Defining Directionality: Use of Directionality Measures with Respect to

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Ind. Eng. Chem. Res. 1996,34, 1244-1252

Defining Directionality: Use of Directionality Measures with Respect to Scaling Jonas B. Wallert and Kurt V. Waller* Process Control Laboratory, Department of Chemical Engineering, Abo Akademi, FIN-20500Abo, Finland

The concepts of process directionality and ill-conditionedness are studied. The definition of these concepts commonly used in the literature is shown to be insufficient, due t o the wide range of different applications connected to directionality studied in the field of process control. The commonly used definition does not provide a general concept, which leads to a state where directionality is used in a very problem-specific way. With the aim of achieving a general concept, a refinement of the definition of process directionality is proposed. The refinement divides the concept of directionality into two parts, which are connected to stability and performance aspects, respectively. The refinement of the definition clarifies the connection between control difficulties and directionality, and it also clarifies the scaling choice when modeling a process with respect to directionality analysis.

Introduction The concepts of process directionality and ill-conditionedness have received an increasing interest in the process control literature during the last few years (Skogestad et al., 1988; Freudenberg, 1989a; Freudenberg, 198913; Andersen and Kummel, 1992a; Andersen and Kummel, 199213). In particular the field of control structures often uses the condition number as a measure when comparing or designing different control structures (Lau et al., 1985; Papastathopoulou and Luyben, 1991). The field of process identification and modeling also studies the directionality as an essential part of the modeling and identification problem (Andersen and Kummel, 1992a; Andersen and Kummel, 1992b;Koung and MacGregor, 1993). It is claimed (Andersen and Kummel, 1992a) that individual identification of the transfer functions in a transfer function matrix is not enough t o capture the directionality of ill-conditioned processes. The commonly used base for directionality analysis stems from the singular value decomposition (SVD) of a matrix. And indeed, there is an intuitive appeal in using SVD when analyzing a multivariable system. The reason for this is that it can be argued that the concept of SISO gain in this way can be extended to MIMO systems (Doyle and Stein, 1981). The gain of a MIMO system is according to this view sandwiched between the largest and the smallest singular value of a process model written as a transfer function matrix. The degree of directionality for a given process described by a transfer function matrix is measured by how much the gain between one set of outputs and one set of inputs may vary. Different sets of inputs may affect the outputs in different ways. Some systems have the same gain for all possible input directions, whereas the gains of other systems vary considerably. Such systems are referred to as being well-conditioned and ill-conditioned, respectively. The common definition of an ill-conditioned process is that it is described by a transfer function model with a large condition number (the ratio between the largest and the smallest singular values). However, the condition number depends upon the Present address: Vasa Polytechnic, PB 6, FIN-65201 Vasa, Finland. Fax: t 3 5 8 61 3230 610. Email: [email protected]. * To whom all correspondence should be addressed. Fax: $358 21 654 479. E-mail: [email protected]. 0888-5885/95/2634-1244$09.00/0

scaling of the process model. This property is due to the scaling dependence of the SVD. Due to the scaling dependence, the definition of ill-conditionedness assumes that the process model is scaled according to the problem at hand. However, the scaling choice (the choice of units) is not a trivial problem, Freudenberg (1989a) states that “the choice of an appropriate set of units may be quite problematic. Indeed even in the context of numerical analysis, scaling appears t o be a very problem-specific procedure.’’ One of the reasons the scaling choice seems difficult is a consequence of the definition of process directionality. Without a general definition, the scaling procedure is difficult to execute. There is a wide range of applications based on the directionality concept studied today, and it seems to us that the use of the concept of directionality might benefit from a more thorough definition. In this paper a refinement of the definition of process directionality is suggested, which aims at clarifying the scaling choice. The proposed refinement also connects the scaling t o the type of problem studied and shows that the, often used, assumption that the process model is properly scaled might result in very conservative measures of the process directionality. One main motive for studying directionality stems from the common belief that plants with a large condition number (plants with high directionality) are difficult to control (Skogestad et al., 1988; Freudenberg, 1989a; Freudenberg, 1989b; Chen, 1992). According to Chen (1992) and Freudenberg (19931, however, no conclusive proof of this connection has yet been presented. The refinement proposed in this paper provides further indications of the connection between ill-conditionedness and control difficulties. Also relationships between the condition number and the (robust) stability and performance of a process controlled with a diagonal controller are investigated. The scaling-dependent condition number is shown to give unreliable information about the effect of the directionality on stability aspects. The minimized condition number (Grosdidier et al., 1985) gives more useful information about stability aspects, but the RGA (Bristol, 1966) o r the Rijnsdorp interaction measure (RIM) (Rijnsdorp, 1965) is shown to provide more thorough information. Concerning performance aspects, the information in RGA, in RIM, or in the minimized condition number is insufficient,

0 1995 American Chemical Society

Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 1245 and for performance analysis the scaling-dependent condition number provides more useful information.

the RGA is defined as

Preliminaries The singular value decomposition (SVD) (Golub and Van Loan, 1983)is the mathematical tool normally used for directionality analysis. Assume a process model of the form y(s) = G(s>m(s>

(1)

the ij'th element of the RGA matrix. For 2 x 2 systems the l-norm becomes

&jis

where

where y are the outputs, m are the manipulators (the inputs), and G is a transfer function matrix. The singular value decomposition states that G, at a given frequency, can be decomposed according to

G = W T

(2)

where U is an orthogonal matrix containing the left singular vectors (the output directions), V an orthogonal matrix containing the right singular vectors (the input directions), and E a real diagonal matrix containing the singular values, u, such that 01 2 uz 2 ... L O n L 0. The singular values are interpreted as bounds on the possible gain of G. u1 is referred t o as the largest singular value (the largest possible gain, commonly denoted 5) and a, as the smallest singular value (the smallest possible gain, denoted g ) . The high and low gain directions for the inputs and the outputs are given by the singular vectors. The condition number is the ratio between the largest and smallest singular values. For a more thorough treatment of SVD in process control, see for example Doyle and Stein (1981) and Moore (1986, 1992). This study mainly concerns itself with multivariable systems with two inputs and two outputs and assumes that the input-output relationships of the process can be described by a linear transfer function model of the type in eq 1. In process control, 2 x 2 systems are probably the most important MIMO systems from a practical point-of-view. The following relationships for 2 x 2 systems are used. Assume a transfer function

:::)

G=

(3)

evaluated at a given frequency, w . The condition number of G at w is defined as

y(G) = aG)/a(G)

(4)

1 = 1 - gl~2l~gllgz2

(10)

The ratio (81gz1/g1gzz) is the RIM, denoted K . The relationship between A l l and K for 2 x 2 systems thus becomes A11

= 1/(1- K )

(11)

For larger systems the following conjecture is presented in Grosdidier et al. (1985):

yminI2 max~llAlll,llAll,.J

(12)

The --norm is defined as

We shall refer to the scaling that minimizes the condition number as the optimal scaling. According to Grosdidier et al. (19851, the optimal scaling of a 2 x 2 system is achieved when

Here the assumption that g y # 0 for i , j = 1 , 2 must be made. Equations 14 imply that the optimal scaling is achieved when lg111 = lgzzl and lg121 = lg211. A diagonal system is optimally scaled (and Ymin = 1) when lglll = lgz21.

The maximum singular value of a matrix is the 2-norm of the matrix:

The minimized condition number, yfin, in turn is defined as

where SI and SZare real, diagonal 2 x 2 matrices (with nonzero diagonal elements). Grosdidier et al. (1985) present the following relationship between ymin and the RGA (A) for 2 x 2 systems:

ymin = IIAlIl

+4

s

= G.*(G-~)~

4 G ) = U3G-l)

(16)

Other important properties of matrix norms can be found in most textbooks on matrix theory.

(6)

The Scaling Issue

where the RGA is defined as

A

where Ihllz is the Euclidian vector norm. The following relationship between the maximum and minimum singular values is useful:

(7)

Here .* denotes the Hadamard product. The l-norm of

Since SVD is scaling dependent, the singular values and the condition number depend on the scaling (the choice of units) used when modeling the plant. To cope with this problem, the minimized condition number

1246 Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995

(ymln) is defined as the smallest possible condition number that can be achieved by varying the scaling (eq 5). Contrary t o the condition number and the singular values, ymm, RGA and RIM are scaling independent. The scaling choice is crucial when using scalingdependent measures. Several different scaling methods have been suggested in the literature, but usually these methods are designed only for the specific situation studied. General scaling procedures have not, to our knowledge, yet been presented. Moore (1986) suggests that a good scaling can be expressed in terms of percent changes, such that the units of G, are approximately % of sensor span % of range of manipulator

Andersen and Kiimmel(1991) state that “SVDis scaling dependent and therefore the outputs should be scaled according to their relative importance with respect to control performance and the inputs should be scaled according t o the physical limits imposed by the process design”. Later the authors use similar formulations (Andersen and Kiimmel, 1992b). Many authors, such as Lau et al. (1985), Andersen et al. (19911, and Papastathopoulou and Luyben (1991) seem to prefer physical scaling. Other papers, like Grosdidier et al. (1985) and Skogestad and Morari (1987) seem to take the view that the scaling that minimizes the condition number is the correct one. Often, the process model used is merely assumed to be properly scaled (Brambilla and D’Elia, 1992; Freudenberg, 1989b) (Freudenberg, 1993). Despite all different suggestions, all the abovementioned papers (except perhaps Moore, 1986) pass the scaling question in a few lines without giving either proofs of their statements or any ultimate solutions. The existence of several different scaling suggestions combined with the scaling dependence of the condition number can easily give rise to an ambiguous situation, where the interpretation of derived results is far from clear. Therefore, it is important to clarify the connection between the scaling choice and process behavior. Further, the (sometimes suggested) “nondimensionalizing” of the process model leads in the ideal case to a process model which is scaled in such a way that the condition number is minimized. However, also the nondimensionalizing can be performed in different ways. It is thus important to realize the impact of scaling on scaling-dependent measures, regardless of what scaling method is used. As illustrated in this paper, the scaling choice is an essential part of the directionality analysis and the concept of directionality cannot be properly used unless the scaling choice is thoroughly performed and connected to the problem studied. To illustrate the effect of the scaling, consider a noninteracting 2 x 2 process:

of for example y1 can be differently chosen when modeling the same plant. Assume for example

Y

G‘

The singular values of these two models differ. The model in eq 18 shows high directionality, y(GS)= 100, even though it describes the same process. The control difficulties have, naturally, not increased (nor have they decreased) merely by changing the units of the model, and for the latter model the condition number indicates control problems not seen in the model in eq 17. This simple example clearly indicates that the variable scaling of the model must be taken into account when using the condition number as a measure of control difficulties. Papastathopoulou and Luyben (1991) use a (3 x 3) model with physical scaling of the elements in G(0)to study different structures. Their structures are scaled according to physical units ((lb mol)/h, mole fractions, etc.). One of their structures is the (R-S-BR) structure with a gain matrix of 3.65 0.170 -6.67 22.6 0.293 -45.2 0.189 0 -0.750 and another is the (RR-S-BR) structure 4.82 -0.824 29.8 -5.86 0.250 -0.052

-4.32 -30.7 -0.628

(19)

i20)

K1 has y = 900 and ymin I9.6 (accordingto conjecture 12), while K2 has y = 750 and ymin 9 32.8. Papastathopoulou and Luyben (1991) use the condition number only, and state that “the smaller this number ( y ) , the better the control”. If we compare K1 and K2 according to this criterion, the conclusion would be than K2 is “better”than )K1.However, if one compares the upper bound of the minimized condition number instead, the opposite conclusion would be appropriate, i.e., K1 is “better” than K2. Another example often used in directionality analysis (Skogestad et al., 1988, Skogestad, 1992; Brambilla and D’Elia, 1992) is a model of a distillation column:

G

1 -0.878 0.014 GDv = -1.082 -0.014 as an example of an ill-conditioned process. Brambilla and D’Elia (1992) state that “the directionality of this column is high” and Skogestad (1992) states “this process is also ill-conditioned as ~ ( G D = v )70.8”. The condition number of this model is (independent of frequency) y(&v) = 70.8. This does not, however, imply that the plant is ill-conditioned, just that the model is. Let us rescale the inputs and the outputs of the model, i.e., consider the following process model:

where y denotes the outputs and m the manipulators. (The units of this model and of all models in the sequel have been left out because the units themselves are irrelevant. The important point is that a different set of units results in a different value of scaling-dependent measures, as is illustrated in the sequel.) This process shows no directionality (y(G)= 1). However, the units

where S1 and S2 are diagonal scaling matrixes. The minimum of the condition number, ymin, can be found by minimizing y(G&,)over all real diagonal SIt 0 and s 2 f 0:

-

m(

Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 1247

(23) One choice of scaling matrices that achieves this is

SIand S2 can also be analytically calculated for 2

x 2 systems (Grosdidier et al., 1985). With this scaling, the process model becomes

-0.878

)

0.9747 (25) -0.878 which has a condition number equal to one. The RGA of GJJVis 1 1 1 = 0.45 and K = -1.23. This indicates that the plant itself should not be considered ill-conditioned, it is the scaling choice that makes the model illconditioned. Our claim that the definition of ill-conditionedness is insufficient is, as stated, motivated by the scaling dependence of the condition number. The problemspecific nature of the concept of directionality can be seen from the claims that “the scaling dependence of the SVD should be viewed as a strength and not as a weakness” (Andersen and Kiimmel, 1992131, while Skogestad (1992) states that “the RGA is another indicator of ill-conditionedness, which is generally better than the condition number because it is scaling independent”. We can conclude that ill-conditionedness has been measured by both scaling-dependent and scalingindependent measures. For many cases, this leads t o totally different values for the directionality of a process, as is illustrated in for example the model of the distillation column (GDv) in eq 21. Concerning the relationship between the condition number and the RGA (eq 6), it is clear that a (2 x 2) matrix with small RGA elements always has a small Ymin. In particular, if 0 5 111 I1, the minimized condition number is always equal to 1. This means that for such systems there exists a scaling which gives a model showing no directionality, if directionality is measured by condition numbers. This holds for all 2 x 2 systems where the input singular vectors are aligned with the single-input vectors or the output singular vectors are aligned with the singleoutput vectors (Sagfors and Waller, 1994,1995),which for example is the case for diagonal systems such as the one described by eq 17. On the other hand, all systems (even diagonal ones) can be scaled to get a very large condition number. The model of the DV structure has an uneven number of negative elements in the transfer function matrix. One consequence of the relationship between ymin and RGA is that such systems always have Ymin = 1. We conclude that the scaling choice is very problem specific when analyzing the directionality of a process. This is, as we see it, due to the definition of directionality commonly used in process control. Since both scaline-dependent (the condition number) and scalingindependent (the minimized condition number, RGA, and RIM) measures can be used, there is an obvious need to connect the scaling to the directionality analysis at hand. As an attempt to clarify the scaling choice, we propose a refinement of the definition of directionality. The proposed refinement divides the concept of directionality

1

=

756+1(-0.9747

into two parts. On one hand, the scaling-independent measures (Ymin, RGA, and RIM) provide information about the stability aspects of directionality. On the other hand, the (scaling dependent) condition number can provide useful information about performance aspects, information not present in the scaling independent measures. When the behavior of a control system is analyzed, a common classification is to divide the problem into two parts, (robust) stability analysis and (robust) performance analysis. The proposed refinement is in accord with this distinction. Further the refinement clarifies the connection between directionality and control difficulties and also clarifies and guides in the scaling choice. The relevance of this refinement of the definition is treated in the sequel.

Stability Since the condition number is scaling dependent, even totally decoupled systems can have an arbitrarily large condition number, while the minimized condition number for decoupled systems always equals 1. Recall the example in eq 18. The model shows high directionality in terms of a large condition number. The high gain direction equals a change in the manipulator ml, whereas the low gain direction equals a change in the manipulator m2. This is thus an example of a system where the input singular vectors are aligned uith the single-input vectors and the output singular vectors with the single-output vectors. The large condition number of the model in eq 18 might seem peculiar since the same process can be modeled as in eq 17 (which shows no directionality). Let us first focus on the stability aspects of this process. If we assume that the process is controlled by two SISO controllers and that the two loops are equally tightly tuned, it is clear that the high-gain direction has the same effect on the stability of the system as the low-gain direction. The stability is neither more nor less endangered for changes in the high- and low-gain directions. The obvious conclusion is thus that the description in eq 17 is more appropriate to use for stability analysis, since it corresponds to the actual behavior of the process. Mathematically, this can be explained in the following way. One version of the small-gain theorem (Morari and Zafiriou, 1989) states that stability of a multivariable system is guaranteed if the maximum singular value of the open-loop transfer function matrix CG is smaller than 1 at all frequencies. A s is well-known, this measure might be very conservative. There are several reasons for conservativeness. One of the reasons is the scaling choice. The minimized condition number, y-, is independent of scaling and can analytically be calculated from the relationships with the RGA. It is clear that for a diagonal controller

(26)

1248 Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995

This gives

A useful relationship for a complex matrix A can be derived according to

(36)

Iclg11cg,2I < 0.25

must hold to guarantee stability for extremely illconditioned processes. 2. On the other hand, as K 0 (which means that ymin(G) 1)

-

-

ymin(P) 1

-

-

IIAIl,

--1

K

(30) Since $A) = $adj(A)) (straightforward to prove for 2 x 2 matrices), we get

$4= Jldet(A)ly(A)

(32)

This relationship is useful when analyzing the relationship between control difficulties and condition numbers. Consider input and output scaling of the process model, GS = S1GSz. The diagonal controller, C, is scaled according to the units chosen for the process model G, i.e., Cs = Sp-lCS1-l. Let P = CG. Ps then becomes

Ps = S2-1CSl-1SlGS2= S2-1CGS, = S,-1PS2= (s22is2l)Pl2) (33) ~::l~s22)P21P22 The condition number of Ps depends on the scaling, whereas the determinant of Psdoes not, since

pllpZ2- p12pZ1 = det(P) (34)

Equation 32 shows that the maximum singular value of the open-loop transfer function P reaches its smallest value when y(CG) = ymi,(CG), i.e., when CG is optimally scaled. If a diagonal controller with both loops equally tightly tuned is used, CG will be optimally scaled when G is. Further, eq 32 shows that with an unfortunate choice of scaling, the condition number of CG and thus also $CG) can be very large, and thus the small-gain theorem very conservative. To illustrate this relationship for the case with lplll = Ipzzl,consider the following extreme values for ymin(G) at a given frequency. 1. As ymin(G) 00, we get the following expression from eq 6:

-

Pi 1P22 k

Obviously, if ymin(G) is large, K is close to 1. Since det(P) =pilpzz -pi2pzi = p i l p 2 2 ( 1 - K ) , the expression for aP) becomes

0

(37)

(38)

-

which means that as ymin(G) 1 we get I C B l l C ~ 2 2 I
= J ~ c ~ ~ -~K ) I~= 0.7068 c ~ ~ ~ (48) ( I and the stability is thus guaranteed. We note that the stability margins in the controllers for the system in eq 47 are larger, although the elements in eq 47 and in eq 45 are of the same size. This is a consequence of the uneven number of negative elements in eq 47. Since Ps = SZ-~PSZ, y(PS) = ymin(P) only when lpiil = lp22l. If l p l l t lp221 (the two loops are unequally tuned) y(P) > ymin(P) and, consequently, 3P) will be larger than the expressions presented above. We have thus been able t o connect ymi,(G) t o 3 C G ) in such a way that the sallest condition number produces the least conservative estimate of the nominal stability. Further, it is clear that processes with high directionality demand more detuning than processes which show a low degree of directionality. When ymin(G) = 1,3 C G ) still depends on K (the individual elements in the matrix), and ymin(G) does not provide the information needed to draw conclusions concerning the need for detuning the controller. For such a case, the information in RGA or in RIM provides complementary information.

Performance Performance of a multivariable system is commonly analyzed by measuring a matrix norm of a weighted sensitivity function. Normally, the 2-norm or the -norm is used. Morari and Zafiriou (1989) express the control objective of linear quadratic control as minllW2EWl1122 C

(49)

where W1 is the (frequency-dependent)input weight, WZ the (frequency-dependent) output weight, and E = (I GC1-l (the sensitivity function). In the same fashion the H, control objective can be expressed as

+

minl IW2EWlIIm c

(50)

The analysis is thus very similar to the stability analysis. If the performance weights are included in the process model, the model will normally not be optimally scaled, y(G) > ymin(G). The condition number of the process model might provide useful indications of possible performance problems, especially for diagonal controllers. Here, we shall only briefly illustrate the effect of output weighting on performance. As an example of the dependence of performance on the directionality, consider the diagonal example in eqs 17 and 18. Above, we showed how ymin(G) is connected to a stability analysis where the directions of the input were of no significance for the stability. The model in eq 18 can be said to consider y1 100 times more important than y2. From a performance point of view, it is clear that a change in the high-gain direction (ml) moves y1 from the set-point, whereas a change in the low-gain direction (m2) moves y2 from the set-point. Since y1 is much more important than y2, the high-gain and low-gain directions have a physical relevance for this system when it comes to performance. The highgain direction causes a more severe deviation from the desired values than the low-gain direction. Consider a (2 x 2) model which is optimally scaled (lg11l = lg22l and lg12l = lg21l) controlled with a diagonal controller with equally tight tuning. If this model represents the importance of the variables, the output weighting matrix has diagonal elements of the same size. On the other hand, if one of the outputs is much more important than the other one, the output weighting matrix will have diagonal elements of different size. If we assume that we have the possibility to choose valves and sensors freely, an uneven scaling of the manipulators or of the outputs is motivated only if it represents the weights (the importance) of the variables. Naturally, the importance of the variables does not affect the stability of a process, merely some performance aspects. The example in eq 45 serves as an illustration. If WZ = W1 = I, the norm IIW2EW1112 becomes 0.9995 and the condition number y(G) = 1999. On the other hand, if w2=(i0

;)

then I lWzEWll12 = 7.92 and the scaled condition number (with Si = Wp) y(W2G) = 10095. From a performance point-of-view, the condition number of a matrix scaled according to the weight of the variables might provide useful information. A process can be ill-conditioned with respect to performance, even if it is well-conditioned with respect to stability.

Robustness To examine the robust stability of the system, let us rewrite the system in the familiar M - A fashion. If we, for the sake of simplicity, focus on input multiplicative uncertainty, M equals the complementary input sensitivity function, HI = CG(1 CG1-l (Skogestad et al.,1988). A represents the uncertainty and the restriction on A is $A) I 1, where A is a full matrix (the unstructured case) or a diagonal matrix (the structured case). $(HI) is then a measure of the robust stability of the system. For the structured uncertainty case, HI) (Doyle, 1982) replaces $HI). HI), which is scaling independent, is always smaller than or equal to 3 H d . The difference between p and iT is usually explained as due to the difference in the uncertainty descriptions.

+

1250 Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995

However, the difference can also be explained as due t o the scaling dependence of the process model, since depends on the scaling, whereas p does not. The effect of the scaling on HI gives H! = S ~ - ' H I S ~ . p is defined in Doyle (1982). For 2 x 2 systems p can be expressed as (Skogestad, 1992)

We have

+

HI = P(1 P)-l =

p(HI)= min 8DHID-l)

I h 2 l - lpl2(1 + P11) - P12Plll --

D

With the connection between B and y for 2 x 2 systems in mind, we notice that

,u(HI)= min 8DH1D-l) = D

Since det(DH1D-l) = det(HI), HI) equals $(HI) when HI is optimally scaled at a given frequency. Assume that G is optimally scaled and assume that both loops are equally tightly tuned (Iclglll = lc2gzzI). Then P = CG is also optimally scaled. It is straightforward t o show that for such a case also HI will be optimally scaled, that is, lhlll = lhzzl and lh12l = Ihzll. This connects p to ymin and shows how the optimal scaling produces the least-conservative estimate of the robust stability of a 2 x 2 system if a diagonal controller with equally tight tuning is used. On the other hand, if the scaling is not chosen to minimize the condition number, we might get arbitrarily conservative measures of the robust stability. The example in eqs 44 and 45 can be used to illustrate the conservativeness in the robust stability measured by $HI). First, let us focus on the nonoptimally scaled system description (eq 44). HI will then also be nonoptimally scaled. B(H1) = 0.8326 for this system, whereas HI) = 0.4999. The difference between b and p indicates large conservativeness in the unstructured uncertainty description. If, however, the optimally scaled system description is used we get $HI) = 0.4999, which is exactly equal to HI). The conservativeness in this example is thus completely removed by an optimal scaling of the process description. Next, we aim at connecting ymin(G) to $(HI) and examining what can be said about the robust stability of a system for different values of the minimized condition number. For the sake of simplicity, we assume that the individual loops are equally tightly tuned, that is, lclglll = lc2gzzI. The condition number of HI will then equal ymin(H1) and

will hold. The ratio

is a measure of ymin(H1) and therefore of central importance. If K(H)> 0 and under the above-mentioned simplification, we might as well measure the ratio lh1~lllhlll.

Pill

lpl2l lP11

lpll(1

+ P22) - P12P21l

-

+ PllP22 - P12P211-

lp121 lpll

+ PllP22(1 - K(P)I

(53)

We notice that when the minimum condition number of G is large, K(G)will be close to 1, and, consequently, the minimum condition number of HI will be large. We have thus been able to show that ymin(H1)depends on ymin(P) = ymin(G)in such a way that when ymin(G)is large, so is ymin(H1)and, contrary, when ymin(G)is small, SO is ymin(H1). The approximate expression for ymin(H1) equivalent to the approximations in eqs 35,38, and 41 become the following. If ymin(H1) m +

(54)

Since we already assumed that CG and thus also HI are optimally scaled, our problem is reduced to the following comparison. Contrary to the nominal stability problem studied above, lhllh221 also depends on the the condition number of G. We thus have to compare lhllh221, or, since we have assumed that lhlll = lh221, we can compare lhlll for the different situations: = I1

PllP22 - P12P21 + pP11+ 11 + P22 + PllP22 - P12P21 I =

lhlll = I

P11+

P112

1 + 2P1, + P l ,

21,

K ( P ) =0

(5%)

If we compare these expressions, bearing in mind that 0 < pii < 1, it is clear that the robust stability is highest when the process is well-conditioned with K(P)x 0, whereas it decreases when the condition number increases. For systems with ymin = 1the robustness still depends on the value of K. For the ill-conditioned case in the example in eqs 44 and 45 we get the approximation

Ind. Eng. Chem. Res., Vol. 34,No. 4,1995 1251

Discussion and Conclusions

I"

1o4 1o.2 1oo 1' 0 Figure 1. Robust stability plot for the process in eqs 21 and 25 with the controller in eqs 60 and 61. Full line shows HI) versus frequency. Dashed line shows G(H1)for the nonoptimally scaled system description and dash-dot line (which almost coincides with the full line) shows G(H1)for the optimally scaled system. Dotted line is the uncertainty weight used by Skogestad et al. (1988).

Since the process in the example is very ill-conditioned, this approximation is good. (Compare with p(Hd = 0.4999.) Concerning the robust stability of the DV model (eq 21), Skogestad et al. (1988) reported that this process is insensitive to diagonal input uncertainty but sensitive to unstructured uncertainty and they draw the conclusion that ill-conditioned plants not necessarily are sensitive to uncertainty. Skogestad et al. (1988) suggested the following diagonal PI controller based on the nonoptimally scaled model: (60) -7.5 With the units that give the optimally scaled model, the controller becomes S

-( +s75s o-0.15

s-1

)

0 (61) -0.12 This system does not have both loops equally tightly tuned, c g l l > cgzz. A robust stability plot is presented in Figure 1. As noted in Skogestad et al. (19881,Figure 1indicates large conservativeness in using as compared top as a robust stability measure. However, if we use the optimally scaled model in the analysis, a much less conservative ii is achieved; see Figure 1. The slight difference between HI) for the optimally scaled system and p(Hd is due to the unequal tuning of the controllers. If we scale the model so that y(CG)is minimized with respect to the scalig matrix Sz,this slight difference would be removed. This is, of course, equal to calculatingp, since y is independent of frequency for this model. However, the conservativeness is not due to the structure of the uncertainty, since we have not put any restrictions on the structure of the uncertainty. The only change made is the use of a different scaling in the system description. These relationships connect p to ymin and show how the optimal scaling produces the least conservative estimate of the robust stability of a 2 x 2 system. On the other hand, if the scaling is not chosen to minimize the condition number, we might get arbitrarily conservative measures of the robust stability.

c

In our opinion the definition that a plant is illconditioned when it is described by a model with a large condition number is unsatisfactory, due to the scaling dependence of the condition number. If one makes an unfortunate choice of scaling for the variables when modeling the plant, it can hardly be considered a characteristic feature of the plant itself. A refinement of the definition of process directionality is thus motivated. The concept of process directionality should be possible to connect to both stability analysis and performance analysis. The common definition does not allow this distinction. Our proposed refinement uses the minimum condition number (or, preferablyh, RGA or RIM) as a directionality measure from a stability point-of-view and the (scaling dependent) condition number as a measure of directionality from a performance point of view. For performance analysis, the weighting of the variables should be incorporated in the scaling choice. This definition is consistent with (most of) the applications previously studied in the literature. By use of the proposed definition it is thus possible t o connect the concept of directionality to both stability and performance aspects. This makes it possible to define general scaling procedures for directionality analysis, as illustrated in the paper. We further illustrate how the scaling that minimizes the condition number also provides the least conservative estimate of the (robust) stability of a process with a given controller. For certain cases this least conservative ii aligns with the structured singular value, p. Thus, for some cases the conservativeness in using ii as compared to using p (usually explained as due to the difference in the uncertainty descriptions) is due t o the scaling of the process model. Further, we have established connections between the degree of ill-conditionedness and the nominal stability as well as the robust stability of a given plant and a diagonal controller. Ill-conditioned plants, defined as having a large minimized condition number, are less robust with respect t o stability than well-conditioned plants if a diagonal controller is used. This result shows that the common belief that ill-conditioned plants are difficult t o control actually has a theoretical (and physical) relevance. We conclude that the common definition of an illconditioned plant and plant directionality is unclear in the literature and so is the common use of condition numbers as indicators of possible control problems. The confusion in the scaling choice is probably one reason why conclusions and suggestions based on SVD measures often seem ambiguous.

Acknowledgment The results reported in this paper have been achieved during a long-range research project on multivariable process control. Financial support from the Academy of Finland, Nordisk Industrifond, the Neste Foundation, Neste Oy, and Tekes is gratefully acknowledged.

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Received for review May 18, 1994 Revised manuscript received December 28, 1994 Accepted January 5, 1995@ IE940318Y

Abstract published in Advance A C S Abstracts, March 1, 1995. @