Variable Selection for Decentralized Control

Decentralized controllers single-loop controllers applied to multivariable plants are often preferred in practice because they are robust and relatively simple to understand and to change. The design of such a control system starts with pairing inputs manipulated variables and outputs controlled variables. For a n n plant there are n! possible pairings, and there is a great need for screening techniques to quickly eliminate undesirable pairings. In this paper we present several tests for eliminating pairings which are not decentralized integral controllable DIC. A system is DIC if there exists a stabilizing decentralized controller with integral action such that the gains of the individual loops may be reduced independtly without introducing instability. Note that DIC is a property of the plant and the chosen pairings. The tests presented are in terms of diierent measures of the sign of steady state gain matrix; including the RGA, the determinant and eigenvalues. The relationship to previously presented results is discussed in detail.


INTRODUCTION
Decentralized control implies the use of single-loop controllers to control multivariable processes.This means that for any particular choice of pairing of controlled and manipulated variables we can rearrange the plant G such that the controller C is diagonal C = diagfc i g The constraints on the controller structure invariably lead to performance deterioration when compared to the case with a full controller matrix.Still, decentralized controllers are very common in practice, for the following reasons: i ease of implementation, ii simpli ed design, iii decentralized tuning, iv robustness with respect to model error, and v ease of making failure tolerant.In short, single loop controllers are preferred by the operators because they are robust and realtively simple to understand and to change.
The designer of decentralized controllers is faced with the issues of 1 pairing outputs and inputs, and 2 controller design tuning of the individual loops.This paper adresses the pairing problem.Even for relatively small plants, there are many decentralized control systems to choose from.Consider pairing of single loops.Then f o r a 2 2 plant there are two alternative s , a 3 3 plant o ers 6, a 4 4 plant 2 4 , a 5 5 plant 120, etc.Thus e cient screening techniques are needed which are capable of eliminating quickly inapropriate control structures.
In this paper the criterium chosen is that the controller structure should be Decentralized Integral Controllable" DIC.A plant G corresponding to a particular choice of pairings is DIC if there exists a stabilizing decentralized controller with intergral action no o set such that the gains of the individual loops can be reduced independently without introducing instability.In particular, DIC implies that any subset of loops can be detuned or taken out of service put in manual" without a ecting stability.Note that DIC is a property of the plant and the particular control structure pairing chosen.
Necessary conditions for DIC are of particular interest, since a violation of such a condition means that DIC is not possible and the corresponding pairing may be eliminated.For most plants the majority of the alternatives may be eliminated using such conditions.To select the best of the few remaining alternatives, su cient conditions for DIC prove more useful.In this paper several necessary conditions in terms of the steady state gain matrix are presented.Some of the results have been presented elsewere, but their interpretation in terms of DIC is new.It is stressed that only steady state data are needed.
The main reason for the problems encountered with decentralized controllers are the interactions" caused by the o diagonal elements in the plant G.If these elements are small" then interactions are weak and decentralized control is simple.If the interactions are large, then it might happen that the sign of the plant gain between a speci c plant input and output changes sign as other loops are closed.Integral control, which i s k n o wn to dependent o n k n o wing the plant gain, is then not possible.All of the conditions presented are therefore in terms of avoiding pairings where the plant gain may c hange sign as other loops are changed.
A good discussion of the importance of the pairing problem is presented by Nett andSpang 1987. Bristol 1966 introduced the relative gain array R GA as a criterion for choosing the best variable pairing, and this measure continues to be the one most often used.Niederlinski 1971 proposed considering the sign of the determinant of the plant as a screening tool.McAvoy 1983 andGrosdidier et al. 1985 discuss the use of the RGA in more detail and provide theoretical justi cation for Bristols rule of avoiding pairings corresponding to negative relative gains.Grosdidier et al. 1985 also present several conditions for a plants to be integral stabilizable IS or integral controllable IC, which upon reformulation turn out to be useful tools for eliminating pairings.Mijares et al. 1986 introduced the Jacobi Eigenvalue criterion" as a tool for selecting the best pairing.This criterion is closely related to the SSV -interaction measure introduced by Grosdidier andMorari 1986. Yu andLuyben 1986 present three rules for eliminating unworkable pairings.The rst is based on the RGA, the second on Niederlinski's result, and the third involving MIC is based on Thm.7 determinant condition for IS in Grosdidier et al. 1985. Grosdidier andMorari 1987 introduced the property of block-IC.This property is easier to satisfy than DIC, since DIC implies block-IC, but not conversely.
The objective of this paper is to show that all the above-mentioned conditions are related rigorously to DIC and IC, and to derive som new conditions for DIC.
Throughout the paper we assume that the plant Gs is a square, open-loop stable, strictly proper transfer matrix.The steady state value of this matrix is G0.A general decentralized control system is shown in Fig. 1.The notation is summarized at the end of the paper.All rules and theorems are given for the case of single loop pairings, but most of them are easily extended to blocks see Manousiouthakis et al., 1986, andGrosdidier andMorari, 1987.

SUMMARY O F R ULES FOR ELIMINATING UNDESIRABLE PAIRINGS
Below w e present a summary of rules for pairing selection.The rules are subsequently proved in Section 4.
Yu and Luyben 1986 present three rules for eliminating what they call unworkable variable pairings: Rule 1. Eliminate pairings with negative R GA's.
Rule 2. Eliminate pairings with negative Niederlinski Indexes, N I = detG0 Q n i=1 g ii .
Rule 3. Eliminate pairings with negative Morari Indexes of Integral Controllability, MIC = R e fG + 0g.
In fact, we will show that violation of any of these three rules imply that we the plant is not decentralized integral controllable DIC with this choice of variable pairings.With respect to rule 3, Yu and Luyben claim that a negative MIC-value shows that the variable pairing will produce an unstable closed-loop system".This is not necessarily correct see Example 2 below, but it might happen if one of the loops is detuned since the system is not DIC.In Section 4 we establish rule 3 rigorously in terms of DIC.We also show that rule 2 involving NI is redundant, because rule 3 always implies rule 2 as special case.
Furthermore, the following new rules for eliminating pairings for which DIC is not possible are established : Rule 4. Eliminate pairings with Re fE0g ,1; E = GG diag G ,1 diag .Rule 5. Eliminate pairings for which there exists a K diagonal matrix with positive e n tries which yields Re fG + 0Kg 0.
Rule 3 and 4 are special cases of rule 5.They are derived from rule 5 by choosing K equal to I the identity matrix and G + diag 0 ,1 , respectively.Rule 4 i n volves the matrix E used in the interaction measures derived by Grosdidier and Morari 1986.Rule 4 is equivalent to eliminating pairings with eigenvalues of the Jacobi iteration matrix" greater than one Mijares et al. 1986.Numerical evidence prove that neither of rules 1,3 or 4 are mutually redundant, and all three are therefore useful.
One important advantage with the RGA rule 1 is that it is very simple to compute and does not have to be recomputed to investigate alternative pairings.This follows since a permutation of the rows or columns in the plant G, corresponding to a change in pairings, results in the same permutation in the RGA Bristol, 1966.Consequently, one should always start by eliminating undesirable pairings according to rule 1 RGA, and subsequently use rules 3 and 4 to screen the remaining alternatives.
Rules 1-5 above m a y all be reformulated as necessary conditions for DIC.This means that a plant that does not pass these tests is not DIC, but there may b e other plants that pass the tests, but still turn out not to be DIC.
There also exist su cient conditions for DIC.One of these is in terms of the structured singular value Doyle, 1982 of E, and yields the rule: Rule 6. Prefer pairings with E0 1.
here is computed with respect to the structure of C. Note that E E jEj 1 and we therefore have that the eigenvalues of E0 should always be greater than -1 rule 4, and their magnitude E0 preferably less than 1 rule 6 and eq. 1.
The criterion that the spectral radius of E0, E0, should be less than one is equivalent to the Jacobi Eigenvalue Condition" of Mijares et al 1986.This condition is rigorously related to IC see below, but not to DIC.For DIC, E0 is the right measure.
There exists no simple necessary and su cient condition for DIC.If it is not possible to nd any K which satis es the criterion for elimination in rule 5, then, under some minor technical conditions, we m a y conclude that the system is DIC.
That is, DIC is equivalent t o min K min i Ref i G + 0Kg 0 However, this last condition is di cult to test, and therefore of limited practical value.

DEFINITIONS
Decentralized Integral Controllability.A plant G corresponding to a particular pairing is decentralized integral controllable DIC if it possible to design a diagonal controller for this plant which 1 has integral action no o set for tracking, 2 yields stable individual loops, 3 is such that the system remains stable when all loops are closed simultaneously and 4 has the property that each loop gain may be reduced independtly with a factor i 0 i 1 without introducing instability.
Decentralized controllers are frequently used in process control, and it is obviously desirable that they satisfy the above requirements for DIC.Note that the property of DIC depends on the particular pairings chosen: the plant m a y satisfy DIC for one choice of pairings, but not for another.To satisfy condition 2 the controller must be such that the individual loop gains g ii c i are all positive.Also note that property 4 implies 2 since one particular choice of loop gains is to have all but one loop with zero gain.
The de nition of DIC is similar to that of Integral Controllability" IC introduced by Grosdidier et al. 1985.Integral Controllability.A system plant and controller is integral controllable IC if 1 the controller has integral action, 2 the overall system is stable, and 3 all controller gains may be reduced by the same factor 0 1 without introducing instability.
Note from the de nitions that for IC all the gains are reduced by the same amount, while for DIC each loop gain may be reduced at a di erent rate.Consequently, there may be a particular decentralized controller which satis es IC, but this does not necessarily imply that the plant with this choice of pairings satis es DIC.However, the reverse holds: DIC ICwith any decentralized controller with positive loop gains 2 Thus control structures which fail the IC test can be eliminated when searching for a plant which is DIC.Another somewhat subtle point is that whereas IC is a property that depends on both the plant G and the controlller C, the property of DIC depends only on the plant.This follows because we are allowing each loop gain to be reduced by an arbitrary amount which is equivalent to allowing any ratio between the elements in the controller and we are therefore considering all possible diagonal controllers at least at steady state.From this point of view DIC is a much more useful property than IC since it is an inherent plant property independent of the particular choice of controller.

Necessary conditions for DIC
The basis for all the results presented below is that negative feedback is needed to have stability under integral control, that is, we m ust know the sign of the plant gain.We will see that all of the results involve di erent expressions for the plant gains, either in terms of the determinant, eigenvalues or relative gains.We will rst recall three results for DIC and IC given by Grosdier et al. 1985 -though they are not explicitly written in this form.
Theorem 1 basis for rule 1. Assume Cs is a diagonal controller and that GsCS is proper always satis ed for any real system.Then RGA ii G0 0 for some i notDIC 3a or equivalently see Appendix DIC RGA ii G0 0; 8i 3b Here RGA ii G denotes the i'th diagonal element of the RGA of G. Proof: Follows from Theorem 6 in Grosdidier et al. 1985.
The rule of avoiding pairings corresponding to negative R GA-elements goes back to Bristol 1966, but it was proved rigorously only recently.Note that ij'th element of the RGA is de ned as RGA ij = @y i =@u j u k;k6 =j @y i =@u j y l;l6 =i = g OL g CL that is, it represents the ratio of the gain from u j to y i in open loop other u's constant and closed-loop other y's constant.If the sign of this gain changes as we c hange or close other loops, then we are not able to apply negative feedback i n all cases, and the plant is not DIC.
Theorem 2 basis for rule 2. Assume Cs is a diagonal controller, Gs i s stable and that GsCs is strictly proper always satis ed for any real system.Then detG0 Q n i=n g ii 0 not DIC 4a or equivalently detG + 0 0 not DIC 4b Proof: F ollows from Thm. 3 in Grosdidier et al. 1985. 4a is Niederlinskis result which tells us that we should avoid using decentralized control on pairings which h a ve the sign of the plant given in terms of its determinant di erent from the product of the plant gains for the loops.Again, this is a condition for avoiding the use of positive feeback.
Most of the new results in this paper Theorems 4, 5 and 6 are based on the following theorem in terms of IC: Theorem 3 eigenvalue condition for IC.Write the controller Cs with integral action as Cs = k s Ĉs.Then there exists a k 0 such that the system is stable for all 0 k k ie., the system is IC if Ref i G Ĉ0g 0; 8i 5a and there does not exist such a k ie., the system is not IC if Ref i G Ĉ0g 0; 8i 5b Proof: See Theorem 7 in Grosdidier et al. 1985.The proof is based MacFarlanes generalized Nyquist theorem in terms of the characteristic loci.
In words, the real part Re of all the eigenvalues of G Ĉ0 must be positive t o have IC, ie., the eigenvalues must all be in the right half plane.Furthermore, if we disregard the few cases where the eigenvalues of G Ĉ0 are one the j!-axis purely complex, this is a necessary and su cient condition.The following condition in terms of DIC when C is diagonal is easily derived from 5b: Theorem 4 basis for rule 5. Let K be a diagonal matrix with real, positive nonzero entries.Then min i Ref i G + 0K 0; for some K not DIC 6 Proof.Consider a speci c diagonal controller C which yields positive individual loop gains g ii c i needed to satisfy property 2 in the de nition of DIC.Write GC0 = G + 0K where K = jC0j has only positive elements.Then from Theorem 3, eq.5b: RefG0 + Kg 0 not IC for this diagonal controller not DIC the last implication follows from eq. 2.
Theorem 4 by itself is not too useful because it requires specifying a controller.However, the following two results are obtained by c hoosing the diagonal controller gains K as I and G + diag 0 ,1 , respectively.Theorem 5. basis for rule 3 min i Ref i G + 0g 0 not DIC 7 Theorem 6. basis for rule 4 Theorem 5 is the basis for the MIC-rule rule 3 which has been presented previously by Y u and Luyben 1986, but without any proof.Furthermore, Yu and Luyben interpret the MIC's in terms IC and not in terms of DIC as they should.
Note that detG + which appears in Thm. 3 is the product of the eigenvalues of G + MIC's which appear in Thm. 5.An even number of negative eigenvalues of G + will result in a positive detG + , but the reverse is not possible ie., negative detG + cannot yield all positive eigenvalues; this follows since any complex eigenvalues come in pairs.Consequently, Thm.4 yields Thm.2 as a special case but not vice versa, and the NI therefore contains less information than the MIC's.Use of rule 3 therefore makes rule 2 redundant.
Rule 4 follows from Thm. 6 since E0 = G0G ,1 diag 0 , 1.A similar result to Theorem 6, but in terms of RefE0g ,1 as a necessary and sucient condition for IC our result is that it is a necessary condition for DIC, has been derived by Mijares et al. 1986 eq. 37 in their paper.They consider the eigenvalues of the Jacobi Iteration Matrix" A = I , G ,1 diag , but this is essentially the same matrix as E since i E = , i A. For an alternative proof of Mijares' result see Skogestad and Morari 1987.Other theorems similar to Theorem 5 and 6 can be derived by making other more or less arbitrary choices for the matrix K: I f w e can show for a particular diagonal controller that IC is not possible, then we know that DIC is not possible for this plant.However, the two c hoices made above seem to be the most reasonable, and also tie in very nicely with results presented previously.
4.2 Su cient conditions for DIC.
-conditions.The matrix E = G , G diag G ,1 diag in rule 4 appears in the interaction measures introduced by Grosdidier and Morari 1986.From Corollary 2.1 in their paper we derive that a su cient condition for having IC is that E0 magnitude of largest eigenvalue is less than one, or equivalently E0 1 IC 9 Note that this does not guarantee DIC since one requirement of using E i s that all loops g ii c i are identical ie., Ĉ = G ,1 diag and the loops cannot be detuned independently.H o wever, an equivalent condition in terms of DIC results if E i s replaced by E, that is Theorem 7 in Grosdidier and Morari, 1986, E0 1 DIC 10 The generalization to DIC follows since the use of E allows the individual loops to be di erent.E0 can be used to tell that DIC is satis ed for a particular pairing.However, it cannot be used to eliminate variable pairings since it may b e possible to achieve DIC for a plant e v en though E0 and thereby also E0 recall eq. 1 is greater than 1.This is illustrated in the discussion on 2 2 plants below.
However, the main advantage with E0 1 is that interactions are small and the controllers for each l o o p m a y easily be designed independtly that is, on the basis of G diag only see Grosdidier andMorari, 1986, andSkogestad andMorari, 1988, who provide guidelines for the design.Consequently, w e prefer pairings with E and E less than 1 because we i are guaranteed DIC and IC, and ii may easily design the loops independently.This is the basis for rule 6.
Block-IC.The de nition of DIC is similar to that of block-IC here denoted loop-IC since we only consider single loops and not blocks introduced by Grosdidier and Morari1987.Loop-IC implies that one loop at the time may be detuned, but DIC is stricter since it all allows all loops to be detuned simultaneosly in an arbitrary fashion.Thm.6 in Grosdidier and Morari 1987 say that, provided the system is IC with a diagonal controller in the rst place, loop-IC of all loops is guranteed if and only if the RGA has positive diagonal elements.Note that a separate test is needed in addition to test IC.The only if part is not too surprising from the DIC-condition in Theorem 1.The main new information in this result is therefore RGAG0 ii 0; 8i loop , IC 11 That is, positive R GA-elements gurantees that loops may be detuned one at the time provided the system is IC in the st place.We h a ve another result eq. 10 which also guarantees the same E0 1 DIC loop , IC 12 However, 11, which is necessary and su cient, is of course more useful less conservative than 12.
4.3 Necessary and su cient conditions for DIC.Theorem 7. Let K be any diagonal matrix with real, positive nonzero Disregard the case when of G0 or any of its subsystems obtained by c hoosing some c i = 0 is exactly zero eg., caused by G0 or any of its subsystems being singular.Then Theorem 3 is necessary and su cient for DIC, that is G0 0 , DIC 14 Proof: This result follows from the de nition of DIC and Theorem 3: From Theorem 3 w e know that conditions 1 to 3 in the de nition of DIC will be satis ed if RefG + 0Kg 0 15 Condition 4 with 0 i 1 is satis ed if eq. 6 is satis ed for all possible K's Follows from Thm. 3, 5a by considering all possible diagonal controllers, that is, each gain may be reduced independtly.Condition 4 with some i = 0 is not covered by this test eq.13.This case corresponds to deleting rows and corresponding columns in G0, and considering instability of the remaining subsystem under decentralized control.Simple limiting arguments show that stability of these susbsystems is also guaranteed by eq. 15 provided neither of the submatrices yield exactly equal to zero eg., caused by singular submatrices.For example, this means that pairing on elements with zero gain is disregarded.Summing up we h a ve under this condition that min K min i Ref i G + 0Kg 0 DIC 16 Combining this with 6 and assuming in addition that G0 is not exactly zero eg., caused by G0 being singular we arrive at condition 14.
As mentioned before condition 14 is of limited usefullness since it is di cult to test.In particular, if the plant is DIC then G0 !0 + choose small elements in K, and if it is not DIC then G0 !, 1 choose large elements in K.In our numerical studies we h a ve used a general purpose optimization routine which seems to have w orked satisfactory.The optimization is stopped as soon as a K which yields negative eigenvalues og G + 0K is found.4.4 2 2 plants Theorem 8. Consider 2 2 plants with G0 nonsingular and both diagonal elements nonzero.Then theorems 1-5 and rules 1-5 are all equivalent and are all necessary and su cient for DIC.DIC , RGA 11 0 , NI 0 , MIC 0 , RefE0g ,1 17 Proof: This follows from the following facts: 1. Eq. 14 in is necessary and su cient for DIC provided G0 is nonsingular and the diagonal elements are both nonzero.
2. For 2 2 plants G + 0K will have all its eigenvalues in the right half plane if and only if G + 0 has all its eigenvalues in the right half plane this fact is easily established by applying the Routh test to the charcteristic polynomial.Consequently, eq. 14 and Theorems 3-5 are equivalent in this case.3. For 22 plants G + 0 has all it's eigenvalues in the right half plane if and only if detG + 0 is positive.Consequently, theorems 5 and 2 are equivalent.
4. For 2 2 plants both the diagonal elements of the RGA are equal and furthermore RGA 11 = 1 =NI.Consequently, Theorems 1 and 2 are equivalent.
-conditions.F or 2 2 plants E = E and condition 12 becomes Grosdidier and Morari, 1986 E0 1 DIC 18 Furthermore, 18 is equivalent to Grosdidier and Morari, 1986 RGA 11 G0 0:5 DIC 19 However, from Theorem 8 we h a ve another necessary ans su cient condition for DIC RGA 11 G0 0 , DIC 20 Consequently, whereas we know from 20 that it is possible to design a controller which is DIC for 2 2 plants with postive diagonal RGA elements, condition 19 indicates that the RGA-elements should be greater than 0.5.Conditions 18 and 19 are therefore conservative su cient only.This does not mean that there might not be factors other than DIC that may f a vor choosing pairings with RGAelements larger than 0.5.For example, closed-loop performance may be better because of less interactions which m a y make it possible to use a higher gain The de nition of DIC just says there will exist some detunable diagonal controller with integral action that yields stability; it does not guarantee good performance.This is a 2 2 plant and rules 1-5, which are necessary and su cient for DIC in this case, all tell us that this plant is DIC.This is the case even though E0=E0 = 1:41 which means that the su cient condition 10 for DIC is not satis ed.Note that the plant is DIC also with the reverse pairing diagonal RGA-elements are 0.67, and in addition E0 = 0:71 1 in this case.The reverse pairing is therefore preferable according to rule 6.Here E0 and NI are inconclusive, the MIC-and RGA-values tell that the plant is not DIC with this pairing.However, this does not necessarily mean the plant is not integral controllable IC.Consider the following diagonal controller consisting of three SISO controllers; C = k s Ĉ; Ĉ = diagf0:1; 1; 0:1g.This controller yields stable individual loops since the loop gains are positive.Furthermore, the matrix G Ĉ0 has all eigenvalues in the right half plane G Ĉ0 = f0:410:23j; 2:19g, and we know from Theorem 3 that the controller yields a system which i s i n tegral controllable IC.This means that the system will remain stable if all the gains in the controller are detuned by the same amount.However, if each loop is detuned in an arbitrary fashion, the the system may become unstable.For example, we know from the negative MIC-values, that if we detune the controller gain in Ĉ for the second loop from 1 to 0.1, and keep the other controller gains xed at 0.1, then the system will become unstable.This kind of conditional stability is clearly undesirable and this is the reason why one in practice prefers plants which are DIC and not only IC.

EXAMPLES
Example 3.This model of a sidestream column is given by Elaahi and Luyben 1985.Here only the RGA allows us to conclude that this pairing is not DIC; all the other tests are inconclusive.In fact, from the RGA we see that it is impossible to rearrange the plant such that all diagonal RGA-elements are positive.Consequently, this plant is not DIC for any c hoice of pairings.Example 5.This example was rst presented by Niederlinski 1971 and is also used by Mijares et al. 1986.G0 = 0 @ 1:0 ,0:1 1:0 ,0:5 0:6 0:1 ,0:2 ,0:8 0:3 1 A RGA = 0 @ 0:34 ,0:02 0:68 0:50 0:39 0:11 0:16 0:63 0:21 1 A NI= 4 :26; MIC= f0:27 0:70j; 1:35g E0 = f,0:52 1:36j; 1:05g This choice of pairings is not excluded by a n y of the pairing rules, and we m a y therefore expect it to be DIC.It is certainly IC, for example with a controller with C0 = k=sI since the MIC's are postive.Furtthermore, we know from eq. 11 that this plant is loop-IC, that is , we m a y detune one loop at the time in an arbitrary fashion.It then seems extremely likely that the plant also is DIC.Indeed, a n umerical search g a ve G + 0 0, and we conclude from Theorem 7 that the plant decentralized integral controllable DIC with this choice of pairings. Notation.

Cs = Ĉs=s -transfer function of decentralized diagonal controller with integral action
Ĉs -transfer function of controller excluding intergral action G + 0 -plant steady-state gain matrix with the signs adjusted so that all diagonal elements have positive signs.