Improved branch and bound method for control structure screening

Chemical Engineering Science 60 (2005) 1555–1564

www.elsevier.com/locate/ces

Improved branch and bound method for control structure screening

Yi Cao∗, Prabirkumar Saha1

School of Engineering, Cranfield University, Cranfield, Bedford MK43 0AL, UK

Received 15 October 2004; received in revised form 26 October 2004; accepted 26 October 2004Available online 7 January 2005

Abstract

The main aim of this paper is to present an improved algorithm of “Branch and Bound” method for control structure screening. Thenew algorithm uses abest-first searchapproach, which is more efficient than other algorithms based ondepth-first searchapproaches.Detailed explanation of the algorithms is provided in this paper along with a case study on Tennessee–Eastman process to justify thetheory of branch and bound method. The case study uses the Hankel singular value to screen control structure for stabilization. The branchand bound method provides a global ranking to all possible input and output combinations. Based on this ranking an efficient controlstructure with least complexity for stabilizing control is detected which leads to a decentralized proportional controller.� 2004 Elsevier Ltd. All rights reserved.

Keywords:Control structure screening; Branch and bound method; Hankel singular value; Tennessee–Eastman process; Process control; Stability;Systems engineering; Optimization

1. Introduction

Control structure screening has been a challenging re-search topic among control engineers during the last decade(Heath et al., 2000; Kookos and Perkins, 2001). It is moredifficult than synthesizing a controller, specially for largesets of measurements and actuators, because design ofcontrol structure is a combinatorial problem. With a largenumber of candidate measurements and manipulations thenumber of possible combinations of inputs and outputs havea combinatorial growth. Thus, an approach consisting ofperforming a controllability analysis for each possible com-bination becomes time consuming (Havre and Skogestad,1996). Tennessee–Eastman test-bed problem (Downs andVogel, 1993) can be referred as an exemplary process dueto its umpteen I/O combinations (41 measurements out ofwhich 22 are potential outputs; 12 manipulated variables).In this case, the total number of possible input–output

∗ Corresponding author. Tel.: +44 1234 750111; fax: +44 1234 750728.E-mail address:[email protected](Y. Cao).

1 Current address. Department of Chemical Engineering, Indian Instituteof Technology Guwahati, North Guwahati 781039, India.

0009-2509/$ - see front matter� 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2004.10.025

combinations is∑12j=1C

12j

∑22i=1C

22i = (212−1)(222−1)=

17,175,670,785. Suppose it takes only 0.01 s to evaluateone I/O combination, it would still take more than 5 years foran exhaustive search to reach a global conclusion. Thereforean efficient algorithm is essential to tackle the combinatorialproblem.

Before discussing the importance of control structurescreening, it is necessary to recapitulate the basics of con-trol structure selection problem.Fig. 1 shows the generalstructure of a feedback process. For a set of exogenousinputs (w) to the process, a set of inputs (u) and out-puts (y) are to be selected such that a compensatorK canbe designed to achieve cost functionz within a specifiedlimit. It is understandable that more number ofu and yselected will generally yield better performance. However,it is intended to minimize the number ofu and y (onlynecessary ones to be selected) to restrict the complex-ity of the feedback controller. In this perspective, controlstructure screening is important because of the followingissues:

• Usually control structure selection is a multiobjectiveproblem.

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1556 Y. Cao, P. Saha / Chemical Engineering Science 60 (2005) 1555–1564

P

K

w

u

z

y

Fig. 1. Basic structure of a feedback system.

• Some objectives may not be suitable to be considered to-gether mathematically.

• Even a mixed cost function can be formulated, but it maynot be computationally tractable.

A possible solution to the above problems is to separate theobjectives into two groups: firstly only use computationallytractable indices suitable for control structure screeningto reduce the candidate set, and then perform exhaustivesearch in the reduced set for the rest objectives.

Branch and bound (BAB) as a general approach to solvecombinatorial problem is well-known. It has been widelyused in numerical algorithms for solving mixed-integer lin-ear and nonlinear programming (MILP and MINLP) prob-lems (e.g.Luyben and Floudas, 1994a,b; Braatz et al., 1996)and global optimization softwares such as BARON computa-tional system for solving non-convex optimization problemsto global optimality (Gutierrez and Sahinidis, 1996; Ryooand Sahinidis, 1996). Several researchers have demonstratedapplications of BAB algorithm in chemical engineering andprocess control as well as in control structure selection(Raman and Grossmann, 1991; Skrifvars et al., 1996). How-ever, most of the available BAB methods are integrated ingeneral MINLP and MILP software, and therefore are diffi-cult to be tailored for control structure screening.

A particular BAB method for subset selection was pro-posed byNarendra and Fukunaga (1977)in the context ofpattern recognition.Roberts (1984)presented a FORTRANcode for the algorithm. Over the years, the algorithm was im-proved by several authors (Ridout, 1988;Yu andYuan, 1993).Search speed has been considerably increased through theImproved BAB method byChen (2003). Basically, it is basedon an asymmetricalsolution tree, whose leaves represent tar-get subsets of optimal features, while the root represents theset of all features. The algorithm is based on the assumptionthat the adopted criterion function fulfills the monotonicitycondition to simplify the bound estimation. Thus, any subsetscreening problem, with criterion that satisfies monotonicityproperty, can be efficiently solved using this method. In otherwords, this method is particularly suitable for control struc-ture screening based on simple controllability indices.Caoet al. (1998)modified this method to use with the minimum

singular value for control structure selection. This work ex-tends this method to address the following screening issues:

(1) The screening procedure involves multiobjective search,some objectives may not satisfy monotonicity requiredby BAB algorithm, for example, the right-half-planezeros. To reduce the search space for these objectives,BAB based on a particular criterion which satisfiesmonotonicity should have the screening feature, viz.,

(a) searching all subsets that satisfies��a, where�is the criterion function anda is a bound (Braatz etal., 1996),

(b) ranking all possible subsets to find a certain numberof best subsets.

(2) There should not be any restriction over either thesubset-size or number of input(s) and/or output(s),i.e., searching algorithm should be able to identify thebest possible subset(s) for any series of pre-selectedsubset-sizes irrespective of whether it is square systemor nonsquare.

(3) There should be a provision for joint I/O screening.

The paper is organized as follows. The theoretical detailsof BAB method are discussed in the next section. It in-cludes the basic principle of the theory, newly improved al-gorithm and the discussions about the special requirementsof control structure screening, which are met by the newalgorithm. Section 3 presents the focused criteria function,viz. the minimum Hankel singular value, which is used forcontrollability analysis for unstable plants. Application ofBAB method is shown in Section 4 through a case study onTennessee–Eastman test bed process. A brief conclusion onthe research finding is presented in Section 5.

2. Branch and bound method

2.1. Principle

Let XN = {x1, . . . , xN } be anN-element set andXn bean n-element subset selected fromXN . There areCnN =N !/(N − n)!n! candidaten-element subsetsXn ∈ XN forscreening with criterion cost function�. The objective is tofind ann-element subset,X∗

n, which satisfies

�(X∗n)= max

Xn⊆XN�(Xn). (1)

Let�(XN) be an upper bound estimation of the cost functionover all possible subsets ofXN , i.e.,

�(XN)� maxXn⊆XN

�(Xn) (2)

Therefore,

�(Xs)��(Xt ), if Xs ⊆ Xt . (3)

Y. Cao, P. Saha / Chemical Engineering Science 60 (2005) 1555–1564 1557

Root Level 1

Level 2

Level 3

Level 4

Level 5

4 1 6

1 5 6 2 6 2

3 6 2 2 3 3 6 3 5 5

5 6 2 5 2 5 3 6 6 2 3 5 5 3 3

Fig. 2. The solution tree for aC26 problem.

Particularly, if� satisfies monotonicity property, i.e.,

�(Xs)��(Xt ), if Xs ⊆ Xt, (4)

then estimation can be simplified as,

�(XN)= �(XN). (5)

Let B be a lower bound on the optimal value of�(X∗n), i.e.,

B��(X∗n). (6)

If a k-element subset,Xk, with k >n satisfies

�(Xk)<B, (7)

then, it can be shown by (3),

�(X′n)<B, ∀ X′

n ⊆ Xk. (8)

This means that none of the subsets ofXk can be the opti-mal solution, thus they can be discarded without any furtherevaluation. By applying this principle properly, the compu-tational load can be reduced considerably.

The above BAB principle can be applied to any criterion,where an algorithm is required to generate an upper-boundestimation of the criterion over all possible subsets of thegiven set, i.e., (2). The simplest case is where the criteriasatisfy monotonicity (4). Hence, no upper-bound estimationis required in this case. For simplicity, the monotonicitycondition is presumed in the rest of the section.

2.2. Branch rule

An efficient BAB algorithm requires a proper branchingstrategy. In a traditional BAB algorithm, branching is repre-sented as a solution tree where each subset belongs to oneand only one branch (Fig.2). However, structure of the so-lution tree is depend on the target subset-size. Thus, it is notsuitable for a varying subset-size problem. By introducingthe concepts of fixed-set and candidate-set, a new branchingrule, which suitable for both fixed and variable subset-sizeproblems, is proposed.

Consider a simple example withX2 = {x1, x2}, X3,1 ={x1, x2, x3} andX3,2 ={x1, x2, x4}. Clearly,X2 can be gen-erated from bothX3,1 andX3,2. If X2 can only be generated

C = {xi | i = 1,2, ...,nc}X = (ϕ ,FC)

X3= X - {x3}(ϕ3,F∪ { x1,x2}, { x4

...Xnc})

Xnc= X - {xnc}(ϕnc,F∪ { x1,x2}, {})

X2= X - {x2}

X1= X - {x1}

(ϕ2,F∪ { x1}, { x3...Xnc})

(ϕ1,F, {x2...Xnc})

Fig. 3. Fixed-set and candidate-set propagation in subsets.

from X3,1, then,x4 cannot be eliminated fromX3,2 (fixedelement), whilstx3 must be removable fromX3,1 (candidateelement). More generally, for ann-element setX, the fixed-set and candidate-set ofX are defined as follows.

Definition 1 (Fixed set F). The fixed set,F ⊆ X is thesmallest subset ofX, whose elements are always included inall subsets generated fromX. It has number ofnf elements.

Definition 2 (Candidate set C). The candidate set is the setof elements which are elements ofX, but not elements ofF,i.e.,C =X − F . It hasnc = n− nf number of elements.

According to the definitions, a setX can be representedby a three-tuple,

X = (�, F, C).The following branch rule, illustrated inFig. 3, determineshow F andC are propagated into subsets.

Definition 3 (Branch rule). AssumeX=(�, F, C) andC={x1, x2, . . . , xnc }. LetXk =X − {xk}. Then,

Xk = (�k, F k, Ck), F k := F ∪ {x1, . . . , xk−1},Ck := {xk+1, . . . , xnc }. (9)

Starting from the full set,XN = (�(XN), {}, XN) andrepeatedly applying the branch rule, all subsets are properlybranched.

Lemma 4 (Branch completeness). The branch rule of(9) iscomplete without any redundancy, i.e.,each subset belongsto one and only one branch.

Proof. Let S(X) denote all subsets generated byX. Con-sider, 1� i < j�nc. Then, according to the branch rule,xi ∈Xs if Xs ⊆ S(Xj ). However,xi /∈Xt if Xt ⊆ S(Xi). Thus,


S(Xi) ∩ S(Xj ) = 0, i.e., no subset belongs to more thanone branch. On the other hand, the total number of second-level subsets ofX is (nc − 1) + (nc − 2) + · · · + 1 + 0 =nc(nc− 1)/2=C2

nc. Thus, the second-level subsets ofX are

complete and without any redundancy. It infers that all sub-sets ofX are complete without redundancy.�

Lemma 5. For an m-element subset selection problem, letnb be the number of expandable branches of X. Then,

nb =m− nf + 1. (10)

Proof. Any set with more thanm elements fixed cannotgenerate anm-element subset, thus can be discarded. Hence,the number of branches of a set is equal to the number of itssubsets, which has their fixed-set sizes less or equal tom.If a set hasnb branches, then, according to (9), the sizeof Fnb is m, i.e.,m = nf + nb − 1, which leads tonb =m− nf + 1. �

Remark 6. If a node has only one branch, there is no needto evaluate intermediate nodes one by one until the terminalnode. Based on this observation,Yu and Yuan (1993)de-veloped their BB+ algorithm, which has been shown to befaster than the basic BAB algorithm. To implement the samestrategy with the above branch rule, a subset withnf = mshould be directly converted to the terminal subset by elim-inatingC.

Remark 7. The branch rule (9) is based on the element-index ofC. In theory, the index-sequence is local (only rel-evant to the first-level subsets) and can be arbitrary. For ef-ficiency (Ridout, 1988), the sequence of subset is orderedas�(X1)��(X2)� · · · ��(Xnc) so that the set with leastcriterion value has the largest candidate set, which are morelikely to be discarded although this will increase the numberof criterion evaluations (number of subsets evaluated morethan number of branches). For some problems, where thefirst-level subset criterion functions can be efficiently eval-uated using an iterative formulation, subset criterion sortingwill significantly reduce computational load.

Remark 8. In the data structure of a three-tuple(�, F, C),F andC are fixed-length integers, where each bit representsan element in the set(bit = 1) or not in the set(bit = 0).Thus, the branch rule can be efficiently programmed usinginteger operations.

2.3. Pool of subsets

To solve a variable subset-size selection problem, the con-cept ofpool of subsetis adopted fromYu and Yuan (1992).The pool,P is a dynamic queue, which consists of all ac-tive subsets to be visited. All subsets are stored in the three-tuple structure and sorted according to their criterion values.

Initially, P0 = {XN }. Assume atkth iteration,

Pk = {X1, X2, . . . , Xn},Xi = (�i , F i, Ci), for i = 1, . . . , n,

�1 = max(�1, . . . ,�n). (11)

Then, the pool is dynamically updated by the following ex-panding and contracting rules.

Definition 9 (Pool expanding). Let S(X1) be the set ofall first-level subsets ofX1 generated by eliminating oneelement fromC1 using the branch rule (9). Then, the poolis expanded as

Pk+1 = S(X1) ∪ (Pk − {X1}). (12)

Definition 10 (Pool contracting). Assume the target subsetsize ism. Let nf,i be the size ofF i , D= {Xi |nf,i�m} andT={(�(F i), F i, {})|nf,i=m}. Then, the pool is contractedas

Pk+1 = T ∪ (Pk − D). (13)

The optimal subset of a pool is determined by the follow-ing theorem.

Theorem 11 (Global optimum). For Pk in (11), let ni bethe size ofXi , for i= 1, . . . , n. ThenX1 is globally optimalamong alln1-element subsets ifn1 =min(n1, n2, . . . , nn).

Proof. SincePk consists all active subsets andn1 is theminimum subset size among all pool members, anyn1-element subset will be a subset of one of the pool members.However,X1 has the largest criterion among all pool mem-bers. Hence, it is globally optimal among alln1-elementsubsets. �

Remark 12. The pool algorithm can be applied to both fixedand variable subset-size selection problems. For a fixed-sizeproblem, the size of the first subset,n1 in the current pool iscompared with the target size,m. The pool will be continu-ously updated untiln1 =m. For a variable-size problem, themultiple target sizes,m1,m2, . . . , mk are sorted in descend-ing order. Then, the pool is dynamically updated to findn1matchingm1, thenm2, until all target sizes being matched.

Remark 13. The pool algorithm can also be applied to findl bestm-element subsets. In this case, when the first memberof a pool is size ofm, it will be fixed. The pool is continu-ously updated to find the second member with sizem. Andso on, until the firstl members are all of sizem. Alterna-tively, if the screening problem is to find allm-element sub-sets which have criterion value larger than a given thresh-old, then any pool members whose criterion value less thanthe threshold will be removed from the pool and only sub-sets with size larger thanmwill be expanded. The final poolcontains the solution sets.


Remark 14. The pool algorithm described above is a best-first-search approach. It is normally more efficient than tra-ditional BAB algorithms, which are based on a depth-first-search solution tree (e.g.Cao et al., 1998) in terms of thenumber of subsets expanded. In a depth-first-search algo-rithm, to determine if a subset should be expanded, the cri-terion value of the subset is compared with the current bestbound of the optimal solution,B in (6). However, in thebest-first-search algorithm, the subset criterion value is com-pared with those of all pool members (including B). Hence,the subset is less likely to be expanded than in a depth-first-search algorithm.

Remark 15. Recently,Chen (2003)spotted some redundantevaluations in BAB algorithms, which can be removed toimprove efficiency. That is, when a non-terminal subset,Xis discarded (�(X)�B), all natural subsets ofX, which maynot belong to the branch ofX, can also be discarded. In thepool-based BAB algorithm, this is implemented by checkingthe first pool-set,X1 before expanding. IfC1 has an element,xk, which, when eliminated, will make the generated subsetbe a natural subset of a pool member ranked afterB, thenxk will be moved fromC1 to F 1.

The pool-based BAB algorithm is summarized in the fol-lowing procedures:

Step1: Define XN = (�, {}, XN), a target-size array,M = {m1,m2, . . . , mk} sorted in descending order, andthe number of best subsets required,l. Initialize a subsetpool, P = {XN } and a pointer of the target-size array,I = 1.

Step2: If I > k stop. Otherwise, get current target sizem=M(I), I = I +1, and initialize a pointer,L=0 to indicatethe number of best subsets obtained.

Step3: Let P = {Xp1, . . . , Xpn} be the current pool. Sort�(Xpi ) for i = L+ 1, . . . , n in descending order.

Step4: If pL+1 =m, thenXpL+1 is a new best-subset,L=L+ 1. If L= l, then all best-subsets required have beenfound, goto Step 2. Otherwise, continue.

Step5: CheckXpL+1 and adjust itsF and C according toRemark 15.

Step6: Perform pool expanding using (12).Step7: Perform pool contracting using (13).Step8: Return to Step 3.

2.4. Joint I/O screening

In most cases, due to lack of suitable tools, the importanceof inputs and outputs have to be assessed separately i.e.,inputs have to be screened with a output set fixed, whilst toscreen outputs, an input set has to be presumed. Obviously,such a way to screen inputs and outputs cannot reveal theeffect of cross-combination between inputs and outputs. Asa result, global solution of the best I/O combination is notguaranteed. To solve this problem, input and output variables

are considered together in the BAB search procedure via anindex translation scheme. More specifically, thekth input isthekth variable of the joint I/O set, whilst thekth output is theM+kth variable in the I/O set (whereM is the total numberof inputs). In this way, any input–output combination can betranslated as a subset of the total I/O set. To avoid a subsetwith pure inputs or pure output being selected, the criterionfunction of such a subset is defined as zero. Therefore, sucha subset will never be selected by the BAB algorithm.

Finally the screening result can be represented graphicallyfor criterion function vs. I/O subset-size. In this way, thescreening is done both vertically (a group of subsets withthe same size) as well as horizontally (subsets with differentsize). Wherein the best I/O combination could be determinedat the point where the increase of criterion value against theincrease of subset-size becomes negligible, or at the pointwhere the controllability criterion is at a satisfactory levelbut the I/O subset-size is minimum.

3. Stabilizing control structure selection

Stabilization of an unstable plant is mandatory before de-signing higher level control structure of the system. There-fore, as argued byHavre and Skogestad (2003), stabilizingcontrol usually belongs to a secondary control layer, whereoutput performance is not an important issue because theoutput setpoint is normally determined by a higher level con-troller. Instead, the input usage is more important for sta-bilizing control because it directly relates to possible inputsaturation, which may lead to instability.Glover (1986)hasproven that the minimum input usage to stabilize an unsta-ble plant can be quantified by the minimum Hankel singularvalue (HSV) of the anti-stable projection of the plant. Thisresult has been applied to stabilizing control structure selec-tion Cao and Saha (2003). However, global optimality is notguaranteed in the direct application of the minimum HSVbecause of the combinatorial nature of the evaluation.

In this section, an algorithm is proposed to evaluate theminimum HSV based on the BAB approach described inSection 2. This algorithm gives global ranking of minimumHSV for all possible I/O combinations without exhaustiveevaluation of the index.

3.1. Hankel singular value

Consider a linear system withnx states,ny outputs andnu inputs, denoted byG= {A,B,C}, whereA, B andC arestate, input and output matrices of the state space equations.

The controllability and observability gramians,Lc andLo of the system are symmetric matrices and satisfy thefollowing Lyapunov equations:

ALc + LcAT + BBT = 0, (14)

ATLo + LoA+ CTC = 0. (15)


The HSV,�=�1��2 · · · ��nx =� of the system is definedas the square roots of the eigenvalues ofLoLc, i.e.,

�k(G)=√

�k(LoLc), k = 1,2, . . . , nx. (16)

For a subsystem with theith output andjth input, its control-lability and observability gramians and HSV are denoted asLcj , Loi and�ijk , respectively. Then according to (14) and(15) the following equations can be obtained:

Lc =nu∑

j=1

Lcj , Lo =ny∑

i=1

Loi. (17)

If the system,G possesses unstable poles, then it is possibleto decompose the system into stable (Gs ) and anti-stable(Ga) projections, i.e.,G=Gs +Ga (Safanov et al., 1987).Glover (1986)studied the robust stabilization problem of alinear multivariable open-loop unstable systemG. The taskfor a controllerK to stabilize (G + �) for all allowableperturbations� with minimum control effort is to minimize‖KS‖∞, whereS := (I −GK)−1 is the sensitivity matrix.It has been proved byGlover (1986)that

min‖KS‖∞ = 1

�([Ga(−s)]∗) . (18)

According to Glover’s theorem, it is desired to select aninput and output combination which has the largest mini-mum HSV. Understandably, it is possible to extract subsys-tems with m selected inputs andn selected outputs fromG. Suppose the selected inputs and outputs have index sets,J = {j1, j2, . . . , jm} and I = {i1, i2, . . . , in}, respectively.Controllability and observability gramians for any such sub-system will be

LcJ =∑

j∈JLcj , LoI =

∑

i∈ILoi . (19)

The HSV can be calculated in a similar manner as in (16).Hence, the best combination of inputs and outputs are to bedetected by comparing the minimum HSV.

There are a couple of issues involved with the HSV whichare discussed below:

• Pure integrator: One particular issue with the HSV is thatit is not well-defined for systems which have pure integra-tors, i.e., poles at zero. Presence of pure integrators in thesystem will invalidate the above theory as the matrixA in(14) and (15) will be singular. To counter this problem, adiagonal matrix, having a small constant as elements, canbe added to the system matrixA before decomposing itinto stable and anti-stable parts. Through this operation,the poles will be shifted slightly towards right without dis-turbing overall characteristics of the process. However, thepure integrators will move to the right-half-plane (RHP)of imaginary axis and will be converted into small positivepoles which can later be extracted inGa .

• Combinatorial problem: The selection of inputs and out-puts by using the HSV is a combinatorial problem. Thisproblem can be tackled by using the BAB algorithm de-scribed previously.

The following theorem shows that the minimum HSV sat-isfies the monotonicity condition (4). Therefore the BAB ap-proach can be simplified to take this advantage as explainedin Section 2.

Theorem 16. If I1 ⊆ I2 andJ1 ⊆ J2 then�I1J1 ��I2J2.

Proof. If �I2J2=0, then �I1J1=0. For the case�I2J2 �=0,LoI2 and LcJ 2 are symmetric and positive definite. LetJ1⊆J2 and I1=I2=I , then LcJ 2=LcJ 1+Lc(J2−J1). Thus,

�(LoILcJ 2) = �(L1/2oI LoJ 2L

1/2oI ) = �(L1/2

oI LcJ 1L1/2oI +

L1/2oI Lc(J2−J1)L

1/2oI ). AsL1/2

oI LcJ 1L1/2oI andL1/2

oI Lc(J2−J1)L1/2oI

are both symmetric, according toWilkinson (1965),�(LoILcJ 2)��(L1/2

oI LcJ 1L1/2oI )=�(LoILcJ 1). Similarly, it

can be proved that�(LoI2LcJ )��(LoI1LcJ ), if I1 ⊆ I2.Hence�I2J2 ��I2J1 ��I1J1. �

3.2. I/O screening procedure

In spite of these advantages, BAB method with the HSVdoes have a demerit, because only part of effect of RHP Ze-ros is identified with this method. If an RHP zero is close toan RHP pole then the HSV will be very small indicating thesystem to be either uncontrollable or unobservable. How-ever, RHP zeros at other locations have little effect on theHankel SV. Such RHP zeros do cause difficulty in stabiliza-tion of system. According to the principle of Root Locus,closed-loop poles start from open-loop poles and finish atopen-loop zeros as the closed-loop gain varying. Therefore,RHP zeros will normally cause an unstable plant not possi-ble to be stabilized by proportional-only controllers becauseextra poles and zeros from the controller are required toshape the Root Locus. Thus, RHP zeros should be avoidedin control structure selection, if possible. However, RHPzeros cannot be directly handled by BAB method. Neverthe-less, we can use BAB method to secure an overall rankingon the Hankel SV and then examine the existence of RHPzeros in those ranked structures starting from top towardsbottom (considering the fact that the “mathematically” beststructure is at the top) until the most top structure with-out any RHP zeros is found. Thus the following is thestep by step procedure of control structure screening withBAB method using the minimum HSV as controllabilitycriterion:

Step1: Input and output scaling is essential for reliable anal-ysis. Here inputs and outputs should be scaled to theirpossible range, e.g. inputs could be scaled to their nearestbound (Cao et al., 1996) whilst output might be scaled totheir noise level (Skogestad and Postlethwaite, 1996).


Step2: Shift poles to convert pure integrator(s) to smallRHP pole(s).

Step3: All unstable poles are extracted out of the processthrough projection method.

Step4: BAB method is applied to the anti-stable sub-systemto screen the best possible subsets of I/O base on theminimum HSV.

Step5: The top ranked subsets are examined for possibleexistence of RHP zeros.

Step6: A decision is reached about the best I/O combina-tion.

4. Application of branch and bound algorithm

The BAB technique is applied to a highly-integratedchemical plant, viz. Tennessee-Eastman (TE) process. Thisprocess has been chosen with an intention of demonstratingusage of controllability measure discussed in the previoussection, i.e., the minimumHankel singular values.

4.1. Tennessee–Eastman process

The Tennessee–Eastman test-bed problem (Downs andVogel, 1993) involves the control of five unit operations: anexothermic two-phase reactor, a water-cooled condenser, acentrifugal compressor, a flash drum and a product strip-per. The simulated plant has 41 process variables and12 manipulated variables which are modelled with 50state variables. The 12 manipulated variables are the fourfeed rates, the purge rate, the agitation rate, steam rate,condenser coolant rate, reactor coolant rate, compressorrecycle, flash drum discharge rate and the stripper pro-duction rate. Out of 41 process variables there are 22controllable outputs including level, pressure, tempera-ture, flow and 19 composition indicators. Complete listsof input and output variables are shown inTables 1and2respectively.

Table 1Tennessee–Eastman process manipulated variables

Variable description Variable number Base case value (%) Scaling (%)

D feed flow XMV 1 63.053 36.947E feed flow XMV 2 53.980 46.020A feed flow XMV 3 24.644 24.644A and C feed flow XMV 4 61.302 38.698Compressor recycle valve XMV 5 22.210 22.210Purge valve XMV 6 40.064 40.064Separator pot liquid flow XMV 7 38.100 38.100Stripper liquid product flow XMV 8 46.534 46.534Stripper steam valve XMV 9 47.446 47.446Reactor cooling water flow XMV 10 41.106 41.106Condenser cooling water flow XMV 11 18.114 18.114Agitator speed XMV 12 50.000 50.000

Details of the process are available elsewhere (Downsand Vogel, 1993). A model of the process is generated byRicker (2002)in Simulink software. Open loop simulationof the model indicates that the process is unstable in nature.The objective is to select the inputs and outputs in order tostabilize the complete system.

4.2. Input–output selection

The entire analysis has been carried out through MATLABsoftware. The step-by-step procedure, described in Section3.2, is followed while doing the analysis. Scaling factorsfor each I/O subset-size are obtained from the model itself.Inputs are already scaled in the model within 0–100% range.However, an extra scaling factor, viz. the maximum possibledeviation from its steady state value for each input variable,is incorporated. More specifically, scaling=min(x,100−x),wherex is the base case value, as shown inTable 1. Whereas,the output variables are scaled using noise level found fromthe model, as shown inTables 2.

The scaled process is decomposed into stable and anti-stable subsystems usingSTABPROJsubroutine of the ro-bust control toolbox (Safanov et al., 1987). The anti-stablesubsystem has two pairs of positive complex poles, 3.0648±5.0837i and 0.024973± 0.15521i and two pure integrators(numerical value is of the order of 10−9).

Using the BAB method, 10 sets of best I/O combinationsare computed as inFig. 4. The figure shows the minimumHSV against the subset-size. Although the calculation isperformed for subset-sizes from 1 through 34, HSV does notincrease notably over subset-size of 20. It can be noted fromFig. 4 that the minimum HSV for up to a subset-size of 4 isnearly zero. By mathematical derivation, it can be provedthat it is actually zero. The significance of this result is aconclusion that, it is impossible to stabilize the plant withany combination of a total number of inputs and outputs lessthan 4. That means a minimum total of 5 inputs and out-puts are necessary to achieve that stabilization. However, in


Table 2Tennessee–Eastman process output variables

Variable description Variable number Base case value Scaling (noise level) Unit

A feed XMEAS 1 0.2502 0.0012 Kscm/hD feed XMEAS 2 3664 18 Kg/hE feed XMEAS 3 4509.3 22 Kg/hA and C feed XMEAS 4 9.3477 0.05 Kscm/hRecycle flow XMEAS 5 26.902 0.2 Kscm/hReactor feed rate XMEAS 6 42.339 0.21 Kscm/hReactor pressure XMEAS 7 2705 0.3 KPa gaugeReactor level XMEAS 8 75 0.5 %Reactor temperature XMEAS 9 120.4 0.01 ◦CPurge rate XMEAS 10 0.33712 0.0017 Kscm/hProduct separator temperature XMEAS 11 80.109 0.01 ◦CProduct separator level XMEAS 12 50 1 %Product separator pressure XMEAS 13 2633.7 0.3 KPa gaugeProduct separator underflow XMEAS 14 25.16 0.125 M3/hStripper level XMEAS 15 50 1 %Stripper pressure XMEAS 16 3102.2 0.3 KPa gaugeStripper underflow XMEAS 17 22.949 0.115 M3/hStripper temperature XMEAS 18 65.731 0.01 ◦CStripper steam flow XMEAS 19 230.31 1.15 Kg/hCompressor work XMEAS 20 341.43 0.2 KWReactor cooling water outlet temp XMEAS 21 94.599 0.01 ◦CSeparator cooling water outlet temp XMEAS 22 77.297 0.01 ◦C

0 5 10 15 200

200

400

600

800

1000

1200

1400

1600

1800

Min

imum

Han

kel S

ingu

lar

Val

ue

Subset size (Number of Inputs and Outputs)

Fig. 4. HSV vs. subset-size for 10 best cases of I/O combination.

terms of decentralized control a 3× 2 or 2× 3 system hasno advantage at all compared to a 3× 3 system as far asthe complexity of controller design is concerned. Hence, theminimum HSVs for subset-size 6 are ranked for the best 10sets inTable 3.

Based on the discussions about RHP zeros in Section 3.1,it is observed (Table3) that best control structure withoutthe presence of RHP zeros is the 4th structure inTable 3,i.e., with liquid levels in stripper and the separator (outputs12 and 15), reactor temperature (output 9), separator andstripper outflows (inputs 7 and 8) and reactor cooling water

Table 3Controllability indices of 10 best I/O set of size 6

Set Input No. Output No. � RHP zeros

1 1, 8, 10 9, 12, 15 926.65 0.152 and 109.5222 4, 8, 10 9, 12, 15 878.57 0.1103 and 5.6057± 10.5721i3 2, 8, 10 9, 12, 15 868.89 0.1571 and 29.22484 7, 8, 10 9, 12, 15 721.47 None5 4, 8, 10 11, 12, 15 688.346 2, 8, 10 11, 12, 15 681.307 7, 8, 10 11, 12, 15 673.458 2, 4, 10 11, 12, 15 660.259 2, 4, 10 9, 12, 15 658.35

10 1, 8, 10 12, 15, 21 610.55

0 5 10 15 20 25 30 350

5000

10000

15000

Num

ber

of E

valu

atio

ns

Subset Size

Pool inheritedPool independentDepth-first BAB

Fig. 5. Total number of evaluations executed by different algorithms.


0 0.2 0.4 0.6 0.8 140

45

50

55

60

Sep

arat

or le

vel (

%) Controlled output profiles

0 0.2 0.4 0.6 0.8 137

37.5

38

38.5

39

Sep

. pot

liq.

flow

(%

) Manipulated Input profiles

0 0.2 0.4 0.6 0.8 140

45

50

55

60

Str

ippe

r le

vel (

%)

0 0.2 0.4 0.6 0.8 145.5

46.5

47.5

Str

. liq

. pro

d. fl

ow (

%)

0 0.2 0.4 0.6 0.8 1120.3

120.4

120.5

Time (hr)

Rea

ctor

tem

pera

ture

(°C)

0 0.2 0.4 0.6 0.8 140

40.5

41

41.5

42

Time (hr)R

eact

or w

ater

flow

(%

)

Fig. 6. Closed-loop response of TE plant using stabilizing P-only controllers.

flow (input 10). The observation is identical with the resultsobtained by (Cao and Saha, 2003) and similar to those sug-gested byHavre and Skogestad (1998)andMcAvoy (1998).However, the guarantee for best I/O combination is obtainedin this work through BAB technique. Moreover, this guar-antee is obtained with only 16160 computations instead ofpossible 17175670785 in an exhaustive search. The effi-ciency of the improved BAB algorithm is compared witha depth-first search based algorithm (Cao et al., 1998) inFig. 5. The main advantage of the new algorithm is for avariable subset-size selection problem, where the total num-ber of evaluations is significantly reduced due to the reuseof the pool of subsets. However, even for a fixed subset-size problems, the new algorithm is still more efficient thandepth-first search approaches due to the reason explained inRemark 14. This is also shown in the comparison figure,where the total number of evaluations of the pool-based al-gorithm using independent pools (every time performing afixed subset-size selection, a new pool is initialized) is stillless than depth-first search algorithm.

The closed loop simulation results are shown inFig. 6. Itis observed that stabilization of the process is achieved withthree unit-gain proportional controllers. Simulation resultsare obtained without any disturbance but with measurementnoise.

5. Conclusion

An improved BAB algorithm based on the best-first searchapproach is proposed. The new algorithm is more efficient

than other depth-first search based algorithms. Applicationof the improved Branch and Bound method is demonstratedin this work for I/O screening using a controllability cri-terion, viz., the minimum Hankel singular value. The re-sults, obtained through simulation, show that the proposedmethod inherits all the advantages of the original branch andbound method. Nevertheless, it demonstrates a few moremerits than the original in terms of efficiency and subset-sizescreening. A screened set of I/O combinations is more usefulthan a single set of best combination because the screenedset provides more options for control structure selection ifthe mathematically best combination leads to RHP zeros. Acase study with Tennessee–Eastman test bed problem showsthat it is possible to stabilize the open-loop unstable plantwith minimum control effort, by using three decentralizedP-controllers with unit gains. It has also been proved thatit is impossible to stabilize the plant with total number ofinputs and outputs less than four.

Acknowledgements

The authors gratefully acknowledge the financial supportprovided by the EPSRC, UK (Grant number GR/R 57324)for this research work.

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Improved branch and bound method for control structure screening

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