Decentralized Control of a Multivariable Flow-Level plant based onRobust Control Approach

Yashar Kouhi, Batool Labibi, Alireza Fatehi, and Rahman Adlgostar

Advance Process Automation and Control, K. N. Toosi University of Technology, Tehran, Iran

where xER n, ueRm

, yeRP, Be Rnxm, AeRnxn

,

C e RPxn. This system is composed of N sub-system asfollows

II. THEORY

In this method, interactions between sub-systems areregarded as uncertainty of the corresponded sub-system.Then using robust methods, we tried to design a suitablecontroller for each sub-system by minimizing the weightedsensitivity functions and aftermath the effect of sub-systemswould drop off.

Suppose G(s) is a large-scale system, and the state spaceequation of this system is displayed as follow.

combination of these results and checking the stability ofoverall system with interactions. In this paper the secondmethod steps is employed.Another problem which we regularly face up in practicalsystems is the deviations referred to un-modeled dynamicsor the dissimilarity of model in different operating pointswhich may cause to instability of closed loop controller. Tosurvey the robust stability we should check whether thestability is hold by assuming worst case uncertainty or not.Robust stability analysis can be reviewed by using the smallgain theorem and Jl-Analysis.In this paper, first the decentralized control problem isdiscussed for a general plant and solved by output controlfeedback. Then the MIMO Flow-Level control plant isintroduced as a practical system to serve the desiredcontroller. Identification of this plant in various operatingpoints is explained briefly. To design a proposed controlleran unstructured uncertainty is extracted from parametricuncertainty and a nominal model and a set of block diagonalmultiplicative uncertainty weight is obtained. The stabilityof the design is automatically guaranteed by outcome of thetheorem, and robust stability is checked to consider how thisdesign has reliability in different operating points.

Abstract- In this paper a novel approach is proposed tosolve a decentralized control problem to stabilize amultivariable system and attenuate the interconnectionsbetween its subsystems. To satisfy these conditions, an utfeedback controller is designed by solving an Boo controlproblem. The designed controller is applied to a practicalmultivariable Flow-Level plant to show the effectivenessof the proposed methodology. The time delays,transmission zero, two different time constants, and themodel uncertainties are the main problems in this plant.

I. INTRODUCTION

I N nowadays modem technology, infrastructures of manysystems are combination of many diverse processes and

communication of many machines. Analysis, design andperfonning accrete controllers for such systems which arecalled large-scale systems are a heavy task. Using highpower and high-speed computers is not the alternative eventhough they can empower our computation capabilities.Subsequently developing some methods to alter the problemto some simpler related one can be always beneficial.

In a centralized controlling case, whole information of asystem is gathered in a single point, and then controlcommand will be calculated and will be sent to actuators.Theories of classic control, modem control andmulti variable control are all dealing with this kind ofcontrolling approaches. Actually using whole information indesigning of these controllers is required in many cases.Therefore executions of such theories are not easilyapplicable because of multiplicity of inputs and outputs andmaybe large distance between sub-systems.

Decentralized control has been attended more in boththeory and application because of its simplicity inimplementation [1]. In recent decades, significant trials areestablished to introduce the new methods of decentralizedcontroller design for multivariable plants [2,3]. Twotechniques in frequency domain based on Rosenbrock'sInverse Nyquist Array method and MacFarlane's character­istic locus method is presented in [4].In this paper, the desired approach is to obtain an outputfeedback controller by solving some H" problems toguarantee the stability of system.Hoc controller has a good robustness in presence ofuncertainties and numerously is used in MIMO and robustcontrol design.To analyze the stability of decentralized controllers, manyinvestigations have been done [2]. All these endeavors aresummarized in two steps: Stability analysis of eachsubsystem by using classic stability theorems and the

978-1-4244-2746-8/08/$25.00 © 2008 IEEE

x(t) =Ax(t) + Hu(t)

yet) = Cx(t)

N N

Xi = A;iXi + BUui + LAyXj + LBijujj=l j=l1'1'i j*i

N

Y i =CUXi + L CijXjj=lj*i

(1)

(2)

(14)

AiN J. It

-CIl~AIN"""' """' """'

-C22P2A2N

-CN2PNAN2

CII~BllKI(S)RI"""' """' """'C22P2B22K2 (s)R2

CNNPNBNNKN(S)RN

-CNNPNANI

(13)

where Xi =[~;]. Outputs of the j'h sub-system would be

calculated as following:

lo .·· Cii . · · 0jX - c);/i;i = CiiP;B;;KiRi

Ilc)=:iJl < aiO'max (Cu), a i < I (15)

interaction among sub-systems will reduce and outputs ofthe i'h sub-system would be calculated as following:

"""' "'" ""''''''~=CiiP;BiiKi(S)Ri i=l, ...,N (16)

It can be shown that

where if; = lAil A;;_I 0 AMIcan be achieved from the (14) that by minimizing

(4)

(3)

(5)

(6)

i=l, ...,N

i=l, ...,N

K =diag{K;(s)}

{X. = A..x. +B..u.

Gd; (s) : l II l II l

y;=Cjjx;

Accordingly, by using the decentralized control in the formof equation (5) the poles of the closed loop system aresettled in the desired intervals. Therefore, the decentralizedcontroller

stabilizes the overall system and also provides desireddynamical characteristics for the overall system, if somesufficient conditions are satisfied.Theorem: Interaction between sub-systems in the wholesystem would be minimized if

where X. E R ", , u. E R tn, , y. E R 1>, , A .. E R "i XIIi ,I I I /I

B ii ERn, Xnl. , C ii E R P, XII, • It is assumed all (Au, Bu) are

controllable, all (Au, Cu) are observable, and Bii and C;;N N N

are full rank. ~A x ' ~ B..u. and ~ C..x. represent the~ijj~lJJ L.JlJJJ=l J=l J=lj:¢:; J*i J*i

interactions between subsystems. Our purpose is to designlocal controllers for each sub-system

in the form of

W Ii in equation(6) is acquired by following inequality

i=l, ...,N

where

(7)

IICuP;H1= IICiiP;HilLwhere if; = [Ail Aii_1 A;;+1 ... AiN ].

Considering

P; = lP;(1 , :) p;(2 , :) p;(3 , :)JWhere

(17)

(18)

[

0 c.. 0]o O],Aij = 0 ~ Bij

o 0 0

(8)

(9)

(10)

1-C;;(sl-4; +B;;I\G;)-lB;;1\

~(1 , :) = -(sI-Au+!3;;I\C;)-lBul\-1\(1-C;;(sl-4; +B;;I\C;;)-lB;;I\)

[

Cu(sl-Au+BuK;Cu)-1 ]11(2 , :)= (sl-Au+BuK;Cu)-1

-K;Cu(sl -Au +BuK;Cu)-1

(19)

(20)

And the sensitivity function of the i'h sub-system is

Si = I - Cu(sl - Au + BuK;Cu)-1 BuKi (11 )

(21)

Proof: Defining

P;(s) = (sEi -A;; +B;;Ki(s)C;;r1(12)

the overall system under decentralized control will be asfollows:

defining sensitivity function of the ith sub-system as

S; =1 -C;i(sl -Au + Bi;K;Cu)-IBuKiwe can have the following calculations:

Cu(sl - Au +BuKiCU )-1 =SiCjj (sl - Ajj )-1

(22)

(23)

here, c. is the ith row of the output matrix C, and n. is theI I

degree of the /h subsystem. Since the isolated subsystemsare observable by appropriately decomposing the inputmatrix B, it is possible to have controllable isolatedsubsystems. By using this transformation the followingdescription of system is obtained

The former equation shows that minimizing of the weightedsensitivity function of the sub-systems given in (24) caneliminate the interaction among sub-systems. However,obtaining the sensitivity function via equation (7) is hard andwe use following method.

Since the large scale system given in (1) is observable, it isalways possible to find a similarity transformation totransform the system into the output decentralized formwhere matrix C is block diagonal and c .. *' 0 for i *' J. . This

lJ

transformation is based on the observability matrix of theoriginal system given in (1) and is defined by

Fig. 1. The schematic of interconction of Flow and Level plants andfonnation ofMIMO systems

(28)

k'2 -8."S][ ](2: +I) e _- U I

_22_ e (}22 S U2

(72s+l)

Where, uland Y1 are respectively the input and output offlow channel. U2 and Y2 are respectively the input and outputcorresponds to level channel. The variation range ofparameters is:

The identification of this system is achieved with two MISOsub-systems. The input signals used for identification arechirp signals with an adequate range of frequencies and largemagnitude to be distinguished from output noise [8]. Theidentified models in various operating points have fixedpoles and variable gain and time delays. Therefore, thegeneral model is suggested as follows

(24)

(25)

C I

CIA

F = CIA 2

C 2

C 2A

and

CU~Hi =[Si SiCu(sI-Au)-1 SiCu(sI-AiJ-1]Hi =

Si[I Cu(sI-Au)-1 Cu(sI-Au)-I]Hi

(26) And

4.1 $; 011 $; 4.5 ,5 $; °12

$; 6.8

5.5 $; °21

$; 6.6 ,4.3 $; °22

$; 5.3(29)

And the sensitivity function for all subsystems can be easilyobtained by fitting a function as the upper bound of the maxsingular value of following equation:

3.23 $; KII~ 14.81 ,4.48 $; K

I2$; 15.54

0.11 $; K 21 ~ 0.39 , 5.4 $; K 22 $; 7.28(30)

in each frequency. Hand BH are respectively corres­pondences to the interactions of matrix A and B, and Ad isthe block diagonal of matrix without considering interaction.

C(sI - Adfl[H BH] (27)From these equations, it is concluded that the system has

parametric uncertainty in various operating points. Toconvert parametric uncertainty into unstructured form foreach element of transfer function the following rule is used.

The following notation is chosen for each element oftransfer function:

In order to represent a unstructured uncertainty for (22) anominal model and multiplicative uncertainty is consideredfor each element as follows

III. The MIMO Flow-Level control plant description andIdentification

The schematic of interconnection between RT522 ProcessTrainer Flow and RT522 Process Trainer Level is illustratedon Fig. 1. The pneumatic valve No.1 in level system andvalve No.2 in flow system are seemed as the inputs ofMIMO system. The level input valve and the flow valve canhave inputs from 0 to 100 and -100 to 100 respectively.There are two outputs for the system, the level in level tankand the flow passing through the flow system [6&7].

G (s) = k e-~ .jSG (s)[Ii,j [Ii.j 0i.j

k i .j min < kPi.j < k i .j tmX ' 0; .j min < O;.j < 0; .j tmX

(31 )

(32)

In this equation wT

(s) is a stable transfer function

For the plant with time delay given in (31) and parametricuncertainty in delay and gain of the system, the uncertaintyprofile can be assumed in the general form of:

indicating the upper bound of uncertainty and ~ , (s)

indicates the admissible uncertainty block, which is a stable

but an unknown transfer function with II~ij IL < 1. In this

general representation, since II~u IL < 1; hence [9],

(39)

(37)

(38)

10.02 1(2s +1)

6.34(72s +1)

[

9.02

G = (2.57s +1)2n .201

(42s + 1)

Where

IV. Scaling

From equations (19),(20), and (21) since interaction frominput flow has less effect on output level and the matrix isclosed to triangular, we scale system from right and left tohave a system with less interaction in both channel

The nominal model that is delay free has a transmission zeroat z = 4.7454, which causes limitation on the performanceof system [1].

(34)

(33)

rK

(1 +--t-)Bmax/./ + r k, ,j

B(~s+l)

2

G

I I Pi ,jWr. . (s) > ---1

I,j Gn..

I,j

And the nominal model can selected

40r

Singular Values

(42)

(41)

o 0

o 0.0002

o 0.0002

o 1

-0.0003 -0.0377 (40)

0 0.45

1.3657 -0.225

B= -1.0628 0.1125

0.0476 0.0881

-0.0011 -0.0012

c= [~0 0 0 0] D=00 0 1 o '

o 1 0

001

A = -0.0757 -0.5405 -1.2782

000

000

By using the mentioned rule in section (II) we find aminimal presentation of the system in the form of (17), asfollows

V. Hoo design

As mentioned in section (II), to design a decentralizedcontroller for the system it is necessary to design some Hoo

controller for each subsystem. The performance weight foreach channel is selected as

(35)

In this representation ~i is a diagonal block such thatII ~i II 00<1. WT 1and WT 2 are respectively the upper bound of

(WT11

' WTl2

) and (WT21

,WT22

).To show the validation of this

model the singular values of the system in various operatingpoints and the singular values of perturbed plant given in(27) are shown in Fig. 2. It is seen that the selectedweighting function covers all the singular values ofperturbed model.

20=:~

- Sigular Values of Mldel in different Working points

+ +Singular Values of MJltiplicative Mldel

12010 10"Frequency (rad/sec)

Where for each element: k = k min + k max r = (k max -kmm )/ 2 .[1]2 'k k

The obtained model for this plant is

G =[(1+WTI~I) 0 ][gnll gnl~.. ] =[(l+WTI~I) O]G (36)p 0 (l+WT2~2) gil" gn" 0 (l+WT2 L\2) n

Fig. 2. Singular values at different operating points and singular valuesof multiplicative model (43)

The nominal model obtained for this system is In this structure, M; is the maximum value of sensitivityfunction in all frequencies. In our design we selected M; less

Singular Values

Fig. 3. Appropirate performance weight for both subsystems

Oulpulflow

: 1 , , I I

-,- - --t - - r - -1- - --t - -1 1 , I 1

-,- - --! - - l- - -,- - --! - -1 1 I , ,

- _I __ ...!. __ ~ __ ' __ ...!. __1 1 I , 1

1 1 1 , ,

- -1- - "1 - - I - -,- - -,- - -, , , , I

- -,- - -+ - - r- - -,- - -+ - -, , 1 I ,

16oo'------L..:--~~300--'------J'------:-'

Time(sec)

€

1260

Oulpullevel

100'---~----::-:-:----L.300-...,-L:-------,-L.,..---:-,

Time(sec)

I1 , , ,

20 !- -'- - ...i __ L __,__ ...i __! ' 1 , , I

I I , , , I

ii' , 1 1

18 : - -I - - -+ - - I- - -I - - -+ - -I , , I ,

E __:__ i__ ~ J~~~ ;~;~ I_t 16 , 1 , 1 I

, I , 1 ,

, 1 I 1 ,

- -, - - "1 - - I - -I - - "1 - -, , , 1 ,

I , , 1 ,

12 - _1- _ ...!. __ ~ __' __ ...!. __1 , , , I

1 , , , I

, 1 , 1 I

Fig. 5. The step response of real system(solid) and model ofsystem(dashed) to step response

It is obvious that two outputs are settled down after150(sec), which is relatively good. And the control signal isshown in Fig. 5.

Ws

/

10 10"

Frequency (rad/sec)

10

80

Gor

40 ~

20 ~

~ 0;

~

! -'w:

-GO,

-80'

-100 i

-120'10'(

than 2. A is a constant less than one. WBi* is the systembandwidth and is chosen based on the limitations in robustcontroller design [9]. By such performance the tracking alsoholds. In Fig. 3 the singular values of the system in equation(18) is sketched and an appropriate weight in the form of(34) is fitted on it as Wp .

Output level Output flow

In order to avoid large control signals, the weightingfunction Wu is selected as follows.

~Ci is the controller bandwidth in each channel which is

obtained by applying a sinusoidal with varying frequency.The controller bandwidth is selected as the highestfrequency that the system is unable to track the inputs after

that, M is the maximum value of the control signal in

, I- - - - "1 - - T - - ,- - -, - -

, , 1 1 ,

- -, - - "I - - t" - - \- - I - -

I I , 1 ,

- -1- - -t - - +- - -,- - -, - -, , I , I

_I _'-l,

_ __ J __ 1 , __I 1 I

I 1 ,- - - I - - I" - - - - -, - -

1 1 1 1 1- -,- - I - - T - -1- - -I - -

1 I 1 , ,

- -I - - "1 - - r - - ,- - -, - -, 1 1 , 1

- -, - - -l - - +- - - 1- - -l - -, , , 1 ,

1600'------'--~----J300----'--~----J

Tlme(sec)

100'------'--~----'-300----'---'-----J

Time(sec)

,

! I , I , 1

20 :- -I - - -l - - +- - - 1- - -, - -ill 1 I ,i , I I 1 ,I , I 1 , ,

18 I - -, - - I - - T - - ,- - -, - -, 1 1 , 1

, 1 1 1----- Modelout!i 16 __ , __ J __ 1 _ =-=-~ea~ s~. _

~ 'I, 1

I , , , ,

14 - -I - - -t - - +- - - 1- - -, - -I , 1 I ,

I , 1 1 ,

, , 1 , ,

12 - -, - - "I - - T - - ,- - -, - -, 1 1 , ,

, 1 I , ,

(44)s+{j)""IM II ,

e; S +(j)hc

each channel and c; is a free small constant [9]. The bode

diagram of WpSis shown in fig

Bode Diagram Bode Diagram

Fig. 6. Control signal of step input

With these weighting functions, two Hoo controllers withand without considering disturbance model are designed.The robust stability is guaranteed if

VI. Robust stability

As represented in robust theory for example In [10], therobust stability is guaranteed if

(45)

Fig. 4. W S for each subsytem

10

Frequency (rad/sec)

10"

Frequency (rad/sec)

In this equation, T is the complementary sensitivity functionof the system. This bound is shown in Fig. 7.

By considering Fig. 4, it is clear that stability is guaranteedfor this design. The step responses when the input vector is

R =[\O~] ofthis design has shown in fig

}10.

10·

10 "'-:;-------L----J......L-L...L...l...L.I-:--"-.L...J...J...L.l..L.U....;----L.......L......L-.l--W-JL.I..L;;--..L.-..I-.J.....I...1..L.LJ.'-:----'----'---'--'-~------'--L.......I....J-.u..u10' 10" 10' 10° 10' 10' 10'

Frequency(l3d1sec)

Fig. 7. The max singular value of WrTto check the robust stability

Since this quantity is less one in each frequency, therobust stability is satisfied.

VII. Conclusion

In this paper, a decentralized control based on robustcontrol approach applied to a practical MIMO Flow-Levelcontrol plant. The utilized method was an output feedbackcontrol based on minimizing weighted sensitivity functions.In this method, the interactions between sub-systems wereconsidered as uncertainty. Minimizing the weightedsensitivity function would help us to make the systemdiagonal dominance for a better control action on each sub­system. Sine the identified model of plant has parametricuncertainty we derived an unstructured uncertainty set and anominal model for the system. Then we designed a localcontroller for each subsystem of scaled model of nominalmodel. Because of the existence of uncertainty, the robuststability of system was checked and the reliability of thedesign in terms of stability in each operating point wasproved.

VIII. REFRENCES

[1] Sigurd Skogestad Ian Postlethwaite, "MultivariableFeedback Control Analysis and Design" John Wileyand Sons. Second Edition 2001 CH.5

[2] J.M.Maciejowski, Multivariable Feedback Design,

Cambridge University, Addison-Wesley Pub (Sd),1989

[3] B.Labibi, B.Lohmann, A.K.Sedigh, Output feedbackdecentralized control of large-scale systems usingweighted sensitivity functions minimization, ,Systems& Control Letters 47(2002) pp191-198

[4] R.V.Pate, N.Munro, Multivariable System Theory andDesign, Pergammon Press, 1982

[5] B.Labibi, H. J. MARQUEZ and T. CHEN, Diagonaldominance via eigenstructure assignment, InternationalJournal of Control, Vol. 79, No.7, July 2006,707-718

[6] U.N.T. Geratebau GmbH, Technical Discription ofRT512 Process Trainer Level, 2003

[7] U.N.T. Geratebau GmbH, Technical Discription ofRT522 Process Trainer Flow, 2003

[8] O.Nelles, Nonlinear System Identification, pringer,2001.

[9] K.Zhou, J.C.Doyle, Essentials of Robust Control,Printice Hall,1998

[10] J.C.Doyle, B.A.Francis, A.R.Tannenbaum, FeedbackControl Theory, Macmillan Publishing Company, 1992

[11] B. Labibi, B. Lohmann, A. K. Sedigh and P. JabedarMaralani, "Robust Decentralized Control of Large ­Scale Systems via Hoo Theory and Using DescriptorSystem Representation", Int. 1. Systems Sciences, Vol.34, no. 12-13,pp. 705-715,2003.