Pole placement self tuning control for packed distillation column

Chemical Engineering and Processing 36 (1997) 309 – 315

Pole placement self tuning control for packed distillation column S. Karacan *, H. Hapoglu, Y. Cabbar, M. Alpbaz Ankara Uni6ersity, Faculty of Science, Chemical Engineering Department, 06100, Tandogan, Ankara, Turkey

Abstract In this paper, we present results from the successful application of pole placement self-tuning control for a packed distillation column in a pilot plant. The steady-state and dynamic behaviour of a binary packed distillation column has been simulated using a back mixing model. The model solution has been obtained employing orthogonal collocation on finite elements. A number of comparisons are made with results obtained both theoretically and experimentally. After a brief description of the pole placement self tuning algorithm the results are compared for the application at a SISO plant. A pseudo-random binary sequence and Bierman algorithm are used to estimate the relevant parameters of the system model. Pole-placement technique is applied to self-tuning proportional-integral-derivative (PID) control. Both experimental and theoretical works were carried out and results were compared. © 1997 Elsevier Science S.A. Keywords: Pole placement technique; Self-tuning control; Orthogonal collocation; Finite element; Packed distillation column

1. Introduction The most significant task for a distillation column control strategy is to maintain the major design and operational variables (for example, product quality) at the expense of others, whilst preserving smooth and stable operation. Thus the operating conditions of the column are fixed by the control action in such a way as to attain these objectives in the most effective manner possible. If process conditions alter, than the controllers involved must be tuned to obtain satisfactory control. The conventional method controlling process is to apply a multiplicity of supposedly independent feedback control loops. Process control systems integrate adjustable controller settings to promote process operation over a wide range of conditions. The simple three term (proportional-integral-derivative: PID) or two term (PI) controller remains the most generally applied industrial process controller today. This is mainly due to the ease of operation, the robustness and the lack of specific process knowledge which is required for the initial controller design. Abbre6iations: ARMA, auto regressive moving average; ARMAX, auto regressive moving average with external input; CARMA, controlled auto regressive moving average; PID, proportional-integralderivative; PPSTC, pole placement self tuning control. * Corresponding author. E-mail: [email protected]. edu.tr 0255-2701/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S 0 2 5 5 - 2 7 0 1 ( 9 6 ) 0 4 1 9 2 - X

Conventional controller design procedures generate constant coefficient algorithms based upon a linear time-invariant assumption for the mode under consideration. The philosophy of self-tuning control is that the coefficients within the control algorithms are considered to be time dependent. More particularly, the strategy is to construct an algorithm that will automatically alter its parameters to meet a specific requirement or condition [8]. This is managed by the addition of an adjustment mechanism which monitors the system and tunes the coefficients of the corresponding controller in order to maintain a desired performance. The terms ‘self-tuning ’ and ‘adaptive’ are, to a large extent, equivalent. Bearing in mind the considerable use of PI and PID controllers in the process industries, it is not surprising that three term control is employed in association with pole-placement techniques. Due to the low order of the PID control law, assumptions have to be made concerning the order of the controlled process to be applied to the PID framework [1]. Although the application of fixed parameter PID controllers to packed distillation columns has received some attention [2,3], there do not appear to have been any studies concerning the application of self tuning controllers to this type of equipment. Wittenmark and Astrom [4] observed that a self-tuning PID controller acts simply as a well-tuned PID controller.

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Table 1 Physical properties of packed column Packing height (mm) Inside diameter of packed column (mm) Packing type Packing diameter (mm) Feed tank volume (l) Reboiler volume (l) Heater oil volume (l) Total pressure (mmHg)

1400 80 Rasching 20/15 60 13 25 690

2. Process description To check the systematic models and solution results, the pilot plant packed column was used to distillate the binary, methanol-water mixture. The physical properties of the packed column used are demonstrated in Table 1. All experimental equipment are shown in Fig. 1. In

the present experimental works, overhead product composition and temperature changes with time were observed at steady-state and dynamic conditions. The experimental method was summarised as below. In the initial work, the reboiler was filled with a methanol-water mixture at the feed composition. When reboiler temperature reached the boiling temperature of feed composition, cooling water was sent to the condenser. The column was operated for approximately 1 h at the total reflux. In this case, there were no feed and product flows. Temperature profiles observed on the computer were recorded and samples were taken regularly from the top and bottom of the column. Absorbances of the samples were determined by using a UV-visible recording spectrophotometer (UV-160A). When the absorbances and temperatures were constant, the system was at a steady-state condition for total reflux. After the system reached the steady-state condition, a preheated mixture was fed to the reboiler and a continuous

Fig. 1. All experimental equipment. 1, Reboiler; 2, packed column; 3, condenser; 4, temperature converter; 5, D/D converter; 6, refluxer; 7, magnetic valve; 8, computer; 9, D/A converter; 10, transducer; 11, control valve; 12, heat exchanger; 13, rotameter; 14, feed vessel; 15, pump; 16, triyac module; 17, oil tank; 18, cooling tank; 19, heat exhancer; and 20, bottom product valve.

S. Karacan et al. / Chemical Engineering and Processing 36 (1997) 309–315

−V

311

dyi d 2yi + HVDy 2 + KOGaS(y1 − y*)= 0 1 dz dz

(2)

In both phases the mass transfer rate of component i is equal at the steady-state condition in the differential volume. An unsteady-state mass balance overall element of packing leads to: (i) In the liquid phase HL

system was obtained. At the same time, the reflux ratio was adjusted to get the required value. Within the short time intervals, product samples were taken and their compositions were recorded by computer onto a floopy disc, creating a data bank. When the system reached a steady-state condition, temperature profiles and compositions were achieved constantly. After this steady-state condition was maintained, step disturbance was given to input variables. Therefore the system became an unsteady-state again and then the second steady-state condition was observed by checking whether temperature profiles and compositions were constant. By using dynamic data, operating conditions for control purposes were determined. For temperature control of the packed distillation column, two different experimental control modes were used. Pole placement self tuning control (PPSTC) was achieved for the first control work. For the comparison of effectiveness of PPSTC, the experimental PID control was also done. The results are shown in Section 5.

3. Modeling and simulation Two versions of the model based on the theory of mass transfer have been known for dynamic and steady-state behaviour of the column, i.e. back-mixing and plug flow models. In this work back mixing model was used. Related model and descriptions are given below. Consider the differential volume of packing of the height (z and area of cross-section S as shown in Fig. 2. If the changes in molar flow rates V and L are neglected, the use of fictitious molar masses of component transferred between phases over section (z is valid [9]. A steady-state mass balance over an element of packing gives: (i) In the liquid phase dxi d 2x +HLDx 2i +KOGaS(yi −y*) i =0 dz dz (ii) In the vapor phase

(3)

(ii) In the vapor phase

Fig. 2. Mass-transfer mechanism in a differential height.

L

dxi d 2x i dxi =L + HLDx 2 + KOGaS(y1 − y*)=0 1 dt dz dz

(1)

HV

dyi d 2yi dyi =L + HVDy 2 + KOGaS(y1 − y*)=0 1 dt dz dz

(4)

Across the packed column for vapor phase all mass transfer coefficient, Kya= KOGa= b(V)m(L)n

(5) −5

Where, for packed type b= 1.28× 10 , m=0.64 and n =0.48 [5]. The model solutions have been obtained employing orthogonal collocation on finite elements. The use of Jacobi, Legendre or Hermite polynomials within the finite element procedure were compared with an approach employing the Galerkin criterion. Results from the three polynomial procedures are very similar. However in cases where more than three collocation points are required the most efficient routine in terms of CPU time on a computer. The Galerkin procedure proved to be the least efficient. In the majority of cases the predictions of behavior using the finite element procedures of this work give better agreement with experiment than analytical and finite difference solutions reported by other researchers. Orthogonal collocation is seen also to give stable solutions with similar CPU times to those of the finite element procedures but it is not possible to place the grid points at desired locations. The steady-state profile of the simulated column is shown in Table 2.

4. Pole placement self tuning control algorithm In order to convert the velocity form of the PID algorithm into a self-tuning equivalent, consider the discrete time PID control algorithm. ut =

S (rt − yt ) R

(6)

where rt represents the set point, and S= s0 + s1z − 1 + s2z − 2

(7)

DT tD + 2t1 DT

(8)



s0 = KC 1+



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312

Table 2 Comparison of the steady-state mole fraction of methanol in liquid phase experimental and model Normalized distance (z)

Bottom 0.250 0.500 0.750 1.00 Top

s1 =KC s2 =



DT 2tD −1− 2t1 DT

KCtD DT

and

Mole fraction of methanol in liquid phase

Temperature (°C)

Experimental data

Model results

Experimental data

Model results

0.153 — — — — 0.932

0.153 0.312 0.491 0.714 0.861 0.932

84.0 79.23 74.12 69.08 65.11 64.22

84.22 78.54 73.91 69.17 65.26 64.20



R= (1 −z − 1)

(9) (10)

Hence ut =

(rt − yt )(s0 +s1z − 1 +s2z − 2) (1−z − 1)

(11)

Derivations of KC, tI and tD from Eqs. (8)–(10). yt =

B C ut − 1 + et A A

(12)

This represents an application of the auto regressive moving average (ARMA) type with an added control (or exogenous input) and therefore Eq. (12) is thus said to be a controlled auto regressive moving average (CARMA) or an auto regressive moving average with external input (ARMAX) model of the system. Substituting Eq. (5) into Eq. (12) and rearranging yields the following closed-loop equation: yt =

z − 1BS RC r+ e AR+z − 1BS t AR +z − 1BS t

(15)

Where the degrees of the S and R polynomials must be two (ns = 2) and one (nr =1) respectively because of

(16)

s2 = (t3b 20 − a3b 20 − a2b 20 − t2b1b0 + b0a2b1 + a1b0b1 +t1b 21 − a1b 21 + b 21)/b 30

(17)

To obtain a unique value tD for the PID controller, Eqs. (16) and (17) must be equated. As a result the coefficients of the T polynomial must be selected according to; t4 = a3b 30 + b1t3b 20 − a3b1b 20 − a2b1b 20 − b0t2b 21 +a2b0b 21 + t1b 31 − a1b 31 + b 31)/b 30

(14)

The coefficients of the A and B polynomials are estimated from the Bierman UDUT algorithm [6] and the coefficients of the T polynomial are defined by user. Then the parameters of the S polynomial (s0, s1 and s2) can be determined by solving the set of the simultaneous equations obtained from Eq. (13) that constitute the characteristic equation. The degrees of the polynomials in the characteristic equation are; na + nr =nb +ns +1 =nt

s2 = (t4 − a3)/b1

(13)

The properties of this closed-loop can be varied by placing the poles of the characteristic equation i.e. the denominator of Eq. (13) utilizing a ‘tailoring polynomial’ T, where the poles of T are chosen by the system designer. Thus the characteristic equation is: T= AR +z − 1BS

the polynomial representation of velocity from of the PID algorithm Then na must be equal to nb +2 and nt must be equal to nb + 3 (= na + 1). If all the coefficients of the T polynomial are user-defined then we must select a second order A polynomial (na = 2 and thus nb = 0 and nt = 3) to make sure that a unique set of PID controller coefficients can be obtained from the design. If the estimated model reduced the first order, then the new form of the algorithm is called pole placement self tuning where tD is defined by the user. When the order of the estimated model is more than two (na \2), the user is not able to define the coefficients of the T polynomial unequally, i.e. to obtain a single set of PID controller coefficients. For instance if the order of the A polynomial is three, i.e. na = 3, nb = 1 and nt =4, and two different equations will be obtained from the pole placement self tuning procedure for the determination of s2, viz;

(18)

Where coefficients t1, t2 and t3 can be defined by user. A similar argument applies with higher order A polynomials. Thus, if the poles of the characteristic equation are to be easily placed, all the coefficients of the T polynomial should be user defined and hence we must choose a third order T polynomial. Therefore we must use a system transfer function of the form; yt =

b0z − 1 u 1+ a1z − 1 + a2z − 2 t

(19)

The closed loop set point following relationship is obtained by combining the system model Eq. (19) and controller Eq. (11), i.e.

S. Karacan et al. / Chemical Engineering and Processing 36 (1997) 309–315

313

yt =

(1−z

−1

b0z − 1(s0 +s1 +s2) ut )(1+ a1z − 1 +a2z − 2) + b0z − 1(s0 +s1 + s2) (20)

The equivalent chosen closed loop T polynomial is of the form: T= 1+ t1z − 1 + t2z − 2 +t3z − 3

(21)

Controller coefficients can be found by equating the real denominator of the closed loop Eq. (20) with Eq. (21) thus: (1 − z − 1)(1+ a1z − 1 +a2z − 2)

Fig. 3. Open-loop step response obtained using simulation programs and identified models: – " – , simulation programs and –  –, identified models.

+ b0z − 1(s0 + s1z − 1 +s2z − 2) = 1+t1z − 1 + t2z − 2 +t3z − 3 By comparing coefficients of powers of z obtain:

(22) −1

, we

s0 =

(t1 −a1 +1) b0

(23)

s1 =

(t2 −a2 +a1) b0

(24)

s2 =

(t3 +a2) b0

(25)

It is significant to note that integral action in the PID controller provides a steady-state following without offset even if the values of the parameters of the system or of the controller change. The necessary increment in the control signal can now be obtained from [7]: Du = s0o(t)+s1o(t − 1) +s2o(t −2)

(26)

In present work, the steps in the operation of the pole placement self tuning algorithm may be given as: (a) apply a prbs to the system as a forcing function and attain the plant output; (b) estimate A and B from the CARMA model using the Bierman U-D update algorithm; (c) calculate s0, s1 and s2 from Eqs. (23)–(25); (d) find KC, tI and tD from Eqs. (7) – (9); (e) use Eq. (26) to obtain the incremental control signal; (f) output the updated control signal to the process; (g) return to (a).

5. Theoretical and experimental results

5.1. Steady-state results In this part steady-state solution of model was shown and compared with experimental results. The variation of temperature and composition profiles with height of packing were given in Table 2. By using pseudo random binary squence (prbs), system parameters were determined and a1 = −0.323, a2 = − 0.222 and b0 =0.108 were found, and then these

parameters were used to find control parameters. The model of the pilot plant column employed in the identification stage of the controller design was: yt =

0.108 u 1−0.323z − 1 − 0.222z − 2 t − 1

(27)

The response of the top product temperature obtained from the computer simulation program and identified models to a unit step increase in manipulated variable (reboiler heat duty) is shown in Fig. 3. Agreement is sufficiently close for the identified models to be used for controller design in the cases studied. In the present work, many dynamic and control experiment were done. Hence among them, only one was chosen as an example. When negative step change was given to feed composition, temperature changes with time in a packed distillation column were observed by experimentally and theoretically. Under the steadystate operation conditions are as shown in Table 3, 15.2% methanol was feeded. When the system was reached to the steady-state condition, feed composition was suddenly reduced to 11.7% methanol. Overhead and bottom product temperature were changed with time at the dynamic condition. These changes were tested by measuring compositions and temperatures. Table 3 First and second steady-state operation conditions Input – output variables of dis- I. Steady state tillation column

II. Steady state

XF (% mol) XD (% mol) XB (% mol) F (mol min−1) D (mol min−1) B (mol min−1) R (reflux ratio) TF (°C) QR (Cal min−1) TD (°C)

0.117 0.804 0.065 4.314 0.723 4.891 3 65.0 36000 67.22

0.152 0.884 0.094 4.314 0.743 4.456 3 65.0 36000 64.88

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Fig. 4. Open-loop response to a step decrease in feed composition from 0.153 to 0.11: – " –, experimental and — – — , theoritical.

As a result of a step change given, product composition decreased when product temperature increased. At the same operating conditions, mathematical model of the system was solved by using orthogonal collocation on finite elements on computer and it was shown in Fig. 4. Experimental and theoretical results are in good agreement. At similar operating conditions which is given in Table 3, negative step change was given to feed composition. In that case, overhead temperature was controlled by PID control system. Control parameters which were estimated by using Cohen-Coon technique [10] where KC = − 28.8, tI =6.82 and tD =5.8. Overhead temperature set point was determined as 64.2°C. Experimental and theoretical work which are done by using these parameters are shown in Fig. 5. Theoretical and experimental control results were in good agreement. At the similar operating conditions, overhead temperature control with PPSTC was achieved. For this reason, experimental and theoretical work were done. By using pseudo random binary sequence, system parameters were determined and it was shown in Eq.

Fig. 5. PID control results for step change in feed composition (xF1 =0.152 and xF2 = 0.117): – " –, open loop; – –, experimental; –  – , theoretical; and — – —, set point.

Fig. 6. Pole placement self tuning control results for step change in feed composition (xF1 =0.152 and xF2 =0.117): – · –, open loop; – – , theoretical; –  – , experimental; and — – —, set point.

(27) and then these parameters were use to find control parameters. Control of overhead temperature with PPSTC was achieved by using relevant parameter. Experimental and theoretical results were shown in Fig. 6.

6. Conclusion In the present work, the steady-state properties and control of pilot plant packed distillation column were investigated experimentally and theoretically. A mathematical model was developed and this model result was compared with experimental data. Related model was solved by orthogonal collocation on finite element. For steady-state solution, the best solution was obtained using various combination of number of finite element (NE) and number of collocation points (NC). Feed composition was chosen as on load variable and feedback PID control of overhead temperature was done by utilizing dynamic data. By using parameters which are obtained from three different method according to load variable, control was achieved. Similar procedure was also done theoretically. Reboiler model developed for dynamic calculations with orthogonal collocation on finite element was also used in this work. Comparison between theoretical and experimental results were done and good agreement was observed. In this work, self-tuning system was applied to control overhead temperature of packed distillation column. Heat duty was used as a manipulated variable like PID control systems. Feed composition was used as a load variable. Pole placement self tuning control algorithm was developed by using on-line computer. This program was written in basic and used for control purpose. In addition to this, pole placement self tuning control algorithm was written in Fortran 77 and it was added to dynamic program as a sub program. This computer program was also run by using distillation column’s

S. Karacan et al. / Chemical Engineering and Processing 36 (1997) 309–315

operating data. Experimental control data which is obtained by using on-line computer, were compared and it was note that these two result were in good agreement. It is concluded that pole placement self tuning control mechanism was controlling overhead temperature very well under the load effects. It is observed that pole placement self tuning control showed better result compared to PID control.

7. Notation Az−1 monic polynomial in the z domain representing the poles of the discrete-time system a area available for mass transfer (m2 m−3) Bz−1 monic polynomial in the z domain representing the zeros of the discrete-time system Cz−1 monic polynomial in the z domain representing the poles of the process noise c total number of components Dx liquid phase eddy diffusivity coefficient (m2 h− 1) Dy vapor phase eddy diffusivity coefficient (m2 h− 1) et white noise HL liquid hold-up in packing (kmol m−1) HV vapor hold-up in packing (kmol m−1) KOG overall gas phase mass transfer coefficient (kmol m−2 h−1) KC steady-state gain for three-term controller L liquid flow rate (kmol h−1) Rz−1 weighting polynomial acting on set point rt set point at time t S cross-sectional area of packing (m2) t time (min) ut input variable at time t V vapor flow rate (kmol h−1)

.

.

x yt y* z−1 z

315

mole fraction in liquid phase output variable at time t mole fraction in vapor phase in equilibrium with x backward shift operator spatial variable (m)

Greek symbols derivative time (min) tD tI integral time (min) D first difference operator Subscripts and superscripts ai parameters of A polynomial bi parameters of B polynomial i component number over the domain 1BiBc

References [1] P.E. Wellstead and M.B. Zarrop, Self-Tuning Systems —Control and Signal Processing, Wiley, Chichester, UK, 1991. [2] D.T. Lee, Dynamic and Control of Packed Multicomponent Distillation Columns. PhD Thesis, University of Maryland, USA, 1976. [3] M. Molender and C. Breitholtz, Proc. Bias. Int. Conf. Control of Industrial Processes, (1987) 429 – 447. [4] B. Wittenmark and K.J. Astrom, Automatica, 20 (1984) 595– 605. [5] B.N. Sahay and M.M. Sharma, Chem. Eng. Sci., 28 (1973) 41 – 47. [6] G.J. Bierman, Automatica, 12 (1977) 375 – 382. [7] Jacquot, Modern Digital Control Systems, Dekker, New York, USA, 1981. [8] K.J. Astrom, U. Borrison and L. Jung, Automatica, 13 (1977) 457 – 467. [9] J. Bravo, A. Patwardhan and T. Edgar, Ind. Eng. Chem. Res., 31 (1992) 604 – 608. [10] G.H. Cohen and G.A. Coon, Automatic Control of Processes, Vol. 17, International Texbook Company, Pennsylvania, 1967, pp. 345 – 365.