A New Pairing Method for Multivariable Processes - Industrial

Ind. Eng. Chem. Res. 2010, 49, 6115–6124

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A New Pairing Method for Multivariable Processes Yongfang Cheng and Shaoyuan Li* Department of Automation, Shanghai Jiao Tong UniVersity, 800 Dong Chuan Road, Shanghai 200240, China

As far as the pairing of multivariable processes is concerned, with the existing method based on the relative gain array (RGA) and Niederlinski index (NI), the interactions among loops are analyzed by the open-loop gains change when all the other loops change from open states to closed states. In this method, the internal interactions among the other loops are hidden, that is, several groups of other loops with different internal interactions can result in the same effect toward the loop considered. Therefore, according to the existing method, there may be more than one feasible way of pairing. To analyze the interactions under these feasible pairings further and get one pairing way with relatively weak interactions, in this paper, a new method of pairing is proposed based on the analysis of interactions between any two loops. By means of relative gain table and relative gain graph, the interactions between two random loops of a n × n system can be analyzed vividly when the number of closed loops within the (n - 1)-dimensional subsystem decreases gradually. On the definition of a new criterion to weigh the mutual interaction between loops, a new pairing method is brought forward. Finally, three examples are used to validate the effectiveness of the proposed method. 1. Introduction Most large-scale complex industrial processes are multi-input and multi-output (MIMO) systems. Compared with single-input and single-output (SISO) systems, MIMO systems are more difficult to control due to the interactions between inputs and outputs.1 MIMO systems with interactions can be controlled by two kinds of controllers, that is, a multivariable controller and a set of decentralized controllers.2 Decentralized control means decomposing the whole system into many low dimensional interactive subsystems, that is, the controller of the original system can be written in a block-diagonal form. Because of fewer parameters to adjust and ease of use, decentralized controllers are applied extensively in industrial processes control. However, given the limited controller structure for decentralized controllers, the interactions among loops can deteriorate the closed-loop performances, because of the nonzero off-diagonal elements in the transfer function matrix. Therefore, the first task for the design of a decentralized control system is to select the structures of loops, that is, to pair manipulated variables and controlled variables effectively, decreasing the interactions among loops to the least. The relative gain array (RGA)3 proposed by Bristol is extensively used for the pairing of manipulated variables and controlled variables. In RGA, the interactions to a certain loop from all the other loops are measured by the ratio of the static open-loop gain when the other loops are totally open to that when the other loops are all closed. RGA embraces advantages as follows. It is dependent on process models only, independent of the scaling of inputs or outputs, and can include all ways of pairing in a single matrix. With a consideration of the stability of the decentralized control systems, the Niederlinski index (NI)4 is used in conjunction with the RGA-based pairing rules.5–9 With the increase of the systems’ dimension, denoted by n, there may be more than one feasible way of pairing, only judged by the pairing rules based on RGA and NI, which makes it necessary to use other criteria to further analyze the interactions among loops in order to select a way with relatively weak interactions. Fatehi and Shariati12 proposed the definition of * To whom correspondence should be addressed. E-mail: syli@ sjtu.edu.cn.

normalized RGA (NRGA) and an automatic pairing method based on NRGA and the Hungarian algorithm. The NRGA projects the RGA elements onto the interval [0, 1], and when its element is closer to 1.0, the interactions among loops are weaker. For systems with high dimensions, the NRGA elements can be ordered by size by the Hungarian algorithm. However, in nature, NRGA merely makes some numerical transformation to RGA, considering the interactions among loops from the same perspective with RGA. Therefore, it cannot differentiate several ways of paring with similar RGA values. From a viewpoint contrary to that of RGA, He and Cai13 introduced a new method of pairing based on the decomposed relative interaction array (DRIA) and generalized interaction array (GIA), which considered the interaction from a certain loop to the other n - 1 closed loops. Although DRIA analyzed interactions between two random loops, it still hid the internal interactions among these n - 1 loops, due to its precondition that all these n - 1 loops were closed. To analyze the interactions among loops in multivariable processes more comprehensively, in this paper, interactions are considered in a loop-to-loop view based on the point of He and Cai.13 Considering a certain loop with all the loops comprising a (n - 1)-dimensional subsystem, the interactions between this loop and any other loop is measured by the changes of static open-loop gain of this loop as the number of closed loops in the subsystems is changed. As is known, the interaction among the subsystem is increasingly complex with the increase of the number of closed loops. If the changes of the open-loop gain of the loop concerned is small when the number of closed loops among the subsystem changes, then the interactions among loops are weak. Thus, in this paper, a new index is introduced denoting the interactions between loops with a different number of closed loops in the subsystem, and a new method of variable pairing is proposed based on this new index. 2. Problem Descriptions 2.1. The Basic Problem of Variable Pairing. Throughout this paper, it is assumed that the system to be dealt with is square (n × n), open-loop stable, and has a multivariable feedback control structure as shown in Figure 1. The reference inputs

10.1021/ie901787u  2010 American Chemical Society Published on Web 06/07/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 13, 2010

Figure 1. Multivariable control system.

Figure 3. Interaction from loop 1 to loop 2.

Figure 2. Interaction from loop 2 to loop 1.

vector, manipulated variables vector, and output variables vector are represented by r ){ri}, u ){ui}, and y ){yi}, respectively, with i ) 1, 2,..., n. G(s) ) {gij}n×n and G(0) denote the transfer function matrix and the static-state gain matrix of the process. The controller is denoted by Gc(s), including all kinds of structures such as a multivariable controller, decentralized controller, and sparse controller. For a general multivariable system, the transfer function G(s) cannot be transformed into a diagonal or block-diagonal form through row and column interchanges, that is, reordering the manipulated variables and controlled variables. Because of the nonzero off-diagonal elements of G(s), every manipulated variable is affected by more than one controlled variable, and every controlled variable affects more than one manipulated variable. In order that the design of controller Gc(s) is as simple as possible, variable pairing is done before the controller design, that is, pairing a certain manipulated variable with the controlled variable which affects it most to form a loop, and the interaction among loops being as small as possible. Therefore, how to pair the variables properly has become a problem concerned by many scholars. In this paper, the problem is considered from the interactions between any two loops in the system. 2.2. Analysis of Interactions between Loops for 2 × 2 Systems. Consider the impact from loop 2 to loop 1, as shown in Figure 2. When loop 2 changes from being open to being closed, the static open-loop gain of loop 1 changes from g11 to g11 - g12g22-1g21; Consider the impact from loop 1 to loop 2, as shown in Figure 3. When loop 1 changes from being open to being closed, the static open-loop gain of loop 1 changes from g22 to g22 g21g11-1g12. According to the definition of RGA,3,10 the elements in RGA describe the gain changes analyzed above properly, that is, RGA can be used to signify the interactions between two loops in 2 × 2 systems. The more closer to 1.0 the RGA elements are, the weaker are the interactions that the system embraces.

2.3. Analysis of Interactions between Loops for n × n (n > 2) Systems. For n × n (n > 2) systems, each loop is affected by the other n - 1 loops; take a 3 × 3 system as the example. Assume the transfer function matrix and the static open-loop gain matrix of a 3 × 3 process are denoted by G(s) ) {gij(s)}3×3 and G(0) ) {gij}3×3, respectively. Under different ways of pairing, through row and column interchanges of the transfer function, it can always be realizable to pair the variables diagonally. Therefore, without the loss of generality, assume the original diagonal variables are paired, that is, 1/1, 2/2, 3/3, where “/” denotes controlled variable/ manipulated variable; for example, 1/1 means y1 and u1 are paired and constitute a loop. Considering loop 1/1, analyze the interactions from loop 2 and loop 3 to loop 1, loop 2 and loop 3 constitute a 2 × 2 subsystem, and assume G(s) )

(

g11 G12 G21 G22

)

with G12 ) (g12 g13 ),

G21 )

( )

g21 , g31

G22 )

(

g22 g23 g32 g33

)

When loop 2/2 is kept open, loop 3/3 changes from being open to being closed, the static open-loop gain of loop 1 changes from g11 to g11 - g13g33-1g31, which results from the effect from loop 3/3. When loop 3/3 is kept closed, loop 2/2 changes from being open to being closed, the static open-loop gain of loop 1 changes from g11 - g13g33-1g31 to g11 - G12G22-1G21, which results from the effect of loop 2/2. When loop 3/3 is kept open, loop 2/2 changes from being open to being closed, the static open-loop gain of loop 1 changes from g11to g11 - g12g22-1g21, which results from the effect of loop 2/2. When loop 2/2 is kept closed, loop 3/3 changes from being open to being closed, the static open-loop gain of loop 1 changes from g11 - g12g22-1g21 to g11 - g12G22-1G21, which results from the effect from loop 3/3. The analysis above can be shown as Figure 4, where the ovals denote the states of loop 2/2 and loop 3/3, Qi ){O ()Open), C ()Closed)}. In Figure 4, the solid arrows present the processes of gain change analyzed above. The more gain changes, the larger the interaction is on loop 1 from the loop which experiences state change. The figure shows the effect of each loop upon loop 1/1 individually. However, as the definition of RGA shows, RGA elements only reflect the process shown by the dashed arrow in Figure 4, that is, the synthetic effect from both loop 2 and loop 3. The relation


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Figure 4. Interaction from loop 2/2 and loop 3/3 to loop 1/1. The ovals denote the states of loop 2/2 and loop 3/3, Qi ) {O () Open), C () Closed)}.

between the process shown by the solid arrows and the process shown by the dashed arrows can be expressed by equation (1) as follows: λ11 ) ) )

g11 g11 - g13g33-1g31 g11 g11 - g12g22-1g21 g11

× ×

Table 1. Relative Gain Table for Loop 1 of a 3 × 3 System

g11 - g13g33-1g31 g11 - G12G22-1G21 g11 - g12g22-1g21 g11 - G12G22-1G21

g11 - G12G22-1G21

(1)

In eq (1), the multiplier factors flanking “×” mean the effects of different loops upon loop 1/1. Because each factor equals the ratio of the static open-loop gains of loop 1/1 before and after the state changes of other loops, when the ratio is farther away from 1.0, the interaction from the loop with state change upon loop 1/1 is stronger. However, different groups of factors can lead to the same RGA element λ11, therefore, for a n × n(n > 2) process, the RGA element of a certain loop signifies the synthetic effects from all the other n - 1 loops. On the basis of the analysis above, compared with 2 × 2 systems, in n × n (n > 2) systems, RGA cannot embody interactions between any loop within the (n - 1)-dimensional subsystem and the loop concerned.11 As for this problem, a new index is defined in this paper to measure the mutual interactions between any two loops. This index can be applied to choose a way of pairing with relatively weak interactions when there exists more than one way of pairing that satisfies the pairing rules based on RGA and NI. 3. An Index to Measure the Interaction between Any Two Loops In this part, a new index is introduced to analyze interactions among loops in multivariable processes. This new index is obtained from relative gain table and relative gain graph of each loop defined in this paper. 3.1. Relative Gain Table. For n × n systems, RGA defines the synthetic impact of all the other n - 1 loops on the concerned loop. This paper attempts to define a new index to measure the interactions among two random loops. To achieve it, a relative gain table is designed for each loop. For an n × n system, there are 2n-1 states for the (n - 1)-dimensional subsystem which affects a certain loop l, therefore, the dimension of the relative gain graph is (2n-1 + 1). In this table, the elements of column 0, Ri0(i ) 1,2,...,2n-1), and the elements of row 0, R0j(j ) 1,2,...,2n-1), denote the states of the (n - 1)-

dimensional subsystem. Element Rij(i,j ) 1,2,...,2n-1) denotes the change of open-loop gain of loop l when the subsystem changes from state i to state j, which is defined by the ratio of static open-loop gains of loop l under two states, referring to the definition of RGA. Take a 3 × 3 system as the example, without loss of generalization. Consider the impact from loop 2 and loop 3 to loop 1, with the relative gain table of loop 1 being shown as Table 1. The elements in row 0 and column 0 present states of loop 2 and loop 3, with an open loop being denoted by “O”, and a closed one being denoted by “C”. The elements, Rij(i,j ) 1,2,...,2n-1), in relative gain table of loop l can be calculated step by step as follows: (1) Ri0 ) R0i,i ) 1, 2,...,2n-1; (2) Rii ) 1(i ) 1,2,...,2n-1), since the open-loop gain of loop l keeps the same when the subsystem maintains the same state; (3) R1j(j ) 2,...,2n-1). This means the change of open-loop gain of loop l, when the subsystem changes from the state where all its loops are open to state j. Because only those closed loops in the subsystem can impose interactions on loop l, then R1j can be obtained through the calculation of RGA for a new matrix consisting of the elements in transfer function G(s) relevant to the controlled variables and manipulated variables corresponding to loop l and the closed loops in the subsystem; (4) Ri1 ) 1/R1i(i ) 2,...,2n-1), since the transition of the states of the subsystem denoted by R1i(i ) 2,...,2n-1) is just inverse to that by Ri1(i ) 2,...,2n-1); (5) Rij ) Ri1 × R1j(i,j ) 2,...,2n-1), because Rij is defined by a ratio. It is apparent, from the steps above, that the key to filling in the whole table is to calculate the 2n-1 - 1 elements in row 1 through the calculation of RGA of different matrices. 3.2. Relative Gain Graph. For a certain loop l of a n × n system, the (n - 1)-dimensional subsystem experiences n - 1 state transitions when all its n - 1 loops turn from open state to closed state sequentially. The mth transition means that one loop of the subsystem changes from open state to closed state under the premise that m - 1 loops have been closed. During this transition, the loop undergoing state transition affects the open-loop gain of loop l. This process can be vividly presented

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Figure 5. Relative gain graphs of loop 1 of n × n (n ) 2,3,4) systems.

by a relative gain graph, as shown in Figure 5. The definition of the relative gain graph is described as follows: Definition 1 (Relative Gain Graph). The relative gain graph of loop l is defined as a graph which can reflect the impact on loop l from each loop undertaking the state transition within the (n - 1)-dimensional subsystem, during which only the state of a loop changes. The elements consisting of the relative gain graph are defined as follow: Definition 2 (Node). A node in the relative gain graph is defined as a state of the subsystem. Definition 3 (Layer). Layer m of the relative gain graph is defined as a layer containing all the nodes in which there are m closed loops within the subsystem. Therefore, there are n layers m in a relative gain graph totally, with Cn-1 nodes on layer m (m ) {0,1,2,...,n - 1}). Definition 4 (Directed Arc). A directed arc is defined as an arrow between nodes of adjacent layers with a number on it, denoting the transition of the subsystem from the state signified by the node at the beginning of the arc to the state by the node at the end of the arc, and the number on the arc means the only loop experiencing state change during this transition. There is no directed arc between nodes on nonadjacent layers, signifying that only one loop of the subsystem changes states in a directed arc. Definition 5 (Path). A path from node u to node V is defined as a chain, V0,e1,V1,e2,V2,...,ek,Vk, with the directed arc ei beginning at Vi-1 and ending at Vi, satisfying V0 ) u and Vk ) V. When the subsystem transfers from the state denoted by u to that by V, the change of open-loop gain of loop l is equal to the multiplication of all the changes experienced by loop l when the subsystem undertakes transitions denoted by e1,e2,...,ek. There are two loops at least having state transitions simultaneously on a path between the nodes on nonadjacent layers. These loops constitute a new subsystem; therefore, the analysis of gain changes of loop l on the path can be used to analyze the interactions between loops within this new subsystem. From the structure of the relative gain graph, it can be seen that with layers going down, more loops are closed and interactions among loops are more complex. RGA takes into account the change from the peak layer to the valley layer without considering the changes on each directed arc of each path. When the RGA elements are close to each other under several ways of pairing, it becomes necessary to find a new index to analyze the interactions among loops. Obviously, it is beneficial to the interaction analysis to consider the gain changes on each directed arc of a path. If these gain changes are close

to each other, then the loops whose states are changed on directed arcs of the path exert weak interaction on the concerned loop. In a relative gain graph, as the layer goes down, more loops are closed, and the state of the subsystem is more approximate to the original system with all loops being closed, therefore, the interaction analysis in the lower layers should be given higher priority, that is, if the interaction analysis on layer m is close, then go on to analyze that on layer m - 1. 3.3. An Index to Measure the Interaction from a Loop to the Other. From the structure of the relative gain table of loop l, Rij(i,j ) 1,2,...,2n-1) means the ratio of static open-loop gain of loop l under state i and state j of the (n - 1)-dimensional subsystem. If Rij satisfies Rij < 0, the sign of the static gain changes, which may lead to positive feedback fatal to the stability of the whole system, so the pairing with negative Rij should be avoided. If Rij satisfies 0 < Rij < 1, the gain is enlarged to 1/Rij times of that before the subsystem’s state change. If Rij satisfies Rij > 1, the gain is reduced to one Rijth of that before the subsystem’s state change. Therefore, define Rij as the amplitude of gain change and set -

Rij ) -

{

max{Rij, 1/Rij}, Rij > 0 +∞, Rij < 0

(2)

A larger Rij means a larger change of the gain, indicating a stronger interaction between loops. Accordingly, here given are some definitions describing interactions among loops as the number of closed loops within various (n - 1)-dimensional systems of a n-dimensional system increasing from m - 1 to m. Definition 6 (Interaction from Loop k to Loop l). In the relative gain graph of loop l, find all the directed arcs with k on them between layer m - 1 and layer m, and then find those relative gains Rij in the relative gain table of loop l, which denote gain changes when the subsystem endures the state transitions signified by these selected arcs. Calculate Rij corresponding to those Rij above in accordance with expression (2), and the sum of all these Rij are defined as the interaction from loop k to (m) loop l, denoted by I(m) kl . A larger Ikl indicates stronger interaction from loop k to loop l. In the relative gain graphs of loops in the same system, states denoted by nodes on the same layer have the same number of closed loops. More closed loops indicate more complex interactions among loops. All the directed arcs between nodes on two adjacent layers embody interactions of the same complexity. Therefore, the sum of all the amplitudes of gain changes Rij


presented on all the directed arcs between layer m - 1 and layer m, that is, the sum of interactions between loops I(m) kl , can be used to measure the interactions among loops of the whole system with the same complexity of the subsystems. Thus, we define the mutual interactions between loops of a n × n system as follows: Definition 7 (Mutual Interactions between Loops of a System). When the number of closed loops within the subsystem of a n × n system increases from m - 1 to m, the mutual interactions between loops of a system are defined as n

I(m) )

n

∑ ∑I

(m) kl

(3)

l)1 k)1 k*l

(m+1)

Under different ways of pairing, if they have similar I , interactions among loops are stronger under the way of pairing with the larger I(m). 4. A New Pairing Method Based on the Mutual Interactions between Loops I(m) For a n × n system, in this paper, a new pairing method is proposed as follows, in accordance with the analysis of the mutual interactions between any two loops. Step 1 For the transfer function of a given n × n process, set s ) 0 to get the static open-loop gain matrix G(0). Step 2 Calculate the RGA and NI. Step 3 Eliminate the pairing ways with negative RGA elements and negative NI. If there is only one way of pairing left, then stop; otherwise, turn to Step 4. Step 4 For those ways of pairing left, design the relative gain table and relative gain graph for each loop of each way of pairing, respectively, set m :) n - 1. Step 5 From the relative gain table of each loop, find all the corresponding relative gain Rij when the subsystem changes from states denoted by the nodes on layer m - 1 to states denoted by the nodes on layer m; calculate the corresponding gain-changing amplitude Rij- and then calculate the mutual interactions I (m) kl between loops. Step 6 Calculate mutual interactions between loops I(m) of the system under all ways of pairing in Step 4, and make a comparison to choose the ways of pairing with the smaller I(m). Step 7 If there is more than one way of pairing after Step 6, set m :) m - 1; turn to Step 5; otherwise, stop. From the steps above, it can be seen that when the system is a 2 × 2 system, the pairing method is the same with the existing method based on RGA and NI, and the advantage of this method is that it can be applied to systems with dimensions larger than 2. 5. Case Studies In this part, three examples are considered to substantiate the effectiveness of the pairing method proposed in this paper. The major procedures of validation are as follows. First, single out all the ways of pairing satisfying the pairing rules based on RGA and NI. Second, under each way of pairing selected through the first step, neglecting the elements in the transfer function matrix which result in interactions among loops, design decentralized controllers for each loop. Third, apply the decentralized controllers without any adjustment of parameters to the original process with interactions and check the stability

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of the closed multivariable system under each feasible way of pairing by the stability conditions proposed in ref 7 for multivariable systems. Finally, compare the output responses of systems with interactions and those of systems without interactions. The more the former diverges from the latter under a certain way of pairing, the stronger are the interactions among loops under this way of pairing. Case 1. Consider a 3 × 3 process (based on the example in ref 15) given by

[

]

0.66e-2.6s -0.61e-3.5s -0.0049e-s 6.7s + 1 8.64s + 1 9.06s + 1 -6.5s -3s -1.2s -2.36e -0.01e 1.11e G(s) ) 3.25s + 1 5s + 1 7.09s + 1 0.87(11.61s + 1)e-s 34.68e-9.2s 46.2e-9.4s 8.15s + 1 10.9s + 1 (3.89s + 1)(18.8s + 1) (4) Pairing Analysis. Calculate RGA and obtain 1.1644 -0.8877 0.7233 -0.5727 Λ ) -0.3745 1.9472 0.2101 -0.0595 0.8494

[

]

The following two ways of pairing satisfy both RGA > 0 and NI > 0: Pairing 1: 1/1, 2/2, 3/3 (RGA: 1.1644, 1.9472, 0.8494. NI: 0.6655 > 0) Pairing 2: 1/3, 2/2, 3/1 (RGA: 0.7233, 1.9472, 0.2101. NI: 2.2489 > 0) According to the pairing method proposed in this paper, make a relative gain table and a relative gain graph for each loop under each way of pairing, respectively. Since the dimension of the process is 3, all the other loops exerting interactions on a certain loop constitute a 2-dimensional subsystem. Calculate the mutual interactions between loops I (2) embodied by the changes of the subsystem from the states denoted by nodes on layer 2 to the states denoted by nodes on layer 3. Under pairing 1, I(2) ) 8.2581, and under pairing 2, I(2) ) 14.3646, the latter is bigger than the former, so the interactions among loops under pairing 2 are stronger than those under pairing 1. Validation of Results. Under the two selected ways of pairing, design decentralized controllers for the original process as follows: First, through row and column interchanges, the transfer function G(s) is transferred into G′(s) whose variables are paired diagonally. Set the off-diagonal elements of G′(s) to zero and obtain a diagonal transfer function matrix G′d(s). Second, by means of the controller designing method proposed in ref 14, design ideal controller Gci(s) for each diagonal element, respectively, with requirements of φm ) 3π/8 and Am ) 4. The parameters of the controllers are as shown in Table 2. Third, apply the decentralized controller Gc(s) to G′(s), draw the Nyquist curve of det(I + E(s)Hd(s)) as shown in Figure 6, with E(s) ) (G′(s) - g′d(s))G′d(s)-1, and Hd(s) ) (I + G′d(s)Gc(s))-1G′d(s)Gc(s)). From the curve, both curves have no encirclement of the original point, so both of the two decentralized controllers can stabilize the original system, judged by the stability conditions proposed in ref 7. Finally, make a comparison between the outputs responses under two ways of pairing, as shown in Figure 7. Obviously, under pairing 2, the dashed line diverges from the solid line more than that under pairing 1. Therefore, the

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Table 2. Parameters of Decentralized Controllers under Two Ways of Pairing (Case 1) {Gd′ (s)}ii pairing 1

(

0.66e 6.7s + 1

0.4577 × 6.7 +

-2.36e-3s 5s + 1

-0.05547 × 5 +

(

0.87(11.61s + 1)e-s (3.89s + 1)(18.8s + 1) pairing 2

0.4514 ×

[

0.1510 ×

e-2.6s s

1 s

0.1309 ×

e-3s s

0.3927 ×

e-s s

0.3927 ×

e-s s

0.1309 ×

e-3s s

0.04268 ×

e-9.2s s

)

(3.89s + 1)(18.8s + 1) (11.61s + 1)

-80.1427 × 9.06 +

1 s

-2.36e-3s 5s + 1

-0.05547 × 5 +

1 s

)

34.68e-9.2s 8.15s + 1

0.001231 × 8.15 +

(

(

(

1.5e-s s+1 e-s s+1 -2e-s 10s + 1

e-s s+1 -2e-s 10s + 1 1.5e-s s+1

Pairing Analysis. Calculate RGA and obtain -0.9302 1.1860 0.7442 0.7442 -0.9302 Λ ) 1.1860 0.7442 -0.9302 1.1860

[

)

1 s

-0.0049e-s 9.06s + 1

interactions among loops under pairing 2 are stronger, substantiating the effectiveness of the method of pairing proposed in this paper. Case 2. Consider a 3 × 3 process16 given by -2e-s 10s + 1 -s G(s) ) 1.5e s+1 e-s s+1

{Gd′ (s)Gc(s)}ii

{Gc(s)}ii

-2.6s

]

]

(5)

The following two ways of pairing satisfy both RGA > 0 and NI > 0: Pairing 1: 1/2, 2/1, 3/3 (RGA: 1.1860, 1.1860, 1.1860. NI ) 1.5926 > 0) Pairing 2: 1/3, 2/2, 3/1 (RGA: 0.7442, 0.7442, 0.7422. NI ) 5.3750 > 0).

1 s

)

)

According to the pairing method proposed in this paper, we make a relative gain table and a relative gain graph for each loop under each way of pairing, respectively. Since the dimension of the process is 3, all the other loops exerting interactions on a certain loop constitute a 2-dimensional subsystem. Calculate the mutual interactions between loops I(2) embodied by the changes of the subsystem from the states denoted by nodes on layer 2 to the states denoted by nodes on layer 3. Under pairing 1, I(2) ) 8.9612, and under pairing 2, I(2) ) 11.8752; the latter is bigger than the former, so the interactions among loops under pairing 2 are stronger than those under pairing 1. Validation of Results. Under the two selected ways of pairing, design decentralized controllers for the original process as follows: First, through row and column interchanges, the transfer function G(s) is transferred into G′(s) whose variables are paired diagonally. Set the off-diagonal elements of G′(s) to zero and obtain a diagonal transfer function matrix G′d(s). Second, by means of the controller designing method proposed in ref 14, design ideal controller Gci(s) for each diagonal element, respectively, with requirements of φm ) 3π/8 and Am ) 4. The parameters of the controllers are as shown in Table 3. Third, apply the decentralized controller Gc(s) to G′(s), and draw the Nyquist curve of det(I + E(s)Hd(s)) as shown in

Figure 6. Nyquist curves of the original system with decentralized controllers under two ways of pairing.


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Figure 7. Outputs responses in case 1: (A) Output responses y1 with reference inputs r1 ) 1(t), r2 ) r3 ) 0. (B) Output responses y2 with reference inputs r2 ) 1(t), r1 ) r3 ) 0. (C) Output responses y3 with reference inputs r3 ) 1(t), r1 ) r2 ) 0. Table 3. Parameters of Decentralized Controllers under Two Ways of Pairing (Case 2) {Gd′ (s)}ii -s

{Gd′ (s)Gc(s)}ii

{Gc(s)}ii

)

0.3927 ×

e-s s

1 s

)

0.3927 ×

e-s s

pairing 1

1.5e s+1

1 0.2618 × 1 + s

pairing 2

e-s s+1

0.3927 × 1 +

( (

Figure 8, with E(s) ) (G′(s) - g′d(s))G′d(s)-1, and Hd(s) ) (I + G′d(s)Gc(s))-1G′d(s)Gc(s)). From Figure 8, the curve has no encirclement of the original point under pairing 1, while the curve encircles the original point clockwise twice; therefore, both of the two decentralized controllers can stabilize the original system. Judging by the stability conditions proposed in ref 7, the decentralized controller can stabilize the original system under pairing 1, while the system is unstable under the decentralized controller designed above. Finally, make a comparison between the outputs responses under two ways of pairing, as shown in Figure 9. Since the transfer function matrix is symmetric, under either way of pairing, three diagonal elements in G′(s) are the same and so

are the interactions from the other two loops. Therefore, the three outputs respond in the same way to the reference inputs; the input corresponding to the output paired to it is a unit step signal while the other two are zero. In Figure 9, yi is used to represent any of the outputs. From Figure 9, obviously, the outputs converge to the set point 1 finally under pairing 1, while the outputs diverge under pairing 2, indicating the decentralized controller designed above cannot stabilize the original system with interactions. Therefore, the interactions under pairing 2 is relatively strong, validating the judgment by the method of pairing proposed in this paper. Case 3. Consider a 3 × 3 process11 given by

(

1 -4.19 -25.96 1-s 6.19 1 -25.96 (1 + 5s)2 1 1 1 Pairing Analysis. G(s) )

Calculate RGA and obtain 1.0009 Λ ) -5.0028 5.0019 The following two ways and NI > 0:

(

5.0010 1.0009 -5.0019 of pairing

)

)

(6)

-5.0019 5.0019 1.0000 satisfy both RGA > 0

6122



Figure 9. Outputs responses in case 2 with reference inputs r1 ) 1(t), r2 ) r3 ) 0.

Pairing 1: 1/1, 2/2, 3/3 (RGA: 1.1860, 1.1860, 1.1860. NI ) 26.9361 > 0) Pairing 2: 1/2, 2/3, 3/1 (RGA: 0.7442, 0.7442, 0.7422. NI ) 0.2476 > 0). According to the pairing method proposed in this paper, make a relative gain table and a relative gain graph for each loop under each way of pairing, respectively. Since the dimension of the process is 3, all the other loops exerting interactions on a certain loop constitute a 2-dimensional subsystem. Calculate the mutual interactions between loops I(2) embodied by the changes of the subsystem from the states denoted by nodes on layer 2 to the states denoted by nodes on layer 3. Under pairing 1, I(2) ) 329.5236, and under pairing 2, I(2) ) 60.6047, the former is bigger than the latter, so the interactions among loops under pairing 1 are stronger than those under pairing 2. Validation of Results. Under the two selected ways of pairing, design decentralized controllers for the original process as follows: First, through row and column interchanges, the transfer function G(s) is transferred into G′(s) whose variables are paired diagonally. Set the off-diagonal elements of G′(s) to zero and obtain a diagonal transfer function matrix G′d(s). Second, by means of the Ziegler-Nichols method in ref 17, design a SISO controller for each diagonal element of G′d(s) independently. The parameters of the controllers are shown in Table 4. Third, the set of SISO controllers comprise the diagonal elements of the decentralized controller Gc(s). Apply Gc(s) to G′(s) and draw the Nyquist curve of det(I + E(s)Hd(s)) as shown in Figure 10, with E(s) ) (G′(s) - g′d(s))G′d(s)-1, and Hd(s) ) (I + G′d(s)Gc(s))-1G′d(s)Gc(s)). From Figure 10, both curves have encirclements of the original point under pairing 1 and pairing

2. Judging by the stability conditions proposed in ref 7, the decentralized controllers cannot stabilize the original system under either way of pairing. Finally, introduce a factor F to adjust the parameters of controllers in the way similar to the biggest log-modulus tuning (BLT), that is, controller kp(1 + 1/τis) is adjusted to kp/F(1 + 1/Fτis). Under pairing 1, the multivariable system cannot be stabilized until F approaches 100 while under pairing 2, the Table 4. Parameters of Decentralized Controllers under Two Ways of Pairing (Case 3) {Gd′ (s)}ii pairing 1

pairing 2

{Gc(s)}ii

1-s (1 + 5s)2

3.7877 × 1 +

1-s (1 + 5s)2

3.7877 × 1 +

1-s (1 + 5s)2

3.7877 × 1 +

-4.19(1 - s) (1 + 5s)2

-0.9040 × 1 +

-25.96(1 - s) (1 + 5s)2

-0.1459 × 1 +

1-s (1 + 5s)2

3.7877 × 1 +

(

1 7.8938s

)

(

1 7.8938s

)

(

1 7.8938s

)

(

(

1 7.8938s

)

(

1 7.8938s

)

1 7.8938s

)


6123


Figure 11. Outputs responses in case 3. F ) 100 under pairing 1, F ) 5 under pairing 2. (A) Output responses y1 with reference inputs r1 ) 1(t), r2 ) 1(t - 4000), r3 ) 1(t - 8000). (B) Output responses y2 with reference inputs r1 ) 1(t), r2 ) 1(t - 4000), r3 ) 1(t - 8000). (C) Output responses y3 with reference inputs r1 ) 1(t), r2 ) 1(t - 4000), r3 ) 1(t - 8000).

6124


multivariable system can be stabilized when F approaches 5. The outputs responses under two ways of pairing are shown in Figure 11. Obviously, under pairing 1, the dashed line diverges from the solid line more than that under pairing 2. Therefore, the interactions among loops under pairing 1 are stronger, substantiating the effectiveness of the method of pairing proposed in this paper. 6. Conclusions In this paper, as for n-dimensional multivariable processes, a new method of pairing is proposed based on the analysis of mutual interactions between loops, improving the method based on RGA in the interaction analysis of higher-dimensional processes. The relative gain graph for a certain loop, from the perspective of the internal states changes of the subsystem consisting of the other n - 1 loops, vividly describes a process obtaining the RGA elements and reflects the role played by each loop during each state change, revealing the interactions between loops hidden by the RGA. The relative gain graph includes the impact on the open-loop gains of the other closed or open loops from a certain loop, which may be beneficial to the decision of controller-designing sequences. However, the interaction analysis between loops in this paper is considered from the viewpoint of static gain under the assumption of perfect controllers whose static gain is infinite. This way of thinking, while enjoying the convenience and simplicity of calculation, also leads to some offset from the accurate interactions. Therefore, further research on the interaction analysis from the dynamic perspective is still needed. Acknowledgment The work was supported by the National Nature Science Foundation of China under Grant 60825302, 60774015, the High Technology Research and Development Program of China (Grant: 2007AA041403), and Sponsored by the Program of Shanghai Subject Chief Scientist, and “Shu Guang” project

supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation. Literature Cited (1) Xiong, Q.; Cai, W.-J.; He, M.-J. A Practical Loop Pairing Criterion for Multivariable Processes. J. Process Control 2005, 15, 741. (2) He, M.-J.; Cai, W.-J.; Ni, W.; Xie, L.-H. RNGA-Based Control System Configuration for Multivariable Processes. J. Process Control 2009, 19, 1036. (3) Bristol, E. H. On a New Measure of Interaction for Multivariable Process Control. IEEE Trans. Autom. Control 1966, 133 CA-11. (4) Niederlinski, A. A Heuristic Approach to the Design of Linear Multivariable Interacting Subsystems. Automatica 1971, 7, 69. (5) McAvoy, T. J. Interaction Analysis; ISA: Research Triangle Park, NC, 1983. (6) Grosdidier, P.; Morari, M. Closed-Loop Properties from SteadyState Gain Information. Ind. Eng. Chem. Fundam. 1985, 24, 221. (7) Grosdidier, P.; Morari, M. Interaction Measures for Systems under Decentralized Control. Automatica 1986, 22, 309. (8) Seborg, D. E.; Edgar, T. T.; Mellichamp, D. A. Process Dynamics and Control; John Wiley and Sons: New York, 1989. (9) Zhu, Z.-X. Stability and Integrity Enforcement by Integrating Variable Pairing and Controller Design. Chem. Eng. Sci. 1998, 53, 1009. (10) Shinskey, F. G. Process Control Systems; McGraw-Hill: New York, 1988. (11) Schmidt, H.; Jacoben, E. W. Selecting Control Configuration for Performance with Independent Design. Comput. Chem. Eng. 2003, 27, 101. (12) Fatehi, A.; Shariati, A. Automatic Pairing of MIMO Plants Using Normalized RGA, 2007 Mediterranean Conference on Control and Automation, July 27-29, 2007, Athens, Greece. (13) He, M.-J.; Cai, W.-J. New Criterion for Control-Loop Configuration of Multivariable Processes. Ind. Eng. Chem. Res. 2004, 43, 7057. (14) Wang, Y.-G.; Cai, W.-J. Advanced Proportional-Integral-Derivative Tuning for Integrating and Unstable Processes with Gain and Phase Margin Specifications. Ind. Eng. Chem. Res. 2002, 41, 2910. (15) Ogunnaike, B. A.; Lemaire, J. P.; Morari, M.; Ray, W. H. Advanced Multivariable Control of a Pilot-Plant Distillation Column. Am. Inst. Chem. Eng. J. 1983, 295, 632. (16) Huang, H.-P.; Ohshima, M.; Hashingmoto, I. Dynamic Interaction and Multiloop Control System Design. J. Process Control 1994, 4, 15. (17) Deshpande, P. B. MultiVariable Process Control; Instrument Society of America: Research Triangle Park, NC, 1989.

ReceiVed for reView November 11, 2009 ReVised manuscript receiVed February 22, 2010 Accepted May 20, 2010 IE901787U

A New Pairing Method for Multivariable Processes - Industrial

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