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A practical loop pairing criterion for multivariable processes
Journal of Process Control 15 (2005) 741–747 www.elsevier.com/locate/jprocont
A practical loop pairing criterion for multivariable processes Qiang Xiong, Wen-Jian Cai *, Mao-Jun He School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore Received 21 September 2004; received in revised form 7 February 2005; accepted 23 March 2005
Abstract Utilizing both steady state gain and bandwidth information of the process open loop transfer function elements, this paper presents a new dynamic loop pairing criterion for decentralized control of multivariable processes. Through defining an effective gain matrix, the loop pairing procedures of popular relative gain array method is directly extended to the new method which can reflect dynamic loop interactions under finite bandwidth control. Compared with existing methods, this method is simple, effective, and easy to be understood and applied by control engineers. Several examples, for which the RGA based loop pairing criterion gives an inaccurate interaction assessment, are employed to demonstrate the effectiveness of the proposed method. 2005 Elsevier Ltd. All rights reserved. Keywords: Multivariable process; Decentralized control; Interaction measures; Loop pairing; Effective gain; Relative gain array; Effective relative gain array
1. Introduction Most of the large and complex industrial processes are naturally multi-input multi-output (MIMO) systems. Compared with single-input single-output (SISO) counterparts, MIMO systems are more difficult to control due to the existence of interactions among input and output variables. Although considerable effort has been dedicated to this problem and many design techniques have been proposed over the decades, control system design and implementation for MIMO processes is still a big challenge for practical control engineers. The interactive multivariable systems can be either controlled by (1) a multivariable or centralized MIMO controller or by (2) a set of SISO decentralized controllers. Algebraic decoupling methods or optimal multivariable control theory are usually applied to obtain centralized MIMO controllers. While centralized multi-
*
Corresponding author. Tel.: +65 6790 6862; fax: +65 6793 3318. E-mail address: [email protected] (W.-J. Cai).
0959-1524/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2005.03.008
variable controllers are complex and lack integrity, the decentralized control system enjoys certain advantages: (1) it requires fewer parameters to tune which are easier to be understood and implemented; and (2) loop failure tolerance of the resulting control system can be assured during the design phase. Therefore, they are more often used in process control applications [1,2]. However, the potential disadvantage of using the limited control structure is the deteriorated closed loop performance caused by interactions among loops as a result of the existence of non-zero off-diagonal elements in the transfer function matrix [3,4]. Thus, the primary task in the design of decentralized control systems is to determine loop configuration, i.e. pair the manipulated variables and controlled variables to achieve the minimum interactions among loops so that the resulting multivariable control system mostly resembles its single-input singleoutput counterparts and the subsequent controller tuning is largely facilitated by SISO design techniques [5]. Since the pioneering work of Bristol [6], the relative gain array (RGA) based techniques for control loop configuration have been widely used in industries,
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Q. Xiong et al. / Journal of Process Control 15 (2005) 741–747
including blending, energy conservation, and distillation columns, etc. [7–9]. The most important advantage of RGA based techniques is its simple calculation since only process steady state gains are involved and scaling is independent due to its ratio nature [10]. However, using steady state gain alone may result in incorrect interaction measures and consequently loop pairing decisions, since no dynamic information of the process is taken into consideration. To overcome the limitations of RGA based loop pairing criterion, several pairing methods have later been proposed by using the dynamic RGA (DRGA) to consider the effects of process dynamics, which employ the transfer function model instead of the steady state gain matrix to calculate RGA [11–13]. In DRGA, the denominator involved achieving perfect control at all frequencies, while the numerator was simply the open loop transfer function. Recently, McAvoy et al. proposed a significant DRGA approach [14]. Using the available dynamic process model, a proportional output optimal controller is designed based on the state space approach and the resulting controller gain matrix is used to define a DRGA. Several examples in which the normal RGA gives the inaccurate interaction measure and wrong pairings were studied and in all cases the new DRGA method gives more accurate interaction assessment and the best pairings. However, DRGA is often controller dependent [14], which makes it more difficult to calculate and to be understood by practical control engineers. To combine the advantages of both RGA and DRGA, this work employs the steady state gain and bandwidth of the process transfer function elements to provide a more comprehensive description for loop interactions. Then, a new loop pairing criterion based on the new interaction measure which results in minimum loop interactions are proposed in terms of effective relative gain array (ERGA) of the process. The main advantages of this method are: (1) compared with DRGA method, it also provides a comprehensive description of dynamic interaction among individual loops without requiring the specification of the controller type and with much less computation; (2) it results in a less conservative controller when the detuning factor design method is used; and (3) it is very simple for field engineers to understand and work out pairing decisions in practical applications. Several examples, for which the RGA based loop pairing criterion gives an inaccurate interaction assessment, are employed to demonstrate the effectiveness of the proposed interaction measure and loop pairing criterion.
r1 r2
+ e ⊗ 1 _ + e ⊗ 2 _
rn
y1 y2
u1 u2 …
Gc(s)
G(s)
… un
+ en _⊗
Fig. 1. Closed loop multivariable control system.
reference inputs; ui, i = 1, 2, . . ., n are the manipulated variables; yi, i = 1, 2, . . ., n are the system outputs, G(s) and Gc(s) are the process matrix and the decentralized controller matrix, respectively, G(s) and Gc(s) are with compatible dimensions and expressed by: 2 3 g11 ðsÞ g12 ðsÞ . . . g1n ðsÞ 6 g ðsÞ g ðsÞ . . . g ðsÞ 7 6 21 7 22 2n GðsÞ ¼ 6 7 4 ... ... ... ... 5 gn1 ðsÞ gn2 ðsÞ . . .
gnn ðsÞ
and 2
gc1 ðsÞ 0 ... 6 0 gc2 ðsÞ . . . 6 Gc ðsÞ ¼ 6 4 ... ... ... 0
0
...
0 0
3
7 7 7. ... 5 gcn ðsÞ
Assume that the system is open loop stable and the process matrix G(s) is non-singular at steady state. The loop pairing problem defines the control system structure, i.e., which of the available plant inputs are to be used to control each of the plant outputs. The most popular loop pairing method is the RGA and NI based pairing rules as follows [6,10,15]: Define relative gain kij ¼
ðoy i =ouj Þall loops open ðoy i =ouj Þall other loops close except for
and RGA 2 k11 6k 6 21 K¼6 4... kn1
loopy i uj
as k12 k22
... ...
...
...
3 k1n k2n 7 7 7 ...5
kn2
...
knn
which can be calculated by [10] K ¼ G GT ;
ð1Þ
where the operator is the Hadamard product and GT represents the transpose of inverse of G. The RGA based pairing criterion is:
2. Preliminaries Consider a multivariable system with n-inputs and noutputs as shown in Fig. 1, where ri, i = 1, 2, . . ., n are the
… yn
• the paired RGA elements are closest to 1.0; • all the paired RGA elements are positive; • large RGA elements should be avoided.
Q. Xiong et al. / Journal of Process Control 15 (2005) 741–747
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Furthermore, if all n loops are closed, the multi-loop system will be unstable for all possible (any) values of controller parameters (i.e., it will be ‘‘structurally monotonic unstable’’), if the NI is negative, i.e.
and 40. In such a case, pairing the faster loops (even with smaller steady state gains) takes advantage of the time scale decoupling such that seriousness of the interactions from the slower loop would be reduced.
jGð0Þj NI ¼ Qn < 0; i¼1 g ii ð0Þ
3. A modified loop pairing rule
where jG(0)j denotes the determinant of matrix G(0). The sign of NI, i.e. NI > 0, provides a necessary stability condition and consequently, constitutes a complementary tool to the RGA in variable pairing selection. One of the main advantages of these methods is that the interaction depends on only the steady state gains. This information is easily obtained from simple identification experiments or steady state design models. A potential weakness of these methods, however, is the same fact that they only use the steady state gains which are based on the assumption of perfect loop control to determine loop pairing. We use the following example to illustrate this point. Example 1. Consider a process given by [14] 2 3 5e40s e4s 6 100s þ 1 10s þ 1 7 7. GðsÞ ¼ 6 4 5e4s 5e40s 5 10s þ 1 100s þ 1 The RGA is 0.8333 KðGð0ÞÞ ¼ 0.1667
0.1667 0.8333
.
This result implies the diagonal pairing 1–1/2–2 is a good choice. Hence, according to the RGA based loop pairing criterion, the pairing of 1–1/2–2 should be preferred for the smaller interaction to any one loop from another closed loop. However, McAvoy et al. used DRGA and optimal decentralized PI controllers for various configurations, and found that the diagonal pairing resulted in a poor closed loop performance [14]. The offdiagonal pairing 1–2/2–1 takes advantage of the fast g21 (s) and g12(s) transfer functions to achieve better response for y2: 1. there is very little interaction from y1 to y2; 2. the off-diagonal pairing y2 response is significantly better than that of y2 with diagonal pairing; 3. the y1 response for the diagonal pairing is somewhat better than that of the off-diagonal pairing when y2 is given a step set point change. The main reason for the poor performance of the diagonal pairing is the dynamic properties of the transfer functions. It can be easily seen that the time constants and delays of 10 and 4 of the off-diagonal elements are 10 times smaller than diagonal ones of 100
In designing decentralized controllers for multivariable processes, it is desired that the interaction measures and loop pairing will address the following issues: (1) The interaction measure should consider the finite bandwidth control, since the assumption of perfect control is only valid for very low frequency range. (2) The loop pairing decision should be controller independent such that any controller type could be designed after loop pairing. (3) The pairing results in minimal interaction within the interested frequency range not only statically but also dynamically. (4) It should be simple and easy to use for practical engineers. In a decentralized control system design, the individual loop is tuned around the critical frequency region of the transfer function which is the region around the control system bandwidth. Thus, this is the frequency region that should be focused upon when considering the effect of interactions. Therefore, two factors in the open loop transfer functions will affect the loop pairing decision: 1. Steady state gain: the steady state gain gij (0) of the transfer function reflects the effect of the manipulated variable uj to the controlled variable yi. 2. Response speed: response speed is accountable for the sensitivity of the controlled variable yi to manipulated variable uj and, consequently, the ability to reject the interactions from other loops. Since the response speed is proportional to the bandwidth in frequency domain, we can use the bandwidth to reflect both interactions from finite bandwidth control and pairing loops to result in a fast response. Let gij ðjxÞ ¼ gij ð0Þg0ij ðjxÞ; where gij (0) and g0ij ðjxÞ are the steady state gain and normalized transfer function of gij(jx), i.e. g0ij ð0Þ ¼ 1, respectively. In order to use both steady state gain and response speed information for interaction measure and loop pairing, we now define the effective gain eij for a particular transfer function as Z xB;ij eij ¼ gij ð0Þ jg0ij ðjxÞj dx; ð2Þ 0
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Q. Xiong et al. / Journal of Process Control 15 (2005) 741–747
effective relative gain between output variable yi and input variable uj, /ij, as the ratio of two effective gains: eij /ij ¼ ; ^eij
Fig. 2. Response curve and effective energy of gij (jx).
where xB,ij for i, j = 1, 2, . . ., n are the bandwidths of transfer function g0ij ðjxÞ and |•| is the absolute value of •. For the frequency response of gij(jx) as shown in Fig. 2, eij is the area covered by gij(jx) up to xB,ij, where xB,ij is determined by the frequency where the magnitude of frequency response reduced to 0.707gij(0), i.e. pffiffiffi gij ðjxB;ij Þ ¼ gij ð0Þ= 2. Since jg0ij ðjxÞj represents the magnitude of the transfer function at various frequencies, eij of Eq. (2) can be considered the effective energy output of gij(jx) and the effective gain matrix can be expressed as 3 2 e11 e12 . . . e1n 7 6 6 e21 e22 . . . e2n 7 7 6 E¼6 7. 6... ... ... ...7 5 4 en1
en2
. . . enn
i; j ¼ 1; 2; . . . ; n.
Then, the effective gain matrix is given as: E ¼ Gð0Þ X;
ð3Þ
where 2
g11 ð0Þ g12 ð0Þ
6 g ð0Þ g ð0Þ 22 6 21 Gð0Þ ¼ 6 4 ... ... gn1 ð0Þ gn2 ð0Þ
. . . g1n ð0Þ
3
. . . g2n ð0Þ 7 7 7 ... ... 5 . . . gnn ð0Þ
and 2
xB;11 6x 6 B;21 X¼6 4 ... xB;n1
xB;12 xB;22
... ...
... xB;n2
... ...
/n1
3 xB;1n xB;2n 7 7 7 ... 5 xB;nn
are the steady state gain matrix and the bandwidth matrix, respectively. Since eij is an indication of interaction energy to other loops when loop yi uj is closed, the bigger the eij value is, the more dominant the loop will be. Replacing the steady state gain matrix G(0) of Eq. (1) by the effective gain matrix E of Eq. (3), we define the
/n2
...
/nn
which can be calculated by U ¼ E ET . Since both RGA and ERGA use relative gains, the properties of RGA can be directly extended to ERGA: 1. The value of /ij is a measure of the effective interaction expected in the ith loop if its output yi is paired with uj. 2. The elements of the ERGA across any row, or any column, sum up to 1, i.e. n n X X /ij ¼ /ij ¼ 1. i¼1
To simplify the calculation, we approximate the integration of eij by a rectangular area, i.e. eij gij ð0ÞxB;ij ;
where ^eij is the effective gain between output variable yi and input variable uj when all other loops are closed. When the effective relative gains are calculated for all the input/output combinations of a multivariable process, it results in an array of the form similar to that of RGA, we call it as effective RGA (ERGA): 2 3 /11 /12 . . . /1n 6/ 7 6 21 /22 . . . /2n 7 U¼6 7 4 ... ... ... ... 5
j¼1
3. Let ^eij represent the loop i effective gain when all the other loops are closed, whereas /ij represents the normal, open loop effective gain, then: 1 ^eij ¼ eij . /ij 4. When /ij is negative, with other loops open, it produces a change in yi in response to a change in uj, totally opposite in direction to that when the other loops are closed. As NI < 0, based on steady state information, a sufficient condition for the loop configuration to be unstable is provided and it can be used to eliminate those structures with unstable pairing options. Similar to RGA and NI based pairing rules, the ERGA and NI based loop pairing rules requires that manipulated and controlled variables in a decentralized control system be paired: (1) (2) (3) (4)
Corresponding ERGA elements are closest to 1.0. The NI is positive. All paired ERGA elements are positive. Large ERGA elements should be avoided.
Here, both ERGA and NI play important roles for control structure selection. ERGA is used to measure
Q. Xiong et al. / Journal of Process Control 15 (2005) 741–747
interactions, while NI is used as a sufficient condition to rule out the closed loop unstable pairings. Since we are using effective gains (energy) instead of simply, steady state gains, dynamic interactions up to the critical frequency can be effectively reflected. Therefore, comparing ERGA method with RGA and DRGA, we may expect that: 1. In addition to steady state gains, only bandwidth need to be calculated in ERGA method, it is far easier to calculate than those in DRGA methods. 2. ERGA combines both steady state gain and bandwidth (effective energy) in measuring the loop interactions, it should provide better pairing results than that of RGA based pairing and comparable with DRGA ones. 3. Since ERGA only uses information of open loop process transfer functions, it is controller type independent.
4. Case studies In this section, we use various examples to show the effectiveness of ERGA method in both simplicity and correctness. Example 2 (Continue with Example 1). For this example, k11 = 0.8333 implies diagonal pairing. The four bandwidths are calculated: xB,11 = 0.01, xB,12 = 0.1, xB,21 = 0.1 and xB,22 = 0.01. Hence the effect gain matrix is " # 0.0500 0.1000 E¼ ; 0.5000 0.0500 and ERGA: /11 = 0.0476. It strongly suggests off-diagonal pairing. In order to test the pairing results for this process, McAvoy et al. [14] designed two sets of optimal PI controllers for diagonal and off-diagonal pairings, respectively. For each configuration, two controller gains and two integral gains are optimized for a step change in y1 followed by a step change in y2 using the same objective function that was used for the optimal control calculation. Equal weighting is given to the measurements and the manipulated variables. In calculating the errors for y1 and y2 the difference between these measurements and their set points is used. Since the minimization involved in tuning the PI controllers is non-convex, several different starting points are used to determine the best values of the tuning parameters. After all, it is concluded that the DRGA correctly indicates that an off-diagonal pairing produces a better overall control system response than that of the diagonal pairing.
745
Example 3. Consider a process given by [3] 2 3 5 2.5e5s 6 4s þ 1 ð2s þ 1Þð15s þ 1Þ 7 6 7 GðsÞ ¼ 6 7. 4 4e6s 5 1 3s þ 1 20s þ 1 RGA: k11 = 0.3333 implies off-diagonal pairing. The bandwidth of each element is xB,11 = 0.25, xB,12 = 0.066, xB,21 = 0.05 and xB,22 = 0.333. Hence 1.2500 0.1650 E¼ ; 0.2000 0.3330 and ERGA: /11 = 0.9265 implies diagonal pairing. This loop pairing decision was obtained by Grosdidier and Morari [3] through analyzing both magnitude and phase characteristics of the interaction between the two loops. Example 4. Consider a 3 · 3 process [16] given by 2
2es 6 10s þ 1 6 6 6 1.5es GðsÞ ¼ 6 6 sþ1 6 4 es sþ1
1.5es sþ1 es sþ1 2es 10s þ 1
3 es sþ1 7 7 7 2es 7 7. 10s þ 1 7 7 1.5es 5 sþ1
The RGA of the system is 2
0.9302
6 K ¼ 4 1.1860 0.7442
1.1860 0.7442 0.9302
0.7442
3
7 0.9302 5. 1.1860
Obviously, two possible parings 1–2/2–1/3–3 and 1–3/ 2–2/3–1 are comparable because all related RGA elements are close to 1. Therefore, RGA pairing approach cannot determine which pairing is better. The bandwidth matrix of the system is 2 3 0.1 1 1 6 7 X¼4 1 1 0.1 5. 1
0.1
1
The ERGA is then obtained as 2 3 0.0554 0.6977 0.2468 6 7 U ¼ 4 0.6977 0.2468 0.0554 5. 0.2468
0.0554
0.6977
This indicates the best paring is 1–2/2–1/3–3 because the corresponding ERGA elements are much closer to 1 than the other option. This was confirmed by the generalized dynamic relative gain (GDRG) approach [16]. However, the proposed method is much simpler.
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Q. Xiong et al. / Journal of Process Control 15 (2005) 741–747
Example 5. Consider a 4 · 4 industrial reactor/recycle system [17] given by 3 2 3.96ð197sþ1Þ 0.536ð258sþ1Þ 9.7 0 24.3sþ1 49.4s2 þ14.1sþ1 83.3s2 þ18.3sþ1 7 6 6 0.00111 0.044 0.0152e32s 0.039e20s 7 7 6 s s s 46.9s2 þs 7. GðsÞ ¼ 6 7 6 15.96ð529sþ1Þ 232.2 139.2 7 6 0 32.2sþ1 7.27sþ1 5 10 417s2 þ204sþ1 4 35s 0.582 2.54ð8.11sþ1Þ 0.0462ð45.6sþ1Þe 0.0358e30s s s 6.25s3 þ5s2 þs 306s3 þ35s2 þs Since interaction analysis via the RGA involves steady state information, for processes that contain pure integrator elements that have no steady state, the steady state gain and RGA of integrating processes can be calculated using a special method proposed by Arkun and Downs [18]. Let us consider I¼
1 s
and substitute I into the original transfer function matrix to obtain K ¼ lim GðsÞ. s!0
Finally determine RGA by RGA ¼ lim K ðK 1 Þ
T
to those produced by the pairing option suggested by RGA. Nevertheless, the proposed method is much more direct and the computation is much easier. 5. Conclusions In this paper, both steady state gain and bandwidth of the process were used to provide a simple yet comprehensive description of loop interactions for MIMO processes. The ERGA can be conveniently calculated by control engineers since the bandwidth can be easily obtained from the given transfer function matrix. The effectiveness of the method was demonstrated by several examples, for which the RGA based loop pairing criterion gives an inaccurate interaction assessment, while the proposed interaction measure and loop pairing criterion provides accurate results and they are very easy to be calculated. Furthermore, the ERGA method is not only an effective tool for loop pairing, but also very useful in helping to design the decentralized and decoupling control systems. The design of the decentralized controller, especially, for high dimensional processes using ERGA information as a detuning factor is currently under investigation, the results will be reported later.
I!1
which is calculated as [17] 2 0 0.107 0.148 6 6 0.072 0.196 0.829 K¼6 6 0 0.723 0.067 4 1.072
0.027
0.745
References
3
7 0.046 7 7 0.210 7 5
0.044
0
indicates parings: 1–4/2–3/3–2/4–1. To use ERGA method, the bandwidth matrix of G(s) and ERGA are obtained, respectively, as 2
0 6 1.4500 6 X¼6 4 0
5.6400 0.1750
4.3800 1.4500
0.0310
0.0710
3 0.0410 1.4500 7 7 7 0.1380 5
1.4500
1.2980
0.4550
1.4500
and 2
0
6 0.0562 6 U¼6 4 0 0.9438
0.7856
0.2461
0.5078
1.2895
0.3174
0.1600
0.4048
0.3486
0.0047
3
0.1621 7 7 7. 0.8426 5 0
This indicates parings: 1–4/2–1/3–2/4–3. This is the same as the pairing option given by the DRGA approaches [14,17]. Robinson et al. [17] confirmed this conclusion and they showed that the transients produced by this pairing option are significantly superior
[1] P. Grosdidier, M. Morari, A computer aided methodology for the design of decentralized controllers, Comput. Chem. Eng. 11 (1987) 423–433. [2] M.S. Chiu, Y. Arkun, Decentralized control structure selection based on integrity considerations, Ind. Eng. Chem. Res. 29 (1990) 369–373. [3] P. Grosdidier, M. Morari, Interaction measures for systems under decentralized control, Automatica 22 (1986) 309–319. [4] M.-J. He, W.-J. Cai, New criterion for control loop configuration of multivariable processes, Ind. Eng. Chem. Res. in press. [5] D.E. Seborg, T.F. Edgar, D.A. Mellichamp, Process Dynamics and Control, John Wiley and Sons, New York, 1989. [6] E.H. Bristol, On a new measure of interactions for multivariable process control, IEEE Trans. Automat. Control 11 (1966) 133– 134. [7] F.G. Shinskey, Process Control Systems, McGraw-Hill, New York, 1988. [8] E.A. Wolff, S. Skogestad, Operation of integrated three-product (Petlyuk) distillation columns, Ind. Eng. Chem. Res. 34 (1995) 2094–2103. [9] J.E. Hansen, S.B. Jørgensen, J. Heath, J.D. Perkins, Control structure selection for energy integrated distillation column, J. Process Control 8 (1998) 185–195. [10] P. Grosdidier, M. Morari, Closed-loop properties from steadystate gain information, Ind. Eng. Chem. Fund. 24 (1985) 221– 235. [11] M. Witcher, T.J. McAvoy, Interacting control systems: steady state and dynamic measurement of interaction, ISA Trans. 16 (1977) 83–90. [12] E.H. Bristol, Recent results on interactions in multivariable process control, in: Proceedings of the 71st Annual AIChE Meeting, Houston, TX, USA, 1979.
Q. Xiong et al. / Journal of Process Control 15 (2005) 741–747 [13] L. Tung, T. Edgar, Analysis of control–output interactions in dynamic systems, AIChE J. 27 (1981) 690–693. [14] T. McAvoy, Y. Arkun, R. Chen, D. Robinson, P.D. Schnelle, A new approach to defining a dynamic relative gain, Control Eng. Practice 11 (2003) 907–914. [15] A. Niederlinski, A heuristic approach to the design of linear multivariable interacting subsystems, Automatica 7 (1971) 691– 701.
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[16] H.-P. Huang, M. Ohshima, I. Hashimoto, Dynamic interaction and multiloop control system design, J. Process Control 4 (1994) 15–27. [17] D. Robinson, R. Chen, T. McAvoy, D. Schnelle, An optimal control based approach to designing plantwide control system architectures, J. Process Control 11 (2001) 223–236. [18] Y. Arkun, J. Downs, A general method to calculate input-output gains and the RGA for integrating processes, Comput. Chem. Eng. 14 (1990) 1101–1110.
A practical loop pairing criterion for multivariable processes Qiang Xiong, Wen-Jian Cai *, Mao-Jun He School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore Received 21 September 2004; received in revised form 7 February 2005; accepted 23 March 2005
Abstract Utilizing both steady state gain and bandwidth information of the process open loop transfer function elements, this paper presents a new dynamic loop pairing criterion for decentralized control of multivariable processes. Through defining an effective gain matrix, the loop pairing procedures of popular relative gain array method is directly extended to the new method which can reflect dynamic loop interactions under finite bandwidth control. Compared with existing methods, this method is simple, effective, and easy to be understood and applied by control engineers. Several examples, for which the RGA based loop pairing criterion gives an inaccurate interaction assessment, are employed to demonstrate the effectiveness of the proposed method. 2005 Elsevier Ltd. All rights reserved. Keywords: Multivariable process; Decentralized control; Interaction measures; Loop pairing; Effective gain; Relative gain array; Effective relative gain array
1. Introduction Most of the large and complex industrial processes are naturally multi-input multi-output (MIMO) systems. Compared with single-input single-output (SISO) counterparts, MIMO systems are more difficult to control due to the existence of interactions among input and output variables. Although considerable effort has been dedicated to this problem and many design techniques have been proposed over the decades, control system design and implementation for MIMO processes is still a big challenge for practical control engineers. The interactive multivariable systems can be either controlled by (1) a multivariable or centralized MIMO controller or by (2) a set of SISO decentralized controllers. Algebraic decoupling methods or optimal multivariable control theory are usually applied to obtain centralized MIMO controllers. While centralized multi-
*
Corresponding author. Tel.: +65 6790 6862; fax: +65 6793 3318. E-mail address: [email protected] (W.-J. Cai).
0959-1524/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jprocont.2005.03.008
variable controllers are complex and lack integrity, the decentralized control system enjoys certain advantages: (1) it requires fewer parameters to tune which are easier to be understood and implemented; and (2) loop failure tolerance of the resulting control system can be assured during the design phase. Therefore, they are more often used in process control applications [1,2]. However, the potential disadvantage of using the limited control structure is the deteriorated closed loop performance caused by interactions among loops as a result of the existence of non-zero off-diagonal elements in the transfer function matrix [3,4]. Thus, the primary task in the design of decentralized control systems is to determine loop configuration, i.e. pair the manipulated variables and controlled variables to achieve the minimum interactions among loops so that the resulting multivariable control system mostly resembles its single-input singleoutput counterparts and the subsequent controller tuning is largely facilitated by SISO design techniques [5]. Since the pioneering work of Bristol [6], the relative gain array (RGA) based techniques for control loop configuration have been widely used in industries,
742
Q. Xiong et al. / Journal of Process Control 15 (2005) 741–747
including blending, energy conservation, and distillation columns, etc. [7–9]. The most important advantage of RGA based techniques is its simple calculation since only process steady state gains are involved and scaling is independent due to its ratio nature [10]. However, using steady state gain alone may result in incorrect interaction measures and consequently loop pairing decisions, since no dynamic information of the process is taken into consideration. To overcome the limitations of RGA based loop pairing criterion, several pairing methods have later been proposed by using the dynamic RGA (DRGA) to consider the effects of process dynamics, which employ the transfer function model instead of the steady state gain matrix to calculate RGA [11–13]. In DRGA, the denominator involved achieving perfect control at all frequencies, while the numerator was simply the open loop transfer function. Recently, McAvoy et al. proposed a significant DRGA approach [14]. Using the available dynamic process model, a proportional output optimal controller is designed based on the state space approach and the resulting controller gain matrix is used to define a DRGA. Several examples in which the normal RGA gives the inaccurate interaction measure and wrong pairings were studied and in all cases the new DRGA method gives more accurate interaction assessment and the best pairings. However, DRGA is often controller dependent [14], which makes it more difficult to calculate and to be understood by practical control engineers. To combine the advantages of both RGA and DRGA, this work employs the steady state gain and bandwidth of the process transfer function elements to provide a more comprehensive description for loop interactions. Then, a new loop pairing criterion based on the new interaction measure which results in minimum loop interactions are proposed in terms of effective relative gain array (ERGA) of the process. The main advantages of this method are: (1) compared with DRGA method, it also provides a comprehensive description of dynamic interaction among individual loops without requiring the specification of the controller type and with much less computation; (2) it results in a less conservative controller when the detuning factor design method is used; and (3) it is very simple for field engineers to understand and work out pairing decisions in practical applications. Several examples, for which the RGA based loop pairing criterion gives an inaccurate interaction assessment, are employed to demonstrate the effectiveness of the proposed interaction measure and loop pairing criterion.
r1 r2
+ e ⊗ 1 _ + e ⊗ 2 _
rn
y1 y2
u1 u2 …
Gc(s)
G(s)
… un
+ en _⊗
Fig. 1. Closed loop multivariable control system.
reference inputs; ui, i = 1, 2, . . ., n are the manipulated variables; yi, i = 1, 2, . . ., n are the system outputs, G(s) and Gc(s) are the process matrix and the decentralized controller matrix, respectively, G(s) and Gc(s) are with compatible dimensions and expressed by: 2 3 g11 ðsÞ g12 ðsÞ . . . g1n ðsÞ 6 g ðsÞ g ðsÞ . . . g ðsÞ 7 6 21 7 22 2n GðsÞ ¼ 6 7 4 ... ... ... ... 5 gn1 ðsÞ gn2 ðsÞ . . .
gnn ðsÞ
and 2
gc1 ðsÞ 0 ... 6 0 gc2 ðsÞ . . . 6 Gc ðsÞ ¼ 6 4 ... ... ... 0
0
...
0 0
3
7 7 7. ... 5 gcn ðsÞ
Assume that the system is open loop stable and the process matrix G(s) is non-singular at steady state. The loop pairing problem defines the control system structure, i.e., which of the available plant inputs are to be used to control each of the plant outputs. The most popular loop pairing method is the RGA and NI based pairing rules as follows [6,10,15]: Define relative gain kij ¼
ðoy i =ouj Þall loops open ðoy i =ouj Þall other loops close except for
and RGA 2 k11 6k 6 21 K¼6 4... kn1
loopy i uj
as k12 k22
... ...
...
...
3 k1n k2n 7 7 7 ...5
kn2
...
knn
which can be calculated by [10] K ¼ G GT ;
ð1Þ
where the operator is the Hadamard product and GT represents the transpose of inverse of G. The RGA based pairing criterion is:
2. Preliminaries Consider a multivariable system with n-inputs and noutputs as shown in Fig. 1, where ri, i = 1, 2, . . ., n are the
… yn
• the paired RGA elements are closest to 1.0; • all the paired RGA elements are positive; • large RGA elements should be avoided.
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Furthermore, if all n loops are closed, the multi-loop system will be unstable for all possible (any) values of controller parameters (i.e., it will be ‘‘structurally monotonic unstable’’), if the NI is negative, i.e.
and 40. In such a case, pairing the faster loops (even with smaller steady state gains) takes advantage of the time scale decoupling such that seriousness of the interactions from the slower loop would be reduced.
jGð0Þj NI ¼ Qn < 0; i¼1 g ii ð0Þ
3. A modified loop pairing rule
where jG(0)j denotes the determinant of matrix G(0). The sign of NI, i.e. NI > 0, provides a necessary stability condition and consequently, constitutes a complementary tool to the RGA in variable pairing selection. One of the main advantages of these methods is that the interaction depends on only the steady state gains. This information is easily obtained from simple identification experiments or steady state design models. A potential weakness of these methods, however, is the same fact that they only use the steady state gains which are based on the assumption of perfect loop control to determine loop pairing. We use the following example to illustrate this point. Example 1. Consider a process given by [14] 2 3 5e40s e4s 6 100s þ 1 10s þ 1 7 7. GðsÞ ¼ 6 4 5e4s 5e40s 5 10s þ 1 100s þ 1 The RGA is 0.8333 KðGð0ÞÞ ¼ 0.1667
0.1667 0.8333
.
This result implies the diagonal pairing 1–1/2–2 is a good choice. Hence, according to the RGA based loop pairing criterion, the pairing of 1–1/2–2 should be preferred for the smaller interaction to any one loop from another closed loop. However, McAvoy et al. used DRGA and optimal decentralized PI controllers for various configurations, and found that the diagonal pairing resulted in a poor closed loop performance [14]. The offdiagonal pairing 1–2/2–1 takes advantage of the fast g21 (s) and g12(s) transfer functions to achieve better response for y2: 1. there is very little interaction from y1 to y2; 2. the off-diagonal pairing y2 response is significantly better than that of y2 with diagonal pairing; 3. the y1 response for the diagonal pairing is somewhat better than that of the off-diagonal pairing when y2 is given a step set point change. The main reason for the poor performance of the diagonal pairing is the dynamic properties of the transfer functions. It can be easily seen that the time constants and delays of 10 and 4 of the off-diagonal elements are 10 times smaller than diagonal ones of 100
In designing decentralized controllers for multivariable processes, it is desired that the interaction measures and loop pairing will address the following issues: (1) The interaction measure should consider the finite bandwidth control, since the assumption of perfect control is only valid for very low frequency range. (2) The loop pairing decision should be controller independent such that any controller type could be designed after loop pairing. (3) The pairing results in minimal interaction within the interested frequency range not only statically but also dynamically. (4) It should be simple and easy to use for practical engineers. In a decentralized control system design, the individual loop is tuned around the critical frequency region of the transfer function which is the region around the control system bandwidth. Thus, this is the frequency region that should be focused upon when considering the effect of interactions. Therefore, two factors in the open loop transfer functions will affect the loop pairing decision: 1. Steady state gain: the steady state gain gij (0) of the transfer function reflects the effect of the manipulated variable uj to the controlled variable yi. 2. Response speed: response speed is accountable for the sensitivity of the controlled variable yi to manipulated variable uj and, consequently, the ability to reject the interactions from other loops. Since the response speed is proportional to the bandwidth in frequency domain, we can use the bandwidth to reflect both interactions from finite bandwidth control and pairing loops to result in a fast response. Let gij ðjxÞ ¼ gij ð0Þg0ij ðjxÞ; where gij (0) and g0ij ðjxÞ are the steady state gain and normalized transfer function of gij(jx), i.e. g0ij ð0Þ ¼ 1, respectively. In order to use both steady state gain and response speed information for interaction measure and loop pairing, we now define the effective gain eij for a particular transfer function as Z xB;ij eij ¼ gij ð0Þ jg0ij ðjxÞj dx; ð2Þ 0
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effective relative gain between output variable yi and input variable uj, /ij, as the ratio of two effective gains: eij /ij ¼ ; ^eij
Fig. 2. Response curve and effective energy of gij (jx).
where xB,ij for i, j = 1, 2, . . ., n are the bandwidths of transfer function g0ij ðjxÞ and |•| is the absolute value of •. For the frequency response of gij(jx) as shown in Fig. 2, eij is the area covered by gij(jx) up to xB,ij, where xB,ij is determined by the frequency where the magnitude of frequency response reduced to 0.707gij(0), i.e. pffiffiffi gij ðjxB;ij Þ ¼ gij ð0Þ= 2. Since jg0ij ðjxÞj represents the magnitude of the transfer function at various frequencies, eij of Eq. (2) can be considered the effective energy output of gij(jx) and the effective gain matrix can be expressed as 3 2 e11 e12 . . . e1n 7 6 6 e21 e22 . . . e2n 7 7 6 E¼6 7. 6... ... ... ...7 5 4 en1
en2
. . . enn
i; j ¼ 1; 2; . . . ; n.
Then, the effective gain matrix is given as: E ¼ Gð0Þ X;
ð3Þ
where 2
g11 ð0Þ g12 ð0Þ
6 g ð0Þ g ð0Þ 22 6 21 Gð0Þ ¼ 6 4 ... ... gn1 ð0Þ gn2 ð0Þ
. . . g1n ð0Þ
3
. . . g2n ð0Þ 7 7 7 ... ... 5 . . . gnn ð0Þ
and 2
xB;11 6x 6 B;21 X¼6 4 ... xB;n1
xB;12 xB;22
... ...
... xB;n2
... ...
/n1
3 xB;1n xB;2n 7 7 7 ... 5 xB;nn
are the steady state gain matrix and the bandwidth matrix, respectively. Since eij is an indication of interaction energy to other loops when loop yi uj is closed, the bigger the eij value is, the more dominant the loop will be. Replacing the steady state gain matrix G(0) of Eq. (1) by the effective gain matrix E of Eq. (3), we define the
/n2
...
/nn
which can be calculated by U ¼ E ET . Since both RGA and ERGA use relative gains, the properties of RGA can be directly extended to ERGA: 1. The value of /ij is a measure of the effective interaction expected in the ith loop if its output yi is paired with uj. 2. The elements of the ERGA across any row, or any column, sum up to 1, i.e. n n X X /ij ¼ /ij ¼ 1. i¼1
To simplify the calculation, we approximate the integration of eij by a rectangular area, i.e. eij gij ð0ÞxB;ij ;
where ^eij is the effective gain between output variable yi and input variable uj when all other loops are closed. When the effective relative gains are calculated for all the input/output combinations of a multivariable process, it results in an array of the form similar to that of RGA, we call it as effective RGA (ERGA): 2 3 /11 /12 . . . /1n 6/ 7 6 21 /22 . . . /2n 7 U¼6 7 4 ... ... ... ... 5
j¼1
3. Let ^eij represent the loop i effective gain when all the other loops are closed, whereas /ij represents the normal, open loop effective gain, then: 1 ^eij ¼ eij . /ij 4. When /ij is negative, with other loops open, it produces a change in yi in response to a change in uj, totally opposite in direction to that when the other loops are closed. As NI < 0, based on steady state information, a sufficient condition for the loop configuration to be unstable is provided and it can be used to eliminate those structures with unstable pairing options. Similar to RGA and NI based pairing rules, the ERGA and NI based loop pairing rules requires that manipulated and controlled variables in a decentralized control system be paired: (1) (2) (3) (4)
Corresponding ERGA elements are closest to 1.0. The NI is positive. All paired ERGA elements are positive. Large ERGA elements should be avoided.
Here, both ERGA and NI play important roles for control structure selection. ERGA is used to measure
Q. Xiong et al. / Journal of Process Control 15 (2005) 741–747
interactions, while NI is used as a sufficient condition to rule out the closed loop unstable pairings. Since we are using effective gains (energy) instead of simply, steady state gains, dynamic interactions up to the critical frequency can be effectively reflected. Therefore, comparing ERGA method with RGA and DRGA, we may expect that: 1. In addition to steady state gains, only bandwidth need to be calculated in ERGA method, it is far easier to calculate than those in DRGA methods. 2. ERGA combines both steady state gain and bandwidth (effective energy) in measuring the loop interactions, it should provide better pairing results than that of RGA based pairing and comparable with DRGA ones. 3. Since ERGA only uses information of open loop process transfer functions, it is controller type independent.
4. Case studies In this section, we use various examples to show the effectiveness of ERGA method in both simplicity and correctness. Example 2 (Continue with Example 1). For this example, k11 = 0.8333 implies diagonal pairing. The four bandwidths are calculated: xB,11 = 0.01, xB,12 = 0.1, xB,21 = 0.1 and xB,22 = 0.01. Hence the effect gain matrix is " # 0.0500 0.1000 E¼ ; 0.5000 0.0500 and ERGA: /11 = 0.0476. It strongly suggests off-diagonal pairing. In order to test the pairing results for this process, McAvoy et al. [14] designed two sets of optimal PI controllers for diagonal and off-diagonal pairings, respectively. For each configuration, two controller gains and two integral gains are optimized for a step change in y1 followed by a step change in y2 using the same objective function that was used for the optimal control calculation. Equal weighting is given to the measurements and the manipulated variables. In calculating the errors for y1 and y2 the difference between these measurements and their set points is used. Since the minimization involved in tuning the PI controllers is non-convex, several different starting points are used to determine the best values of the tuning parameters. After all, it is concluded that the DRGA correctly indicates that an off-diagonal pairing produces a better overall control system response than that of the diagonal pairing.
745
Example 3. Consider a process given by [3] 2 3 5 2.5e5s 6 4s þ 1 ð2s þ 1Þð15s þ 1Þ 7 6 7 GðsÞ ¼ 6 7. 4 4e6s 5 1 3s þ 1 20s þ 1 RGA: k11 = 0.3333 implies off-diagonal pairing. The bandwidth of each element is xB,11 = 0.25, xB,12 = 0.066, xB,21 = 0.05 and xB,22 = 0.333. Hence 1.2500 0.1650 E¼ ; 0.2000 0.3330 and ERGA: /11 = 0.9265 implies diagonal pairing. This loop pairing decision was obtained by Grosdidier and Morari [3] through analyzing both magnitude and phase characteristics of the interaction between the two loops. Example 4. Consider a 3 · 3 process [16] given by 2
2es 6 10s þ 1 6 6 6 1.5es GðsÞ ¼ 6 6 sþ1 6 4 es sþ1
1.5es sþ1 es sþ1 2es 10s þ 1
3 es sþ1 7 7 7 2es 7 7. 10s þ 1 7 7 1.5es 5 sþ1
The RGA of the system is 2
0.9302
6 K ¼ 4 1.1860 0.7442
1.1860 0.7442 0.9302
0.7442
3
7 0.9302 5. 1.1860
Obviously, two possible parings 1–2/2–1/3–3 and 1–3/ 2–2/3–1 are comparable because all related RGA elements are close to 1. Therefore, RGA pairing approach cannot determine which pairing is better. The bandwidth matrix of the system is 2 3 0.1 1 1 6 7 X¼4 1 1 0.1 5. 1
0.1
1
The ERGA is then obtained as 2 3 0.0554 0.6977 0.2468 6 7 U ¼ 4 0.6977 0.2468 0.0554 5. 0.2468
0.0554
0.6977
This indicates the best paring is 1–2/2–1/3–3 because the corresponding ERGA elements are much closer to 1 than the other option. This was confirmed by the generalized dynamic relative gain (GDRG) approach [16]. However, the proposed method is much simpler.
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Q. Xiong et al. / Journal of Process Control 15 (2005) 741–747
Example 5. Consider a 4 · 4 industrial reactor/recycle system [17] given by 3 2 3.96ð197sþ1Þ 0.536ð258sþ1Þ 9.7 0 24.3sþ1 49.4s2 þ14.1sþ1 83.3s2 þ18.3sþ1 7 6 6 0.00111 0.044 0.0152e32s 0.039e20s 7 7 6 s s s 46.9s2 þs 7. GðsÞ ¼ 6 7 6 15.96ð529sþ1Þ 232.2 139.2 7 6 0 32.2sþ1 7.27sþ1 5 10 417s2 þ204sþ1 4 35s 0.582 2.54ð8.11sþ1Þ 0.0462ð45.6sþ1Þe 0.0358e30s s s 6.25s3 þ5s2 þs 306s3 þ35s2 þs Since interaction analysis via the RGA involves steady state information, for processes that contain pure integrator elements that have no steady state, the steady state gain and RGA of integrating processes can be calculated using a special method proposed by Arkun and Downs [18]. Let us consider I¼
1 s
and substitute I into the original transfer function matrix to obtain K ¼ lim GðsÞ. s!0
Finally determine RGA by RGA ¼ lim K ðK 1 Þ
T
to those produced by the pairing option suggested by RGA. Nevertheless, the proposed method is much more direct and the computation is much easier. 5. Conclusions In this paper, both steady state gain and bandwidth of the process were used to provide a simple yet comprehensive description of loop interactions for MIMO processes. The ERGA can be conveniently calculated by control engineers since the bandwidth can be easily obtained from the given transfer function matrix. The effectiveness of the method was demonstrated by several examples, for which the RGA based loop pairing criterion gives an inaccurate interaction assessment, while the proposed interaction measure and loop pairing criterion provides accurate results and they are very easy to be calculated. Furthermore, the ERGA method is not only an effective tool for loop pairing, but also very useful in helping to design the decentralized and decoupling control systems. The design of the decentralized controller, especially, for high dimensional processes using ERGA information as a detuning factor is currently under investigation, the results will be reported later.
I!1
which is calculated as [17] 2 0 0.107 0.148 6 6 0.072 0.196 0.829 K¼6 6 0 0.723 0.067 4 1.072
0.027
0.745
References
3
7 0.046 7 7 0.210 7 5
0.044
0
indicates parings: 1–4/2–3/3–2/4–1. To use ERGA method, the bandwidth matrix of G(s) and ERGA are obtained, respectively, as 2
0 6 1.4500 6 X¼6 4 0
5.6400 0.1750
4.3800 1.4500
0.0310
0.0710
3 0.0410 1.4500 7 7 7 0.1380 5
1.4500
1.2980
0.4550
1.4500
and 2
0
6 0.0562 6 U¼6 4 0 0.9438
0.7856
0.2461
0.5078
1.2895
0.3174
0.1600
0.4048
0.3486
0.0047
3
0.1621 7 7 7. 0.8426 5 0
This indicates parings: 1–4/2–1/3–2/4–3. This is the same as the pairing option given by the DRGA approaches [14,17]. Robinson et al. [17] confirmed this conclusion and they showed that the transients produced by this pairing option are significantly superior
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