Simplified, ideal or inverted decoupling?
Introduction
The choice of a decoupling method is a relatively complex task since all techniques have their advantages and limitations. Simplified decoupling is by far the most popular method. Its main advantage is the simplicity of its elements. Ideal decoupling, which is rarely used in practice, greatly facilitates the tuning of the controller transfer matrix. Inverted decoupling, which is also rarely implemented, presents at the same time the main advantage of both the simplified and ideal decoupling methods.
Some authors have already compared simplified, ideal and inverted decoupling. Luyben[1]and Weischedel and McAvoy[2]have compared ideal and simplified decoupling methods using distillation column simulators. They concluded that simplified decoupling is more robust than ideal decoupling. According to Waller[3], stability problems encountered by Luyben[1]with ideal decoupling are explained by the fact that he used the same controller tuning for both decoupling methods. Weischedel and McAvoy[2]also kept the same controller tuning for both decoupling techniques, therefore leading to the same conclusion about robustness. Following these studies, McAvoy[4]concluded that ideal decoupling is very sensitive to modeling errors.
To evaluate control systems robustness, Arkun et al.[5]proposed a general analysis procedure based on the singular values. To illustrate their methodology, they studied decoupling control systems applied to several distillation columns. The distillation columns used came from the literature2, 6, 7. They compared ideal decoupling and simplified decoupling. However, to be able to carry out direct analysis and comparisons with results already presented, they used the same decouplers and controllers parameters as found in the literature. As in the preceding studies, they also concluded that ideal decoupling can be less robust than simplified decoupling.
In his book, Shinskey[8]detailed both simplified and inverted decoupling structures. He explains why the initialization problem, which consists in finding the right controller outputs values to allow bumpless switches between the manual and automatic modes, is easier to solve with inverted decoupling technique. He also describes why it is much more easier to take into account saturation of manipulated variables when using inverted decoupling. Furthermore, it can be added that ideal decoupling presents the same deficiencies as simplified decoupling when analyzing initialization and saturation.
Simplified and inverted decoupling methods are also described by Seborg et al.[9]. Referring respectively to Shinskey[8]and to Luyben[1], Waller[3]and Weischedel and McAvoy[2], they concluded that inverted decoupling is appropriate to take into account the saturation of manipulated variables, however it is more sensitive to modeling errors.
Recently, Wade[10]discussed implementation issues for the inverted decoupling method. In most commercial distributed control systems (DCS), the PID function block has an auxiliary input called “feedforward input”. The feedforward input is summed, within the PID function block, with the output of the PID algorithm. Consequently, such a PID function block is appropriate for direct implementation of inverted decouplers, allowing, without any programming, correct initialization, which facilitates bumpless switches between manual and automatic modes.
The purpose of this paper is to show that the three decoupling methods present the same robust stability and robust performance when the controllers are tuned to obtain equal closed loop nominal performance. An example, which uses the structured singular value, depicts this important point. However, the three decoupling methods may not have the same nominal stability. In fact, a relation is established between the presence of right-half plane (RHP) zeros of a process in series with its simplified decoupler and the nominal instability of the ideal and inverted decouplers for the same process. A potential implementation problem with inverted decoupling, which can deteriorate performance, is also explained. It is shown, with an example, that this problem can even destabilize the control system. Finally, this paper summarizes in a simple table the main advantages and limitations of each decoupling method.
Section snippets
Decoupling methods
Decoupling at the input of a two input–two output (TITO) process P(s) requires the design of a transfer matrix D(s), such that P(s)D(s) is a diagonal transfer matrix T(s):
andFig. 1 shows a decoupling control system for a TITO process. The variables r1 and r2 are the set points, c1 and c2 are the controller outputs, u1 and u2 are the manipulated variables and y1 and y2 are the process outputs. The
Robustness and stability
For equivalent nominal performance, the robust performance and robust stability of nominally stable control systems with simplified, ideal and inverted decoupling are identical. In fact, equivalent nominal performance implies that the closed loop transfer matrices are identical. The process being the same in each case, the controllers transfer matrices in series with the decoupling transfer matrices are also identical for all three decoupling methods. Therefore, they all present the same
Implementation
When a system is nominally stable with ideal or inverted decoupler, it is as robust as a simplified decoupled system. Since the robustness study deals with process variations and not with controller or decoupler variations, it is independent of the decoupler proximity to instability. In practice, however, it is recommended to make sure that some variations to the controller and decoupler parameters do not lead to instability[14]. In fact, slight differences between calculated and implemented
Summary table
Table 1 summarizes the advantages and disadvantages of each decoupling method. The robustness is not discussed since it is identical for all three decoupling techniques.
Conclusion
This paper gives some guidelines for the selection of a decoupler. It is shown that robust performance and robust stability of nominally stable control systems, using simplified, ideal and inverted techniques, are the same if the controllers are tuned to obtain the same nominal performance. Therefore, the selection of one of the three methods must not be based on robustness considerations. A relation has also been established between the presence of RHP zeros of a process in series with its
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