal Binary Distillation Column, Proc. 1992 American Control Conf.‘, Chicago, IL. Soroush, M. and C. Kravaris (1992). Nonlinear Control of a Batch Polymerization Reactor: An Experimental Study, AIChEJl, 38, 1429-1448. Shen, S. H. and C. C. Yu (1994). Use of Relay-Feedback Test for Automatic Tuning of Multivariable Systems, AIChE Jl, 40, 627-646.