Variable Pairing Selection Based on Individual and OverallInteraction Measures

Zhong-Xiang Zhu*

Honeywell Hi-Spec Solutions, 343 Dundas Street, London, Ontario, Canada N6B 1V5

A new steady-state interaction measure, the relative interaction array (RIA), is explored in termsof its features, properties, and relationship to the relative gain array (RGA). It is demonstratedthat the RIA offers a better measure of individual interactions than the RGA. Moreover, it isshown that the RIA also contains direct information about closed stability, integrity, androbustness. Taking advantage of its definiteness in measuring the amount of loop interactionby its size, the RIA is further extended to propose a new overall interaction measure which isable to avoid ambiguities associated with the RGA pairing criteria and to overcome problemsassociated with other overall interaction measures. Interestingly, the RIA-based overallinteraction measure leads to a “corrected” version of the intuitive RGA-based one. The finalRIA-based pairing criteria provide a comprehensive and reliable solution to the selection of thebest variable pairing and thus represent a promising tool. Also, variable pairing criteria on thebasis of interaction measurement, stability, and integrity as well as robustness considerationsare systematically addressed with new insights provided.

1. Introduction

Decentralized control is commonly used to tacklemultivariable processes. Control structure selection orvariable pairing choice usually represents the first mainissue control engineers face in designing decentralizedcontrol systems. The major objective in selecting thebest variable pairing has been primarily to minimizeloop interaction so that the resulting multivariablecontrol system mostly resembles its SISO counterpartsand the subsequent controller tuning is largely facili-tated by independent design (Seborg et al., 1989). Thisobjective is usually accomplished by utilizing variousinteraction measures, predominantly the steady stateones, as tools to screen possible pairing alternatives(Bristol, 1966; McAvoy, 1983; Majares et al., 1986; Zhuand Jutan, 1993a). In particular, the relative gain array(RGA) (Bristol, 1966) has been the most widely used one.Originally proposed as an empirical measure of

interaction, the RGA pairing rule has found widespreadacceptance, particularly after the improvement on closed-loop stability considerations by using the Niederlinskiindex (NI) (Niederlinski, 1971) as a stability rule (seeMcAvoy, 1983; Grosdidier et al., 1985; Shinskey, 1988;Seborg et al., 1989; Skogestad et al., 1990). The RGA-based tool is further enhanced by taking into accountintegrity (Chiu and Arkun, 1990; Zhu and Jutan, 1993a)as well as robustness (Yu and Luyben, 1987; Zhu andJutan, 1993b) considerations, with the joint use of theRGA and the NI. The following briefly summarizes theRGA-based criteria and the perspective roles the RGAand the NI play in variable pairing choices:The original RGA offers an interaction rule by its size.The NI provides a necessary stability condition by its

sign.The signs of the RGA elements lead to the integrity

rules.The sensitivity of the RGA elements to gain uncer-

tainties presents the robustness rule.The RGA has been extended to other unconventional

situations such as disturbance rejection (Chang and Yu,

1992), nonsquare systems (Reeves and Arkun, 1989;Chang and Yu, 1990), block structures (Manousiouthakiset al., 1986), and nonlinear processes (Manousiouthakisand Nikolaou, 1989). The NI has also been extendedto provide an overall measure of steady-state interaction(Zhu and Jutan, 1993a) and to include controller designduring variable pairing decisions (Zhu and Jutan, 1995).However, the RGA only measures loop interaction in

individual control loops since it is defined on the basisof individual loops. Moreover, the amount of interactionindicated by the distance of the size of a RGA elementand the interaction-free reference value (1.0), i.e., thecloseness of RGA elements to 1.0 required by the RGApairing rules, is rather qualitative. As a result, ambi-guity may arise when several alternatives satisfy theRGA-based rules (Zhu and Jutan, 1993a). Hence, anoverall interaction measure, which can be used toidentify a particular pairing as the final one exhibitingtruly the minimum interaction, is required.Majares et al. (1986) proposed a measure of the

overall interaction in decentralized control systems froman algebraic perspective. However, in comparison to theRGA-based rules, this measure lacks sufficient informa-tion about system stability and integrity (Majares et al.,1986; Zhu and Jutan, 1993a). An intuitive overallinteraction measure based on the distance of the RGAand 1.0 was proposed by Zhu and Jutan (1993a). Thismeasure is found to be inadequate under some circum-stances. Zhu and Jutan (1993a) suggested using theNI, actually the size of the NI, as an overall interactionmeasure to avoid possible ambiguities associated withthe RGA. Hence, with the overall interaction measureby its size, in addition to the traditional stability andintegrity indications by its sign (as in the conventionalRGA-based rules), the NI is proposed as a comprehen-sive screening tool for variable pairing choices. This NI-based pairing criterion leverages heavily on the rela-tionship between the NI and the RGA. It representsan empirical rule and thus may lead to incorrectdecision on the best pairing (see Zhu, 1993; McAvoy,1993).This article starts with a brief review and critical

assessment of the existing interaction measures andvariable pairing rules. New insights into the RGA and

* Email: john.x.zhu@ontario.honeywell.com. Telephone: (519)640 6617. Fax: (519) 679 3977.

4091Ind. Eng. Chem. Res. 1996, 35, 4091-4099

S0888-5885(96)00143-1 CCC: $12.00 © 1996 American Chemical Society

the RGA-based pairing rules are offered. Variablepairing selection is systematically addressed from thefollowing aspects: amount of interaction; closed-loopstability; closed-loop integrity; robust stability againstgain errors. An updated RGA-based pairing criterionwith individual rules for interaction, stability, integrity,and robustness requirements integrated is presented.Variable pairing rules based on the existing measuresof overall interaction are also evaluated.Using the loop decomposition approach to decompose

a decentralized control system into equivalent SISOloops with loop interaction structurally embedded (Zhu,1993, 1996; Zhu and Jutan, 1996), a new steady-stateinteraction measure, called the relative interactionarray (RIA), is presented and further elaborated byclosely exploring its features, properties, and relation-ship to the RGA. It is shown that the RIA provides abetter measure of interaction than the RGA. Informa-tion about closed-loop stability and integrity as well asrobustness contained in the RIA is also discovered,giving further rise to the RIA as a comprehensive toolfor variable pairing choices. Like the RGA, however,RIA lends itself as an interaction measure also on thebasis of individual loops and, hence, may also lead toambiguities regarding the best variable pairing choice.A new measure of the overall interaction on the basisof the RIA is subsequently developed. The new RIA-based overall interaction measure is capable of avoidingpossible ambiguities involved in individual interactionmeasures and overcoming problems associated withother overall interaction measures. Noticeably, theproposed overall interaction measure leads to a cor-rected version of the intuitive overall interaction mea-sure using the RGA by Zhu and Jutan (1993a). The bestvariable pairing is suggested to be the one exhibitingthe minimum overall interaction and meanwhile satis-fying stability and integrity, as well as robust stabilityrequirements, governed by the proposed RIA basedcriteria. Consequently, the RIA-based pairing criteriarepresent a comprehensive and reliable tool in makingvariable pairing decisions.

2. RGA-Based Pairing Rules

2.1. RGA as an Interaction Measure. As origi-nally introduced by Bristol (1966), a relative gain, λij,in a control loop corresponding to a given paired elementof the process, gij(s) ∈ G(s), i.e., uj-yi pairing, in adecentralized control system is defined as

where gij denotes the gain of the process element in theindependent SISO subsystem and gij represents the gainof the equivalent process in the interactive environmentwith all the loops closed. The following remarks areworth noting about the definition of the relative gain.Remarks. 1. gij is meaningful only if the subsystem,

composed of the original control system with the uj-yiloop removed, is stable; i.e., the system possessesintegrity against single-loop failure.2. gij is obtained by assuming that all the outputs

except yi in response to a step change in uj converge tozero at steady state, i.e., the subsystem with the uj-yiloop removed achieves perfect (offset free) steady-statecontrol. This is the case when integral control isadopted in each control loop.

3. gij reflects the effects of all the other loops on theuj-yi loop due to the existence of the hidden loops (seeShinskey, 1988).4. gij indicates how much the process gain deviates

from the original gain, gij, due to interaction. Hence,the difference between the two quantities reflects theseverity of loop interaction.5. Equivalently, λij constitutes a measure of interac-

tion in the uj-yi loop by its deviation from unity.All the relative gains corresponding to all the ele-

ments of a square process constitute the RGA, whichcan be conveniently calculated as (Shinskey, 1988)

where X denotes the element by element multiplicationof matrices and superscript T denotes the transposeoperation of a matrix.The RGA possesses the following main properties:1. It is independent of input and output scaling.2. Each row and each column sums up to 1.0.3. λij is independent of how the other n - 1 loops are

paired.4. RGA ) I for diagonal and triangular processes.Property 3 above, which is often overlooked, is the

result of the perfect control assumption. It makes theRGA particularly attractive in making variable pairingdecisions, since, unlike other tools such as the NI(defined later), all the possible pairings can be scannedsimultaneously without a prior requirement for a givenpairing. Property 4 has traditionally been attributedto the RGA as a limitation of being unable to measurethe one-way interaction. In fact, property 4 reveals theinherent property of the system, i.e., there exists no one-way interaction in triangular systems. This is clear bynoticing that interaction is caused by the hidden loops(Shinskey, 1988) which disappear in triangular systems,since signal can no longer loop back to the original loop.As an interaction measure, the RGA leads to the

following well-known variable pairing rule.RGA Interaction Rule. Input and output variables

of a process should be paired in such a way that all thecorresponding RGA elements are closest to 1.0.Note that the closeness to 1.0 of RGA elements is

rather qualitative, and hence, ambiguity arises as tohow to select the RGA elements closest to 1.0 in theabove interaction rule. Apparently, the pairing rule isaimed at achieving the minimum interaction in eachcontrol loop in the system. However, the minimuminteraction requirement is not sufficient enough todecide on the desired pairings. Other requirements,particularly the closed-loop properties such as stability,integrity, and robust stability, should be simultaneouslyconsidered. The RGA alone provides no completeanswer in this regard, and hence, other tools such asthe NI have to be jointly employed.2.2. NI as a Stability Condition. The NI is

defined, also using steady-state gains of a process, as(Niederlinski, 1971)

where det(A) denotes the determinant of matrix A andGh ) diag(G). The sign of the NI, i.e., NI > 0, providesa necessary stability condition (Niederlinski, 1971;Grosdidier et al., 1985; Zhu and Jutan, 1995) and,consequently, constitutes a complementary tool to theRGA in variable pairing selection, as stated below.

λij )gijgij

(1)

RGA ) G X (G-1)T (2)

NI )deg(G)deg(Gh )

(3)

4092 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996

NI Stability Rule. Input and output variablesshould be paired in such a way that the resulting systemleads to a positive NI.Notice that the stability rule is stated independent

of the interaction rule. Various rules will be integratedat the end of this section to give the final RGA-basedpairing criteria. Observations regarding the definitionof the NI and its use as a pairing tool are given below.Remarks. 1. The calculation of the NI requires

diagonal pairings. Hence, nondiagonal pairings arerequired to be rearranged as diagonal ones by placingthe paired elements to diagonal positions.2. G(s) is assumed to be rational and strictly proper,

particularly for PI-type controllers.3. The independent subsystem is required to be

stable.4. Originally, the same controller designed for the

independent subsystem is required when closing all theloops. Zhu and Jutan (1995) alleviated this restrictionto allow for tuning of the independently designedcontroller.5. The NI is traditionally used merely as a stability

condition by its sign only. Zhu and Jutan (1993a)utilized its size as an interaction measure as well (seelater).2.3. RGA, NI, and Integrity. Integrity represents

an important property of a decentralized control systemdesired during variable pairing choices. The followingrelationship between the NI and the RGA offers asolution to the integrity issue (Chiu and Arkun, 1990;Zhu and Jutan, 1993a):

where NI(i) denotes the NI of subsystem of G with ithrow and ith column removed.Clearly, NI(i) > 0 provides a necessary stability

condition for the above subsystem, i.e., a necessarycondition for integrity against a single loop (ith loop)failure. Chiu and Arkun (1990) also considered systemintegrity against any combination of loop failure. Ap-parently, integrity against single-loop failure is of morepractical importance. Naturally, the RGA lends to anintegrity condition (Zhu and Jutan, 1995); i.e.,

Integrity Rule. Input and output variables of aprocess should be paired in such a way that all the RGAelements corresponding to the paired elements arepositive.Implications of the sign of the RGA are studied, and

pairing on positive RGA elements is also suggested byBristol (1966), McAvoy (1983), Grosdidier et al. (1985),and Morari and Zafiriou (1989). Again, the integrityrule is stated independently and will be combined withother rules to define the final criteria for variablepairing.Remarks. 1. The same assumptions required for NI

discussed previously apply to NI(i) and hence to λii aswell.2. Only paired elements, i.e., diagonal pairings, are

concerned. Hence, alternative pairings need to bearranged as diagonal ones one at a time.3. Only integrity against single-loop failure is con-

sidered.4. The integrity rule is subject to the stability rule.2.4. RGA, NI, and Robust Stability. Robustness

in the face of model uncertainties also represents an

important issue in any control system and, hence,deserves close investigation during variable pairingdecisions. The RGA is found to contain very usefulinformation in this regard. The sensitivity of the RGAelements to model error (single gain error) is given by

The above equation was initially demonstrated byGrosdidier et al. (1985) and Yu and Luyben (1987). Zhuand Jutan (1993b) provided a rigorous derivation andgeometrical interpretations. Clearly, large RGA ele-ments imply that loop interaction under the pairing isvery sensitive to uncertainties in process gains.A quantitative measure of stability robustness, i.e.,

a necessary condition for robust stability, in face of gainerrors is given by

where ∆gij denotes the gain error in gij. Yu and Luyben(1987) developed the above relation but in a qualitativemanner. Zhu and Jutan (1993b) offered a theoreticalderivation. Apparently, eq 7 indicates that pairing onlarge RGA elements results in a system sensitive to gainerrors, i.e., lack of stability. Skogestad and Morari(1987) also suggested avoiding large RGA elements fromother perspectives. Thus, we have the following.Robust Stability Rule. Input and output variables

should be paired in such a way that large RGA elementscorresponding to the paired process elements should beavoided.Remarks. 1. Only paired elements are concerned.2. Both the size and direction of model errors are

important in characterizing robust stability. In par-ticular, only one-way errors lead to stability concerns(see Zhu and Jutan, 1993b).3. Only independent model errors in single gains are

addressed. However, the rule is likely applicable tomultiple correlated errors as well (see Zhu and Jutan,1993b).4. Nominal stability and integrity against single-loop

failure, i.e., stability and integrity rules, are required.5. Although the above robustness measure is ex-

pressed in terms of RGA elements, the NI stabilitycondition actually provides the theoretical basis (see Zhuand Jutan, 1993b).2.5. Final RGA-Based Pairing Criteria. When

making decisions on the final variable pairing choice,requirements for minimum interaction, stability, integ-rity, and robustness should be jointly considered. Thecombination of the interaction rule, the stability rule,and the integrity rule as well as robustness rule leadsto the final RGA-based pairing criterion, as below.Final RGA-Based Pairing Criterion. Manipulated

and controlled variables in a decentralized controlsystem should be paired in such a way that (a) thepaired RGA elements are closest to 1.0, (b) the NI ispositive, (c) all the paired RGA elements are positive,and (d) large RGA elements should be avoided.Notice that the robustness requirement coincides with

the minimum interaction requirement. As a result,large RGA elements will be excluded by the RGAinteraction rule. The redundancy of the robustness ruleactually reveals a limitation of the RGA as an interac-tion measure. Specifically, only large RGA elements are

NIλii ) NI(i) ∀ i (4)

λii > 0 ∀ i (5)

dλijdgij

)(1 - λij)λij

gij(6)

∆gijgij

< - 1λij

(7)

Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4093

dismissed but by both interaction and robustness rules.Very small RGA elements, also known as harmful,would be favored. We will show later that the interac-tion rule needs to be enhanced, while the robustnessrule deserves its position in making pairing decisions.In the RGA-based pairing criteria, requirement a is

solely a result of the RGA, whereas other requirementsessentially result from the NI. Noticeably, the stabilityrule, the integrity rule, and the robust stability ruleappear to remain largely unchanged, whereas alterna-tives to the (steady-state) interaction rule have beenconstantly proposed (e.g., Bruans and Smith, 1982;Mijares et al., 1986; Zhu and Jutan, 1993a). Anothermajor limitation of the RGA as an interaction measurelies in the fact that ambiguities arise when severalpairings individually satisfy the interaction rule, sincethere exists no criterion to identify the best pairing. Thefollowing section will discuss some of the overall inter-action measures proposed to avoid the ambiguity.

3. Existing Overall Interaction Measures

3.1. Intuitive RGA-Based Measure. Zhu andJutan (1993a) proposed an intuitive measure of theoverall interaction on the basis of the distance of theRGA and 1.0 as follows.Intuitive RGA-Based Overall Measure. The final

pairing among alternatives satisfying the RGA interac-tion rule may be decided by the minimum overalldistance of the paired RGA elements and 1.0; i.e.,

where λijk denotes the paired RGA elements correspond-ing to the kth alternative.The intuitive overall interaction measure is aimed at

avoiding ambiguities when using the RGA interactionrule by identifying the final pairing as the one showingminimum overall interaction. However, the overallinteraction measure inherits the same limitations fromthe RGA. For instance, while large RGA elements areexplicitly penalized (this coincides with the interactionrule and the robustness rule), very small positive RGAelements, which indicate a severe shift of the processgains due to loop interaction, would be favored. This isdue to the fact that the distance between RGA elementsand 1.0 does not quantify the amount of interaction.Hence, the intuitive RGA-based overall interaction isinadequate for practical use.3.2. NI Interaction Measure. The NI has been

extended to measure the overall interaction by its size,in addition to its stability and integrity rules by its sign,as stated below (Zhu and Jutan, 1993a).NI Interaction Pairing Rule. Variables should be

paired in such a way that the resulting pairing corre-sponds to an NI closest to 1.0.The NI interaction rule is based on empirical observa-

tions of the definition of the NI and is justified largelyfrom the relationship between the size of the RGA andthat of the NI. The NI interaction rule has been foundto be capable of avoiding ambiguities in using the RGAinteraction rule. Together with the stability rule by theNI and the integrity rule by the NI applied to sub-systems with loops removed (Zhu and Jutan, 1993a) aswell as the robustness rule (Zhu and Jutan, 1993b), theresulting NI-based pairing criteria appear to provide aself-contained comprehensive tool for variable pairingchoices. However, in addition to its lack of theoreticalbasis and physical significance, the NI interaction rule

has to rely on the RGA for preliminary screening ofalternatives, taking the calculational advantage of theRGA. More importantly, due to its empirical nature,the NI overall interaction measure may result in incor-rect choice of the best pairing (Zhu, 1993; McAvoy, 1993;also see examples later).3.3. Jacobi Eigenvalue Criterion. Majares et al.

(1986) proposed an overall interaction measure frompurely algebraic perspectives in terms of the difficultiescaused by the off-diagonal elements in performing theinverse of the process gain matrix. The resultingpairing rule, referred to as the Jacobi eigenvaluecriterion, is given as follows.Jacobi Eigenvalue Criterion. Variables should be

paired so that the Jacobi eigenvalue matrix defined as

has the smallest spectral radius, i.e.,

The Jacobi eigenvalue interaction measure is foundto be equivalent to the NI interaction measure in mostcases in terms of the amount of overall interaction. Thisis clear by noticing the following relationship (see Zhuand Jutan, 1993a),

where θi ∈{θ1, θ2, ..., θn} denote the eigenvalues of A.Obviously, min F(A) implies that all the eigenvalues areclose to 0, and thus, NI is close to 1.0. Notice that bothare empirical measures without theoretical justifica-tions. However, the Jacobi eigenvalue interaction mea-sure lacks sufficient information about stability, integ-rity, and robustness, in comparison to the NI measure(Zhu and Jutan, 1993a).

4. Loop Decomposition and Definition ofInteraction

4.1. Dynamic Decomposition. Following Zhu andJutan (1996), loop interaction can be exposed by inves-tigating the response of an output to its input in thepresence of interaction from other loops through thehidden loops described by Shinskey (1988). This situ-ation can be best depicted by Figure 1. Focusing on anarbitrary control loop with uj paired with yi, output inresponse to any change in the input consists of twoparallel paths: the direct forward path via the interac-tion-free process and the parallel path via the hiddenloops due to off-diagonal elements of the process.

min ∑|λijk - 1.0| ∀ k (8)

Figure 1. Signal transmittances in a control loop.

A ) I - Gh -1G (9)

min F(A) (10)

NI ) det(I - A) ) ∏i)1

n

(1 - θi) (11)

4094 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996

Mathematically, we have

Notice that a change in ui would cause disturbances toother outputs as well; i.e.,

The parallel path in eq 12 clearly represents theinteraction in the loop. And since it represents theabsolute amount of loop interaction, it is thus called theabsolute interaction in the loop. The detailed expressionfor aij(s) can be obtained by the signal flow technique(see Zhu and Jutan, 1996). A relative interaction canbe then defined as the ratio of the absolute interactionand the independent interaction-free process as follows:

Hence, the equivalent process with interaction embed-ded is given by

Clearly, eq 15 implies that the original interaction-free process is deviated by an amplification or attenu-ation factor of 1 + φij(s) due to loop interaction. Inparticular, the relative interaction represents the mul-tiplicative model error associated with the interaction-free process. Clearly, the definition of loop interactionis physically meaningful, as shown in Figure 2. It isworthwhile to emphasize that the definition of loopinteraction above follows a rigorous derivation with norestrictive assumption made about the process and theclosed-loop system for any standard decentralized con-trol system.4.2. Steady-State Decomposition. At steady state,

assuming (1) the closed-loop system possesses integrityagainst any single-loop failure and (2) each controllerin each control loop contains integral action, we obtainthe following (see Shinskey, 1988):

and

Notice that the assumptions required above arestrikingly similar to those in defining the RGA (see eq1). It will be seen later that the interaction calculatedis indeed intimately related to the RGA.

5. RIA and Pairing Rules

5.1. Relative Interaction Matrix and Relationto RGA. From eqs 14 and 17, the relative interactionin a loop can be easily found to be

All the individual relative interaction terms constitutea relative interaction array (RIA) as follows:

Hence, we have

Notice that the division in the above equation is definedon an element-by-element basis.It is clear from eq 2 that the relative interaction is

directly related to the corresponding RGA element by

or

Overall, one has

Again, the division in the above equation is defined onan element-by-element basis.Clearly, the RIA provides an equivalent interaction

measure to the RGA. The equivalent process gain in acontrol loop becomes

Interestingly, the inverse of a RGA element should havebeen defined to better characterize loop interaction moredirectly as an amplifying or attenuating factor imposedupon the independent gain and to avoid some problemsin using the RGA (see later). Obviously, the steady-state relative interaction represents the gain errorassociated with the independent gain in a loop.Loop interaction and implications can be briefly

characterized by the RIA as follows: φij ) 0 w nointeraction; φij > 0 w interaction acts in the samedirection as interaction-free process; φij > 1.0 w interac-tion dominates over interaction-free process gain; φij <0 w interaction acts in the reverse direction as interac-tion-free process gain; φij < -1.0 w reverse interactiondominates over interaction-free process gain.

Figure 2. Definition of dynamic interaction in the uj-yi loop: (a)absolute interaction; (b) relative interaction.

yi(s) ) [gij(s) + aij(s)]uj(s) (12)

yk(s) ) dkj(s)uj(s) ∀ k, k * i (13)

φij(s) )aij(s)

gij(s)(14)

gij(s) ) gij(s)(1 + φij(s)) (15)

dkj(0) ) 0 ∀ k (16)

aij ) 1[G-1]ji

- gij (17)

φij ) 1gij[G

-1]ji- 1 (18)

φij ) [RIA]ij (19)

RIA ) 1G X (G-1)T

- 1 (20)

φij ) 1λij

- 1 (21)

λij ) 1φij + 1

(22)

RIA ) 1RGA

- 1 (23)

gij ) (1 + φij)gij ) 1λijgij (24)

Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4095

The relationship between the RIA and RGA can bebest described by Figure 3. The following can beobserved from Figure 3: λij ) 1.0 S φij ) 0; λij f 0+ Sφij f +∞; λij f 0- S φij f -∞; λij f +∞ S φij f -1.0; λijf -∞ S φij f -1.0.It is important to notice that both very large positive

and very large negative RGA elements drive loopinteraction toward the inverse direction of the originalinteraction-free process gain and that both very smallpositive and very small negative RGA elements are also,actually even more, harmful to process control. Appar-ently, in contrast to the RIA, the RGA exhibits discon-tinuous behavior when used to characterize loop inter-action. Similar discontinuous behavior was also observedin characterizing stability robustness by the RGA incontrast to the NI, which shows no discontinuity.Obviously, we have the following.RIA Interaction Rule. Variables should be paired

in such a way that all the paired RIA elements areclosest to 0.Notice that the above states the interaction rule only.

Other considerations will be addressed later. Althoughthe above interaction rule seems to be equivalent to theRGA interaction rule, the closeness to a value (0 in RIArule, 1.0 in RGA rule) has different implications.Specifically, the distance of the RGA elements from 1.0may not realistically reflect the amount of interactionas noted previously, whereas the distance of the RIAdoes. In particular, the RGA interaction rule mainlytargets at dismissing large RGA elements, which makesthe robustness rule redundant, whereas the RIA inter-action rule essentially aims at eliminating small RGAelements (large interaction; see eq 22), which comple-ments the robustness rule (see later).5.2. Stability, Integrity, and Robustness by the

RIA. Applying the stability consistency principle forSISO systems to any uj-yi loop after decomposition(Grosdidier et al., 1985; Zhu and Jutan, 1995), oneobtains the necessary stability condition below,

where cj denote the steady-state gain of the compensatorobtained from the controller with the integrator explic-itly separated; i.e., cj(s) ) 1/sgcj(s). Assuming that eachof the corresponding independent loops is designed tobe stable, one has

Combining eqs 25 and 26 and using eq 24, one obtainsthe following necessary stability condition for variablepairing in terms of the RIA:

It is interesting to note that from eq 22, the conditiongiven in eq 27 is equivalent to that given in eq 5, whichprovides a necessary condition for integrity, rather thanstability. With the RIA, one is able to clarify thisseeming mismatch. In fact, integrity and stability areclosely related to each other (see Zhu, 1996). Thecondition in eq 27 is actually consistent with thestability condition by the NI. This is clear from thefollowing relationship between the RIA and the NIobtainable from eqs 4 and 22 (note that rearrangementfor diagonal pairing is required):

Since the definition of the RIA requires the systemto possess integrity against single-loop failure, i.e., NI(i)> 0 (actually, so does the definition of the RGA), φii andthe NI provide equivalently a necessary stability condi-tion. However, if a system lacks integrity, the RIA nolonger measures loop interaction (simply because thesteady-state value of φii(s) is not defined from the finalvalue theorem). On the other hand, nevertheless, anegative φii does indicate lack of integrity subject to apositive NI. This is why eq 5 (or eq 27) provides anintegrity condition that is subject to stability require-ment. More strictly, eq 5 (or eq 27) provides a stabilitycondition subject to integrity requirement as assumedin the definition of the RIA (and the RGA). Since theintegrity check using NI involves calculation of all theNI(i), practically, NI and φii can be used for stability andintegrity checks, respectively. Therefore, the stabilityrule and the integrity rule in the RGA-based criteriacan be inherited but with RGA (eq 5) replaced by theRIA (eq 27).It can be shown that the sensitivity of φij to the

independent gain is proportional to itself; i.e.,

Equation 29 coincides with the fact that φij is muchsmoother than the corresponding λij.A robust stability measure can be expressed in terms

of φij as

Clearly, the maximum relative gain error allowed tomaintain stability is proportional to the amplifying orattenuating factor imposed by loop interaction on theoriginal gain. Again, although eq 30 only gives a boundon the independent gain with a single gain error, it isalso likely applicable to multiple gain errors (see Zhuand Jutan, 1993a). Apparently, relative interactionclose to -1 (large RGA elements) should be avoided forrobust stability.Combining the RIA interaction rule with stability,

integrity, and robustness requirements, we have thefollowing RIA-based pairing criterion.RIA-Based Pairing Criterion. Variables should be

paired in such a way so that (a) all the RIA elementsare closest to 0, (b) NI is positive, (c) all the RIAelements are greater than -1, and (d) RIA elementsclose to -1 are avoided.

Figure 3. Relationship between RIA and RGA.

gijcj > 0 ∀ j (25)

gijcj > 0 (26)

φij > -1 ∀ j (27)

(φii + 1)NI(i) ) NI ∀ i (28)

dφijdgij

) -φij

gij(29)

∆gijgij

< -(φij + 1) (30)

4096 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996

In contrast to the RGA criterion, both large (smallRGA elements) and close to inverse (large RGA ele-ments) interactions are explicitly identified as undesir-able pairing elements for interaction and robustnessconsiderations, respectively. In particular, interactionand robustness rules both play a role and complementeach other.Clearly, the RIA-based pairing criteria also represent

a comprehensive tool for variable pairing choices. Forcomparison, the RIA-based pairing rules and the cor-responding RGA-based ones are summarized in Table1.5.3. A New Overall Interaction Measure and

Pairing Criterion. Like the RGA, the RIA measuresonly individual interactions in individual control loops.Hence, ambiguity may still arise, since there may existseveral alternatives satisfying the RIA-based pairingcriterion. By the RIA, the distance between a RIAelement and 0 actually implies the amount of interactionin a loop. Therefore, an overall interaction measure canbe readily proposed as

where φijk denotes pairing elements corresponding to thekth alternative. The proposed overall interaction mea-sure can identify the best pairing choice as the oneshowing minimum overall interaction among pairingcandidates that satisfy stability, integrity, and robust-ness requirements.In comparison to the NI-based overall interaction

measure, the new one has the following advantages: (1)it has a more rigorous basis; (2) it is calculationallyconvenient without requiring diagonal pairings; (3) itovercomes problems associated with other overall mea-sures (see examples).Expressing eq 31 in terms of the RGA elements, one

obtains

Noticeably, the above rule can be viewed as a “cor-rected” version of the intuitive RGA-based overallinteraction measure given in eq 8 due to the correcteddefinition of the RGA as an interaction measure fromthe RIA. A major advantage of the rule in eq 32 lies inthe fact that very small RGA elements (very largeinteraction) are explicitly penealized, while very largeRGA elements (close to inverse interaction) are dis-missed explicitly by robustness requirements.Finally, a new pairing criterion based on the new

overall interaction measure is given below.New Pairing Criteria. Variables should be paired

so that (a) NI > 0 (stability rule), (b) φij > -1 (integrityrule), (c) φij close to -1 is avoided (robustness rule), and(d) min ∑|φijk| (overall interaction rule).

Notice that, although the overall interaction can bedirectly applied to all possible pairings, in practice, theinteraction rule based on the individual interactionmeasure may be first used to reduce the number ofalternatives before using the overall interaction rule todistinguish desirable alternatives.In conclusion, the RIA-based pairing criteria devel-

oped offer a comprehensive and reliable solution to thevariable pairing problem and consequently represent apromising tool.

6. Examples

This section includes examples to demonstrate thatthe proposed RIA-based pairing criteria constitute acomprehensive and reliable screening tool, by showingits effectiveness in comparison with the RGA-based aswell as the NI and Jacobi eigenvalue criteria.6.1. Example 1. This example shows that the RGA-

based criteria, more precisely, the RGA interaction rule,lead to more than one choice and are unable to distin-guish them, whereas all three overall interaction mea-sures agree on the final decision.Consider the following plant gain matrix studied by

Majares et al. (1986) and Zhu and Jutan (1993a):

The RGA and RIA can be calculated as

and

The NI’s, the Jacobi eigenvalues, and the absolutevalues of the RIA elements corresponding to all possiblepairings are shown in Table 2.According to the new RIA-based rules, we shall first

eliminate pairings corresponding to NI > 0 and φij <-1 by using the stability and integrity rules. As aresult, pairings 2, 4, 5, and 6 should be dismissed,leaving pairing 1 and 3 as the final two candidates. Note

Table 1. Comparison between RGA-Based and RIA-Based Pairing Criteria

RIA criteria

RGA criteria by φij by λij comments

loop interaction λij f 1.0 φij f 0.0 λij f 1.0 closeness of λij in RGA rule ambiguousstability NI > 0 φij > -1 T NI > 0 λij > 0 T NI > 0 integrity required in RIA ruleintegrity λij > 0 NI(i) > 0 NI(i) > 0 stability required in RGA rule; integrity

assumed in RIA rulerobustness avoid large λij avoid φij close to -1.0 avoid large λijoverall interaction min |φijk| min |1/λijk -1.0| overall interaction not available in original RGA rule

Table 2. Example 1: Overall Interaction by DifferentMeasures

no. pairings NI {θi} ∑|æij|1 (1,1)-(2,2)-(3,3) 0.62 -0.1, -0.53, 0.63 1.772 (1,1)-(2,3)-(3,2) -0.93 -0.37, -0.98, 1.34 4.73 (1,2)-(2,1)-(3,3) 1.87 0.62, -0.31 ( 1.78i 2.574 (1,2)-(2,3)-(3,1) 18.7 -4.13, 2.02 ( 1.58i 28.395 (1,3)-(2,1)-(3,2) -9.35 1.83, -0.91 ( 2.77I 12.326 (1,3)-(2,2)-(3,1) -62.3 0.34, 9.65, -10.0 35.21

G(0) ) [1.0 1.0 -0.11.0 -3.0 1.00.1 2.0 -1.0 ]

RGA ) [0.53 0.59 -0.120.43 1.59 -1.020.04 -1.18 2.14 ]

RIA ) [0.87 0.70 -9.131.34 -0.37 -1.9825.71 -1.85 -0.53 ]

min ∑|φijk| (31)

min ∑| 1λijk - 1| (32)

Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4097

that the RGA-based criteria always give the sameresults up to this stage (with NI > 0 and λij > 0).Between the two final alternatives, the RGA fails to

clearly point out the best one, and hence, ambiguityarises. The intuitive overall RGA rule would identifypairing 3 as the final choice. However, pairing 3contains a relative gain of 0.43 which is less than 0.5and this is undesirable according to Majares et al.(1986). Hence, no decision can be clearly made usingthe RGA.However, all three overall interaction measures, the

NI measure, the Jacobi eigenvalue measure, and theRIA overall measure, suggest that pairing 1 should bethe final choice showing the minimum overall interac-tion. This is obvious by examining the values listed inTable 2.6.2. Example 2. This example is intended to dem-

onstrate that the RIA-based pairing criteria, moreprecisely, the RIA overall interaction measure, suggestsuniquely the right final pairing, whereas the NI and theJacobi eigenvalue overall interaction measures lead toincorrect answers and ambiguities arise from the RGAmeasure.Consider the gain matrix studied by Zhu (1993) below:

The RGA and RIA are obtained as

and

The NI’s, the Jacobi eigenvalues, and the absolute sizeof the RIA elements corresponding to all possiblepairings are calculated as shown in Table 3.By the stability and integrity rules, equivalently in

terms of the RGA or RIA, pairings 1 and 6 are accept-able alternatives deserving further consideration. Again,the RGA is unable to distinguish between the twocandidates.The Jacobi eigenvalue criterion suggests pairing 4 as

the most desirable pairing. It is known that this pairingresults in a system lacking integrity. This is becausethe Jacobi eigenvalue criterion contains no sufficientinformation about system stability and integrity.The NI interaction rule identifies pairing 1 as the best

choice which exhibits minimum overall interaction.However, pairing 1 contains a large RIA element (orvery small RGA element), indicating a large loopinteraction and significant gain change due to interac-tion. Hence, practitioners will lean toward the other

pairing (pairing 6) as the best choice (see Zhu, 1993;McAvoy, 1993).The RIA overall interaction rule still correctly sug-

gests pairing 6 as the final choice, showing the mini-mum overall interaction. Hence, the RIA-based criteriarepresent a more reliable tool than others.

7. Conclusions

Variable pairing choice from interaction measure-ment, stability, integrity, and robustness perspectiveshas been systematically addressed with new insightsoffered. Problems, particularly the ambiguities in iden-tifying the final variable pairing, associated with themost widely used RGA-based pairing criteria, arerevealed. Limitations on some of the existing overallinteraction measures have also been identified. A newinteraction measure, the RIA, has been explored interms of its advantages over the RGA, and its implica-tions for the amount of interaction, closed-loop stability,integrity, and robustness against gain uncertaintieshave also been presented. The RIA has been furtherextended to provide an overall interaction measure. Ithas been shown that the RIA-based overall interactionmeasure is able to avoid potential ambiguities in usingthe individual interaction measures such as the RGAand RIA itself and to overcome problems associated withthe existing overall interaction measures. Conse-quently, the new RIA-based pairing criteria offer acomprehensive and reliable solution to the variablepairing selection problem in decentralized control sys-tems and hence represent a promising tool for industrialuse.Finally, it must be pointed out that the new pairing

criteria as well as other pairing rules studied in thisarticle are concerned only with the steady-state case.Although the use of steady-state pairing rules has beenwidely advocated and justified (see Zhu, 1996), detaileddynamic analysis may be required under some circum-stances (see Rijnsdorp, 1965; Hovd and Skogestad, 1992;Zhu and Jutan, 1996). Ultimately, variable pairingrepresents only the first step, and hence, dynamicsimulation should be performed during the design of anydecentralized control system.

Nomenclature

A ) Jacobi eigenvalue matrixaij(s) ) dynamic absolute interaction in the uj-yi loopc(s) ) controller transfer function (TF) with the integratorseparated in a SISO loop

dij(s) ) perturbation termG(s) ) process TF matrixG ) process gain matrixGh ) diag(G)gij(s) ) ijth element of G(s)gij ) ijth element of G∆gij ) model error in gijgii(s) ) equivalent process TF with interaction included inthe ith loop

gii ) equivalent process gain with interaction included inthe ith loop

gij ) equivalent process gain in the uj-yi loop with all otherloops closed

s ) Laplace variableui ) ith inputyi ) ith output

Greek Lettersφij(s) ) dynamic relative interaction in the uj-yi loop

Table 3. Example 2: Overall Interaction by DifferentMeasures

no. pairings NI {θi} ∑|æij|1 (1,1)-(2,2)-(3,3) 0.91 0.3 ( 0.75i, 0.59 13.882 (1,1)-(2,3)-(3,2) -0.98 -0.65 ( 10.78i, 1.30 18.013 (1,2)-(2,1)-(3,3) -2.28 -0.8 ( 0.66i, 1.62 5.54 (1,2)-(2,3)-(3,1) 1.50 (0.70i, 0 7.755 (1,3)-(2,1)-(3,2) -0.91 -0.63 ( 0.94i, 0.8 3.566 (1,3)-(2,2)-(3,1) 0.56 -0.4 ( 0.92i, 0.80 1.68

G(0) ) [1.5 -1.2 -1.51.5 -3.0 1.4-1.8 2.2 -1.1 ]

RGA ) [0.073 0.23 0.70-1.53 2.88 -0.352.46 -2.12 0.66 ]

RIA ) [12.71 3.33 0.44-1.65 -0.65 -3.83-0.59 -1.47 0.53 ]

4098 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996

φij ) steady-state relative interaction in the uj-yi loopλij ) ijth element of the RGAθi ) eigenvalue of a matrixF ) spectral radius of a matrix

AbbreviationsAIM ) absolute interaction arraydet ) determinant of a matrixNI ) Niederlinski indexNI(i) ) Niederlinski index of the subsystem with the ithloop removed

RGA ) relative gain arrayRIA ) relative interaction arraySISO ) single input, single output

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Received for review March 11, 1996Revised manuscript received August 23, 1996

Accepted August 26, 1996X

X Abstract published in Advance ACS Abstracts,October 15,1996.

Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4099