Structural Analysis and Stability Conditions of Decentralized Control Systems

PROCESS DESIGN AND CONTROL

Structural Analysis and Stability Conditions of DecentralizedControl Systems

Zhong-Xiang Zhu†

SACDA Inc., 343 Dundas Street, London, Ontario, Canada N6B 1V5

Stability analysis in decentralized control systems relies heavily on steady state tools such asthe relative gain array and the Niederlinski index. However, only necessary stability conditionsare provided by these tools and their usefulness lies essentially solely in eliminating unstablepairings. In this paper, upon structurally decomposing a decentralized control system intocompletely equivalent individual dynamic single input-single output loops with interactionsexplicitly embedded, system structure and main properties, such as right half plane (RHP) zeros,RHP poles, integrity, and stability, are analyzed in a systematic and transparent way. Theintrinsic connections among these properties are elucidated. New important insights into theeffects of loop interaction due to the process and the controller on the closed loop system areoffered. Various necessary and sufficient conditions to prevent interaction from inducingundesirable behavior, such as nonminimum phase, lack of integrity, and instability, aredeveloped. Significant implications for variable pairing and controller tuning are presented.

1. Introduction

Full scale multivariable control techniques such asstate space based modern control and, more recently,model predictive control have found increasing applica-tions in the process industries (Eaton and Rawlings,1990; Zhu and Jutan, 1994; Zhu et al., 1995). Never-theless, decentralized control still remains popular andoften predominant, either used alone or as a lower levelcontrol on top of various model based control schemes.In decentralized control, closed loop analysis, to a

large extent, relies heavily on steady state tools suchas the relative gain array (RGA) (Bristol, 1966) and theNiederlinski index (NI) (Niederlinski, 1971) (e.g., Gros-didier et al., 1985; Yu and Luyben, 1987; Seborg et al.,1989; Chiu and Arkun, 1990; Yu and Fan, 1990; Zhuand Jutan, 1993a,b, 1995c). Nevertheless, the RGA andthe NI, together with many other proposed steady stateinteraction measures (Majare et al., 1986; Chang andYu, 1992), ignore the dynamics of the process and thecontroller, thus leading to some fundamental limitations(Zhu and Jutan, 1995b). Take stability analysis, forexample; only necessary (not sufficient) conditions areavailable, and final stability usually has to resort to adhoc approaches, typically by means of detuning thecontrollers independently designed on the basis ofindividual interaction-free loops.Numerous dynamic interaction measures have been

proposed to overcome the limitations of steady stateones (Bristol, 1978; Tung and Edgar, 1981; Gagnepainand Seborg, 1982; Jensen et al., 1986; Bequette andEdgar, 1988; Balchen and Mumme, 1988; Huang et al.,1994). Unfortunately, they have little impact on theanalysis and design of decentralized control systems,mainly due to their unrealistic assumptions. Seborg etal. (1989) presented some general stability conditions,but only for 2 × 2 systems. Manousiouthakis (1993)

attempted to parametrize all stabilizing controllers indecentralized control systems from an overall perspec-tive. However, this approach does not provide a stronglinkage between the overall system and the independentloops, so important issues, such as loop interaction,failure tolerance, and independent design, cannot beeasily addressed.In this paper, upon decomposing a decentralized

control system into separate but equivalent individualsingle input-single output (SISO) loops, dynamic analy-sis is systematically performed. Important insights intothe effects of loop interaction due to both the processand the controller on the closed loop properties, suchas right half plane (RHP) zeros and poles, integrity, andstability, are presented. New necessary and sufficientconditions for preventing interaction from inducingundesirable behavior, such as open instability, non-minimum phase, loss of integrity, and instability, aredeveloped with direct implications for both variablepairing choices and controller tuning.

2. Structural Decomposition

The following derivation follows Zhu (1993) and Zhuand Jutan (1995b). Let us examine the transmittancesin a control loop with the manipulated variable uj(s) ∈{u1(s),...,un(s)} paired with the controlled variable yi(s)∈ {y1(s),...,yn(s)}. When the controller, cj(s) ∈ Rn, has totake action in response to setpoint changes and/ordisturbances, it affects the overall system in the follow-ing sequence: the controller cj(s) attempts to bring itsoutput yi(s) to its target (setpoint) by sending a controlsignal to uj(s); the control action uj(s), meanwhile,creates perturbations in all the other loops in the systemthrough the off-diagonal elements of the plant, forcingother controllers to take actions as well; other loopsfurther influence the original loop by adding additionaldynamics to the control action uj(s) via other off-diagonalelements of the plant. This process of strong couplingin the form of action-cross action-interaction among

† E-mail: [email protected]. FAX: (519) 679-3977.

736 Ind. Eng. Chem. Res. 1996, 35, 736-745

0888-5885/96/2635-0736$12.00/0 © 1996 American Chemical Society

control loops, continues throughout the whole transientuntil a steady state is reached.Mathematically, responses to setpoint changes are

governed by the following equations:

ym(s) ) ∑l)1

n

gml(s) ul(s) ∀ m

ul(s) ) cl(s)[rl(s) - yl(s)] (1)

Note that in the above equation ul is assumed to bepaired with yl, however without loss of generality.Focusing on the uj-yi loop and assuming that only thisloop undergoes setpoint changes, we thus have,

yi(s) ) gij(s) uj(s) - ∑l)1,l*j

n

gil(s) cl(s) yl(s) (2)

Successively using eq 2 to express all the outputs interms of uj(s), we find the transmittance between thecontrol action uj(s) and its own output to be

yi(s) ) gij(s) uj(s) + aij(s) uj(s) (3)

and the perturbations caused by uj to other loops as

yk(s) ) dkj(s) uj(s) ∀ k (4)

Structurally, aij(s) in the above equations representsthe additional dynamics exerted by other loops to theuj-yi loop and is physically separated from the inde-pendent SISO loop and dij(s) represents the perturba-tions to other loop by the underlining loop. Apparently,aij(s) and dij(s) provide measurements of the interactionin a loop and the cross-interaction in all other loops,respectively. Consequently, a decentralized controlsystem can be structurally decomposed into individualSISO loops with the coupling among all the loopsexplicitly exposed and embedded in these separate loops.Figures 1 and 2 show the physical structure of aparticular loop after the structural decomposition.

The detailed expressions of aij(s) and dij(s) can bereadily obtained by using the signal flow graph tech-nique (Ogata, 1990) or by appropriately partitioning theprocess and the controller (see the Appendices). Take2 × 2 systems with diagonal pairing, for example, thefollowing terms for the two loops can be easily derivedand verified:

aii(s) ) -cj(s) gij(s) gji(s)

1 + cj(s) gjj(s)∀ i, j * i (5)

dij(s) )gij(s)

1 + ci(s) gii(s)∀ i, j * i (6)

Similar interaction terms are also demonstrated byother researchers (Seborg et al., 1989; O’Reilly andLeithead, 1991; Shen and Yu, 1994), however, only for2 × 2 systems. Nevertheless, the cross-interactionterms are ignored by most of the above methods andother RGA type of interaction measures (Bristol, 1978;Tung and Edgar, 1981; Gagnepain and Seborg, 1982)as well. This is the main reason why RGA can onlymeasure the one-way interaction. Leithead and O’Reilly(1992) proposed an alternative decomposition approach,however, from a matrix algebraic perspective. Nonethe-less, no implication for interaction measure is explored.We will see that the interaction defined here withphysical significance has a substantial impact on thesubsequent development in this paper.The structural decomposition leads to significant

implications for some important issues in decentralizedcontrol, as stated in the following remarks.Remark 1. More important information about in-

teraction measurement is provided. In particular, two-way interaction and hence the limitations of the existingmethods is clearly revealed. Moreover, in contrast toexisting interaction measures, the influence of both theprocess and the individual controllers on interaction isincluded.Remark 2. Substantial implications for the analysis

of decentralized control systems are offered. Systemproperties such as open loop stability, nonminimumphase behavior, integrity, and stability can be definedand studied on the basis of individual SISO loops. Moreimportantly, the effects of interaction, variable pairing,and controller tuning can be elucidated.Remark 3. The structurally decomposed individual

SISO loops are completely equivalent to the originalsystem with no assumption made. The coupling amongall the loops is distributed into separate SISO loops withdirect comparison to their independent counterparts.Consequently, decentralized system design can be greatlyreduced to that for SISO systems, and independentdesign can be performed with interactions taken intoaccount.

3. Open Loop Structure

3.1. Loop Interaction and Connection to RGA.In the following development, diagonal pairing is as-sumed unless specified otherwise, without loss of gen-erality. The Laplace variable s is omitted for simplicity.Note that nondiagonal pairings can be rearranged asdiagonal ones by properly rearranging the paired ele-ments to diagonal positions (Zhu and Jutan, 1993b).From the structural decomposition shown in Figure 1,the following remark is obvious upon associating loopinteraction with the interaction-free term.

Figure 1. Structure of loop uj-yi by structural decomposition.

Figure 2. Structure of loop 1 after decomposition of 2 × 2Systems.

Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 737

Remark 4. The equivalent process transfer functionof the ith loop, denoted by g̃ii, is given by

where

is defined as the relative interaction in the loop.The significance of remark 4 is that it allows for the

distribution of open loop poles and zeros, thus open loopstability and nonminimum phase behavior, to be exam-ined on the basis of individual process elements andinteraction in individual loops. The interaction and therelative interaction can be viewed as an addictive modeldeviation from the independent process due to loopinteraction (see Figure 1), whereas 1 + φii can be viewedas a multiplicate model deviation. Obviously, the phaseand magnitude of the relative interaction determine thefeasibility and easiness of the fine-tuning process usingindependent design (Seborg et al., 1989).The relative interaction is also closely related to the

Rijnsdorp interaction quotient (Rijnsdorp, 1965) and theRGA (Bristol, 1966) in 2 × 2 systems by

where κ is the Rijnsdorp quotient (Rijnsdorp, 1965)defined by

and hj is the closed loop transfer function of theindependent SISO loop defined by

The Rijnsdorp quotient is related to the (1,1) elementof the dynamic RGA, λ11, by

From eqs 9-12, it is clear that both the RGA and theRijnsdorp quotient provide equivalent but incompleteinformation regarding loop interaction. Specifically,they both contain only the process dynamics and lendthemselves as interaction measures only if perfectdynamic control (which never holds in reality except atsteady state!) is assumed. The relative interactiondefined in remark 4 shows that controllers can in factplay a significant role in loop interaction (see Zhu andJutan, 1995b).In particular, expression 9 allows the interaction

contributed by the process alone, which is referred asprocess interaction, and that by control loop, calledcontrol interaction, to be investigated separately. Dueto its importance in this study, the relative interactionwill be simply referred as loop interaction or interaction.3.2. Open Loop Stability. The following theorem

governs the distribution of open loop poles:Theorem 1. Assuming that individual elements of

the process and the controller do not contain any RHPpoles, all the equivalent processes in all the individualloops do not contain any RHP poles if and only if thesystem possesses integrity against any single loop failure.

Proof. See Appendix A.Theorem 1 gives the necessary and sufficient condi-

tions to prevent the interaction from inducing any polein the right half plane (RHP). For 2 × 2 systems, wehave,Corollary 1. Assume that all individual elements of

the process and the controller in a 2 × 2 system do notcontain any RHP poles. Both equivalent processes areopen loop stable if and only if both independent SISOloops are stable.Proof. The proof is straightforward from theorem 1

upon noticing that integrity is reduced to stability ofthe two independent loops for 2 × 2 systems, sinceremoving one loop leaves a single independent loop.Theorem 1 and corollary 1 imply that local instability,

i.e., lack of integrity, is the only source for open loopinstability in a particular loop in an interactive de-centralized control system.3.3. Nonminimum Phase Behavior. It is well-

known that the existence of RHP zeros may causeinverse responses and may potentially impose limita-tions on the achievable dynamic performance (Seborget al., 1989). The following remark offers a word ofcaution in designing decentralized control systems.Remark 5. RHP zeros may be induced in a loop by

loop interaction, even though individual processes areminimum phase.Significantly, unlike methods for general multi-

variable systems, the existence of the RHP zeros canbe determined by the RHP zeros of the individualelements of the process and the independent loops. Thefollowing theorem provides a necessary and sufficientcondition to prevent interaction from inducing non-minimum phase behavior.Theorem 2. Assume that individual elements of the

process and the controller do not contain any RHP polesand RHP zeros, and the system possesses integrityagainst failure of any single loop. The overall systemremains minimum phase if and only if

where N(-1,φii) denotes the number of clockwise en-circlements of (-1,0) point by the Nyquist plot of φii.Proof. See Appendix B.For 2 × 2 systems, theorem 2 leads to the following.Corollary 2. Assuming that none of the individual

elements of the process and the controller contain anyRHP poles or RHP zeros, and that both independentSISO loops are stable, the overall system remainsminimum phase if and only if

where N(-1,-κhj) represents the number of clockwiseencirclements of (-1,0) point by the Nyquist plots of -κhj.Proof. See proof of theorem 2.Corollary 2 allows for the effects of the process

interaction (i.e., the Rijnsdorp quotient) and the controlinteraction (i.e., controller tuning) on the creation ofnonminimum phase behavior to be explicitly exposedseparately. Theorem 2, corollary 2, and remark 5indicate that care should be exercised, when selectingthe desirable variable pairing and performing controllertuning, to avoid creating open loop RHP zeros in anyloop. The following example elucidates this.Example 1 (Zhu and Jutan, 1995a). Consider the

following open loop stable, individually minimum phaseprocess

g̃ii ) gii(1 + φii) ∀ i (7)

φii ) aii/gii ∀ i (8)

φii ) -κhj ∀ i ) 1, 2, j * i (9)

κ )g12g21g11g22

(10)

hj )cjgjj

1 + cjgjj∀ j ) 1, 2 (11)

κ ) 1 - 1λ11

(12)

N(-1,φii) ) 0 ∀ i (13)

N(-1,-κhj) ) 0 ∀ j (14)

738 Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996

to be controlled by the PI controller below,

where R is the parameter for tuning the amount ofprocess interaction and k1 and k2 are the proportionalgains of the controller for tuning the amount of controlinteraction. The effects of the interaction on the dis-tribution of poles and zeros of the equivalent open loopprocesses are analyzed below.It can be readily verified that the closed loop transfer

functions of both independent SISO loops are

and that both independent loops are stable with anypositive gains. The relative interaction and the multi-plicate deviation of the independent single loop processmodel in each loop as defined by eq 9 are given by

and

As indicated by corollary 1, eq 19 shows that theequivalent processes of both loops are also stable for anypositive kj, j ) 1,2. However, nonminimum phasebehavior or RHP zeros is induced if

or

Figure 3 shows the region (shaded) for the equivalentprocess in a loop to maintain minimum phase. Basedon theorem 2 and corollary 2, Figure 4 also clearlydemonstrates the above conditions by Nyquist plots asfunctions of interactions due to the process and thecontroller tuning. The above two conditions havesignificant implications as discussed below.Clearly, both the process and the controller play

substantial roles in creating nonmimimum phase be-havior. Specifically, larger loop interaction due to alarger Rijnsdorp quotient, i.e., larger R in this example(see Figure 4c in comparison with Figure 4a), or tightercontrol action, i.e., large kj, j ) 1, 2 (see Figure 4b incomparison with Figure 4a), tends to drive the open loopzeros toward and eventually into the nonminimumphase region. Interestingly, if the interaction in a loop,upon closing independently designed loops, tends tocreate a nonminimum phase equivalent process, detun-ing the controller in the other loop counteracts thetendency.

Surprisingly, however, if process interaction is inher-ently large enough, e.g., R > 10, nonminimum phasebehavior becomes inevitable regardless of controllertuning (see Figure 4d). The same situation occurs whenR becomes negative, indicating that not only the mag-nitude but also the phase of the interaction contributesto the occurrence of undesirable behavior.It is important to point out that the limit on the

process interaction for creating RHP zeros implies thatthe nature of the resulting nonminimum phase behaviorhas changed so that the inverse response becomes“permanent” and consequently instability may arise. Inthis case, the main system property is captured byinstability, and nonminimum phase behavior becomesmeaningless. In fact, the maximum process interactioncan be characterized by the steady state value of theRijnsdorp quotient, or the sign of the RGA, as statedby the following well-known result (Grosdidier et al.,1985; Chiu and Arkun, 1990; Zhu and Jutan, 1993a,1995a,c).Theorem 3. Assume in a 2 × 2 system that the

process, G(s), is stable, the controller, C(s), containsintegral action, and G(s) C(s)/s is rational and proper,and that the two loops are independently stable. Theoverall system becomes unstable if

or

where κ(0) is the Rijnsdorp quotient and λ11 (0) is the(1,1) element of the RGA, both at steady state.Proof. See Appendix C.Appendix C provides an alternative proof of theorem

3 based on the structural information directly. Theorem3 indicates that process interaction may cause instabil-

G(s) ) [ 1S + 0.1

1S + 0.1

RS + 1

1S + 0.1

] (15)

C(s) ) s + 0.1s [k1 0

0 k2 ] (16)

hj )kj

s + kjj ) 1, 2 (17)

φii ) - s + 1R(s + 0.1)

k2s + k2

∀ i, j * i (18)

1 + φii )s2 + (1 + (1 - R)kj)s + (1 - 0.1R)kj

(s + 1)(s + kj)i ) 1, 2, j * i (19)

kj > 1R - 1

j ) 1, 2, 1 < R < 10 (20)

R > 10 (21)

Figure 3. Nonminimum phase region in a loop in example 1.

Figure 4. Nyquist plots as functions of interaction in example 1.

κ(0) > 1

λ11(0) < 0


ity by its own (independent of controller design). Also,theorem 3 implies that nonminimum phase behavior notonly causes initial inverse responses, thus imposinglimitations on achievable performance, but also maycreate “permanent” inverse responses thus becoming asource of instability.It is worthwhile to note that, in the case of inherently

large process interaction, the only choice to maintainstability by means of controller design is to reverse thecontrol direction of one of the controllers (Zhu andJutan, 1995a). However, this is undesirable sincesystem integrity has to be sacrificed and, as a result,one of the control loops will become open loop unstableby theorem 1. It thus becomes clear that systemproperties such as RHP zeros, RHP poles, integrity, andstability are actually intrinsically related to each other.It is important to point out that the best approach toavoid undesirable system properties, such as lack ofintegrity, is to avoid large process interaction by meansof variable pairing first and subsequently to fine tunethe controller. Consequently, variable pairing andcontroller tuning must be jointly considered for desirableclosed loop properties.3.4. Structure for Stability Analysis. As previ-

ously noted, the cross-interaction term as shown inFigure 1 constitutes an integrated part of a loop afterdecomposition. However, under some circumstances,this term may be, to a large extent, ignored whenperforming stability analysis, as stated in the followingintuitive remark.Remark 6. The cross-interaction terms contain no

RHP poles if the system possesses integrity against eachsingle loop failure. In particular, they converge to zeroat steady state if individual controllers contain integralaction.Further notice that the cross-interaction terms are

outside the feedback loop (see Figure 1). Therefore, asfar as stability is concerned, we can focus on thefeedback structure composed of the equivalent processand the controller in a particular loop with the cross-interaction ignored. In fact, the effects of the cross-interaction terms on loop stability are reflected in loopinteraction indirectly. This is obvious from eqs 5 and 6and previous discussions, upon noting that the cross-interaction terms are the sources of loop interaction.

4. Stability Conditions

Necessary and sufficient stability conditions can beobtained by investigating the location of the roots of thecharacteristic equation of each individual closed loop,as stated below.Theorem 4. A decentralized control system is stable

if and only if the characteristic equation of each indi-vidual loop below,

where g̃ii is defined in eq 7, does not contain any RHPzeros.Theorem 4 imposes no assumption about the process

and the system, thus representing a general stabilitycondition. Since the interaction, φii, represents theintrinsic action of all other loops on the ith loop, thecombination of interaction and direct control action ina loop should reveal the characteristics of the overallsystem. Hence, one could surmise that the character-istic equations of all the individual loops are actuallythe same, i.e., the characteristic equation of the overall

system. Indeed, for 2 × 2 systems, for which simpleclosed form of interaction terms are available, we have,

Equation 23 can be easily verified from eqs 5, 7, and8 or 7, 9, and 11. Seborg et al. (1989) and Zhu andJutan (1995a) also demonstrated the final form of thecharacteristic equation, i.e., the third term in the aboveequation, by rather lengthy algebraic transformation ofvarious transfer functions. In contrast, eq 23 is readilyobtained directly on the basis of the structural informa-tion of the system.Applying the Nyquist stability criterion to individual

loops yields the following more attractive stabilityconditions.Theorem 5. Assuming that individual elements of

G(s) and independent SISO subsystems do not containany RHP poles, and the system possesses integrityagainst any single loop failure, the decentralized controlsystem is stable if and only if the Nyquist contour of theequivalent open loop transfer function, cigii(1 + φii), ∀ i,does not have any clockwise encirclement of the (-1,0)point, i.e.,

Proof. See Appendix D.Theorem 5 offers more insights into the intrinsic

connection between the properties of various compo-nents and closed loop stability in a system. For in-stance, it allows for the effects of the interaction onstability to be extracted and assessed. Also, the stabilityrobustness of a particular loop against loop interactioncan be evaluated in comparison to the independent SISOsubsystems. Specifically, if the independent SISO loopsare designed to have a large stability margin, i.e., theNyquist contour of cigii is far away from the (-1,0) point,the equivalent open loop transfer function in eq 24 cantolerate a “large” interaction in the loop. On the otherhand, a large loop interaction may cause a deviation ofthe independent open loop transfer function to be largeenough to move the Nyquist contour across the (-1,0)point, leading to instability of the overall system.For 2× 2 systems, theorem 5 reduces to the following.Corollary 3. Assuming that individual elements of

the process do not contain any RHP poles, and that bothindependent subsystems are open loop and closed loopstable, the overall 2 × 2 system is stable if and only if

where

denotes the open loop transfer function of the ith loop, κis the Rijnsdorp quotient, and hj represents the closedloop transfer function of the jth loop.Proof. Corollary 3 can be obtained from theorem 5

by observing that requiring integrity of the system isequivalent to requiring stability of independent SISOloops in 2 × 2 systems.Corollary 3 explicitly exposes the intrinsic connection

between the stability of the overall system and the twoindependent loops as well as the Rijnsdorp quotient. Thefollowing important observations can be extracted fromcorollary 3:

1 + cig̃ii ) 0 ∀ i (22)

1 + c1g̃11 ) 1 + c2g̃22 ) 1 + c1g11 + c2g22 +c1c2(g11g22 - g12g21) ) 0 (23)

N(-1,cigii(1+φii)) ) 0 ∀ i (24)

N(-1,ïi(1-κhj)) ) 0 ∀ i, j * i (25)

ïi ) cigii ∀ i (26)


1. Closed loop stability strongly depends on theinteractions contributed by both the process (repre-sented by the Rijnsdorp quotient) and the controllertuning, and their influence on stability can be examinedseparately.2. For a given process, closed loop stability can

usually be maintained or improved by means of tuning,usually detuning, the other controller.3. When the process interaction itself becomes inher-

ently large, tuning the other controller may not besufficient enough to counteract the interaction in orderto maintain stability (see theorem 3). Instead, thedirection of one of the two controllers may have to bereversed, leading to the loss of integrity (Zhu and Jutan,1995a)!4. It is well-known that process interaction, as

measured by the RGA, can be substantially decreasedby means of proper variable pairing. Theorem 4 alsoimplies this. In particular, an appropriate variablepairing can greatly facilitate controller tuning to main-tain stability, especially help avoid compromising sys-tem integrity for stability during controller design.However, it is the controller tuning, not variable pairing,that takes the decisive control in maintaining stability.Consequently, it advocates a common practicesvariablepairing and controller tuning are jointly considered inthe design of decentralized control systems in order tomaintain stability and integrity. Theorem 5 and corol-lary 3 provide theoretical justifications for this commonpractice.5. The overall stability also relies heavily on the

stability margin of the independent SISO loops. If anindependent loop has a large stability margin, i.e., cigiiis far away from (-1,0) point, the loop can tolerate alarge loop interaction. As a result, stability is insensi-tive to the tuning of the other controller and to variablepairing choices.Furthermore, the following remark offers a significant

result in analyzing decentralized control systems byfrequency plots.Remark 7. The encirclements of (-1,0) point by the

Nyquist plots of the various components in 2 × 2systems and their linkage to important system proper-ties are summarized below:

ïi: closed loop stability of independent ith loop (overallintegrity and robustness)

ïj, j * i: open loop stability of the independent ith loop(RHP poles)

-κhj: nonminimum phase behavior of the equivalentith loop (RHP zeros)

ïi(1 - κhj): closed loop stability of the ith loop withinteraction (overall stability)

According to remark 7, important properties of thesystem, such as open loop stability, existence of RHPzeros, system integrity, stability robustness, and closedloop stability, can be readily extracted and monitoredby plotting the Nyquist contours of the above variousquantities as functions of variable pairing choices andcontroller tuning.Recalling the discussions about the effects of inter-

action and controller tuning on the existence of RHPzeros (nonminimum phase behavior), one may obtainthe following.Remark 8. A 2 × 2 system likely undergoes the

following intrinsic paths from independently stable tooverall unstable due to an increasing process interaction

(large RGA) and corresponding controller tuning tomaintain stability:From independently stable and nonminimum phase

to nonminimum phase with interaction, closed loopunstable with maximum process interaction, open loopunstable (reversing control direction), loss of integrity,and overall stable.However, undesirable behavior in remark 8 (non-

minimum phase, open loop instability, and loss ofintegrity) can be avoided by jointly considering variablepairing and controller tuning. The above observationsand remarks regarding the effects of variable pairingand controller tuning on performance limiting (non-minimum phase), system integrity, and closed loopstability are likely applicable to general n × n systems.Example 2. Consider the same process and control-

ler given in example 1. Example 2 is intended todemonstrate the effects of loop interaction, due to boththe process and controller, and variable pairing onclosed loop stability as well as the RHP zeros byexamining the Nyquist plots and their encirclements of(-1,0) point of various components in the first loop (thesame can be performed for the second loop) in thesystem.Nyquist contours of the independent open loop sys-

tem, c1g11, the interaction, φ11, the multiplicate modelerror, 1 + φ11, and the equivalent open loop system,c1g11(1 + φ11), as functions of process interaction (byadjusting R) and controller tuning (by tuning k2) areplotted in various figures. The observations from thesefigures can be summarized as follows.Case 1. Figure 5 shows that loop 1 is always stable

(Figure 5d) with small process interaction (R ) 2.0) andlarge stability margin (Rk1 ) 1.0) in the independentsubsystem (Figure 5a), regardless of controller tuning(even with k2 ) 20). However, loop 1, and hence theoverall system, becomes nonminimum phase when thecondition given by eq 20 is violated (Figure 5b,c),whereas the loop remains minimum phase when con-troller 2 is less tightly tuned for the same process (notshown here). Moreover, large control interaction (tighttuning of controller 2) also results in an decrease ofstability robustness of the loop in the interactive system(Figure 5a,d).Case 2. When the process interaction is large (R )

10.0), tighter controller 2 drives the loop from minimumphase and stable to stable but at the boundary of

Figure 5. Example 2: Nyquist plots with R ) 2.0, k1 ) 0.5, k2 )20.0.


nonminimum phase (not shown), stable but nonmini-mum phase (Figure 6), and eventually unstable and“nonminimum phase” (Figure 7). In contrast to theabove case, the shape of the Nyquist plot of theequivalent open loop system (Figures 6d and 7d) hasdrastically changed from the independent open loopsystem due to large loop interaction (Figures 6a and 7a).Similar to case 1, the stability robustness decreases asthe loop interaction increases (tuning up controller 2).Case 3. Loop 1, and hence the overall system, can

be made stable by choosing a different pairing ofvariables. Figure 8 shows the same set of Nyquist plotsin loop 1 corresponding to the off-diagonal pairing withthe same stability robustness for the independent loop.It can be seen that loop 1 is stable even with a verytight tuning of the other controller. This case corre-sponds to case 1 with diagonal pairing.Case 4. Loop 1 will be unstable regardless of control-

ler tuning when the process interaction becomes largeenough. Figure 9 shows that loop 1 is always unstableand “nonminimum phase” even though controller 2 isdrastically detuned (k2 ) 0.01), when process interactionbecomes very large (R ) 20.0). The encirclement of(-1,0) point in Figure 9d could be more clearly shownby Bode plot (not presented here). Also note that thedirection of the Nyquist plot of the independent openloop system is reversed due to loop interaction (Figure

9a,d). Clearly, the system can be made stable by propervariable pairing, as in case 3. Otherwise, the directionof controller 2 has to be reversed to maintain stability(see case 5 below).Case 5. Figure 10 shows the Bode plot of the

equivalent open loop transfer function in loop 1. It isclear that loop 1 can be made stable by reversing thedirection of controller 1 (k1 ) -0.5) and properly tuningk2. However, reversing the direction of controller 1leads to the loss of integrity of the overall system againstfailure of loop 2, since a local positive feedback is formed(Zhu and Jutan, 1995a). Also, loop 2 becomes open loopunstable as a result according to theorem 1. Conse-quently, reversing control direction should be avoidedand changing a variable pairing should be resorted.In conclusion, variable pairing and controller tuning

should be jointly considered to achieve desired behavioras demonstrated in case 1 and case 3.

5. Conclusions

Upon structurally decomposing an interactive multi-variable control system into individual SISO loops, newimportant insights into interaction measurement andclosed loop analysis in decentralized control systems arepresented. A systematic analysis of system structureand closed loop properties such as loop interaction,

Figure 6. Example 2: Nyquist plots with R ) 10.0, k1 ) 1.0, k2) 1/8.5.

Figure 7. Example 2: Nyquist plots with R ) 10.0, k1 ) 1.0, k2) 2.0.

Figure 8. Example 2: Nyquist plots with off-diagonal pairing (R) 10, k1 ) 0.1, k2 ) 20).

Figure 9. Example 2: Nyquist plots with R ) 20, k1 ) 0.5, k2 )0.01.


nonminimum phase behavior, open loop stability, sys-tem integrity, and closed loop stability is performed. Theintrinsic linkage among various properties and, moreimportantly, effects of the process and the controller onloop interaction and subsequently on system properties,particularly stability, are elucidated.In particular, various stability conditions, using the

characteristic equations and the Nyquist plots directlyon the basis of the individual SISO loops with interac-tions embedded, are provided in a transparent manner.More important insights into the significance andperspective roles of variable pairing and controllertuning in avoiding undesirable behavior, such as non-minimum phase, open loop instability, and lack ofintegrity, while maintaining closed loop stability, areoffered.The main results are summarized as follows:1. Both the process and the controller play an

important role in contributing to loop interaction.2. Controller tuning or design is the final decisive

factor in maintaining closed loop stability. However,process interaction may impose difficulties in controllertuning to maintain integrity and stability.3. Nonminimum phase behavior may be induced by

loop interaction, thus imposing limitations on achievableperformance.4. RHP zeros not only cause performance problems

but also may become a source for lack of integrity andinstability.5. Inherently severe process interaction may impose

a permanent problem in maintaining integrity, in spiteof controller tuning.6. Lack of integrity is a direct source of open loop

instability.7. Variable pairing and controller tuning complement

each other and consequently should be jointly consid-ered in the design of decentralized control systems. Inparticular, variable pairing can be used to reduceprocess interaction and to alleviate difficulties in con-

troller tuning to avoid creating undesirable behavior,while controller tuning is the final tool responsible forensuring desirable behavior, particularly system stabil-ity.

Nomenclature

aij ) interaction in the uj-yi loop in the addictive formC(s) ) controller transfer function (TF) matrix (diagonal)ci(s) ) controller TF in the ith loopc′i(s) ) ci(s)/sc′i(0) ) steady state value of c′i(s)dij ) perturbation termG(s) ) process TF matrixgij(s) ) the ijth element of G(s)g̃ii(s) ) equivalent process TF with interaction included inthe ith loop

hi ) closed loop TF of the ith independent loopH ) closed loop TF matrix of a multivariable systemki ) controller gain in the ith loops ) Laplace variableui ) ith inputyi ) ith output

Greek LettersR ) parameter for tuning process interaction in examplesφij ) relative interaction in the uj-yi loop defined by eq 12κ ) Rijnsdorp quotientλij ) ijth element of the RGAïi ) open loop TF of the independent ith loop

Appendices

A. Proof of Theorem 1. We prove the theorem forthe first loop only, without loss of generality. Similarproof applies to other loops as well by properly re-arranging the diagonal elements of the process and thecontroller to the (1,1) position respectively. Let uspartition an n × n process and decentralized controller

Figure 10. Example 2: Bode plot of c1g11(1 + φ11) with R ) 20, k1 ) -0.5, k2 ) 0.1.


as follows:

1 n - 1

G(s) ) [g11 ···G12

‚ ‚ ‚···

‚ ‚ ‚

G21···G22

]n - 1

1

(a1)

1 n - 1

C(s) ) [c1 ···0

‚ ‚ ‚···

‚ ‚ ‚

0···C2

]n - 1

1

(a2)

The equivalent process is given by

where

is the relative interaction in the loop, and

is the closed loop transfer function of the systemconstituted by the original closed loop system with loop1 removed.Combining eqs a3-a5 yields,

Clearly, from eq a6, the poles of g̃11 consist of the polesof different process and controller blocks as well as thereduced system with the first loop removed.It is known that the poles of a transfer function

matrix are the poles of the least common denominatorof all non-identically-zero minor of all orders of it(Postlethwaite and MacFarlane, 1979). Clearly, a trans-fer function matrix does not contain any RHP poles ifneither of its individual elements contains any RHPpoles, and vice versa. Since the individual elements ofthe process and the controller are assumed not tocontain any RHP poles, the existence of RHP poles ofthe equivalent process, g̃11, is determined solely by thesubsystem with the first loop removed.Clearly, the subsystem H2 contains no RHP poles if

the system possesses integrity against the failure of thefirst loop. Likewise, g̃11 will inherit any RHP poles inH2, should the system lack integrity against single loopfailure. Therefore, integrity against single loop failureconstitutes the necessary and sufficient condition for g̃11to be stable.B. Proof of Theorem 2. First notice that the

overall system is minimum phase if and only if each andevery individual loop is minimum phase. The open looptransfer function of the ith loop is given by

Apparently, the zeros of g̃11 are the combinations ofthe zeros of the independent process, gii, and themultiplicate model error, (1 + φii), thus,

where Z(‚) denotes the number of RHP zeros of thetransfer function enclosed in the parentheses.The assumption implies that gii contains no RHP

zeros, i.e.,

Consequently, the existence of any RHP zeros of theequivalent open loop transfer function of the ith loopdepends solely on (1 + φii), i.e.,

Further notice that φii can be viewed as the “openloop” transfer function of (1 + φii) in terms of theexistence of RHP zeros. Consequently, the condition intheorem 2 reduces to

Obviously, a necessary and sufficient condition for (1+ φii) not to contain any RHP zeros is given by

where p denotes the number of RHP poles of φii.Without loss of generality, we shall prove the condi-

tion for the first loop only. Similar proof applies to otherloops as well by properly rearranging the respectivediagonal elements. Upon partitioning the process andthe controller as shown in eqs a1 and a2, the expressionfor φ11 is given by eq a4. Clearly, the poles of φii arisefrom the poles of various process blocks and those ofthe reduced subsystem with loop 1 removed. From theproof of theorem 1, one can conclude that the assump-tions guarantees

Finally, it can be concluded that the condition intheorem 2 constitutes a necessary and sufficient condi-tion for the system to maintain minimum phase.C. Proof of Theorem 3. According to Zhu and

Jutan (1995a), when the assumption in theorem 3 holds,a consistency principle for stability (necessary conditionfor stability) for any SISO system is given by

where

In eq c1 and c2, g(s) and c(s) denote the process and thecontroller transfer functions, respectively, and g(0) andc′(0) denote the steady state gain of their perspectivetransfer functions. Condition c1 actually specifies afeedback condition in any SISO system.In an interactive 2 × 2 system, the equivalent open

loop transfer function in either loop is given by, accord-ing to eqs 7 and 9,

Applying the necessary stability condition in eq c1 toeither loop in the 2 × 2 system upon decomposition, one

g̃11 ) g11(1 + φ11) (a3)

φ11 ) -G12G22

-1H2G21

g11(a4)

H2 ) G22C2(I + G22C2)-1 (a5)

g̃11 ) g11 - G12C2(I + G22C2)-1G21 (a6)

g̃ii ) gii(1 + φii) (b1)

Z(g̃ii) ) Z(gii) + Z((1 + φii)) (b2)

Z(gii) ) 0 (b3)

Z(g̃ii) ) Z((1 + φii)) (b4)

N(-1,φii) ) 0 ∀ i (b5)

N(-1,φii) ) p ∀ i (b6)

p ) 0 (b7)

g(0) c′(0) > 0 (c1)

c′(s) ) (1/s)c(s) (c2)

g̃ ) gii(1 - κhj) ∀, j * i (c3)


has the following sufficient condition for instability,

where c′1 ) c1/s.At steady state, we have

since independent loops are assumed to be stable andintegral action is used in the controller. Substitutingeqs c5 and c3 into c4 yields

Since independent loops are assumed stable, the fol-lowing condition holds:

Finally, a sufficient condition for instability is obtainedas follows:

By eq 12, one has

D. Proof of Theorem 5. By the Nyquist criterion,the necessary and sufficient condition for stability of theclosed loop system of the ith loop is

whereN denotes the number of clockwise encirclementsof the point (-1,0) by the Nyquist contour of the openloop transfer function go, which is defined by

and z denotes the number of RHP poles of g°.The assumption ensures, by theorem 1,

where P(‚) denotes the function to evaluate the numberof RHP poles of the transfer function enclosed in thebrackets, and

Consequently, by eqs b2-b5 the RHP poles of the openloop system becomes

Literature Cited

Balchen, J. G.; Mumme, K. I. Process Control: Structure andApplications; Van Nostrand Reinhold Company: New York,1988.

Bequette, B. W.; Edgar, T. F. A Dynamic Interaction Index Basedon Set Point Transmittance. AIChE J. 1988, 34 (5), 849.

Bristol, E. H. On a New Measure of Interaction for MultivariableProcess Control. IEEE Trans. Autom. Control 1966, AC-11, 133.

Bristol, E. H. Recent Results on Interactions in MultivariableProcess Control. Presented at the AIChE Annual Meeting,Chicago, 1978.

Chang, J. W.; Yu, C.-C. Relative Disturbance Gain Array. AIChEJ. 1992, 38, 521.

Chiu, M.-S.; Arkun, Y. Decentralized Control Structure SelectionBased on Integrity Considerations. Ind. Eng. Chem. Res. 1990,29, 360.

Eaton, J. W.; Rawlings, J. B. Feedback Control of ChemicalProcesses Using On-line Optimization Techniques. Comput.Chem. Eng. 1990, 14, 469.

Gagnepain, J. P.; Seborg, D. E. Analysis of Process Interactionswith Applications to Multiloop Control Systems Design. Ind.Eng. Chem. Process Des. Dev. 1982, 21, 5.

Grosdidier, P.; Morari, M.; Holt, B. R. Closed-Loop Properties fromSteady State Gain Information. Ind. Eng. Chem. Fundam. 1985,24, 221.

Jensen, N.; Fisher, D. G.; Shah, S. L. Interaction Analysis inMultivariable Control System. AIChE J. 1986, 32 (6), 959.

Leithead, W. E.; O’Reilly, J. M-Input M-Output Feedback Controlby Individual Channel Design Part 1. Structural Issues. Int. J.Control 1992, 56, 1347.

Mijares, G.; et al. A New Criterion for the Pairing of Control andManipulated Variables. AIChE J. 1986, 32, 1439.

Manousiouthakis, V. On the Parametrization of All DecentralizedStabilizing Controllers. Syst. Control Lett. 1993, 21, 397.

Niederlinski, A. A Heuristic Approach to the Design of LinearMultivariable Interacting Control Systems. Automatica 1971,7, 691.

Ogata, K. Modern Control Engineering; Prentice Hall: EnglewoodCliffs, NJ, 1990.

O’Reilly, J.; Leithead, W. E. Multivariable Control by IndividualChannel Design. Int. J. Control 1991,54, 1.

Postlewaite, I.; MacFarlane, A. G. J. A Complex Variable Approachto the Analysis of Linear Multivariable Feedback Systems;Springer Verlag: Berlin, 1979.

Rijnsdorp, J. E. Interaction in Two-variable Control Systems forDistillation Columns. Automatica 1965, 1, 15.

Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamicsand Control; John Wiley and Sons: New York, 1989.

Shen, S.-H.; Yu, C.-C. Use of Relay-Feedback Test for AutomaticTuning of Multivariable Systems. AIChE J. 1994, 40, 627.

Tung, L. S.; Edgar, T. F. Analysis of Control-Output Interactionsin Dynamic Systems. AIChE J. 1981, 27, 690.

Yu, C.-C.; Luyben, W. L. Robustness with Respect to IntegralControllability. Ind. Eng. Chem. Res. 1987, 26, 1043.

Yu, C.-C.; Fan, M. K. H. Decentralized Integral Controllability andD-stability. Chem. Eng. Sci. 1990, 45, 3299.

Zhu, Z.-X. Modern and Decentralized Control of MultivariableProcesses. Ph.D. Dissertation, University of Western Ontario,London, Canada, 1993.

Zhu, Z.-X.; Jutan, A, A New Variable Pairing Criterion Based onNiederlinski Index. Chem. Eng. Commun. 1993a, 121, 235.

Zhu, Z.-X.; Jutan, A. Stability Robustness for DecentralizedControl Systems. Chem. Eng. Sci. 1993b, 48, 2337.

Zhu, Z.-X.; Jutan, A. Model Identification and Optimal StochasticControl of A Multivariable Pressure Tank System. Chem. Eng.Res. Des. 1994, 72, 64.

Zhu, Z.-X.; Jutan, A. Consistency Principles for Stability inDecentralized Control Systems. Chem. Eng. Commun. 1995a,132, 107.

Zhu, Z.-X.; Jutan, A. Loop Decomposition and Dynamic InteractionAnalysis of Decentralized Control Systems. Chem. Eng. Sci.1995b, in press.

Zhu, Z.-X.; Jutan, A. RGA as A Measure of Integrity. Chem. Eng.Res. Des. 1995c, in press.

Zhu, Z.-X.; Jutan, A.; Krishnapura, V. Robust Design of A PI-TypeState Feedback Controller for Discrete Multivariable Systems.Trans. Inst. Meas. Control 1995, 17, 65.

Received for review July 24, 1995Revised manuscript received November 14, 1995

Accepted November 29, 1995X

IE950455A

X Abstract published in Advance ACS Abstracts, February1, 1996.

g̃11(0) c′1(0) < 0 (c4)

hj ) 1 (c5)

gii(0)(1 - κ)c′1(0) < 0 ∀ i (c6)

gii(0) ci(0) > 0 ∀ i (c7)

κ > 1 (c8)

λ11 < 0 (c9)

N(-1,g°) ) z (d1)

g° ) cig̃ii ∀ i (d2)

g̃ii ) gii(1 + φii) ∀ i (d3)

P(g̃ii) ) 0 (d4)

P(ci) ) 0 (d5)

z ) P(ci) + P(g̃ii) ) 0 (d6)


Structural Analysis and Stability Conditions of Decentralized Control Systems

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