Multivariable Control of Industrial Fractionators

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MULTIVARIABLE CONTROL OF INDUSTRIAL FRACTIONATORS A. Shakouri KUllink/ijkd Shd/-Labo)"{//urilllll. Allls/ndalll (Sh pl/ Resmrrh B. I '.) . BadhllislL'I'g 3, 1031 (.",\1 A IIIs/nda Ill , Th p .\'elhn/a nds

ABSTRACT The multivariable Nyquist Array method offers a concept which enables the classical single-input/single-output Nyquist control design methods to be extended to multivariable systems. This concept is based on partial decoupling using a compensator network to achieve what is known as a "diagonal-dominant" structure, whereby single-loop controllers can be designed independently for the different loops. The method described here offers an approach by which a compensator network is designed automatically, a function minimization algorithm being used to obtain dominance over a specified frequency range by minimizing the ratio of moduli of the off-diagonal terms to the diagonal terms of the appropriate open-loop transfer matrix. Once a suitable compensator has been found, the design of an overall control system is completed by designing single-loop controllers for the different loops separately. The applicability of the method is demonstrated, with the aid of a dynamic model of a high-vacuum distillation unit, by the design of the viscosity control configuration for that unit. Further, attention has been paid to incorporating such a control configuration into modern, distributed-control systems such as Foxboro Spectrum. KEYWORDS. Cascade control, quality estimation, dynamic compensator, frequency domain control design, integrated control system, multivariable control system, industrial control, quality control. INTRODUCTION

For implementation in integrated control systems it is most desirable, in view of the limited availability of arithmetic blocks, to achieve the reduction in the interaction by making use of static or loworder dynamic compensators. The methods available to achieve this reduction (Rosenbrock, 1969, 1974; Leininger, 1979a, 1979b) require a high degree of involvement of the designer and most are only suitable for the design of static compensators.

The process instrumentation in current use has

reached a level of sophistication such that advanced process control schemes are now being introduced more and more to improve process operation. Increasingly, on-line quality measuring instruments, distributed-control systems and process computers are becoming standard refinery equipment, so that a higher level of process optimization is becoming feasible. If we bear in mind that distillation processes are responsible for approximately 40 % of refinery energy consumption, the significance of the improved control of these particular processes becomes apparent. rhe objective is, by means of improved control, to be able to produce much closer to specification, reducing quality give-away and hence also energy consumption. The successful application of advanced process control schemes, including multivariable control, depends on the availability of a reliable control design method.

In this paper a method for designing a dynamic compensator will be introduced that is based on the Direct Nyquist Array (DNA) method. This method utilizes a conjugate direction minimization technique to adjust automatically the compensator parameters so as to satisfy the required dominance cond it ion. The applicability of the resulting control method is demonstrated via the control design of a highvacuum distillation column. Due to the particular structure of the distillation column, models have been used that can be divided into a fast dynamic part, relating process inputs to intermediate variables (e.g. temperature and pressure), and a slower part, relating to the product qualities. This justifies the use of a cascade control system (master-slave control). A dynamic pre-compensator is used in the slave control loop to reduce the process interaction, which allows single-loop control in the master control loop. In the situation where the quality measurements are subject to relatively large analysis times, Smith prediction can be combined with DNA (Smith, 1957). In these cases, very much the same performance is achieved as with computer-based time-domain multivariable

The multivariable Nyquist Array design method (Rosenbrock 1969, 1974) forms a powerful tool in the design and analysis of multivariable control systems. This method tends to break down the multivariable problem into a number of almost non-interacting single-input/sing le-output feedback problems which can be solved using classical control design techniques. Reduction of the interaction is achieved by using a compensator network in such a way that the modified open-loop transfer matrix is diagonal-dominant. Without dominance the multivariable Nyquist Array method neither confirms nor denies stability of the closed-loop system.

49

:\.. Shakouri methods (e.g. linear quadratic control: Kwakernaak and Sivan, 1972; Ten Hacken and Van Wijk, 1984). This paper is organized as follows. Firstly the DNA design method is summarized, after which the decoupling algorithm for the compensator design is presented. The process under consideration is described briefly and the off-line simulation results are presented. Finally, the performance improvement achieved is evaluated and a number of conclusions are drawn.

The diagonal dominance can be graphically interpreted via an Array of the Nyquist plots of the diagonal elements Qii(s) with, in each frequency point, a circle with a radius equal to dCj(Q(s)) or drj(Q(s)) for column and row dominance respectively; these bands of circles are called the Gershgorin bands. If the point (-1,0) does not lie within the Gershgorin band, diagonal dominance is obtained. In case R(s) is (either row or column) diagonal-dominant the Nyquist stability criterion can be used for evaluation of the stability of the closed-loop system by application of the following rule

DNA CONTROLLER DESIGN The notation used in this section corresponds to that adopted previously (Rosenbrock, 1969, 1974). The control configuration considered is given by Fig. 1. The open-loop transfer function matrix Q(s) is defined as Q(s) = p(s) C(s) = L(s) G(s) K(s) C(s)

(1)

with G(s): n x n process transfer function matrix L(s): n x n post-compensator transfer function matrix

K(s): n x n pre-compensator transfer function matrix

C(s): n x n diagonal controller transfer function matrix.

! ·- ------- -- ~~~=~=~---=-:--=-----=-----=-~~==== = ~;~~ H-i I i, y' i ,l. _________________ _ _ ___________ ___ J,

!

L.. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Fig. 1.

_____ _ ___________ •

.

.J

with R(s): n x n return difference matrix can be investigated via the following procedure. Use is made of the property of diagonal dominance, defined by n

> L:

i=1 ijl j

(3)

IQ· . (s)1 1J

n

Q .. ( s) I JJ

> L:

i=1 ijl j

THE DECOUPLING ALGORITHM The reduction of interaction attainable, and thereby ultimately the controller performance, is dictated primarily by two factors the structure of the pre- and/or post-compensator;

The order of the compensator is a compromise: on the one hand the order should be kept limited for implementation reasons; a high compensation order,

on the other hand, gives a better reduction of the interaction and allows better control performance. This emphasizes the need of a method for automatic generation of compensator matrices, resulting in a diagonal-dominant system for a given compensator structure. Thus, different compensator structures can be evaluated in a fast and efficient way, enabling the above compromise to be found. The algorithm proposed therefore utilizes a numerical optimization method to minimize the system interaction, defined as a function of the compensator parameters.

The compensator design procedure in its most general form is characterized by the following phases specification of frequency region { O, wmax } structure determination and parameter initialization parameter optimization

Equation (2) considers "column diagonal dominance"; "row d iagona 1 dominance" is def ined in a 5 imi lar way

+

margin.

evaluation of the interaction reduction design and implementation considerations.

for all columns and over the pre-specified frequency range.

11

If Q(s) is not diagonal-dominant, the above-mentioned rule cannot be applied. The Gershgorin bands and the Ostrowski bands can be plotted in the frequency plots to evaluate th e system interaction, for the open-loop and closed-loop process respectively. For the design of the controllers use is made of classical criteria like gain and/or phase

interaction criterion.

(2)

+ Q . . (s)1 JJ

with "i: the number of encirclements of the Gershgorin band of Qii(S) o f the point (-1,0) in the Nyquist plot nh: number of unstable poles of H(s) nq: number of unstable poles of Q(s)

the ability of the designer to manipulate the compensator parameters with respect to some

Feedback structure of multivariable control system using DNA method

The design of a controller by the DNA method is split into two phases. Firstly, the interaction in the process model G(s) is reduced over a pre-specified frequency region with the use of a pre-compensator and/or post-comp e nsator. In the next section this compensator design is discussed in more detail. The decoupled process p(s) can then be controlled by a diagonal controller C(s). The stability of the closed-loop transfer function

11

(5)

IQ· . (s)1 J1

dr.(Q(s))

(4)

In the first step use is made of an automatic procedure to detect the maximum frequency of interest (Leininger, 1979b). A good initial pre-compensator is the (pseudo) inverse of the open-loop process model gains. The third step is performed automatically by making use of the numerical minimization method of a function of the interaction measure.

J

again over the pre-specified range.

In order to be able to implement the compensator determined in an integrated control system, the following additional constraints must be imposed on the compensator elements

\Iulti\'ariablc Control of Industrial Fractionators

SI

In the optimization the parameters are determined which minimize the following criterion

each element must be stable the order of the denominator must be greater than or equal to the numerator order each element is minimum-phase

n

the ability to specify only real poles and/or zeros in the elements must be present.

max w

These constraints follow immediately from the definition of the different functions in integrated control systems, such as Foxboro Spectrum and Honeywell TDC-2aaa. For reasons of simplicity, in this paper we only consider the case of the design of a pre-compensator K(s) in order to reduce the column interaction of pes). L(s) is taken as the unity matrix. Reduction of interaction in the columns, as opposed to reduction of interaction of the rows, is less

complex in that it limits the number of parameters in the optimization procedure. The columns of the compensator can be designed independently. Other cases, like post-compensator design, follow conceptually the same procedure. The following parametrization is used for each compensator element Kij(S) to satisfy the above

with w k

L

i=l iolj

Iq··(j w)I/jq.·(j w)1 1J

(8)

JJ

{a, wmax l 0, ••• , 3 0, 1, ... , n.

Given a compensator structure for a column, the parameters which minimize the criterion are deter-

mined. If the interaction reduction is not sufficient the order is increased until a satisfactory reduction is found. The procedure is then repeated for all columns, after which one can commence with the controller design. The success of any numerical optimization method depends on the shape of the contours of the performance function and the convergence properties of the algorithm employed. In an attempt to reduce the impact of the contour irregularities the variable-matrix algorithm is used (Powell, 1981). By using this method the constraints on the parameters are translated into

requirements

some analytical function to eliminate the need to use constraint optimization.

K .. (s)

1J

with akij

1 + alijs + a 2ij s

> a,

(6)

2

PROCESS DESCRIPTION

k = 1, 2, 3.

The latter constraints follow from the criterion from Routh (Morris, 1966). The above parametrization allows complex poles and zeros. If, however, only real poles and zeros are allowed the following parametrization is used

K .. (s)

(7)

1J

The process under consideration takes place in a high-vacuum distillation unit (HVU), producing distillates as feedstock for the manufacture of luboils. Figure 2 gives a simplified flow scheme of the unit with the main controls. The feed is long residue produced in a crude distiller. The product streams are: gas oil (GO), spindle oil (SO), light machine oil (LMO), middle machine oil (MMO) and short residue (SR). The main objective is to keep the viscosity of the distillates to within narrow limits in order to arrive at constant luboil viscosities.

with akij

> a, k

1, 2, 3.

S T EA M

LR STEA M

04 L---------------~--

Fig. 2.

__

SR

Simplified scheme of the HVU

PROCESS CHARACTERISTICS AND CONTROL STRATEGY

-

Ql

-Q

From a number of cases of application in practice it has been found that a general structure for a distillation columns exists, which can be expressed as follows:

2

Here the relations between manipulatable variables F (for instance reflux flows, reboiler duties, etc.), intermediate variables T (column tray

:\ . Shakouri temperatures) and qualities Q are given, with Ql and Q2 representing the product qualities SO and LMO, respectively (see Fig. 2). Together with the column pressure, tray temperatures determine the product qualities via vapour/liquid equilibria. This structure proves to have a number of characteristics influencing both the modelling procedure and the final control scheme. The first part has relatively fast dynamics and considerable interaction, in contrast with the second part, which is characterized by slow dynamics and/or time delays and relatively little interaction. To control a process with the above characteristics the following structure is suitable. The interaction in the first part of the process can be reduced by making use of a pre-compensator, thus allowing the use of single-input/single-output PlO controllers. To overcome the adverse effect of time delays on the control performance Smith predictors or quality estimators can be used (Smith, 1957; Alevisakis and Seborg, 1973); intermediate variables are used to predict the actual qualities.

The model used for the control des ign is given by the following equations :

C(S») 3O

T34(S) T37(S)

G(s)

(Ql (s») Q2(S)

H(s)

C(S») 24

F26(S) F27( s)

(T30(S») T37(S)

The exact model parameters are given in the append ix. RESULTS The interaction in the system can be found by inspection of the column interaction indices Fi (i = I, 2, 3) as defined by equat ion (8). Figure 4 shows a decoupled first column; FI close to 0 over the entire frequency region of interest. The second and third loops, however, have a high interaction index. F I REL I B

In this particular application, however, a cascade control structure was chosen. This structure is

\

\

\ \

believed to be more r obust with respect to modelling errors in the delays between the temperatur e s and the qualities. The final control system configuration is given by Fig. 3.

I

QUALITY

: fEMPERATURE

: PRE

CONTROLLERS

: CONTROLLERS

,COMPENSATO" ,I

\

"

PROCESS

o " o

,

Fig. 4.

,--- ~- ....

"

,,----

'-'- ' ~ ' -'-'-

_ .-

025 05 FRE OUE NCY, rod / mln

Column interaction of the uncompensated process

With the use of the method presented a precompensator is designed; the resulting interaction indices F2 and F3 are given by Fig. 5 and the parameters in the appendix. A considerabl e reduction of the interaction is obtained. Fig. 3.

Control strategy of control system 1

CHOICE OF MANIPULATABLE VARIABLES AND CONTROLLED VARIABLES

6

F IR EL I

15

F I RE L I

10

The first step in the modelling procedure of the process is to determine the manipulatable variables, disturbances and controlled variables. All process variables which have a direct influence on the controlled (output) variables are defined as : manipulatable variables if they can be manipulated, and disturbances if they cannot be manipulated. In this case the controlled variables are the properties of the products mentioned above. The choice of the manipulatable variables, the reflux flows F24, F26 and F27 and the controlled variables, the viscosities of SO and LMO, are mainly based on available dynamic models (Ten Hacken and Van Wijk, 1984). A dynamic model between the refluxes and the viscosities showed long dead times which would result in poor controllability. For this reason intermediate variables (temperatures), through which the model can be split into a relatively fast and slow part, have been opted for. From measurements it has been found that the draw-off temperatures T30 and T37 correlate well with the qualities Ql and Q2, and that the draw-off temperatures correlate well with the reflux flows. The temperature T34 is added as a controlled variable to give a square system.

" .", .- . ........ ..........

"" ""

05

0 0

02 5 05 FREO UEN CY, rod /m!n

lal

Fig. 5.

FRE OUENC Y, rod / mln

I bI

Open-loop interactions for (a) loop 2 and (b) loop 3 with compensator

The settings of the controllers were determined off-line using frequency analysis, by making use of phase and gain margin; the final tuning was done on the basis of step responses. The control settings were slightly conservative for reasons of robustness. Figures 6-8 give the results of the off-line tuning via the responses of the system to set point changes in Ql, T34 and Q2 respectively. The different elements of the compensator and the controllers are implemented in control blocks of a Foxboro Universal Control Module (UCM). Special attention has been paid to a correct initialization procedure, needed for smooth switching of the control system from manual to automatic mode; the outputs of the controllers must be determined from the manual flow settings via the inverse of K(O), the static gains of the pre-compensator.

!\Iulti,·ariable Cu ntrul uf Indu stria l Fractiu nators are necessary to enable the compensator to be implemented in an integrated control system; these are defined.

SE T POINT ( REL )

SE'" PO INT (RE L )

The applicability of the design method as a whole has been demonstrated by the design of a system for the control of product stream qualities in a highvacuum unit. The variations in product quality were reduced by a factor of 2.5. -025 L--L__ 120 o

J-~

__-L__~~

240

_10 L--L__

o

360

J-~

__-L__L--" 240 360 SA MPLE No

"120

SA M PLE No

FIG . 6a

10

SET PO INT ( REL)

__ - - - -

, .-,' ,.,,' T34

SET POINT (R EL )

00"10

,, ,,,

06

I

02 0,

,

0

:. ~!. T,o

-00 10L--L__J-~L--L~L-~ o "120 240 360

-02 0

120

240

SA MPLE No

360

SAMPLE No

FIG . 1a

FIG 7b

SET POINT (REL )

,,~-----------

"100

SE T POINT (REL )

/_.-._._-_._----

8

I

i

,' 02 I

/ T37

I

050

I I

I

!

ooo~~'---Q~'L------------

o ~~----------

___

_025 L--L__J -__L--L__L-~

o

120

REFERENCES

FIG . 6b

240

Alevisakis, G., and D.E Seborg (1973). An extension of the Smith predictor method to multivariable linear systems containing time delays. Int. J. Control, 17, 541-551. ------Ten Hacken, G:V., and R.A. van Wijk (1984). Use of multivariable control on a high-vacuum distillation unit. Proc. IFAC 9th World Congress, Budapest, 3, 116-121. Kwakernaak, H.~ and R. Sivan (1972). Linear Optimal Control Systems. Wiley & Sons, New York. Leininger, G.G. (1979a). Diagonal dominance for multivariable Nyquist Array methods using function minimization. Automatica, 15, 339-345. Leininger, G.G. (1979b). New dominance Characteristics for the multivariable Nyquist Array method. Int. J. Control, 30, 459-475. Morris, J. (1966). A simple derivation of Routh's stability criteria. Control , 10, 358. Powell, M.J.D. (1981). Some Properties of the Variable Metric Algorithm. Harwell Subroutine Library, England. Rosenbrock, H.H. (1969). Design of multivariable control systems using the inverse Nyquist Array. Proc. IRE, 116. Rosenbrock, H.H. (1974~Computer-Aided Control System Design. Academic Press, New York. Smith, O.J.M. (1957). Closer control of loops with dead time. Chem. Eng., 21., 217-219.

360

SA M PLE No

SA MPLE No

APPENDIX

FI G. Bb

FIG . Ba

Elements of the process models G(s) and H(s): Figs. 6-8.

Closed-loop responses in (a) quality and (b) temperature to step increases in the set point of : Q1 (Fig. 6), T34 (Fig. 7) and Q2 (Fig. 8). Sample interval 30 s

The control system is now operational. In comparison with the previous situation, in which the flows were adjusted manually to obtain the correct qualities, the temperature and quality variations are reduced considerably, as Table 1 demonstrates. TABLE 1

Improvement of Variation of the Controlled Variables

Variable

Variat ion: manual cont ro 1

Variation:

6.5 4.0 0.2 0.6

1.2 1.7

G () = (0 . 0046 s - 0.014) Exp(-1.5 s) 11 s 2 s + 3.305 s + 0.289 G (s) 12

(-0.00011 s - 0.0216) Exp(-0.5 s) 2 s + 1.65 s + 0.385

G

-0.00039 Exp( -2. 5 s) s + 0.0347

l3

(s)

G (s) 21

0

G

(-0.00122 s - 0.000102) Exp(-3 s) 2 s + 0.1669 s + 0.0216

n (s)

o

automatic

o

cont rol

G32 (s) = 54

0.1 0.2

(4.594 53 - 17.37 52 - 3.265 5 - 0.1489) 0. 284 53 + 0.0673 52 + 0.0069 5 + 0.00057

+

x 0.0000 1 x Exp(-3 5)

G (s) 33

-0.0015 s + 0.0823

H11 (s)

0.00491 Exp(-20 s) s + 0.113

H (s) 12

0

H (s) 21

0

CONCLUSIONS A method is presented for extending the use of function mlnlmlzation algorithms to the design of dynamic compensators, which allows a better decoupling of the separate parts of the process and thus improved control performance. Constraints

54 H (s) 22

A. Shakouri

= 0.022

Exp(-15 s) s + 0.159

Compensator matrix K(s).

11.0

-0.65 - 4.73 s 1 + 1.38 s

-0.46 - 4.13 s 1 + 25 s

0.0

1.0

0.0

0.0

-0.14 - 6.74 s 1 + 21.3 s

1.0

Slave cont ro 11 er s 1 controller T30: Cll(s) = -45(1 + ""fl'S) controller T

34

: C ( s) 22

controller T37: C (s) 33

-25(1 + _1_) 4 s -250( 1 +

1 ns)

Master controllers cont roller Q : Cll (s) 1

1 -5.5(1 + ""fl'S)

controller Q2: C (s) 22

-1.2(1 + _1_) 4 s