Decentralized Control System Design for Multivariable Processes

Ind. Eng. Chem. Res. 2006, 45, 2769-2776

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Decentralized Control System Design for Multivariable ProcessessA Novel Method Based on Effective Relative Gain Array Qiang Xiong, Wen-Jian Cai,* Mao-Jun He, and Ming He School of Electrical and Electronic Engineering, Nanyang Technological UniVersity, Singapore 639798

In this paper, a novel method for design of a decentralized control system for multivariable processes is proposed. On the basis of a new interaction measure, effectiVe relatiVe gain array (ERGA), in terms of energy transmission ratio, loop interactions are quantified by two elements, i.e., relative gain and relative critical frequency. The interaction effects for a particular loop from all other closed loops are analyzed through both steady-state gain and critical frequency variations. Consequently, appropriate detuning factors for decentralized controllers under different interaction conditions can be derived based on the effective relative gain, relative gain, and relative critical frequency. The design method can be effectively used for both normal processes as well as process-loop transfer functions containing unstable zeros resulted from other closed loops. This design method is simple, straightforward, and effective and can be easily understood and implemented by field engineers. Several multivariable industrial processes with different interaction characteristics are employed to demonstrate the effectiveness and simplicity of the design method. 1. Introduction Most of the advanced modern processes are complicated and multi-input multioutput (MIMO) in natural. MIMO processes are more difficult to control, compared with their single-input single-output (SISO) counterparts, because of the existence of interactions among loops. Although considerable effort has been dedicated to this problem and many design techniques have been proposed over the years, the design and implementation of a MIMO control system still poses a big challenge to control researchers and practical engineers. An interactive MIMO process can be controlled by either a centralized multivariable controller or a decentralized multiloop controller which consists of a set of SISO controllers. While centralized controllers are complex and weak in control-system integrity, the decentralized control systems enjoy certain advantages: (1) the structure is easier to understand and implement and requires fewer parameters to tune; (2) loop failure tolerance of the resulting control system can be assured during the design phase; and (3) both hardware and software realization is simple. Therefore, they are more often used in process-control applications.1,2 However, because of the existence of interactions among the control loops, the design of such controllers to meet certain performance specifications would encounter more difficulties than that of single loops. Therefore, it has been an open research topic for years. The current existing design methods for decentralized control systems can be roughly divided into three categories: (1) sequential loop closing methods;3-7 (2) independent design methods;8-10 and (3) detuning methods.11-14 While each method enjoys certain advantages, they suffer a common drawback, which is that only steady-state information is used in the design, resulting in limited control performance. It is well-known that the process open-loop initial or highfrequency response behavior is the most important for controller design purposes.15-17 From frequency asymptote analysis, the initial response of a particular control loop with other loops in manual or automatic status is known to be very similar in * To whom all correspondence should be addressed. Tel.: +65 6790 6862. Fax: +65 6793 3318. E-mail: [email protected].

Figure 1. Closed-loop multivariable control system.

nature.13 In addition, initial response is proportional to the bandwidth or phase-crossover frequency in the frequency domain. Therefore, decentralized controller design based on the corresponding main-loop process model by taking changes in both steady-state gain and crossover frequency into consideration should provide satisfactory overall system performance. Even though control-system design based on dynamic relative gain array (RGA)12 or performance RGA18 can generally improve the control system performances, it is somehow too complicated for practical control engineers, especially when the dimension is high. In this paper, by defining energy transmission ratio for a transfer function, a novel interaction measure for MIMO processes in terms of effective relative gain array (ERGA) is proposed which can provide information on both gain and phase changes when other loops are closed. By detailed analysis of the control-loop interactions, different detuning factors are derived for different classes of interactions. These detuning factors are, then, directly used to detune those proportionalintegral-derivative (PID) controllers without considering interactions among the loops. The main contribution of the work is to provide an effective, universal, and open-loop transfer function based decentralized controller design method with very simple computations. Especially, it is more advantageous for higher-dimensional processes with complicated interaction modes than other existing approaches. Several multivariable industrial processes with different interaction characteristics are employed to demonstrate the effectiveness and simplicity of the design method.

10.1021/ie051331t CCC: $33.50 © 2006 American Chemical Society Published on Web 03/16/2006

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Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006

2. Preliminaries Consider an open-loop stable multivariable system with n inputs and n outputs as shown in Figure 1, where ri, i ) 1, 2, ‚‚‚, n, are the reference inputs; ui, i ) 1, 2, ‚‚‚, n, are the manipulated variables; yi, i ) 1, 2, ‚‚‚, n, are the system outputs; and G(s) and Gc(s) are the process transfer-function matrix and the decentralized controller matrix with compatible dimensions, expressed by

[

g11(s) g (s) G(s) ) 21 ... gn1(s)

g12(s) g22(s) ... gn2(s)

]

g1n(s) g2n(s) , ... gnn(s)

... ... ... ...

[

gc1(s) 0 and Gc(s) ) ... 0

0 gc2(s) ... 0

... ... ... ...

0 0 ... gcn(s)

]

respectively. In the design of a decentralized control system, it is desired that inputs and outputs with dominant transfer function be paired together for effective control. In the frequency domain, two factors in the open-loop transfer functions will affect the looppairing decision and should be focused upon when considering the effect of interactions: 1. Steady-state gain: the steady-state gain gij(0) of the transfer function reflects the effect of manipulated variable uj on controlled variable yi. 2. Critical frequency: critical frequency is accountable for the sensitivity of the controlled variable yi to manipulated variable uj and, consequently, the promptness of a particular output response to an input and the ability to reject the interactions from other loops. Let

Figure 2. Frequency response curve and effective energy of gij(jω).

phase crossover frequencies, such as transfer functions of first order or second order without time delay, it is reasonable to use corresponding bandwidths as critical frequencies to calculate eij. It should be noted, however, that the phase crossover frequency information (ωu,ij) is recommended if applicable for calculation of eij, since it is more closely linked to control-system performance. Since the response speed is proportional to the ultimate frequency in the frequency domain, it reflects both interactions from finite bandwidth control and pairing loops to results fast response; we will use ωu,ij as the basis for the following development without loss of generality. For the frequency response of gij(jω) as shown in Figure 2, eij is the area covered by gij(jω) up to ωu,ij. Since |g0ij(jω)| represents the magnitude of the transfer function at various frequencies, eij is considered to be the energy transmission ratio from the manipulated variable uj to the controlled variable yi. Express the energy transmission ratio array as

[

e11 e E ) 21 ... en1

gij(jω) ) kijg0ij(jω) g0ij(jω)

where kij and are the steady-state gain and the normalized transfer function of gij(jω), i.e., g0ij(0) ) 1, respectively. Similar to ref 19, we define eij of a particular transfer function as

∫0

eij ) kij

ωc,ij

|g0ij(jω)|

dω

where ωc,ij for i, j ) 1, 2, ‚‚‚, n is the critical frequency of the transfer function gij(jω) and |•| is the absolute value of •. In the actual determination of eij, critical frequencies can be defined in two ways: 1. ωc,ij ) ωB,ij, where ωB,ij for i, j ) 1, 2, ‚‚‚, n is the bandwidth of the transfer function g0ij(jω) and is determined by the frequency where the magnitude plot of frequency response is reduced to 0.707gij(0), i.e.,

|gij(jωB,ij)| ) 0.707|gij(0)| 2. ωc,ij ) ωu,ij, where ωu,ij for i, j ) 1, 2, ‚‚‚, n is the ultimate frequency of the transfer function g0ij(jω) and is determined by the frequency where the phase plot of frequency response crossovers equals -π, i.e.,

arg[gij(jωu,ij)] ) -π For the transfer-function matrix with some elements without

e12 e22 ... en2

... ... ... ...

e1n e2n ... enn

]

To simplify the calculation, we approximate the integration of eij by a rectangular area, i.e.,

eij ≈ kijωu,ij i, j ) 1, 2, ‚‚‚, n Since only a relative value other than the absolute value will be concerned in the definition of new interaction measure, as shown in eq 2, the above approximation will not result in significant error. Then, the effective energy transmission ratio array is given as

E ) G(0) X Ω

[

] [

(1)

where the operator X is the hadamard product, and

k11 k G(0) ) 21 ... kn1

k12 k22 ... kn2

... ... ... ...

k1n k2n ... knn

ωu,11 ω and Ω ) u,21 ... ωu,n1

ωu,12 ωu,22 ... ωu,n2

... ... ... ...

ωu,1n ωu,2n ... ωu,nn

]

are the steady-state gain array and the critical frequency array, respectively. Since eij is an indication of the energy transmission ratio for loop yi - uj, the bigger the eij value is, the more dominant the loop will be. Similar to the definition of relative gain,20 by replacing the steady-state gain matrix with the effective gain matrix E of eq

Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006 2771

Figure 3. Interaction with λii < 1, γii < 1.

Figure 4. Interaction with λii < 1, γii > 1.

1, we define the effective relative gain, φij, between output variable yi and input variable ui as the ratio of two effective energy transmission ratios,

φij )

eij eˆ ij

(2)

where eˆ ij is the effective energy transmission ratio between output variable yi and input variable ui 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 ERGA, which can be calculated by

[

φ11 φ Φ ) E X E-T ) 21 ... φn1

φ12 φ22 ... φn2

... ... ... ...

φ1n φ2n ... φnn

]

(3)

Figure 5. Interaction with λii > 1, γii < 1.

Similar to RGA and Niederlinski index (NI) based pairing rules, the ERGA and NI based loop-pairing rules require that manipulated and controlled variables in a decentralized control system be paired: (1) Corresponding ERGA elements are closest to 1.0. (2) The NI is positive. (3) All paired ERGA elements are positive. (4) Large ERGA elements should be avoided. 3. Detuning Factor Suppose that the best loop configuration has been determined and the best pair is placed at a diagonal in the transfer-function matrix, as shown in Figure 1. Similar to the open-loop effective gain, we let the effective gain, eˆ ij, when all other loops are closed be eˆ ij ) gˆ ij(0)ω ˆ u,ij, i, j ) 1, 2, ‚‚‚, n, where gˆ ij(0) and ω ˆ u,ij are the steady-state gain and ultimate frequency between output variable yi and input variable ui when all other loops are closed, respectively. Then, we have from eq 2

ˆ u,ij ) gˆ ij(0)ω

gij(0)ωu,ij φij

(4)

Define

γij )

φij λij

where λij is the RGA value and γij represents the relative ultimate frequency change of loop i-j when other loops are closed, which

Figure 6. Interaction with λii > 1, γii > 1.

is defined here as the relative critical frequency. We suppose that the best pairing is diagonal. Otherwise, we can alter the related columns to make it diagonal. Subsequently, we can write eq 4 for each loop as

gˆ ii(0)ω ˆ u,ii )

gii(0) ωu,ii λij γij

(5)

In terms of ERGA, there are two factors (λii and γii) affecting the control-system performance for an individual loop. There are four different interaction modes as shown in Figures 3-6, and these are discussed as follows: Case 1: λii < 1, φii < λii f γii < 1 • λii < 1, the retaliatory effect from the other loops, magnifies the main effect of ui on yi. Since the magnitude of the frequency

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response when other loops are closed is bigger than that when other loops are open, the gain detuning is required. • γii < 1, the ultimate frequency when other loops are closed, is larger than that when other loops are open. The enlarged ultimate frequency will increase the phase margin. Hence, no phase detuning is needed. In such a case, we only need to detune the gain, implying that the detuning factor only depends on λii and is given as

gˆ c,ii(0) ) λiigc,ii(0)

(6)

Case 2: λii < 1, φii > λii f γii > 1 • λii < 1, the same as in Case 1. The retaliatory effect from the other loops magnifies the main effect of ui on yi, and the magnitude of the frequency response when other loops are closed is bigger than that when other loops are open; the gain detuning is required. • γii > 1, the ultimate frequency when other loops are closed; is smaller than that when other loops are open. The reduced ultimate frequency will reduce the phase margin. Hence, phase detuning is also necessary. As both gain and phase need to be detuned to guarantee the system stability, a reasonable detuning factor is proposed as

gˆ c,ii(s) ) φiigc,ii(s)

For original controller design of the diagonal elements in the transfer-function matrix, many SISO PID tuning methods can be used. Here, we adopt the gain and phase margins design approach.22 This is primarily because the frequency-response method provides good performance in the face of uncertainty in both plant model and disturbances. Without loss of generality, we assume that each element in the transfer-function matrix is represented by a second-order plus dead time (SOPDT) model, which can be used to describe most of the industrial processes,

g(s) )

a2s + a1s + 1

gc(s) ) kp +

e-ds

ki + kds s

The controller can be rewritten as

gc(s) )

kds2 + kps + ki As2 + Bs + C )k s s

where A ) kd/k, B ) kp/k, and C ) ki/k. By selecting A ) a2, B ) a1, and C ) 1, the open-loop transfer function becomes

gc(s)g(s) ) k

b0 -ds e s

(10)

Denoting the gain and phase margins as Am and Φm and their crossover frequencies as ωg and ωp, respectively, we have

arg[gc(jωg)g(jωg)] ) -π

(11)

Am|gc(jωg)g(jωg)| ) 1

(12)

|gc(jωp)g(jωp)| ) 1

(13)

Φm ) π + arg[gc(jωp)g(jωp)]

(14)

Substituting eq 10 into eqs 11-14, we obtain

ωgd )

ωg π Am ) 2 kb0

kb0 ) ωp Φm )

(9)

Remark 1. A unique problem for decentralized control of MIMO processes is the zero crossing:21 stable or unstable zeros might be introduced into a particular control loop when other loops are closed. If an unstable zero is introduced, causing a phase shift to the left in the frequency domain, controller detuning based on steady-state gain information may fail. To guarantee a stable control system, the controllers have to be

b0 2

and the standard PID controller is of the form

(8)

Case 4: λii > 1, φii > λii f γii > 1 • λii > 1, the same as in Case 3. The retaliatory effect from other loops acts in opposition to the main effect of ui on yi, and the magnitude of the frequency response when other loops are closed is smaller than that when other loops are open; no gain detuning is required. • γii > 1, the ultimate frequency when other loops are closed, is smaller than that when other loops are open. The reduced ultimate frequency will reduce the phase margin. Hence, detuning is needed to guarantee the stability. In this case, we only need to consider detuning the controller to account for the phase change; a reasonable detuning factor is proposed as

gˆ c,ii(s) ) gc,ii(s)/γii

4. Decentralized PID Controller Design

(7)

Case 3: λii > 1, φii < λii f γii < 1 • λii > 1, the retaliatory effect from other loops, acts in opposition to the main effect of ui on yi. Since the magnitude of the frequency response when other loops are closed is smaller than that when other loops are open, no gain detuning is required. • γii < 1, the ultimate frequency when other loops are closed, is larger than that when other loops are open. The enlarged ultimate frequency will increase the phase margin. Hence, no phase detuning is needed. The control-loop integrity requires no more aggressive of a controller be allowed when other loop are closed than that when other loops are open. Therefore, the best choice in this case is to keep the controller unchanged, i.e.,

gˆ c,ii(s) ) gc,ii(s)

detuned conservatively.21 By introducing frequency factor γii to indicate phase changes after other loops are closed, the effects of unstable zeros can be accurately estimated in each control loop. Consequently, the designed control system will be much less conservative.

π -ωpd 2

which results in

Φm )

(

)

π 1 π 1, k) 2 Am 2Amdb0

By this formulation, the gain and phase margin are interrelated to each other; some possible selections are given in Table 1.

Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006 2773 Table 1. Typical Gain and Phase Margin Values Am

Φm

k

2 3 4 5

π/4 π/3 3π/8 2π/5

π/4db0 π/6db0 π/8db0 π/10db0

The PID parameters are given by

[]

[]

kp a1 ki ) π 1 2Amdb0 a kd 2

(15)

Applying eqs 6-9 and 15 for the decentralized control-system design by the proposed detuning factor method, we can easily obtain the PID parameter setting for the four cases as follows: Case 1: λii < 1, φii < λii f γii < 1

[]

[]

kp,ii a1,ii πλii ki,ii ) 1 2Am,iidiib0,ii a kd,ii 2,ii

(16)

Case 2: λii < 1, φii > λii f γii > 1

[]

[]

kp,ii a1,ii πφii ki,ii ) 1 2Am,iidiib0,ii a kd,ii 2,ii

(17) Figure 7. Closed-loop responses for Example 1.

Case 3: λii > 1, φii < λii f γii < 1

[]

[]

kp,ii a1,ii π ki,ii ) 1 2Am,iidiib0,ii a kd,ii 2,ii

Case 4: λii > 1, φii > λii f γii > 1

[]

[]

kp,ii a1,ii π ki,ii ) 1 2γiiAm,iidiib0,ii a kd,ii 2,ii

Table 2. Controllers for Example 1 proposed

(18)

Mc Avoy

controller

kp,ii

τi,ii

kp,ii

τi,ii

kp,ii

τi,ii

loop 1 loop 2

0.0850 0.5314

4.0000 5.0000

0.1368 0.8547

12.0000 6.0000

0.0828 0.5828

2.8280 3.1060

[

Example 1. Consider a process given by

]

e-2s 5 e-4s 1 15s + 1 . G(s) ) 4s + 2 e-2s e-s 20s + 1 5s + 1

(19)

5. Case Studies In this section, we apply the proposed design method to a variety of industrial processes. The loop pairing is based on the criterion of ERGA plus Niederlinski index (NI).19 The gain and phase margin for all loops are specified to be 3 and π/3, respectively. The proposed design method is compared with the following decentralized control design approaches to show the effectiveness of the proposed design: (1) the Ziegler-Nichols with detuning factor approach proposed by McAvoy;12 (2) the biggest log modulus (BLT) tuning approach proposed by Luyben;11 (3) the RGA-based tuning approach proposed by Chien et al.;13 (4) the sequential loop-tuning approach proposed by Shen and Yu;7 and (5) the relay-based autotuning approach proposed by Loh et al.5 Example 1 is focused on comparing ERGA-based detuning and RGA-based detuning. The comparison in other examples is extended to other decentralized control designs.

Chien et al.

The RGA, critical frequency array, and ERGA are given, respectively, by

Λ) Ω)

[

[

1.1111 -0.1111 -0.1111 1.1111

0.5236 0.8055 1.5708 0.8976

]

Φ)

[

]

1.3684 -0.3684 -0.3684 1.3684

]

It can be found that, by both RGA (1.1111) and ERGA (1.3684) with NI ()0.9000 > 0), the best pairing is diagonal, i.e., 1-1/ 2-2. Using method (1), we obtain the detuning factor, i.e., λii

- xλii2-λii ) 0.7597. According to method (3), the detuning is not necessary in this case, since the RGA value is >1. Because λii > 1 and γii > 1, by the proposed method the detuning is still necessary in this case, and the detuning factors for both controllers are 1/γii ) 0.8120. The resultant PI controllers by different designs and the closed-loop responses are listed in Table 2 and shown in Figure 7, respectively, where the unit set points change in r1 at t ) 0 and r2 at t ) 200.

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Luyben

Chien et al.

Loh et al.

controller

kp,ii

τi,ii

kp,ii

τi,ii

kp,ii

τi,ii

kp,ii

τi,ii

loop 1 loop 2

0.3762 0.2924

4.5720 1.8010

0.210 0.175

2.26 4.25

0.263 0.163

1.42 1.77

0.620 0.247

0.60 1.78

It can be seen that all designs have a similar interaction decoupling effect, though the detuning method (1) has an unsatisfactory one with y1. However, the proposed method has almost perfect set-point tracking performance, while the other methods give either sluggish or large overshoot responses. Example 2. Consider an industrial-scale polymerization reactor given by13

[

22.89 -11.64 -0.4s e-0.2s e 4.572s + 1 1.807s +1 G(s) ) 4.689 5.80 e-0.2s e-0.4s 2.174s + 1 1.801s + 1

]


Λ)

[

]

[

]

0.7087 0.2913 8.0554 4.1888 , Ω) 0.2913 0.7087 7.8540 4.3036 0.7193 0.2807 Φ) 0.2807 0.7193

[

]

Both ERGA and RGA indicate diagonal pairing (NI ) 1.4111 > 0). We consider the controller design with the same pairing. In this example, we have λii < 1 and γii > 1 for i ) 1, 2. According to the proposed method, the detuning factors for the two original controllers are 0.7193. The resultant PI controllers determined by the proposed method, together with those determined by methods (2), (3), and (5), are listed in Table 3. Figure 8 shows the closed-loop responses for different controllers, where the unit set points change in r1 at t ) 0 and r2 at t ) 20. It can be seen that, even with the same pairing decision, the proposed controller-design method gives the best performance. Example 3. Consider a 3 × 3 process given by23

[

G(s) )

0.66 e-2.6s 6.7s + 1

-0.61 e-3.5s -0.0049 e-s 8.64s + 1 9.06s + 1

1.11 e-6.5s 3.25s + 1

-2.36 e-3s 5s + 1

-34.68 e-9.2s 46.2 e-9.4s 8.15s + 1 10.9s + 1

-0.01 e-1.2s 7.09s + 1 0.87(11.61s + 1) e-s (3.89s + 1)(18.8s + 1)

]


[

]

2.0084 -0.7220 -0.2864 Λ ) -0.6460 1.8246 -0.1786 -0.3624 -0.1026 1.4650 0.6981 0.5108 1.6535 Ω ) 0.3491 0.6283 1.3963 0.2244 0.2094 1.6982 1.2808 -0.1954 -0.0853 Φ ) -0.2001 1.2385 -0.0384 -0.0806 -0.0431 1.1237

[

[

]

Figure 8. Closed-loop responses for Example 2.

The best pairing according to ERGA and NI ()0.3859 > 0) is 1-1/2-2/3-3. For i ) 1, 2, 3, λii > 1 and γii < 1. Hence, there is no need to detune for all three loops. The resultant PI controllers by the proposed method are listed in Table 4, together with those of methods (2), (3), and (4). Figure 9 shows the closed-loop responses, where the unit set points change in r1 at t ) 0, r2 at t ) 500, and r3 at t ) 1000. The controller parameters that resulted from the proposed method are different from the others since no detuning is required for this process. Especially, proportional gain for Loop 3 is about two times that of the others, resulting in a less-conservative design. Example 4. Consider a 4 × 4 process given by24

[

G(s) ) -9.811 e-1.59s 0.374 e-7.75s 11.36s + 1 22.22s + 1

-3.79s -2.368 e-27.33s -11.3 e 33.3s + 1 (21.74s + 1)2

5.984 e-2.24s 14.29s + 1

-8.72s -1.986 e-0.71s 0.422 e 66.67s + 1 (250s + 1)2

5.24 e-60s 400s + 1

2.38 e-0.42s (1.43s + 1)2

0.0204 e-0.59s 0.513 e-s s+1 (7.14s + 1)2

-0.33 e-0.68s (2.38s + 1)2

-11.3 e-3.79s -0.176 e-0.48s 15.54 e-s s+1 (21.74s + 1)2 (6.9s + 1)2

4.48 e-0.52s 11.11s + 1

The RGA, critical frequency array, and ERGA are given by

]

[

0.1264 0.0107 Λ) 0.7264 0.1366

-0.1013 1.0935 0.0025 0.0054

-0.0314 0.0003 0.1630 0.8680

1.0063 -0.1045 0.1081 -0.0099

]

]

[

1.0134 0.7306 Ω) 1.7952 0.1551 and

[

0.5237 - 0.0024 Φ) 0.4719 0.0067

0.2285 2.2440 0.6830 0.7854

-0.0052 1.0035 0.0012 0.0004

0.0757 0.0306 1.9635 2.0268

-0.0222 0.0000 0.2965 0.7256

0.1571 0.0279 1.0833 3.1416

]

0.5036 -0.0012 0.2303 0.2673

Ind. Eng. Chem. Res., Vol. 45, No. 8, 2006 2775

]

respectively. The best pairing according to ERGA and NI ()1.1814 > 0, ideal pairing) is 1-4/2-2/3-1/4-3. This example has complicated interaction modes. For i ) 1, 2, λii > 1 and γii < 1; there is no need to detune those loops by using the proposed method. For i ) 3, 4, λii < 1 and γii < 1; the detuning factors for those loops are corresponding RGA elements, i.e., 0.7264 and 0.8680, respectively. The resultant



PI/PID controllers by the proposed method are listed in Table 5, together with method (2) based on the diagonal pairing. It should be noticed that different designs have much different controller parameters due to different pairing schemes.

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Table 4. Controllers for Example 3 proposed controller loop 1 loop 2 loop 3

τi,ii

kp,ii

Luyben kp,ii

τi,ii

Chien et al. kp,ii

Shen and Yu

τi,ii

kp,ii

τi,ii

1.5333 6.7000 1.51 16.4 1.08 4.25 2.55 16.5 -0.2773 5.0000 -0.295 18.0 -0.233 3.32 -0.235 26.1 5.6039 12.4150 2.63 6.61 2.78 5.24 3.39 7.39


Luyben

controller

kp,ii

τi,ii

τd,ii

kp,ii

τi,ii

loop 1 loop 2 loop 3 loop 4

1.4981 -24.7566 0.0245 -0.4614

2.8600 66.6700 1 43.4800

0.7150 0 0 10.8700

-0.084 -5.16 0.305 0.529

33 15.5 17.0 11.2

Figure 10 shows the closed-loop responses, where the unit set points change in r1 at t ) 0, r2 at t ) 400, r3 at t ) 800, and r4 at t ) 1200. This example illustrates that the proposed method can be easily applied to high-dimensional processes with complicated interaction modes. 6. Conclusions This paper presented a novel decentralized controller-design technique for multivariable interactive processes. The simplicity and effectiveness of the method is basically from the intrinsic property of the ERGA, which results in better loop pairing and provides both gain and critical frequency information. Consequently, the controllers based on a single-loop transfer function can be simply detuned by a set of interaction measures based on the effective relative gain, relative gain, and relative critical frequency. Simulation results for a variety of industrial 2 × 2, 3 × 3, and 4 × 4 processes show that the proposed design method is simple and gives compatible dynamic performance to other reported designs or even better ones, which confirms the effectiveness and universality of the proposed method. The advantage of the proposed method is even more significant when applied to higher-dimensional processes with complicated interaction modes. It can be easily integrated into an autotuning control structure when combined with an on-line parameteridentification module and implemented for real industrial control systems. Literature Cited (1) Grosdidier, P.; Morari, M. A computer aided methodology for the design of decentralized controllers. Comput. Chem. Eng. 1987, 11, 423433. (2) Chiu, M.-S.; Arkun, Y. Decentralized control structure selection based on integrity considerations. Ind. Eng. Chem. Res. 1990, 29, 369373.

(3) Chiu, M.-S.; Arkun, Y. A methodology for sequential design of robust decentralized control systems. Automatica 1992, 28, 997-1001. (4) Lee, J.; Cho, W.; Edgar, T. F. Multiloop PI controller tuning for interacting multivariable processes. Comput. Chem. Eng. 1998, 22, 17111723. (5) Loh, A. P.; Hang, C. C.; Quek, C. K.; Vasnani, V. U. Autotuning of multiloop proportional-integral controllers using relay feedback. Ind. Eng. Chem. Res. 1993, 32, 1102-1107. (6) Mayne, D. Q. The design of linear multivariable systems. Automatica 1973, 9, 201-207. (7) Shen, S.-H.; Yu, C.-C. Use of relay-feedback test for autotuning of multivariable systems. AIChE J. 1994, 40, 627-646. (8) Grosdidier, P.; Morari, M. Interaction measures under decentralized control. Automatica 1986, 22, 309-319. (9) Lee, T. K.; Shen, J.; Chiu, M. S. Independent design of robust partially decentralized controllers. J. Process Control 2001, 11, 419-428. (10) Skogestad, S.; Morari, M. Robust performance of decentralized control systems by independent design. Automatica 1989, 25, 119-125. (11) Luyben, W. L. Simple method for tuning SISO controllers in multivariable systems. Ind. Eng. Chem. Proc. Des. DeV. 1986, 25, 654660. (12) McAvoy, T. J. Interaction Analysis: Principles and Applications; Inst. Soc. of America: North Carolina, 1983. (13) Chien, I.-L.; Huang, H.-P.; Yang, J.-C. A simple multiloop tuning method for PID controllers with no proportional kick. Ind. Eng. Chem. Res. 1999, 38, 1456-1468. (14) Chen, D.; Seborg, D. E. Design of decentralized PI control systems based on Nyquist stability analysis. J. Process Control, 2003, 13, 27-39. (15) Skogestad, S.; Morari, M. LV-Control of a High-Purity Distillation Column. Chem. Eng. Sci. 1988, 43, 33. (16) Chien, I.-L.; Fruehauf, P. S. Consider IMC Tuning to Improve Controller Performance. Chem. Eng. Prog. 1990, 86, 33. (17) Chien, I.-L.; Ogunnaike, B. A. Modeling and control of a temperature-based high-purity distillation column. Chem. Eng. Commun. 1997, 158, 71. (18) Hovd, M.; Skogestad, S. Simple frequency-dependent tools for control system analysis, structure selection and design. Automatica 1992, 28, 989-996. (19) Xiong, Q.; Cai, W.-J.; He, M.-J. A practical loop pairing criterion for multivariable processes. J. Process Control 2005, 15, 741-747. (20) Bristol, E. H. On a new measure of interactions for multivariable process control. IEEE Trans. Autom. Control 1966, 11, 133-134. (21) Cui, H.; Jacobsen, E. W. Performance limitations in decentralized control. J. Process Control 2002, 12, 485-494. (22) 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, 2910-2914. (23) Ogunnaike, B. A.; Ray, W. H. Multivariable controller design for linear systems having multiple time delays. AIChE J. 1979, 25, 1043. (24) Doukas, N.; Luyben, W. L. Control of sidestream columns separating ternary mixtures. Instrum. Technol. 1978, 25, 43.

ReceiVed for reView November 29, 2005 ReVised manuscript receiVed February 12, 2006 Accepted February 20, 2006 IE051331T

Decentralized Control System Design for Multivariable Processes

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