Hankel-norm Based Interaction Measure for Input-output Pairing

8/8/2019 Hankel-norm Based Interaction Measure for Input-output Pairing

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diagonally dominant, at least in the frequency bandsof T 1 , T 2 , . . . , T p .

In this problem it is possible to distinguish severalcrucial elements, such as

The (plant and loop) coupling characteristics, ingeneral, are frequency dependent

The (loop) coupling characteristics, in general,are dependent on the choices made for the SISOcontrol designs.

If the system steady-state gain is known, steady-state decoupling can be implemented leading tolow frequency decoupled control loops.

Different dominance characteristics may lead toa different pairing of inputs and outputs.

The Relative Gain Array (RGA), Bristol (1966), is abasic tool to decide on input-output pairing in multi-variable processes. The advantage of the RGA and, atthe same time, its disadvantage is that it only relieson the steady-state process gain. The RGA gives in-dications about the difculties to control the processwith only SISO controllers and, in many cases, sug-gests the most suitable input-output pairings. Differentmodications have been proposed, see e.g. Niederlin-ski (1971) and Chiu and Arkun (1991). Also, attemptshave been made to take the frequency response of theprocess into account, such as those reported in Gagnonet al. (1999), Zhu (1996), and Yang et al. (1999).

Moore (1986) uses the singular values of K = G(1) =U V T and the vectors of U and V to make the pair-ing. Structured singular values are used in Grosdi-dier and Morari (1986) to analyze interaction. In Stan-ley et al. (1985) the Relative Disturbance Gain is in-troduced, which includes the inuence of the distur-bances of the process. Witcher and McAvoy (1977)proposed a dynamic RGA (DRGA) as a natural ex-tension of Bristols RGA, namely

DRGA(G ) = G (e j ) G (e j ) T

where the operation symbol denotes element byelement multiplication. Although this measure mayreveal some hidden interaction features, it can be hardfor the designer to make decisions regarding input-output pairing based upon the comparative analysis of p p frequency responses. A good survey on input-output pairing can be found in Kinnaert (1995).

In Conley and Salgado (2000) a new gramian basedinteraction measure is introduced. This new measureis based upon the sum of the squared Hankel singularvalues for the elementary subsystems of the process. Inthis paper, a modication to that approach is proposedwhere we only use the Hankel norm of the subsystems.With this modication, we achieve further physicalinsight in terms of the controllability and observabilityof the different subsystems. The new measure dealswith the full frequency range of the plant. However, it

can also be used to investigate suitable bandwidths forthe different control loops.

The idea of using gramians has already been usedin some methods to solve the input/output selectionproblem, as reported, for instance, in van de Wal andde Jager (2001). However, in those cases the gramiansserve to determine which inputs and outputs are to beused to control a process. In this paper, we assume thatthe input/output selection has already been made andwe deal with how to pair those inputs and outputs tobuild a decentralized controller.

2. PROCESS DESCRIPTION AND PROBLEMFORMULATION

Assume that the (completely controllable and com-pletely observable) process is described by its squarepulse transfer function matrix G ( z) given by (2) whereG i j ( z) C denotes the transfer function from the j:thinput to the i:th output. It is further assumed that theprocess is stable.

We note that G i j ( z) describes the completely control-lable and completely observable part of the elemen-tary subsystem (i, j). The (minimal) realization for thewhole process is given by the 4-tuple ( A, B, C , D)while the (not necessarily minimal) realization asso-ciated to the elementary subsystem (i, j) is given bythe 4-tuple ( A, B j , C i, D i j ), where B j is the j:th col-umn of B, C i is the i:th row of C and where D i j is the(i, j) element of matrix D.

The main issue addressed in this paper is how tond the most suitable pairing of inputs and outputs.We also want to consider the dynamic properties of the system, which implies that we are looking for amore sophisticated measure than the RGA, since thisonly considers the zero frequencycharacteristics of theprocess, i.e. deals only with G (1) or at some otherpredened frequency related to the desired closed loopbandwidth. Our goal naturally implies that we must beprepared to perform more elaborate computations todetermine the new interaction index.

When the above dened goal is satisfactorily achieved,a sensible input-output pairing is found. This denes acontroller structure, which can be used to design andto tune a decentralized controller. With some abuseof language this will be called a diagonal structure,since the controller structure can be made diagonalby permutations of the inputs and/or the outputs. Wealso want to have indications if a diagonal structure isinsufcient to provide a good control.

3. HANKEL SINGULAR VALUES

The controllability gramian, c , and the observabilitygramian, o , associated to system (1) satisfy the Lya-punov equations

c A c A T B B T = 0


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o A T o A C T C = 0

Furthermore the corresponding gramians, ( j)c and (i)o , associated to the elementary subsystem (i, j)satisfy

( j)c A ( j)c A T B j B j T = 0 (4)

(i)o AT (i)o A C i

T C i = 0 (5)

The controllability and observability gramians of asystem quantify the difculty to control and to observethe system state. For instance, the ranks of matrices c and o are the dimensions of the controllable andthe observable subspaces, respectively. Gramians varywith state similarity transformations. However, theeigenvalues of the product of the gramians c and o are invariant with respect to those transformations(see, e.g. Glover (1984)).

If 1 , 2 , . . . , n are the eigenvalues of c o , then thesystem Hankel singular values H i are

H i (G ( z)) = i i = 1, 2, . . . , nwhere,by convention, H i H i+ 1 0. The Hankel sin-gular values (HSV) are fundamental invariants that arerelated to both the gain and the dynamic complexity of the system. The number r (r n) of nonzero Hankelsingular values is the dimension of the controllableandobservable subspace. The HSV also play an important

role when making balanced realizations and model re-duction. We will use the HSV for a different purpose.

The Hankel norm of a system with transfer functionG ( z) is dened as

G ( z) H = max ( c o) = H 1The Hankel norm is thus the maximum HSV. TheHankel norm can also be interpreted in the followingway. Consider a system input u[k ] = 0 k > 0 and asystem output y[k ], then

G ( z) H = supu 2 ( ,0 )

y 2 (0 , )u

2 ( ,0 )(6)

where 2 (0, ) is the space of square summable vectorsequences in the interval [0, ). An analogous deni-tion applies to 2 ( , 0).

The Hankel norm gives the 2 gain from past inputsto future outputs. Equation (6) can be interpreted asa measure of how signicant is the effect of an inputon the state and how much that effect is reected inthe output. The Hankel norm can thus be interpretedas a controllability and observability measure of thesystem. Summarizing, the Hankel norm has two fun-damental properties: is a controllability-observabilitymeasure and, secondly is independent of the statespace representation of the system.

4. HANKEL INTERACTION INDEX

We will next use the Hankel norm to build an inter-action index. For each elementary subsystem ( A, B j ,C i , D i j ), we use the Hankel norm to quantify the abil-ity of input u j to control output yi . These norms are

collected into a matrix H , where the element (i, j),[ H ]i j , is dened by

[ H ]i j = G i j ( z) H

The H array is very similar to the Participation Ma-trix, , proposed in Conley and Salgado (2000), wherethe elements of the matrix are the trace of ( j)c

(i)o .

The elements in can easily be bounded using theHankel norm for the corresponding elementary sub-system. Furthermore, the H has a structure which isthe transpose of the structure of the Participation Ma-trix. This and equation (6) allow us to interpret

H as

a gain matrix and we may then write the relation

y H = H u H

which provides an easy way to visualize differentconnections between inputs and outputs. The H arrayis thus a gain measure in the same way as the RGA.

To get rid of difculties arising from scaling we cannormalize the matrix in different ways. One idea tointroduce a scaling is to use the same method as forthe RGA, i.e. to use an index

H T H 1 (7)

where the operator denotes the element-by-elementmultiplication of the two matrices. This normalizationresults in a matrix where the sum of all elements ina row or column is equal to one, although (7) doesnot have the same nice physical interpretation as forthe RGA. Another way of making the normalization isto express every input and every output in percentageof full scale. This normalization supports the idea of scaling all elements in H so that their sum is equalto one. The latter normalization is preferred and we

choose the following normalized Hankel InteractionIndex Array

[ H ]i j =G i j ( z) H

i, j G i j ( z) H (8)

Using the same methodology as for the RGA wedetermine the input-output pairing by nding in eachrow i the largest element (i, j). Input j is then selectedto control output i.

To compute the Hankel Interaction Array we haveto solve the 2 p Lyapunov equations (4) and (5) and

compute the eigenvalues of the p2

products ( j)c

(i)o to

obtain the Hankel norm.

If G i j ( z) = 0 for a given pair (i, j), then ( j)c

(i)o =

0, leading to [ H ]i j = 0. This implies that a block


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diagonal G ( z) gives a block diagonal H matrix, withthe same structure. This is consistent with intuition,since, in those cases, the H matrix will suggest theright controller structure. It is important to observe thatthe H takes the full dynamic effects of the system intoaccount and not only the steady-state performance as

the RGA or the behavior at a single frequency.

5. CLOSED LOOP BANDWIDTH EFFECT

When a measure of dynamic interaction is built, atten-tion should be paid to the relevant frequency range.This has, for instance, been proposed in Witcher andMcAvoy (1977) and Gagnepain and Seborg (1982).Specically, interactions are meaningful in control de-sign only in a frequency band where the plant in-put has signicant energy. One way to introduce thiselement into the building of the interaction index isto observe that the (vector) plant input, u[k ] is con-nected to the reference signals and to the output distur-bances through the control sensitivity Su( z), see Good-win et al. (2000), through the expression

Su( z) = ( G ( z)) 1 T ( z)

The frequency response of Su( z) depends on the re-lationship between the plant bandwidth and the closedloop bandwidth. To gain insight, we consider the SISOcase with a biproper controller. In that simpler situa-tion, we observe that, when the closed loop bandwidthis larger than the plant bandwidth, the control sensi-tivity has larger magnitude at high frequencies thanat low frequencies. In the reverse situation, the con-trol sensitivity has a low-pass characteristic. This alsoapplies to MIMO systems. However, it has to be ap-plied with caution since there may be a non-uniqueclosed loop bandwidth. Even with this caution, thereare many cases when this approach will provide a use-ful information if we apply the Hankel Interaction In-dex to a modied system, G ( z), obtained by lteringthe plant through a (scalar) case-dependent lter, F ( z),i.e.

G ( z) = G ( z)F ( z)

The effect of this ltering will be illustrated in thefollowing section.

6. EXAMPLES

Some examples are used to illustrate the usefulness of the Hankel Interaction Index.

E XAMPLE 1Consider the system with the pulse transfer function

G ( z) =

0.1( z 0.8)( z 0.5)

0.08 z 0.8

0.24 z 0.5

0.1( z 0.1)( z 0.8)( z 0.5)

(9)

The Hankel Interaction Index Array (8) is

H =0.3755 0 .18720.1300 0 .3073

This result suggests that a diagonal controller should

be chosen with the pairs (u1 , y1 ) and (u2 , y2 ), since thelargest elements are the elements (1, 1) and (2, 2).

For the MIMO process (9), the RGA is

RGA (G ) =0.8242 0 .17580.1758 0 .8242

In this example the RGA leads to the same conclusionregarding the pairing of inputs and outputs.

E XAMPLE 2The system in this case has a transfer function G ( z)given by

0.1021 z 0.9048

0.3707 z 0.3535 z2 1.724 z+ 0.7408

0.192 z+ 0.1826 z2 1.869 z+ 0.8781

0.09516 z 0.9048

(10)If the RGA is computed we obtain

RGA (G ) =0.5033 0 .49670.4967 0 .5033

We observe that the RGA suggests, very weakly, thatthe pairing should be (u1 , y1 ) and (u2 , y2 ). On the otherhand, the Hankel Interaction Index Array is

H =0.1936 0 .32810.2978 0 .1805

(11)

The Hankel Interaction Index provides a clear sugges-tion. It directs the designer to pair (u1 , y2 ) and (u2 , y1 ).The reason for this difference is that the Hankel In-teraction Index Array takes into account the dynamicfeatures of the interaction.

A deeper insight can be gained with a simple controldesign. Assume that we want that y1 [k ] tracks a stepreference signal, r 1 [k ] = [k ], and that y2 [k ] tracksanother step reference signal, r 2 [k ] = [k 5]. Wewant to synthesize a dead-beat control which drivesthe error to zero after two time units. The synthesismethod is the Youla parameterization of all stabilizingcontrollers (Goodwin et al. (2000)), i.e. the controllerhas a transfer function, C ( z), given by

C ( z) = ( I G o( z)Q ( z)) 1 Q ( z)

where Q ( z) is any stable transfer function matrix, and

where G o( z) is the nominal model. The dead-beatperformance is nominally achieved if

Q ( z) =1

z2(G o( z))

1


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0 5 10 15 20

1.5

1

0.5

0

0.5

1

1.5

Time [s]

S t e p r e s p o n s e

y1 [k]

y2 [k]

Fig. 1 Closed loop response to reference signals r 1 [k ] = [k ] andr

2 [k ] = [k 5] in Example 2.

provided that G o( z) has all its zeros inside the openunit disk (as it does happen in this example).

If we abide by the RGA (weak) indication, the nominalmodel is G o(s) = diag (G 11 ( z), G 22 ( z)) ; if, instead, wefollow the Hankel Interaction Index suggestion, thenthe nominal model is G o(s) = adiag (G 12 ( z), G 21 ( z))where adiag stands for anti-diagonal .

When the controller is computed in each case, andthe controller is used to control the full MIMO plant(10), we can then verify that for the RGA case, theclosed loop system is unstable. However, when usingthe pairing suggested by the Hankel Interaction Index

the result is a stable closed loop system, with the stepresponse (for the given reference signals) shown inFig. 1. Naturally, the prescribed dead-beat behaviorhas not been achieved, since the controller was appliedto the full MIMO process.

A key issue in this simple design is that we have speci-ed a very high bandwidth. This makes the RGA morelikely to yield a bad control performance than othercriteria where the dynamic nature of the interactions isaccounted for. This is intimately connected to the ideaspresented in Section 5, and it can be illustrated using alter as suggested in that section. Consider rst a low-

pass lter F ( z) with transfer function 0 .01 / ( z 0.99 ).We can then re-compute the Hankel Interaction IndexArray for G ( z). This yields

H =0.2706 0 .24090.2362 0 .2522

We observe that now the Hankel Interaction IndexArray gets closer to the RGA in the sense that no clearconclusion can be drawn.

If we use a high pass lter F ( z) with transfer function1.5 z/ ( z+ 0.5) and re-compute the Hankel InteractionIndex Array for G ( z), we obtain

H =0.1932 0 .32690.2998 0 .1801

This is very close to the case when no lter was used(as seen in equation (11)). If intermediate lters wereused one could observe a transition from the RGA tothe Hankel Interaction Index Array given in (11), aswe increase the frequency response of the lter at highfrequencies.

E XAMPLE 3In this case we deal with a 3 3 MIMO system withthe transfer function

G ( z) = G 1 ( z) G 2 ( z) G 3 ( z)

where

G 1 ( z) =

0.7987 z+ 0.7673 z2 1.575 z+ 0.6065

0.1813

z 0.8187 0.1814 z 0.7408

G 2 ( z) =

0.04758 z 0.9048

1.517 z 1.457 z2 1.489 z+ 0.5488

0.2592 z 0.7408

G 3 ( z) =

0.09516 z 0.9048

0.09063 z 0.8187

2.163 z 2.077 z2 1.411 z+ 0.4966

The Hankel Interaction Index Array is given by

H =0.1429 0 .0617 0 .04510.0295 0 .2437 0 .06450.0589 0 .0309 0 .3228

and the RGA index is

RGA (G ) =0.1739 0 .2348 0 .59130.5217 0 .5913 0.11300.3043 0 .1739 0 .5217

We observe that clear suggestions can be derived fromboth arrays. From the RGA the suggested pairing isanti-diagonal, e.g.( u1 , y3 ), (u2 , y2 ) and ( u3 , y1 ). On theother hand the Hankel Interaction Index suggests adiagonal pairing, e.g. ( u1 , y1 ), (u2 , y2 ) and ( u3 , y3 ).

Again if, as in Example 2, we design a dead-beat con-trol (in one step) we can verify that the suggestedRGA pairing leads to an unstable closed loop system.However, a reasonable control is obtained if the pair-ing suggested by the Hankel Interaction Index is used.Since the dead beat control requires a high-pass con-trol sensitivity, the result is not surprising, since we didnot use any lter. We would like to investigate what


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happen if we apply the Hankel Interaction Index to alow-pass ltered plant, where the lter is chosen asF ( z) = 0.01 / ( z 0.99 ). We then obtain

H =0.1228 0 .1368 0 .09460.0705 0 .1166 0 .13520.1411 0 .0684 0 .1140

It is now unclear what the pairing should be. Tocompare both alternatives we can compare the traceof the matrix H with the sum of the elements in theanti-diagonal. This yields 0.3534 against 0.3519. Thisimplies that for a very low bandwidth, it is not clearhow to do the pairing.

7. CONCLUSIONS

A new interaction measure has been presented. Thisindex is based upon the Hankel norm of the elementarysubsystems. The proposed index includes the dynamicinteraction and it is shown that it can be applied in con- junction with available information on closed loop fre-quency bands. The Hankel Interaction Index has beencompared with the traditional RGA index. The new in-dex exhibits a superior performance when the multi-variable interaction has a non-monotonic behaviour infrequency. It is also simpler than other indices whichmeasure dynamic interaction, since it does not requirecomplex analysis of a (possibly very large) number of

frequency responses.

Acknowledgments The authors gratefully acknowl-edge the nancial support provided by UTFSMthrough grants 230011 and 230111, which funded thevisit of the rst author, when the rst version of thispaper was drafted. The rst author was also partly sup-ported by the Swedish Strategic Foundation throughthe CPDC-project.

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Hankel-norm Based Interaction Measure for Input-output Pairing