Ind. Eng. Chem. Res. 2003, 42, 4575-4577
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Decoupling Revisited† Matias Waller,‡ Jonas B. Waller,§ and Kurt V. Waller*,§ The Åland Institute of Technology, Mariehamn, Åland, Finland, and Åbo Akademi University, FIN-20500 Åbo, Finland
Techniques for decoupling, commonly applied in distillation control, have not gained recognition in all areas of process control. This is illustrated by investigating an example from the literature that presents the design of a multivariable controller for an industrial solid-fuel boiler. The suggested decoupling controller can be improved by the use of simplified decoupling valid for all frequencies using well-known methods. However, the existing methods for simplified decoupling of 2 × 2 systems cannot always be applied because of realizability issues. To solve this problem, a modification to simplified decoupling approaches is introduced. 1. Introduction To reduce the interaction between different control loops, decoupling is commonly used in many industrial applications. Naturally, the main goal of decoupling is to make the design of diagonal proportional-integral (PI) (or similar) control of multiple-input multipleoutput systems possible by eliminating interactions; i.e., the closed-loop response of each loop is the same as it would be if the other loops were on manual control. The design of suitable decoupling elements for 2 × 2 systems is quite well established within the field of distillation,1,2 but it seems that these methods of general applicability have not been recognized within all other fields of control. Using the design of a multivariable controller for an industrial solid-fuel boiler presented in a paper published in a well-known control journal,3 it is shown that possibilities and advantages of “true” decoupling have been neglected. Using the technique presented by Waller (1974),2 a decoupling valid for all frequencies can, indeed, easily be achieved. The corresponding results are compared with the approach of the original paper.3 The present paper also shows that there are cases where the existing methods for simplified decoupling cannot be directly applied without introducing some approximations due to realizability problems. With a modification introduced in the current paper, however, it is shown that such cases can still easily be decoupled for all frequencies. 2. Case Study The feedback control system for a general 2 × 2 system, including decoupling elements, is described in Figure 1, where Ci represents the controllers, Dij the decoupling elements, and Gij the process. The goal behind decoupling is to determine a decoupling matrix D so that GD ) T is diagonal. For the case of a 2 × 2 system,
(
)(
) (
G G D D T 0 GD ) G11 G12 D11 D12 ) 11 T 0 21 22 21 22 22 †
)
(1)
Dedicated to Bill Luyben, pioneer in process control, on his 70th birthday. * To whom correspondence should be addressed. Tel.: +358 2 2154449. Fax: +358 2 2154479. E-mail:
[email protected]. ‡ The A° land Institute of Technology. § A° bo Akademi University.
and, provided that G-1 exists, the decoupling matrix can be obtained as
D)
(
G22T11 -G12T22 1 G11G22 - G12G21 -G21T11 G11T22
)
(2)
To determine D, different suggestions for the choice of T11 and T22 have been made, and the, perhaps, intuitive choice of T11 ) G11 and T22 ) G22 is labeled ideal decoupling by Luyben (1970).1 However, this often yields rather complicated expressions for the elements of D. For a general system with n inputs and n outputs, n elements in the decoupling matrix can be chosen arbitrarily, leading to the suggestion
(
-G12/G11 1 D ) -G /G 1 21 22
)
(3)
for 2 × 2 systems. This approach is known as simplified decoupling.1 From eq 2 it is clear that any two elements in D can be chosen equal to 1 as long as they are not in the same column, and the other three possibilities are thus
( ( (
1 1 D ) -G /G -G /G 21 22 11 12 D) D)
-G22/G21 -G12/G11 1 1 -G22/G21 1 -G11/G12 1
) ) )
(4) (5) (6)
which obviously improves the prospects for obtaining a realizable decoupling matrix. Apparently, these other possibilities were overlooked until presented by Waller.2 The model for the process considered in the current study represents an industrial solid-fuel boiler,3 given by
G11(s) )
-e-2s 10s + 1
G21(s) ) 0
G12(s) )
-1 10s + 1
G22(s) )
e-10s 60s + 1
and as decoupling elements the authors suggest the constant decoupler (precompensator) D11 ) 1, D21 ) 0, D12 ) -1, and D22 ) 1. With this choice, decoupling is
10.1021/ie020911c CCC: $25.00 © 2003 American Chemical Society Published on Web 07/03/2003
4576 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
Figure 1. Block diagram for the feedback control of a general 2 × 2 system with decoupling elements.
Figure 3. First graph showing output 1, second output 2, and third input 1 and the lowest graph showing input 2 for a simulated load disturbance as an additive disturbance to the second process input. Simulations show results with constant precompensator (solid line) and with decoupling for all frequencies (eq 8; dashed line) using the PI controllers suggested in the original paper.3
Figure 2. First graph showing output 1, second output 2, and third input 1 and the lowest graph showing input 2 for a simulated set-point change from zero to one in yr2 at t ) 0 for the solid-fuel boiler with constant precompensator (solid line) and with decoupling for all frequencies (eq 8; dashed line) using the PI controllers suggested in the original paper.3
obtained merely at steady state. Luyben’s simplified decoupling (eq 3) is not applicable because
D)
(
+2s
1 -e 0 1
)
D ˜ ) (7)
is clearly not realizable. Using an alternative “simplified” approach, in this case eq 4, yields
D)
(
1 1 0 -e-2s
)
mance, and significant improvement in the upper loop is obtained, without degradation of the lower loop response. For load disturbances, the two schemes are practically identical, as expected (see Figure 3). Concerning the robustness, there is no obvious difference between the decoupling strategies. Introducing additional dead times in the control loops, as is done in the studied example, might require detuning of the diagonal controller (although not necessary in the current case study nor used in the simulations). Thus, the robustness for all decoupling schemes mainly depends on the accuracy of the process model and the tuning of the diagonal controller. An inspection of the four “simplified” possibilities to choose D (eqs 3-6) reveals that two of the elements are from the matrix
(8)
i.e., decoupling for all frequencies can be achieved with this choice. Here, it can also be noted that the closedloop responses will depend on the choice of D according to eqs 3-6, and this should be taken into account when choosing and tuning the controller (in the example in question, a delay of 2 time units is introduced in the lower control loop). Using the values for PI controllers suggested in the paper3 (using the constant precompensator), the responses to a step change in the reference value yr2 (the interesting case) are provided in Figure 2 for both the constant decoupling scheme3 and “simplified” decoupling (i.e., in this case using eq 8). Clearly, decoupling at all frequencies yields superior perfor-
(
-G22/G21 -G12/G11 -G21/G22 -G11/G12
)
(9)
with two other elements in different columns chosen equal to 1. If none of the simplified decoupling schemes are realizable, no easily applicable strategy seems to have been suggested in the literature. This problem can be solved in the following way. Consider only the realizable parts of the elements of each respective column; i.e., choose D11 as the realizable part of -G22/G21 and D21 as the remaining inverse of G22/G21 and so on. To illustrate the approach, we use the solid-fuel boiler example but with the following modified interaction term:
G12(s) )
-1 (10s + 1)(60s + 1)
It is seen that none of the simplified decoupling approaches can be realized for all frequencies without the introduction of some sort of approximation [e.g., the second approach (eq 4) will yield an order of the numerator larger than the one of the denominator].
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4577
in Figure 4, which illustrates the resulting connection from ui to yi. By also taking possible common factors for G22 and G21, as well as for G11 and G12, into account, simpler expressions may be obtained. The modification of the model of the solid-fuel boiler illustrates this case. 3. Conclusions Figure 4. General decoupling strategy for 2 × 2 systems.
Using the modification introduced in the present paper, however, yields
D)
(
1 0
1 60s + 1 -2s
-e
)
(10)
and a resulting diagonal matrix T, i.e., decoupling for all frequencies, is achieved. Does this approach work for all cases? When eq 2 is examined for the general decoupling matrix, it is found that this modification can be considered a special case of choosing T11 ) T22 ) G11G22 - G12G21, i.e.,
D)
(
G22 -G12 -G21 G11
)
(11)
which is realizable as long as the process is. This strategy for decoupling of 2 × 2 systems is also shown
Different possibilities for simplified decoupling that makes decoupling possible for all frequencies have been discussed. With the use of an example from a paper in the open literature, it is shown that results of general applicability from the area of distillation control have not received proper recognition in process control at large. A modification of the four “established” approaches to “simplified” decoupling is also introduced. This modification is useful for cases when the so far known “simplified” decoupling schemes cannot directly be applied because of realizability issues. Literature Cited (1) Luyben, W. L. Distillation Decoupling. AIChE J. 1970, 16, 198. (2) Waller, K. V. T. Decoupling in Distillation. AIChE J. 1974, 20, 592. (3) Johansson, L.; Koivo, H. N. Inverse Nyquist Array Technique in the Design of a Multivariable Controller for a Solid Fuel Boiler. Int. J. Control 1984, 40, 1077.
Received for review November 13, 2002 Revised manuscript received March 25, 2003 Accepted March 31, 2003 IE020911C
