Ind. Eng. Chem. Res. 1996, 35, 4091-4099
4091
Variable Pairing Selection Based on Individual and Overall Interaction Measures Zhong-Xiang Zhu* Honeywell Hi-Spec Solutions, 343 Dundas Street, London, Ontario, Canada N6B 1V5
A new steady-state interaction measure, the relative interaction array (RIA), is explored in terms of its features, properties, and relationship to the relative gain array (RGA). It is demonstrated that the RIA offers a better measure of individual interactions than the RGA. Moreover, it is shown that the RIA also contains direct information about closed stability, integrity, and robustness. Taking advantage of its definiteness in measuring the amount of loop interaction by its size, the RIA is further extended to propose a new overall interaction measure which is able to avoid ambiguities associated with the RGA pairing criteria and to overcome problems associated with other overall interaction measures. Interestingly, the RIA-based overall interaction measure leads to a “corrected” version of the intuitive RGA-based one. The final RIA-based pairing criteria provide a comprehensive and reliable solution to the selection of the best variable pairing and thus represent a promising tool. Also, variable pairing criteria on the basis of interaction measurement, stability, and integrity as well as robustness considerations are systematically addressed with new insights provided. 1. Introduction Decentralized control is commonly used to tackle multivariable processes. Control structure selection or variable pairing choice usually represents the first main issue control engineers face in designing decentralized control systems. The major objective in selecting the best variable pairing has been primarily to minimize loop interaction so that the resulting multivariable control system mostly resembles its SISO counterparts and the subsequent controller tuning is largely facilitated by independent design (Seborg et al., 1989). This objective is usually accomplished by utilizing various interaction measures, predominantly the steady state ones, as tools to screen possible pairing alternatives (Bristol, 1966; McAvoy, 1983; Majares et al., 1986; Zhu and Jutan, 1993a). In particular, the relative gain array (RGA) (Bristol, 1966) has been the most widely used one. Originally proposed as an empirical measure of interaction, the RGA pairing rule has found widespread acceptance, particularly after the improvement on closedloop stability considerations by using the Niederlinski index (NI) (Niederlinski, 1971) as a stability rule (see McAvoy, 1983; Grosdidier et al., 1985; Shinskey, 1988; Seborg et al., 1989; Skogestad et al., 1990). The RGAbased tool is further enhanced by taking into account integrity (Chiu and Arkun, 1990; Zhu and Jutan, 1993a) as well as robustness (Yu and Luyben, 1987; Zhu and Jutan, 1993b) considerations, with the joint use of the RGA and the NI. The following briefly summarizes the RGA-based criteria and the perspective roles the RGA and the NI play in variable pairing choices: The original RGA offers an interaction rule by its size. The NI provides a necessary stability condition by its sign. The signs of the RGA elements lead to the integrity rules. The sensitivity of the RGA elements to gain uncertainties presents the robustness rule. The RGA has been extended to other unconventional situations such as disturbance rejection (Chang and Yu, * Email:
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1992), nonsquare systems (Reeves and Arkun, 1989; Chang and Yu, 1990), block structures (Manousiouthakis et al., 1986), and nonlinear processes (Manousiouthakis and Nikolaou, 1989). The NI has also been extended to provide an overall measure of steady-state interaction (Zhu and Jutan, 1993a) and to include controller design during variable pairing decisions (Zhu and Jutan, 1995). However, the RGA only measures loop interaction in individual control loops since it is defined on the basis of individual loops. Moreover, the amount of interaction indicated by the distance of the size of a RGA element and the interaction-free reference value (1.0), i.e., the closeness of RGA elements to 1.0 required by the RGA pairing rules, is rather qualitative. As a result, ambiguity may arise when several alternatives satisfy the RGA-based rules (Zhu and Jutan, 1993a). Hence, an overall interaction measure, which can be used to identify a particular pairing as the final one exhibiting truly the minimum interaction, is required. Majares et al. (1986) proposed a measure of the overall interaction in decentralized control systems from an algebraic perspective. However, in comparison to the RGA-based rules, this measure lacks sufficient information about system stability and integrity (Majares et al., 1986; Zhu and Jutan, 1993a). An intuitive overall interaction measure based on the distance of the RGA and 1.0 was proposed by Zhu and Jutan (1993a). This measure is found to be inadequate under some circumstances. Zhu and Jutan (1993a) suggested using the NI, actually the size of the NI, as an overall interaction measure to avoid possible ambiguities associated with the RGA. Hence, with the overall interaction measure by its size, in addition to the traditional stability and integrity indications by its sign (as in the conventional RGA-based rules), the NI is proposed as a comprehensive screening tool for variable pairing choices. This NIbased pairing criterion leverages heavily on the relationship between the NI and the RGA. It represents an empirical rule and thus may lead to incorrect decision on the best pairing (see Zhu, 1993; McAvoy, 1993). This article starts with a brief review and critical assessment of the existing interaction measures and variable pairing rules. New insights into the RGA and © 1996 American Chemical Society
4092 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
the RGA-based pairing rules are offered. Variable pairing selection is systematically addressed from the following aspects: amount of interaction; closed-loop stability; closed-loop integrity; robust stability against gain errors. An updated RGA-based pairing criterion with individual rules for interaction, stability, integrity, and robustness requirements integrated is presented. Variable pairing rules based on the existing measures of overall interaction are also evaluated. Using the loop decomposition approach to decompose a decentralized control system into equivalent SISO loops with loop interaction structurally embedded (Zhu, 1993, 1996; Zhu and Jutan, 1996), a new steady-state interaction measure, called the relative interaction array (RIA), is presented and further elaborated by closely exploring its features, properties, and relationship to the RGA. It is shown that the RIA provides a better measure of interaction than the RGA. Information about closed-loop stability and integrity as well as robustness contained in the RIA is also discovered, giving further rise to the RIA as a comprehensive tool for variable pairing choices. Like the RGA, however, RIA lends itself as an interaction measure also on the basis of individual loops and, hence, may also lead to ambiguities regarding the best variable pairing choice. A new measure of the overall interaction on the basis of the RIA is subsequently developed. The new RIAbased overall interaction measure is capable of avoiding possible ambiguities involved in individual interaction measures and overcoming problems associated with other overall interaction measures. Noticeably, the proposed overall interaction measure leads to a corrected version of the intuitive overall interaction measure using the RGA by Zhu and Jutan (1993a). The best variable pairing is suggested to be the one exhibiting the minimum overall interaction and meanwhile satisfying stability and integrity, as well as robust stability requirements, governed by the proposed RIA based criteria. Consequently, the RIA-based pairing criteria represent a comprehensive and reliable tool in making variable pairing decisions. 2. RGA-Based Pairing Rules 2.1. RGA as an Interaction Measure. As originally introduced by Bristol (1966), a relative gain, λij, in a control loop corresponding to a given paired element of the process, gij(s) ∈ G(s), i.e., uj-yi pairing, in a decentralized control system is defined as
λij )
gij gˆ ij
(1)
where gij denotes the gain of the process element in the independent SISO subsystem and gˆ ij represents the gain of the equivalent process in the interactive environment with all the loops closed. The following remarks are worth noting about the definition of the relative gain. Remarks. 1. gˆ ij is meaningful only if the subsystem, composed of the original control system with the uj-yi loop removed, is stable; i.e., the system possesses integrity against single-loop failure. 2. gˆ ij is obtained by assuming that all the outputs except yi in response to a step change in uj converge to zero at steady state, i.e., the subsystem with the uj-yi loop removed achieves perfect (offset free) steady-state control. This is the case when integral control is adopted in each control loop.
3. gˆ ij reflects the effects of all the other loops on the uj-yi loop due to the existence of the hidden loops (see Shinskey, 1988). 4. gˆ ij indicates how much the process gain deviates from the original gain, gij, due to interaction. Hence, the difference between the two quantities reflects the severity of loop interaction. 5. Equivalently, λij constitutes a measure of interaction in the uj-yi loop by its deviation from unity. All the relative gains corresponding to all the elements of a square process constitute the RGA, which can be conveniently calculated as (Shinskey, 1988)
RGA ) G X (G-1)T
(2)
where X denotes the element by element multiplication of matrices and superscript T denotes the transpose operation of a matrix. The RGA possesses the following main properties: 1. It is independent of input and output scaling. 2. Each row and each column sums up to 1.0. 3. λij is independent of how the other n - 1 loops are paired. 4. RGA ) I for diagonal and triangular processes. Property 3 above, which is often overlooked, is the result of the perfect control assumption. It makes the RGA particularly attractive in making variable pairing decisions, since, unlike other tools such as the NI (defined later), all the possible pairings can be scanned simultaneously without a prior requirement for a given pairing. Property 4 has traditionally been attributed to the RGA as a limitation of being unable to measure the one-way interaction. In fact, property 4 reveals the inherent property of the system, i.e., there exists no oneway interaction in triangular systems. This is clear by noticing that interaction is caused by the hidden loops (Shinskey, 1988) which disappear in triangular systems, since signal can no longer loop back to the original loop. As an interaction measure, the RGA leads to the following well-known variable pairing rule. RGA Interaction Rule. Input and output variables of a process should be paired in such a way that all the corresponding RGA elements are closest to 1.0. Note that the closeness to 1.0 of RGA elements is rather qualitative, and hence, ambiguity arises as to how to select the RGA elements closest to 1.0 in the above interaction rule. Apparently, the pairing rule is aimed at achieving the minimum interaction in each control loop in the system. However, the minimum interaction requirement is not sufficient enough to decide on the desired pairings. Other requirements, particularly the closed-loop properties such as stability, integrity, and robust stability, should be simultaneously considered. The RGA alone provides no complete answer in this regard, and hence, other tools such as the NI have to be jointly employed. 2.2. NI as a Stability Condition. The NI is defined, also using steady-state gains of a process, as (Niederlinski, 1971)
NI )
deg(G) deg(G h)
(3)
where det(A) denotes the determinant of matrix A and G h ) diag(G). The sign of the NI, i.e., NI > 0, provides a necessary stability condition (Niederlinski, 1971; Grosdidier et al., 1985; Zhu and Jutan, 1995) and, consequently, constitutes a complementary tool to the RGA in variable pairing selection, as stated below.
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4093
NI Stability Rule. Input and output variables should be paired in such a way that the resulting system leads to a positive NI. Notice that the stability rule is stated independent of the interaction rule. Various rules will be integrated at the end of this section to give the final RGA-based pairing criteria. Observations regarding the definition of the NI and its use as a pairing tool are given below. Remarks. 1. The calculation of the NI requires diagonal pairings. Hence, nondiagonal pairings are required to be rearranged as diagonal ones by placing the paired elements to diagonal positions. 2. G(s) is assumed to be rational and strictly proper, particularly for PI-type controllers. 3. The independent subsystem is required to be stable. 4. Originally, the same controller designed for the independent subsystem is required when closing all the loops. Zhu and Jutan (1995) alleviated this restriction to allow for tuning of the independently designed controller. 5. The NI is traditionally used merely as a stability condition by its sign only. Zhu and Jutan (1993a) utilized its size as an interaction measure as well (see later). 2.3. RGA, NI, and Integrity. Integrity represents an important property of a decentralized control system desired during variable pairing choices. The following relationship between the NI and the RGA offers a solution to the integrity issue (Chiu and Arkun, 1990; Zhu and Jutan, 1993a):
NIλii ) NI(i)
∀ i
(4)
where NI(i) denotes the NI of subsystem of G with ith row and ith column removed. Clearly, NI(i) > 0 provides a necessary stability condition for the above subsystem, i.e., a necessary condition for integrity against a single loop (ith loop) failure. Chiu and Arkun (1990) also considered system integrity against any combination of loop failure. Apparently, integrity against single-loop failure is of more practical importance. Naturally, the RGA lends to an integrity condition (Zhu and Jutan, 1995); i.e.,
λii > 0
∀ i
(5)
Integrity Rule. Input and output variables of a process should be paired in such a way that all the RGA elements corresponding to the paired elements are positive. Implications of the sign of the RGA are studied, and pairing on positive RGA elements is also suggested by Bristol (1966), McAvoy (1983), Grosdidier et al. (1985), and Morari and Zafiriou (1989). Again, the integrity rule is stated independently and will be combined with other rules to define the final criteria for variable pairing. Remarks. 1. The same assumptions required for NI discussed previously apply to NI(i) and hence to λii as well. 2. Only paired elements, i.e., diagonal pairings, are concerned. Hence, alternative pairings need to be arranged as diagonal ones one at a time. 3. Only integrity against single-loop failure is considered. 4. The integrity rule is subject to the stability rule. 2.4. RGA, NI, and Robust Stability. Robustness in the face of model uncertainties also represents an
important issue in any control system and, hence, deserves close investigation during variable pairing decisions. The RGA is found to contain very useful information in this regard. The sensitivity of the RGA elements to model error (single gain error) is given by
dλij (1 - λij)λij ) dgij gij
(6)
The above equation was initially demonstrated by Grosdidier et al. (1985) and Yu and Luyben (1987). Zhu and Jutan (1993b) provided a rigorous derivation and geometrical interpretations. Clearly, large RGA elements imply that loop interaction under the pairing is very sensitive to uncertainties in process gains. A quantitative measure of stability robustness, i.e., a necessary condition for robust stability, in face of gain errors is given by
∆gij 1 0 w interaction acts in the same direction as interaction-free process; φij > 1.0 w interaction dominates over interaction-free process gain; φij < 0 w interaction acts in the reverse direction as interaction-free process gain; φij < -1.0 w reverse interaction dominates over interaction-free process gain.
4096 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996
φij > -1
∀ j
(27)
It is interesting to note that from eq 22, the condition given in eq 27 is equivalent to that given in eq 5, which provides a necessary condition for integrity, rather than stability. With the RIA, one is able to clarify this seeming mismatch. In fact, integrity and stability are closely related to each other (see Zhu, 1996). The condition in eq 27 is actually consistent with the stability condition by the NI. This is clear from the following relationship between the RIA and the NI obtainable from eqs 4 and 22 (note that rearrangement for diagonal pairing is required): Figure 3. Relationship between RIA and RGA.
The relationship between the RIA and RGA can be best described by Figure 3. The following can be observed from Figure 3: λij ) 1.0 S φij ) 0; λij f 0+ S φij f +∞; λij f 0- S φij f -∞; λij f +∞ S φij f -1.0; λij f -∞ S φij f -1.0. It is important to notice that both very large positive and very large negative RGA elements drive loop interaction toward the inverse direction of the original interaction-free process gain and that both very small positive and very small negative RGA elements are also, actually even more, harmful to process control. Apparently, in contrast to the RIA, the RGA exhibits discontinuous behavior when used to characterize loop interaction. Similar discontinuous behavior was also observed in characterizing stability robustness by the RGA in contrast to the NI, which shows no discontinuity. Obviously, we have the following. RIA Interaction Rule. Variables should be paired in such a way that all the paired RIA elements are closest to 0. Notice that the above states the interaction rule only. Other considerations will be addressed later. Although the above interaction rule seems to be equivalent to the RGA interaction rule, the closeness to a value (0 in RIA rule, 1.0 in RGA rule) has different implications. Specifically, the distance of the RGA elements from 1.0 may not realistically reflect the amount of interaction as noted previously, whereas the distance of the RIA does. In particular, the RGA interaction rule mainly targets at dismissing large RGA elements, which makes the robustness rule redundant, whereas the RIA interaction rule essentially aims at eliminating small RGA elements (large interaction; see eq 22), which complements the robustness rule (see later). 5.2. Stability, Integrity, and Robustness by the RIA. Applying the stability consistency principle for SISO systems to any uj-yi loop after decomposition (Grosdidier et al., 1985; Zhu and Jutan, 1995), one obtains the necessary stability condition below,
g˜ ijcj > 0
∀ j
(25)
where cj denote the steady-state gain of the compensator obtained from the controller with the integrator explicitly separated; i.e., cj(s) ) 1/sgcj(s). Assuming that each of the corresponding independent loops is designed to be stable, one has
gijcj > 0
(26)
Combining eqs 25 and 26 and using eq 24, one obtains the following necessary stability condition for variable pairing in terms of the RIA:
(φii + 1)NI(i) ) NI
∀ i
(28)
Since the definition of the RIA requires the system to possess integrity against single-loop failure, i.e., NI(i) > 0 (actually, so does the definition of the RGA), φii and the NI provide equivalently a necessary stability condition. However, if a system lacks integrity, the RIA no longer measures loop interaction (simply because the steady-state value of φii(s) is not defined from the final value theorem). On the other hand, nevertheless, a negative φii does indicate lack of integrity subject to a positive NI. This is why eq 5 (or eq 27) provides an integrity condition that is subject to stability requirement. More strictly, eq 5 (or eq 27) provides a stability condition subject to integrity requirement as assumed in the definition of the RIA (and the RGA). Since the integrity check using NI involves calculation of all the NI(i), practically, NI and φii can be used for stability and integrity checks, respectively. Therefore, the stability rule and the integrity rule in the RGA-based criteria can be inherited but with RGA (eq 5) replaced by the RIA (eq 27). It can be shown that the sensitivity of φij to the independent gain is proportional to itself; i.e.,
dφij φij )dgij gij
(29)
Equation 29 coincides with the fact that φij is much smoother than the corresponding λij. A robust stability measure can be expressed in terms of φij as
∆gij < -(φij + 1) gij
(30)
Clearly, the maximum relative gain error allowed to maintain stability is proportional to the amplifying or attenuating factor imposed by loop interaction on the original gain. Again, although eq 30 only gives a bound on the independent gain with a single gain error, it is also likely applicable to multiple gain errors (see Zhu and Jutan, 1993a). Apparently, relative interaction close to -1 (large RGA elements) should be avoided for robust stability. Combining the RIA interaction rule with stability, integrity, and robustness requirements, we have the following RIA-based pairing criterion. RIA-Based Pairing Criterion. Variables should be paired in such a way so that (a) all the RIA elements are closest to 0, (b) NI is positive, (c) all the RIA elements are greater than -1, and (d) RIA elements close to -1 are avoided.
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4097 Table 1. Comparison between RGA-Based and RIA-Based Pairing Criteria RIA criteria RGA criteria
by λij
by φij
comments
loop interaction stability integrity
λij f 1.0 NI > 0 λij > 0
φij f 0.0 φij > -1 T NI > 0 NI(i) > 0
λij f 1.0 λij > 0 T NI > 0 NI(i) > 0
robustness overall interaction
avoid large λij
avoid φij close to -1.0 min |φijk|
avoid large λij min |1/λijk -1.0|
In contrast to the RGA criterion, both large (small RGA elements) and close to inverse (large RGA elements) interactions are explicitly identified as undesirable pairing elements for interaction and robustness considerations, respectively. In particular, interaction and robustness rules both play a role and complement each other. Clearly, the RIA-based pairing criteria also represent a comprehensive tool for variable pairing choices. For comparison, the RIA-based pairing rules and the corresponding RGA-based ones are summarized in Table 1. 5.3. A New Overall Interaction Measure and Pairing Criterion. Like the RGA, the RIA measures only individual interactions in individual control loops. Hence, ambiguity may still arise, since there may exist several alternatives satisfying the RIA-based pairing criterion. By the RIA, the distance between a RIA element and 0 actually implies the amount of interaction in a loop. Therefore, an overall interaction measure can be readily proposed as
min
∑|φijk|
(31)
where φijk denotes pairing elements corresponding to the kth alternative. The proposed overall interaction measure can identify the best pairing choice as the one showing minimum overall interaction among pairing candidates that satisfy stability, integrity, and robustness requirements. In comparison to the NI-based overall interaction measure, the new one has the following advantages: (1) it has a more rigorous basis; (2) it is calculationally convenient without requiring diagonal pairings; (3) it overcomes problems associated with other overall measures (see examples). Expressing eq 31 in terms of the RGA elements, one obtains
min
∑|
|
1 -1 λijk
closeness of λij in RGA rule ambiguous integrity required in RIA rule stability required in RGA rule; integrity assumed in RIA rule overall interaction not available in original RGA rule
Table 2. Example 1: Overall Interaction by Different Measures no.
pairings
NI
{θi}
∑|φij|
1 2 3 4 5 6
(1,1)-(2,2)-(3,3) (1,1)-(2,3)-(3,2) (1,2)-(2,1)-(3,3) (1,2)-(2,3)-(3,1) (1,3)-(2,1)-(3,2) (1,3)-(2,2)-(3,1)
0.62 -0.93 1.87 18.7 -9.35 -62.3
-0.1, -0.53, 0.63 -0.37, -0.98, 1.34 0.62, -0.31 ( 1.78i -4.13, 2.02 ( 1.58i 1.83, -0.91 ( 2.77I 0.34, 9.65, -10.0
1.77 4.7 2.57 28.39 12.32 35.21
Notice that, although the overall interaction can be directly applied to all possible pairings, in practice, the interaction rule based on the individual interaction measure may be first used to reduce the number of alternatives before using the overall interaction rule to distinguish desirable alternatives. In conclusion, the RIA-based pairing criteria developed offer a comprehensive and reliable solution to the variable pairing problem and consequently represent a promising tool. 6. Examples This section includes examples to demonstrate that the proposed RIA-based pairing criteria constitute a comprehensive and reliable screening tool, by showing its effectiveness in comparison with the RGA-based as well as the NI and Jacobi eigenvalue criteria. 6.1. Example 1. This example shows that the RGAbased criteria, more precisely, the RGA interaction rule, lead to more than one choice and are unable to distinguish them, whereas all three overall interaction measures agree on the final decision. Consider the following plant gain matrix studied by Majares et al. (1986) and Zhu and Jutan (1993a):
[
1.0 1.0 -0.1 G(0) ) 1.0 -3.0 1.0 0.1 2.0 -1.0
]
The RGA and RIA can be calculated as
[
(32)
Noticeably, the above rule can be viewed as a “corrected” version of the intuitive RGA-based overall interaction measure given in eq 8 due to the corrected definition of the RGA as an interaction measure from the RIA. A major advantage of the rule in eq 32 lies in the fact that very small RGA elements (very large interaction) are explicitly penealized, while very large RGA elements (close to inverse interaction) are dismissed explicitly by robustness requirements. Finally, a new pairing criterion based on the new overall interaction measure is given below. New Pairing Criteria. Variables should be paired so that (a) NI > 0 (stability rule), (b) φij > -1 (integrity rule), (c) φij close to -1 is avoided (robustness rule), and (d) min ∑|φijk| (overall interaction rule).
0.53 0.59 -0.12 RGA ) 0.43 1.59 -1.02 0.04 -1.18 2.14 and
[
0.87 0.70 -9.13 RIA ) 1.34 -0.37 -1.98 25.71 -1.85 -0.53
] ]
The NI’s, the Jacobi eigenvalues, and the absolute values of the RIA elements corresponding to all possible pairings are shown in Table 2. According to the new RIA-based rules, we shall first eliminate pairings corresponding to NI > 0 and φij < -1 by using the stability and integrity rules. As a result, pairings 2, 4, 5, and 6 should be dismissed, leaving pairing 1 and 3 as the final two candidates. Note
4098 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 Table 3. Example 2: Overall Interaction by Different Measures no.
pairings
NI
{θi}
∑|φij|
1 2 3 4 5 6
(1,1)-(2,2)-(3,3) (1,1)-(2,3)-(3,2) (1,2)-(2,1)-(3,3) (1,2)-(2,3)-(3,1) (1,3)-(2,1)-(3,2) (1,3)-(2,2)-(3,1)
0.91 -0.98 -2.28 1.50 -0.91 0.56
0.3 ( 0.75i, 0.59 -0.65 ( 10.78i, 1.30 -0.8 ( 0.66i, 1.62 (0.70i, 0 -0.63 ( 0.94i, 0.8 -0.4 ( 0.92i, 0.80
13.88 18.01 5.5 7.75 3.56 1.68
that the RGA-based criteria always give the same results up to this stage (with NI > 0 and λij > 0). Between the two final alternatives, the RGA fails to clearly point out the best one, and hence, ambiguity arises. The intuitive overall RGA rule would identify pairing 3 as the final choice. However, pairing 3 contains a relative gain of 0.43 which is less than 0.5 and this is undesirable according to Majares et al. (1986). Hence, no decision can be clearly made using the RGA. However, all three overall interaction measures, the NI measure, the Jacobi eigenvalue measure, and the RIA overall measure, suggest that pairing 1 should be the final choice showing the minimum overall interaction. This is obvious by examining the values listed in Table 2. 6.2. Example 2. This example is intended to demonstrate that the RIA-based pairing criteria, more precisely, the RIA overall interaction measure, suggests uniquely the right final pairing, whereas the NI and the Jacobi eigenvalue overall interaction measures lead to incorrect answers and ambiguities arise from the RGA measure. Consider the gain matrix studied by Zhu (1993) below:
[
1.5 -1.2 -1.5 G(0) ) 1.5 -3.0 1.4 -1.8 2.2 -1.1
]
The RGA and RIA are obtained as
[
]
[
]
0.073 0.23 0.70 RGA ) -1.53 2.88 -0.35 2.46 -2.12 0.66 and
12.71 3.33 0.44 RIA ) -1.65 -0.65 -3.83 -0.59 -1.47 0.53
The NI’s, the Jacobi eigenvalues, and the absolute size of the RIA elements corresponding to all possible pairings are calculated as shown in Table 3. By the stability and integrity rules, equivalently in terms of the RGA or RIA, pairings 1 and 6 are acceptable alternatives deserving further consideration. Again, the RGA is unable to distinguish between the two candidates. The Jacobi eigenvalue criterion suggests pairing 4 as the most desirable pairing. It is known that this pairing results in a system lacking integrity. This is because the Jacobi eigenvalue criterion contains no sufficient information about system stability and integrity. The NI interaction rule identifies pairing 1 as the best choice which exhibits minimum overall interaction. However, pairing 1 contains a large RIA element (or very small RGA element), indicating a large loop interaction and significant gain change due to interaction. Hence, practitioners will lean toward the other
pairing (pairing 6) as the best choice (see Zhu, 1993; McAvoy, 1993). The RIA overall interaction rule still correctly suggests pairing 6 as the final choice, showing the minimum overall interaction. Hence, the RIA-based criteria represent a more reliable tool than others. 7. Conclusions Variable pairing choice from interaction measurement, stability, integrity, and robustness perspectives has been systematically addressed with new insights offered. Problems, particularly the ambiguities in identifying the final variable pairing, associated with the most widely used RGA-based pairing criteria, are revealed. Limitations on some of the existing overall interaction measures have also been identified. A new interaction measure, the RIA, has been explored in terms of its advantages over the RGA, and its implications for the amount of interaction, closed-loop stability, integrity, and robustness against gain uncertainties have also been presented. The RIA has been further extended to provide an overall interaction measure. It has been shown that the RIA-based overall interaction measure is able to avoid potential ambiguities in using the individual interaction measures such as the RGA and RIA itself and to overcome problems associated with the existing overall interaction measures. Consequently, the new RIA-based pairing criteria offer a comprehensive and reliable solution to the variable pairing selection problem in decentralized control systems and hence represent a promising tool for industrial use. Finally, it must be pointed out that the new pairing criteria as well as other pairing rules studied in this article are concerned only with the steady-state case. Although the use of steady-state pairing rules has been widely advocated and justified (see Zhu, 1996), detailed dynamic analysis may be required under some circumstances (see Rijnsdorp, 1965; Hovd and Skogestad, 1992; Zhu and Jutan, 1996). Ultimately, variable pairing represents only the first step, and hence, dynamic simulation should be performed during the design of any decentralized control system. Nomenclature A ) Jacobi eigenvalue matrix aij(s) ) dynamic absolute interaction in the uj-yi loop c(s) ) controller transfer function (TF) with the integrator separated in a SISO loop dij(s) ) perturbation term G(s) ) process TF matrix G ) process gain matrix G h ) diag(G) gij(s) ) ijth element of G(s) gij ) ijth element of G ∆gij ) model error in gij g˜ ii(s) ) equivalent process TF with interaction included in the ith loop g˜ ii ) equivalent process gain with interaction included in the ith loop gˆ ij ) equivalent process gain in the uj-yi loop with all other loops closed s ) Laplace variable ui ) ith input yi ) ith output Greek Letters φij(s) ) dynamic relative interaction in the uj-yi loop
Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4099 φij ) steady-state relative interaction in the uj-yi loop λij ) ijth element of the RGA θi ) eigenvalue of a matrix F ) spectral radius of a matrix Abbreviations AIM ) absolute interaction array det ) determinant of a matrix NI ) Niederlinski index NI(i) ) Niederlinski index of the subsystem with the ith loop removed RGA ) relative gain array RIA ) relative interaction array SISO ) single input, single output
Literature Cited Bristol, E. H. On a New Measure of Interaction for Multivariable Process Control. IEEE Trans. Auto. Control 1966, AC-11, 133. Bruans, D. D.; Smith, C. R. Singular Value Analysis: A Geometrical Structure for Multivariable Processes. AIChE Winter Meeting, Orlando, FL,1982. Chang, J. W.; Yu, C.-C. The Relative Gain for Non-square Multivariable Systems. Chem. Eng. Sci. 1990, 45, 1309. Chang, J. W.; Yu, C.-C. Relative Disturbance Gain Array. AIChE J. 1992, 38, 521. Chiu, M.-S.; Arkun, Y. Decentralized Control Structure Selection Based on Integrity Considerations. Ind. Eng. Chem. Res. 1990, 29, 360. Grosdidier, P.; Morari, M.; Holt, B. R. Closed-Loop Properties from Steady-State Gain Information. Ind. Eng. Chem. Fundam. 1985, 24, 221. Hovd, M.; Skogestad, S. Simple Frequency-dependent Tools for Control System Analysis, Structure Selection and Design. Automatica 1992, 28, 989. Manousiouthakis, V.; Nikolaou, M. Analysis of Decentralized Control Structures for Nonlinear Systems. AIChE J. 1989, 35, 549. Manousiouthakis, V.; Savage, R.; Arkun, Y. Synthesis of Decentralized Process Control Structures Using the Concept of Block Relative Gain. AIChE J. 1986, 32, 991. McAvoy, T. Interaction Analysis; ISA: Research Triangle Park, NC, 1983. McAvoy, T. Private Communication, University of Maryland, 1993.
Mijares, G.; Cole, C. D.; Naugle, N. W.; Preisig, H. A.; Holland, C. D. A New Criterion for the Pairing of Control and Manipulated Variables. AIChE J. 1986, 32, 1439. Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. Niederlinski, A. A Heuristic Approach to the Design of Linear Multivariable Interacting Control Systems Automatica 1971, 7, 69. Reeves, D. E.; Arkun, Y. Interaction Measures for Nonsquare Decentralized Control Structures. AIChE J. 1989, 35, 603. Rijnsdorp, J. E. Interaction in Two-variable Control Systems for Distillation Columns. Automatica 1965, 1, 15. Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; John Wiley and Sons: New York, 1989. Shinskey, F. G. Process Control Systems; McGraw: New York, 1988. Skogestad, S.; Morari, M. Implications of Large RGA Elements on Control Performance. Ind. Eng. Chem. Res. 1987, 26, 2323. Skogestad, S.; Lundstrom, P.; Jacobsen, E. W. Selecting the Best Distillation Control Configuration. AIChE J. 1990, 36, 753. Yu, C.-C.; Luyben, W. L. Robustness with Respect to Integral Controllability. Ind. Eng. Chem. Res. 1987, 26, 1043. Zhu, Z.-X. Modern and Decentralized Control of Multivariable Processes. Ph.D. Dissertation, University of Western Ontario, London, Canada, 1993. Zhu, Z.-X. Structural Analysis and Stability Conditions in Decentralized Control Systems. Ind. Eng. Chem. Res. 1996, 35, 736. Zhu, Z.-X.; Jutan, A. A New Variable Pairing Criterion Based on Niederlinski Index. Chem. Eng. Commun. 1993a, 121, 235. Zhu, Z.-X.; Jutan, A. Stability Robustness for Decentralized Control Systems. Chem. Eng. Sci. 1993b, 48, 2337. Zhu, Z.-X.; Jutan, A. Consistency Principles for Stability in Decentralized Control Systems. Chem. Eng. Commun. 1995, 132, 107. Zhu, Z.-X.; Jutan, A. Loop Decomposition and Dynamic Interaction Analysis of Decentralized Control Systems. Chem. Eng. Sci. 1996, 51, 3325.
Received for review March 11, 1996 Revised manuscript received August 23, 1996 Accepted August 26, 1996X X Abstract published in Advance ACS Abstracts, October 15, 1996.
