A Practical Procedure for Multivariable Control Systems Design

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©

IFAC . \ utolllatic Control in

Petroleulll, Petrochemical aTld

DeS~llil1ati()1l

Industries. Kuwail. 19HG

A PRACTICAL PROCEDURE FOR MULTIVARIABLE CONTROL SYSTEMS DESIGN I. M. Alatiqi DfPartlllrllt

of

ChmllC(I/ ElIgill/,Prillg. Kuwait L'lIillnsity. Kuwait

Abstract . A novel procedure is developed for the design of multivariable control systems with two mode 5150 controllers. The procedure involves the selection of the control structure and the design of the control loops using steady state and dynamic models of the transfer function type. The procedure uses the latest state of the art techniques in a systematic way to select, compare and tune the control system. The relative gain array and integral controllability tests are used as a screening tool to eliminate the unfavourable structures. Multivariable Nyquist-Bode plots are then used to tune the favourable structures according to the specified robustness and load rejection criterion. A final selection of the best scheme is made based on the frequency responses to load disturbances . A 4x4 distillation example illustrates the procedure. Keywords. Control system synthesis; multivariable control systems; linear systems; PID control; process control; computer-aided design. INTRODUCTION In control systems design for multivariable processes, two fundamental problems are encountered. The first is input output pairing, that is what manipulative variable m. should be connected with the controlled variable y~. The second problem is how to construct the control: matrix K and what values its parameters should assume. The two problems were often treated separately. Considering the second problem first we find that researchers in control theory developed several multivariable control methods that normally gave rise to complex control structures, often characterized by off - diagonal feedback compensation. Among the most famous of these methods are decoupling (Luyben, 1970), Inverse Nyquist Array procedure (INA), (Rosenbrock, 1974), and Characteristic Loci procedure (Mc Farlane, 1980). One major common factor in these methods is that they tend to select control structures that minimize interaction between the feedback control loops. The resultant structure often involves off-diagonal compensation even in the moderately interactive plants.

problems in a logical systematic steps. Both steady state and dynamic information are u tilized in selecting the control structure or structures. Thus the limitations of variable pairing f r om steady state information are overcome by testing the performance of the selected structures dynamically and choosing the structure that gives the best performance. Performance is defined in terms of robustness and disturbance rejection characteristics. A Bode type plots are utilized to design 5150 proportional integral controllers for multivariable systems thus maintaining simplicity of the control structure.

STEADY STATE ANALYSIS It is assumed that a steady state model is available for the system, from which process gains could be calculated. The steps are as follows

Considering the pairing problem perhaps the Relative Gain Array proposed by Bristol is by now widely accepted as a screening tool, i.e., to eliminate the unfavourable pairings. Singular value analysis and integral controllability concepts can also be used for this purpose. Although some success was reported with these methods , they share the limitation of l acking dynamic information. The pairing problem grows in complexity with the dimension of the system since the number of possible ways to connect controls with mani pulations increases factorially with dimension. This fact makes it a practical necessity to eliminate as much as possible of the unfavoured structures before cons1dering the more involved dynamic analysis . The present work is aimed to combine the two

I l:j

1. Define the controlled variables. Those may be chosen directly, e . g. product quality or indirectly as a temperature or another variable or variables that is related to the variable need to be controlled . This step is mainly an engineering judgement . The outcome is the controlled variables set Y = (Yl' Y2 " .. , Yn)' 2. Specify sets of manipulative variables 51' St' ... . . , Sk' Each set differs from the o hers by at least one manipulative variable m . i 3. Calculate the process gains between every manipulative variable m and controlled variable i Yi' Construct the k process gain matrices G (0)1' G(o)2'····' G(o)k

1. ],,1. Alatiqi

116

4. Calculate the relative gain array S (Bristol, 1966)1 for each set and eliminate the pairings that involve i.

negative elements in the diagonal.

ii.

elements that are very far from unity.

V111. Trim the controller settings to obtain the desired disturbance rejection characteristics (without violating the robustness specifications). For example if one output y . is more important than the others, one could 1tighten the settings on Yi and loosen on the others. ix.

i.e.

Go to i

Sij«l or Sij»l (Mc Avoy, 1983).

Carry on the remaining P structures to the next B step. 5. Perform Morari Integral Controllability (MIC) test (Morari, 1983) on the remaining structures. Eliminate structures that are not integral controllable 2 . Carry on the resultant PM structures to the next step. The resultant number of control structures depends on the original system dimension, number of sets and number of rejected structures. If IT is an array which include all the original structures, then IT will have a dimension of k x n!, i.e. k rows and n! columns. This shows how valuable is the process of structure elimination in limiting the further study to managable number of control

8. Compare the control structures for their load rejection characteristics and select the structure that have "best" responses.

EXAMPLE:

A COMPLEX DISTILLATION SYSTEM

Alatiqi (1985) studied a 4x4 system consisting of a distillation column connected with a sidestripper separating ternary mixture. The results of the procedure are briefly described below. Tables 1 and 2 show the results of steady state analysis for two sets of manipulative variables. The analysis resulted in a recommending two contro structures for further evaluation:the RR scheme and the R scheme. Tables 3 and 4 show the transfer function matrices for these schemes.

structures.

TABLE 1

Steady State Results for RR Set

DYNAMIC ANALYSIS y. 1

6. Obtain the G(s) transfer function matrix for the surviving structures. G(o) 7. Controller tuning (Robustness-Disturbance Rejection Tuning) i.

Select a structure P .. 1J

2.25 -2.39 -1.08 -6.30

-2.96 3.46 3.54 4.49

0.18 -0.53 4.41 -1. 38

-0.67 -1.76 -5.38 4.96

RR

QB

QB

LS

2.10 -2.55 -0.02 1.47

-3.37 2.22 0.03 -0.88

0.39 -0.57 1.60 -0.42

-1.13

9.7

4.0

0.71

+0.56i

m. : 1

Calculate Ziegler-Nichols settings for the individual n loops.

s

XDl XB3 XS2 ['T

11.

S iii. Divide each Z-N controller gain Kc by a detuning factor f, and multiply each Z-N integral time TI by the same factor f. MIC iv. Check stability with multivariable Nyquist Plot (Rosenbrook, 1972).

Recommended structure

v. Calculate the "robustness factor" (Doyle and Stein, 1981)3. vi. Change the value of the detuning factor f until the desired specification on robustness is obtained. 4

1.9 -0.61 0.84

Y

XDl

XB3

XS2

[' T

m

RR

QB

QB

LS

TABLE 2

s

Steady State Resul ts for R set

vii. Calculate the individual "Bode" load rejection curves (Tyreus, 1984) for the plant outputs. 5 G(o)~

1.

S ~ G(o)*G-l(o), where the star implies multiplication element by element.

2.

Integral controllable systems are those having Re Ai (G+(o»>o. VA, where Ai is the eigen value,

m. 1

G+(o)is G(o) with signs of elements adjusted to have all diagonal elements positive. 3.

The robustness factor is the maximum ot O{ (I+GK)-l GK } or 0 {GK(I+GK)-l}, K is the controllers matrix. The (RB) is a measure of the maximum closed loop frequency response besides its

S

MIC

~

4.09 -4.17 -1. 73 -11.18

-6.36 6.93 5.11 14.04

-0.25 -0.05 4.61 -0.1

-0.49 1.53 -5.49 4.49

R

QB

QBs

LS

3.11 -5.03 -0.08 3.01

-.9 4.67 .05 -2.83

-.48 -.04 1.55 -0.03

13.8

4.8

0.75

value as a robustness measure. ~.

5.

The idea of the de tuning factor was originally introduced by Luyben (1985) The load rejection curves are the frequency response plots of the individual y./d 1

(I+GK)-l L; d is a disturbance input and

L is a vector of load transfer functions.

Recommended structure Y

XDl

XB3

XS2

[' T

m

R

QB

QB

LS

s

-.73 1.4 -.52 0.85 ::0.575i

y. 1 XDl XB3 XS2 ['T

A Practical Proced u re TABLE 3

The Transfer Function Matrix for RR 5cheme

XD1

2.22e- 2 • 55

XD1

(365+1) (255+1)

QB

_2.94(7.95+1)e- 1 . 055 (23.75+1)2(25S 2 +25+1) 3.46e -1.015

AA XB3

-2.33e- 65

XB3

AA

(35S+1)2

QB

32S + 1

XS2

1.12e- 235

X52

3.54e- 14S

(175+1)2

QB

AA liT RR

_5.73e- 2 . 55 (85+1) (505+1)

liT QB

XB3 QBs

X52 QBs

(125+1)2 4.32(255+1)e-· 02S (505+1) (20.2S 2 +2.6S+l)

TABLE 4

The Transfer Function Matrix For R 5cheme

XDl

4.0ge -1.35

XD1

R

(335+1)(8.3S+l)

QB

(31.65+1) (20S+l)

XB3

-4.l7e- 55

XB3

6.93e -1.025

R

455 + 1

QB

44.6S+1

XS2

5 .1le -125

XD1 QBs

6.36e-1.2S

117

0.17e-1.25

XB3

1.68e -3S

15

(285 + 1)2

4.4le-1.0l 5 16.25 + 1

XDl QBs

0.25e -1.4S

1.73e- 185

XS2

(13S+ 1) 2

QB

(13.35+1)2

X52 QBs

liT

-11.l8e -2.6

R

(43S+1) (6.5S+l)

liT QB

l4.04(10S+l)e-· 025 2 (45S+l) (17.45 +35+1)

liT QBs

(29S+1)2

(325 + 1)2

1. 25e -2. 85S

R

_.64e- 205

_0.51e- 8 . 55

liT QBs

XB3 QBs

XD1

15 (31.65+1) (7S+1)

(43.65+1) (9S+1)

X52

15 liT LS

XDl

-5 .38e -1.55 l7S + 1 4.78e- 0 • 15S (485+1)(55+1)

0.4ge- 65

IS

(22S+1)2

-0.05e- 65

XB3

1.53e -3.85

(34.55+1)2

IS

215+1

4.6le -1.015 18.55+1

X52

15

0.le- 0 . 05S (3l.6S+1) (55+1)

liT L5

48S + 1 -5.4ge -1.55 l5S + 1 4.4ge- 0 . 6S (48S+1) (6.3S+])

Figures 1&2 show robustness factors for the two schemes. Figure 3 and 4 show the frequency responses of the output for a load disturbance input (feed composition disturbance), for the RR and R schemes respectively. The R scheme outputs have smaller values at low frequencies than those of the RR scheme, indicating faster attenuation for the R scheme.

alternative control structures. It is recommended that this procedure be consulted before going to the more complex methods that involve off-diagonal compensation. The availability of graphic display terminals to the designer is almost necessary to make efficient use of the procedure.

The two schemes were tested on the nonlinear model of the system for the time response of the same disturbance used in creating figures 3 and 4. Figure 5 compares the two schemes for the time response, and shows that indeed the R scheme have faster responses than that of the RR scheme. Thus the R scheme is favoured and the analysis is completed. Figure 6 shows a schematic of the process with the basic feedback loops including the level controls.

Alatiqi, I.M. (1985). Ph.D. Dissertation in Chemical Engineering, Lehigh University,USA. Bristol, E. (1966). IEEE Trans. Autom. Control AC-ll, 133. Doyle, J.C. and G. Stein (1981). IEEE Trans. Autom. Control, AC-26, 1. Luyben, W.L. (1970). AIChE J., 16, 198. Luyben, W.L. (1985). Ind. Eng. Chem. Proc.Des. Dev., In Press McAvoy, T.J. (1983). Interaction Analysis. I5A Monograph Series. McFarlane A.G.J. (1980). Complex Variable Methods for Linear Multivariab1e Feedback 5ystems. Taylor and Francis Ltd. London. Morari, M. (1983). 22nd IEEE Conference on Decision and Control. 5an Antonio, Texas. Dec. 14-16. Rosenbrook, H.H. (1972). IEEE Trans. Autom. Control, AC-17, 105. Rosenbrook, H.H. (1974). Computer Aided Control 5ystem Design. Academic Press, London. Tyreus, B. (1984). Distillation Dynamics and Control Short Course, Lehigh University, U5A.

CONCLUSION A practical procedure have been presented for design of multivariab1e control systems using 5I50 proportional integral controllers. The procedure assumes the availability of input-output gains for the process which are normally available from steady state models. Dynamic transfer function type models are only needed for the few control structures that remains after the preliminary screening process. The RDR tuning method is a consistent controllers design method that could be used as a basis for comparing

REFERENCES

1. M . Alatiqi

118

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Fig. 1. Robustness factor for RR scheme .

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Robustness factor for R scheme.

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A Practical Procedure

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Disturbance rejection plots for a feed composition input. RR scheme.

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Fig. 4. Disturbance rejection plots for a feed composition input. R scheme.

120

I. M. Alatiqi

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RR «(".!flux l'"ni.o) SChotlDol

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Mr.E; ·Mi

Fig. 5.

Controlled variables response to a feed compositon disturbance (20% increase in the intermediate component).

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Xylene p",dty T.llllptutuU tJiCClHenc~ adlux. rue a..boilu duty, lUin C'o 1WIll. laboLler duty, uripper

or a QI

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cc

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Side duw rau Ca.pos ition tontroller Level Controller

40r---~~----~----------~~----

vs

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'-------/ CC \--#---1_ )Cs:).

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Fig. 6. Basic control loops for distillation column with sidestripper.

t .,