Ind. Eng. Chem. Res. 2004, 43, 283-287
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Dynamic Interaction Measures for Decentralized Control of Multivariable Processes Jietae Lee† Department of Chemical Engineering, Kyungpook National University, Taegu 702-701, Korea
Thomas F. Edgar* Department of Chemical Engineering, University of Texas, Austin, Texas 78712
We present some insights on the relative error matrix between the process transfer function matrix and its diagonal matrix, which has been used in analyzing dynamic interactions by several authors. Proposed interaction measures can be interpreted in terms of differences of the complementary sensitivity function matrices when loops are closed, extending the concept of gain changes in the relative gain array. To employ the interaction measures, control systems are designed in advance using internal model control. With changes in the closed-loop time constant, proper pairings based on closed-loop response speeds can be obtained. Introduction Designing decentralized control systems can be difficult because of interactions between the input/output variables; hence, it is important to determine control system structures that have minimal interactions. The relative gain array (RGA) method introduced by Bristol1 has been widely used for this purpose.2 It requires minimal process information, namely, the steady-state process gain matrix, and is very simple to calculate. The RGA can also be used as a sensitivity measure for model uncertainties.3,4 However, because it does not account for process dynamics, it can lead to incorrect loop pairings for some processes. To provide for dynamic interaction measures, the RGA has been extended in various ways by many authors (e.g., refs 5-8). However, they are not as well accepted as the steady-state RGA of Bristol.1 Hovd and Skogestad9 have shown that the frequency-dependent RGA can be used as both performance and stability measures. The steady-state property of decentralized integral controllability (DIC) quantifies difficulties in controlling processes with multiloop controllers.10 The process is said to be DIC if there exists a decentralized controller with integral action in each loop such that the closedloop system is stable and each loop gain can be detuned without introducing instability. Because almost all loops in industry have integral action for offset-free operations, this property can also be used for determining control loop pairings. A design rule that avoids pairings that correspond to negative values of RGA is a necessary condition for DIC. The calculation of DIC requires just the process gain matrix, but it is hard to compute except for low-dimensional processes.11,12 Grosdidier and Morari13 proposed the µ-interaction measure of µ-1[E(jω)] where E(s) ) G(s) diag[G(s)]-1 I is the relative error matrix between the process G(s) and its diagonal matrix. If a multiloop control system is designed so that magnitudes of individual closed-loop * To whom correspondence should be addressed. Tel: 512471-3080. Fax: 512-471-7060. E-mail:
[email protected]. † Tel.: +82-53-950-5620. Fax: +82-53-950-6615. E-mail:
[email protected].
transfer functions for the diagonal process are less than the µ-interaction measure, the multiloop control system is stable. The µ-interaction measure plot can be used as a dynamic interaction measure. Because only the magnitudes of the individual closed-loop transfer functions are considered, the measure can be conservative. To reduce its conservatism, an interaction measure that considers the phase angle of the individual closed-loop transfer functions was proposed by Lee and Edgar.14 Both methods require complex computations of structured singular values. The magnitude of the difference between the closedloop transfer function matrix for the full process transfer functions and its diagonal approximation has been used for dynamic interaction measures by Arkun15 and Huang et al.16 Their measures are functions of the angular frequency and a design parameter for the control system; hence, they are difficult to interpret. To avoid such difficulty, Arkun15 used the open-loop asymptote of his interaction measure equivalent to the above relative error matrix of Grosdidier and Morari,13 but this loses some of its physical meaning. Huang et al.16 integrated the interaction measure in the frequency domain, which also loses some information. In a related development, norms of differences in the sensitivity function matrices and the complementary sensitivity function matrices (closed-loop transfer function matrices) have also been used to determine better models in the context of iterative identification and model/controller reduction.17 These norms of differences in the sensitivity function matrices and the complementary sensitivity function matrices between full and diagonal process transfer function matrices are suggested as dynamic interaction measures in this paper. Some insights on the relative error matrix between the process transfer function matrix and its diagonal matrix, which has been used in analyzing dynamic interactions by several authors, are also obtained. For the proposed interaction measures, control systems are designed in advance, and multiloop internal model controllers with one design parameter, the closed-loop time constant, are used here. The requirement of prior control system design may not be a disadvantage
10.1021/ie020869l CCC: $27.50 © 2004 American Chemical Society Published on Web 05/20/2003
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With this matrix, various interaction measures have been proposed by several authors.13-15 In the model reduction area and the iterative closedloop identification area, some other error matrices are used.17 The complementary sensitivity error matrix Figure 1. Feedback control system.
because designing an internal model controller is straightforward for a given transfer function. With changes in the design parameter, proper pairings based on the desired closed-loop speed of response can be obtained. Besides the process dynamics, the presence of controllers also affects strongly the amount of interactions.8 Interaction analyses independent of controllers may lead to erroneous conclusions on the control system structure for some cases. For example, slow and fast control systems may prefer different pairings for processes where static and dynamic interaction measures recommend different pairings. The proposed interaction measure will provide proper pairings according to the closed-loop speed of response.
EH(s) ) H(s) - H ˜ (s) ) [I + G(s) C(s)]-1G(s) C(s) G ˜ (s) C(s)[I + G ˜ (s) C(s)]-1 is used to quantify how the approximation preserves the performance in tracking. The sensitivity error matrix
ES(s) ) S(s) - S ˜ (s) ˜ (s) C(s)]-1 ) [I + G(s) C(s)]-1 - [I + G is used for disturbance rejection performance in outputs. Both EH(s) and ES(s) reduce to the same expression:17
EH(s) ) ES(s) ) [I + G(s) C(s)]-1[G(s) G ˜ (s)]C(s) [I + G ˜ (s) C(s)]-1
Criteria for Model Accuracy Consider a process with n inputs and n outputs
G(s) ) {gij(s)} and a decentralized (diagonal) controller as shown in Figure 1. For the process and its diagonal subsystem, G ˜ (s) ) diag{gii(s)}, sensitivity function matrices (closedloop transfer function matrices between the disturbances d and the outputs y) are defined for a diagonal controller as
S(s) ) [I + G(s) C(s)]-1 S ˜ (s) ) [I + G ˜ (s) C(s)]-1 ) diag{1/[1 + gii(s) ci(s)]}
) S(s) E(s) H ˜ (s) Dynamic interaction measures based on these error matrices are proposed below. Interaction Measures The norm of EH(s) can be used as a dynamic interaction measure. Because EH(s) contains a controller C(s), first we design a multiloop control system from the diagonal transfer functions gii(s) by specifying the closed-loop transfer functions h ˜ ii(s) using the direct synthesis method,18 which is equivalent to internal model control:19
{
and complementary sensitivity function matrices (closedloop transfer function matrices between the set points r and the outputs y) are defined as
H(s) ) [I + G(s) C(s)]-1G(s) C(s) H ˜ (s) ) [I + G ˜ (s) C(s)]-1 G ˜ (s) C(s) ) diag{gii(s) ci(s)/[1 + gii(s) ci(s)]} For the magnitudes of signals and systems, we used the following norms:20
for a time-domain signal |y(t)|2 )
x
∑i ∫-∞|yi(τ)|2 dτ ∞
C(s) ) diag
h ˜ ii(s)
}
gii(s) [1 - h ˜ ii(s)]
Here
h ˜ ii(s) )
gii+(s) (λs + 1)rii
where gii+(s) is the noninvertible part in gii(s) and rii is the relative order of the numerator and denominator in gii(s). The parameter rii can be 1 less than the relative order when derivative action is allowed in the controller. The parameter λ is a design parameter which represents the time constant of the closed-loop system. We can then obtain a dynamic interaction measure as a function of λ:
q1(λ) ) ||EH(s)||∞ ) ||ES(s)||∞
for a transfer function j [G(jω)] |G(s)|∞ ) max σ ω
where σ j (‚) means the maximum singular value. When the process G(s) is approximated by its diagonal transfer functions G ˜ (s), the relative error matrix becomes
E(s) ) [G(s) - G ˜ (s)]G ˜ -1(s)
This interaction measure implies, for signals d(t) and r(t) with finite norms, that
||y(t) - y˜ (t)||2 e q1(λ) ||d(t)||2 ||y(t) - y˜ (t)||2 e q1(λ) ||r(t)||2 under the assumption that H(s) is stable.20 The stability
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under the stability assumption on H(s). In addition, we have
q2(λ) - ||h ˜ ii(s)||∞ e ||hii(s)||∞ e ||h ˜ ii(s)||∞ + q2(λ) A low value of q2(λ) guarantees a low peak amplitude ratio of diagonal elements of H(s), yielding low overshoot in step responses of y(t) for the paired set-point changes. Hence, systems with smaller q2(λ) are recommended. q2(λ) . 2 means an oscillatory or unstable control system. To control each paired output, only n diagonal elements in H(s) are designed because the multiloop (diagonal) controller consists of n controllers. The above interaction measure q2(λ) will indicate if this objective can be obtained by the multiloop controller. Remarks Figure 2. Amplitude ratios of h ˜ ii(s) (solid line), s˜ ii(s) (dotted line), and h ˜ ii(s) s˜ ii(s) (dashed line) for λ ) 1, gii+(s) ) 1, and rii ) 1.
of H(s) can be checked by applying the Nyquist method to the polynomial of det[I + G(s) C(s)]. Here y˜ (t) means the closed-loop response for each diagonal subsystem without interaction terms. If q1(λ) is small, the set-point responses and disturbance rejection responses do not differ much from what are expected at the design stage. Hence, pairings providing smaller q1(λ) are recommended. We have
q1(λ) ) ||[I + G(s) C(s)]-1[G(s) ˜ (s) C(s)[I + G ˜ (s) C(s)]-1||∞ G ˜ (s)]G ˜ (s)-1G ) ||S ˜ (s) [I + E(s) H ˜ (s)]-1E(s) H ˜ (s)||∞ Because σ j [S ˜ (jω) H ˜ (jω)] shows its peak at around ω ) 1/λ as in Figure 2, it can be approximated as
q1(λ) ≈ σ j ([(j + 1)I + E(jω)]-1E(jω)) ≈ 0.707σ j [E(jω)], ω ) 1/λ if σ j [E(jω)] is small. Hence, a plot of σ j [E(jω)] as in ref 15 may be considered as an approximate dynamic interaction measure of q1(λ). The above dynamic interaction measure q1(λ) indicates how much interaction terms degrade the control performances, and the recommended pairing selects the smaller value. However, q1(λ) is dependent on input and output scaling, unlike the RGA. Hence, it may be very large for processes with one-way interactions and can be conservative. A less conservative interaction measure is considered here. When interactions are weak, the diagonal elements of a closed-loop transfer function matrix will not be much different from the design specification, h ˜ ii(s). As a dynamic interaction measure, we can use the magnitude of differences
q2(λ) ) ||diag(EH(s))||∞ ) ||diag(ES(s))||∞ It can be interpreted in the time domain as, for any i,
||yi(t) - y˜ i(t)||2 e q2(λ) ||di(t)||2 ||yi(t) - y˜ i(t)||2 e q2(λ) ||ri(t)||2
Characteristics of our interaction measures can be summarized as follows. (1) By plotting q1 or q2 versus 1/λ, we can find the best pairing according to the closed-loop speed of response. ˜ (s) can be interpreted (2) The term EH(s) ) H(s) - H as
(closed-loop transfer function when all of the other loops are closed) (closed-loop transfer function when all of the other loops are open) or
(actual closed-loop transfer function) (ideal closed-loop transfer function) (3) Process dynamics at very low and high frequencies are not important for interaction analysis in the sense of our interaction measure. (4) The interaction measure q1(λ) is dependent on input and output scaling. On the other hand, the interaction measure q2(λ) is independent of input and output scaling such as D-1G(s) D for positive diagonal D. The RGA is independent of input and output scaling such as D1G(s) D2 for positive diagonals D1 and D2. (5) Computations of q2(λ) are much simpler than those of q1(λ). (6) Instead of the internal model control method, controller design methods such as the pole assignment method which have adjustable parameters related to control system speed of response can also be used. However, the internal model control method is simple to use and will be sufficient for this control system structure selection purpose. Procedure for the Pairing Problem For selection of control structures in multiloop control systems, steady-state interaction measures are first applied and infeasible pairs are removed. Then dynamic interaction measures are applied. The design procedure can be summarized as follows. (1) Apply the RGA method and select all of the pairs having positive diagonal relative gains. (2) For the pairs selected at step 1, apply the DIC condition and remove pairs failing to meet necessary conditions of DIC.
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Figure 3. Plots of σ j [E(jω)], q1(λ), and q2(λ) for example 1. (Solid lines are for the 1-1/2-2 pairing, and dotted lines are for the 1-2/ 2-1 pairing.)
Figure 4. Plots of σ j [E(jω)], q1(λ), and q2(λ) for example 2. (Solid lines are for the 1-1/2-3/3-2 pairing, and dotted lines are for the 1-3/2-2/3-1 pairing.)
(3) Apply the proposed dynamic interaction measures to those pairs remaining after the first two steps. (4) Check the stability and loop failure tolerance of the multiloop IMC-based proportional-integral-derivative control system. The last step usually requires the computation of real structured singular values, which are difficult to compute for higher order processes.20 This is beyond the scope of this paper. Examples Example 1. Consider the process in work by Grosdidier and Morari:13
[
2.5 exp(-5s) 5 4s + 1 (2s + 1)(15s + 1) G(s) ) -4 exp(-6s) 1 3s + 1 20s + 1
]
Off-diagonal pairing is better at low frequencies but not at high frequencies because of the large time constant and time delay elements. Figure 3 shows plots of σ[E(jω)], q1(λ), and q2(λ). The diagonal pair has a lower value in σ[E(jω)] for ω greater than about 0.06. q1(λ) and q2(λ) for the diagonal pair have lower values for λ that are smaller than 4.5 and 6.5, respectively. The diagonal pairing has no loop failure tolerance problem14 and is recommended. However, the off-diagonal pairing may be used when a slower control system is preferred. Example 2. Consider the process in ref 6:
[
1.5 1 -2 10s + 1 5s + 1 s + 1 1.5 2 1 G(s) ) 5s + 1 s + 1 10s + 1 2 1.5 1 s+1 10s + 1 5s + 1 The steady-state RGA is
[
]
0.75 -0.03 0.28 RGA ) G(0)oG-T(0) ) -0.03 0.28 0.75 0.28 0.75 -0.03
]
where o means the element-by-element product.20 Hence,
Figure 5. Responses of example 2 with the multiloop IMC controller for the unit step change in r1. (a and b) Cases of λ ) -0.1 and 10, respectively. (Solid lines are for the 1-1/2-3/3-2 pairing, and dotted lines are for the 1-3/2-2/3-1 pairing.)
the pairings of 1-1/2-3/3-2 and 1-3/2-2/3-1 pass the RGA and DIC conditions.11 For these pairings, we apply the proposed dynamic interaction measures and determine which pairing is better. Figure 4 shows that the system with the 1-1/2-3/3-2 pairing has lower q1(λ) and q2(λ) for large λ than the system with the 1-3/22/3-1 pairing. For a smaller λ, the latter pairing has a smaller interaction measure. Hence, the 1-1/2-3/3-2 pairing is recommended for slow control with large λ, and the 1-3/2-2/3-1 pairing is recommended for fast control with small λ. However, we should be cautious in choosing the 1-3/2-2/3-1 pair. A large q1(λ) or q2(λ) for large λ indicates that the multiloop control system may be too oscillatory or even unstable. This means that the system with the 1-3/2-2/3-1 pairing can have a loop failure tolerance problem. Figure 5 shows closedloop responses of IMC controllers for unit step changes in r1. As expected, for λ ) 0.1, the system with the 1-3/ 2-2/3-1 pairing shows excellent control responses, whereas the system with the 1-1/2-3/3-2 pairing shows unstable responses. For λ ) 10, the system with the 1-1/2-3/3-2 pairing shows excellent control responses, whereas the system with the 1-3/2-2/3-1 pairing shows unstable responses. Although the 1-3/
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simple and can provide appropriate pairings according to the closed-loop speed of responses. Acknowledgment The first author was supported by Korea Research Foundation Grant KRF-2001-013-E00083. Note Added after ASAP Posting. This article was released ASAP on 5/20/03 with some incorrect fonts throughout the paper. The correct version was posted on 5/28/03. Literature Cited
Figure 6. Plots of σ j [E(jω)], q1(λ), and q2(λ) for example 3. (Solid lines are for the 1-1/2-2/3-3 pairing, dotted lines are for the 1-1/ 2-3/3-2 pairing, and dashed lines are for the 1-3/2-1/3-2 pairing.)
2-2/3-1 pairing is needed for fast control with small λ, it does not allow stable detuning. Example 3. Consider the process in ref 21:
[
1.986e-0.71s 66.67s + 1 -0.0204e-0.59s G(s) ) (7.14s + 1)2 -0.374e-7.75s 22.22s + 1
-5.24e-60s 400s + 1 0.33e-0.68s (2.38s + 1)2 11.3e-3.79s (21.74s + 1)2
-5.984e-2.24s 14.29s + 1 -2.38e-0.42s (1.43s + 1)2 9.811e-1.59s 11.36s + 1
The steady-state RGA is
[
1.092 -0.104 0.012 RGA ) 0.0064 0.104 0.890 -0.0987 1.004 0.0983
]
]
Hence, the pairings of 1-1/2-2/3-3, 1-1/2-3/3-2, and 1-3/2-1/3-2 pass the RGA and DIC conditions. For these pairings, we apply the proposed dynamic interaction measures. The pairing of 1-3/2-1/3-2 shows the worst values in all three interaction measures of σ[E(jω)], q1(λ), and q2(λ). The plot of σ[E(jω)] in Figure 6 shows that the system with the 1-1/2-2/3-3 pairing may give better responses for smaller λ. In addition, compared to g32(s), the small time delays and time constants in g22(s) and g33(s) favor the 1-1/2-3/3-2 pairing. However, plots of q1(λ) and q2(λ) for the 1-1/2-2/3-3 pairing in Figure 6 show that the pairing is not effective even for smaller λ. It will not be feasible to decrease λ to the value that the 1-1/2-2/3-3 pairing will have a lower dynamic interaction measure. Hence, the 1-1/2-3/3-2 pairing is recommended. Conclusion Dynamic interaction measures based on differences between closed-loop transfer function matrices for the full process transfer functions and those for the paired diagonal transfer functions are proposed. They are
(1) Bristol, E. On a New Measure of Interaction for Multivariable Process Control. IEEE Trans. Autom. Control 1966, AC-11, 133. (2) Shinskey, F. G. Process Control Systems, 4th ed.; McGrawHill: New York, 1996. (3) Grosdidier, P.; Morari, M.; Holt, B. R. Closed-Loop Properties from Steady-State Gain Information. Ind. Eng. Chem. Res. 1985, 24, 22. (4) Skogestad, S.; Morari, M. Implications of Large RGA Elements on Control Performance. Ind. Eng. Chem. Res. 1987, 26, 2029. (5) Witcher, M.; McAvoy, T. J. Interacting Control Systems: Steady State and Dynamic Measurement of Interactions. ISA Trans. 1977, 16, 35. (6) Gagnepain, J. P.; Seborg, D. E. Analysis of Process Interactions with Application to Multiloop Control System Design. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 5. (7) Tung, L.; Edgar, T. F. Analysis of Control Output Interactions in Dynamic Systems. AIChE J. 1981, 27, 690. (8) Jensen, N.; Fisher, D. G.; Shah, S. L. Interaction Analysis in Multivariable Control System. AIChE J. 1986, 32, 959. (9) Hovd, M.; Skogestad, S. Simple Frequency-dependent Tools for Control System Analysis, Structure Selection and Design. Automatica 1992, 28, 989. (10) Skogestad, S.; Morari, M. Variable Selection for Decentralized Control. AIChE Annual Meeting, Washington, DC, 1988. (11) Campo, P. J.; Morari, M. Achievable Closed-loop Properties of Systems under Decentralized Control: Conditions Involving the Steady-state Gain. IEEE Trans. Autom. Control 1994, AC-39, 932. (12) Lee, J.; Edgar, T. F. Conditions for Decentralized Integral Controllability. J. Process Control 2002, 12, 797. (13) Grosdidier, P.; Morari, M. Interaction Measures for Systems under Decentralized Control. Automatica 1986, 22, 309. (14) Lee, J.; Edgar, T. F. Phase Conditions for Stability of Multiloop Control Systems. Comput. Chem. Eng. 2000, 23, 1623. (15) Arkun, Y. Relative Sensitivity: A Dynamic Closed-Loop Interaction Measure and Design Tool. AIChE J. 1988, 34, 672. (16) Huang, H. P.; Ohshima, M.; Hashimoto, I. Dynamic Interaction and Multiloop Control System Design. J. Process Control 1994, 4, 15. (17) Landau, I. D.; Karimi, A.; Constantinescu, A. Direct Controller Order Reduction by Identification in Closed Loop. Automatica 2001, 37, 1689. (18) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; John Wiley and Sons: New York, 1989. (19) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. (20) Skogestad, S.; Postlethwaite, I. Multivariable Feedback Control; John Wiley & Sons: New York, 1996. (21) Tyreus, B. D.; Multivariable Control System Design for an Industrial Distillation Column. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 177.
Received for review November 4, 2002 Revised manuscript received March 18, 2003 Accepted March 18, 2003 IE020869L
