Partial Decoupling Control for Multivariable Processes - Industrial

Apr 13, 2011 - A practical guideline to control structure selection for MIMO processes. Yunhui Luo , Hongbo Liu , Lei Jia , Wenjian Cai. 2011,405-410 ...
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Partial Decoupling Control for Multivariable Processes Yuling Shen,† Shaoyuan Li,*,† Ning Li,† and Wen-Jian Cai‡ †

Key Laboratory of System Control and Information Processing, Ministry of Education of China, and Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China ‡ School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore ABSTRACT: Based on the theory of equivalent transfer functions (ETFs), this work proposes a novel partial decoupling control technique for multivariable processes. By measuring the effect of each ETF element in decoupling control, a partial decoupling control structure selection criterion is proposed with the aims that the decoupler has the least complexity and that overall system performance is satisfied. The ETF parameters are then obtained under certain constraints. Consequently, a systematic design procedure is proposed in order to obtain a stable, proper, and causal partial decoupler matrix. Simulation results of several multivariable processes with different interaction characteristics are provided to demonstrate the simplicity and effectiveness of the design method.

1. INTRODUCTION In the process control industry, decoupling control for multivariable processes is a popular technique that uses a decoupler to eliminate the effect of undesirable cross-couplings such that the process can be treated as multiple single loops and less conservative single-loop control design methods can be directly applied. The diagonal decoupling control system is depicted as in Figure 1, where G(s), GI(s), and GC(s) are n-dimensional process, decoupler, and controller transfer function matrices, respectively. The introduction of the additional transfer function block GI(s) between the single-loop controller GC(s) and the process G(s) is to act on the process G(s) such that the transfer function matrix satisfies the relation GR ðsÞ ¼ GðsÞ GI ðsÞ

ð1Þ

Because the controller output is diagonal GR ðsÞ ¼ diagfgR , 11 ðsÞ, gR , 22 ðsÞ, ... , gR , nn ðsÞg

ð2Þ

In the ideal situation, the decoupler causes the control loops to act as if they were completely independent of one another, thereby reducing the controller tuning task to the of tuning several, noninteracting controllers. Generally, three types of decoupling control schemes are currently available, namely, ideal decoupling, simplified decoupling, and inverted decoupling.13 In an ideal decoupling control scheme, the decoupler can be obtained by specifying GR(s) as a diagonal matrix GI ðsÞ ¼ G1 ðsÞ GR ðsÞ

ð3Þ

Because the ideal decoupler is calculated from G1(s), the decoupled transfer function elements can end up with very complicated structures. When the system is of high dimension with nonminimum phase elements in the transfer functions, it is difficult for the inverses to be both causal and stable. Simplified decoupling, on the other hand, has simpler decoupler transfer functions. However, it results in complex decoupled transfer function elements. The inverted decoupling scheme has the advantages of both the simplified and ideal decoupling schemes, r 2011 American Chemical Society

Figure 1. Block diagram of the decoupling control system.

but it is difficult to extend to higher-dimensional processes and is more sensitive to modeling errors. Because of these difficulties, research activities so far have focused primarily on two-inputtwo-output (TITO) processes.4,5 For high-dimensional processes, Wang et al. developed a systematic decoupling design method with full-dimensional non-proportional integralderivative (non-PID) controllers.6 Liu et al. also proposed an analytical decoupling control strategy for high-dimensional processes.7 Through the specification of a desirable system response transfer function matrix, high-order non-PID controllers were derived in both decoupling schemes. Even though some significant performance improvements were demonstrated, extensive numerical computation efforts are required for the model reduction and approximation of G1(s). Recently, Cai et al. proposed a normalized decoupling scheme based on relative normalized gain array (RNGA) and equivalent transfer function (ETF) concepts.5,8 By establishing a simple relation between the ETF matrix and G1(s), the calculation effort to approximate G1(s) is greatly reduced. For high-dimensional processes, however, the full normalized decoupling control scheme might not always guarantee a satisfactory performance because of the improper selection of ETF parameters. In this work, two Received: October 11, 2010 Accepted: April 13, 2011 Revised: April 13, 2011 Published: April 13, 2011 7380

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constraints on ETF parameters are given after analysis of the dynamic characteristics of the true decoupled transfer function matrix. In addition, a partial decoupling control scheme is presented to find a fine balance between computational efforts and control performance. The main contributions of this work include the following: (1) Based on a compromise between performance improvement and control system structure complexity, an engineeroriented structure selection criterion is proposed. (2) Through analysis of the dynamic characteristics of the decoupled process, a constraint on the ETF parameters is derived. (3) The ETF parameters are calculated under the given constraint. (4) A partial decoupling control technique is proposed that can guarantee the stability, appropriateness, and causality of the decoupler. This method is simple, straightforward, and easy to understand and implement. Several multivariable processes with different interaction characteristics are employed to demonstrate the design procedure and effectiveness in performance improvement.

2. PRELIMINARIES Because most industry processes are open-loop stable, the higher-order transfer function element can be simplified to a firstorder plus time delay (FOPTD) model for interaction analysis and control system design.9,10 We thus assume that all process transfer function elements can be described by a FOPDT transfer function kij eθij s ð4Þ gij ðsÞ ¼ τij s þ 1 where kij, τij, and θij are the open-loop gain, time constant, and dead time, respectively, of the simplified process. Before presenting the main results of this work, we first introduce some concepts that are important to their later development. 2.1. RGA, RNGA, and RARTA. Relative Gain Array (RGA). The relative gain array (RGA) is used to measure the steady-state gain changes when all other loops are open and all other loops are closed and is defined as11 2 3 λ11 λ12 3 3 3 λ1n 6λ 7 6 21 λ22 3 3 3 λ2n 7 6 7 Λ ¼ 6l 3 l 33 l 7 4 5 λn1 λn2 3 3 3 λnn 3 2 k11 k12 k1n 333 ^ 7 6 6 ^k11 ^k12 k1n 7 7 6 k2n 7 6 k21 k22 7 6 3 3 3 ^ ^ ^k2n 7 ð5Þ ¼6 7 6 k21 k22 6l 3 l 33 l 7 7 6 7 6 knn 5 4 kn1 kn2 ^kn1 ^kn2 3 3 3 ^knn which can be calculated as Λ¼K X K

T

ð6Þ ^ where the operator X is the Hadamard product, kij is the closedloop gain, K is the steady-state gain matrix of G(s). Relative Normalized Gain Array (RNGA). The relative normalized gain array (RNGA) is used to describe both steady-state gain and dynamic changes when all other loops are open and all other loops are closed by defining a normalized gain, KN, as12 KN ¼ ½kn, ij nn ¼ K . Tar

where the operator . represents element-by-element division and Tar = [τar,ij]nn, for i, j = 1, 2, ..., n, in which τar,ij is the average resident time of gij(s). For FOPTD models, τar,ij can be obtained as ð8Þ τar, ij ¼ τij þ θij Similarly to RGA, RNGA can then be expressed as 2 3 λn, 11 λn, 12 3 3 3 λn, 1n 6 7 6 λn, 21 λn, 22 3 3 3 λn, 2n 7 7 ΛN ¼ 6 3 6l 7 l 33 l 4 5 λn, n1 λn, n2 3 3 3 λn, nn 3 2 kn, 11 kn, 12 kn, 1n 6 7 6 ^kn, 11 ^kn, 12 3 3 3 ^kn, 1n 7 7 6 6 k21 k22 k2n 7 7 6 333 ^ 7 6 kn, 2n 7 ¼ 6 ^kn, 21 ^kn, 22 7 6 3 l 7 6l 33 l 7 6 6 kn, n1 kn, n2 kn, nn 7 5 4 ^kn, n1 ^kn, n2 3 3 3 ^kn, nn

ð9Þ

which can be calculated as ΛN ¼ KN X KN T

ð10Þ

where kn,ij, ^kn,ij, and KN are the open-loop normalized gain, the closed-loop normalized gain, and the open-loop normalized gain matrix, respectively. Relative Average Resident Time Array (RARTA). The relative average resident time array (RARTA) is used to describe the dynamic changes when all other loops are open and all other loops are closed and is defined as5 2 3 γ11 γ12 3 3 3 γ1n 6 6 γ21 γ22 3 3 3 γ2n 7 7 7 Γ¼6 6l 7 3 l l 4 5 33 γn1 γn2 3 3 3 γnn 3 2 ^τar, 11 ^τar, 12 ^τar, 1n 6 7 6 τar, 11 τar, 12 3 3 3 τar, 1n 7 7 6 6 ^τar, 21 ^τar, 22 ^τar, 2n 7 7 6 7 6 333 τ ð11Þ ¼ 6 τar, 21 τar, 22 ar, 2n 7 7 6l 3 l l 7 6 3 3 7 6 6 ^τ ^τar, nn 7 5 4 ar, n1 ^τar, n2 τar, n1 τar, n2 3 3 3 τar, nn where ^τar,ij is the closed-loop average resident time. Because ΛN ¼ Λ X Γ 2 λ11 λ12 6 6 λ21 λ22 ¼6 6l l 4 λn1 λn2

333 333 3 33 333

3 2 λ1n γ11 7 6 6 γ21 λ2n 7 6 7 l 7 X6 4l 5 γn1 λnn

γ12 γ22 l γn2

333 333 3 33 333

3 γ1n 7 γ2n 7 7 l 7 5 γnn ð12Þ

Γ can be calculated as

ð7Þ

Γ ¼ ΛN . Λ 7381

ð13Þ

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2.2. ETF and ETF Matrix. Equivalent Transfer Function (ETF). The ETF is the equivalent transfer function of gij(s) when all other loops are closed. Because control loop transfer functions when all other loops are closed should have frequency properties similar to those when other loops are open after being wellpaired, we assume that ETFs have the same structures as the corresponding open loop transfer functions but with different parameters.13 For the FOPDT model, the ETF is expressed as

1 ^ ^g ij ðsÞ ¼ ^kij eθij s ^τij s þ 1

The relationship between the ETF matrix and the inverse matrix of G1(s) is derived as14

ð17Þ

ð18Þ

^ (s) to replace the inverse of the process transfer matrix, Using G we have T

^ T ðsÞ GR ðsÞ GI ðsÞ ¼ G

ð19Þ

~g 1n ðsÞ

333

7 ~g 2n ðsÞ 7 7 7 7 l 7 5 ~g nn ðsÞ

333 3

3

3

3

333

ð22Þ

where ~g ij ðsÞ ¼

n

g ðsÞ

ik , ∑ ^ g k ¼ 1 jk ðsÞ

i, j ¼ 1, 2, ... , n

ð23Þ

Substituting eqs 4 and 14 into eq 23 results in the expression ~g ij ðsÞ ¼

n

k ^τ s þ 1

ik jk eðθ ∑ ^ τ s þ 1 k ¼ 1 kjk ik

ik

^ Þs θ jk

,

i, j ¼ 1, 2, ... , n

ð24Þ

which can be further simplified to ~g ij ðsÞ ¼

ð16Þ

3. DETERMINATION OF PARTIAL DECOUPLING CONTROL STRUCTURE Define the structure of the ETF matrix as 2 3 k11^g 11 1 ðsÞ k12^g 12 1 ðsÞ 3 3 3 k1n^g 1n 1 ðsÞ 6 7 1 1 1 6 k21^g 21 ðsÞ k22^g 22 ðsÞ 3 3 3 k2n^g 2n ðsÞ 7 ^ ðsÞ ¼ 6 7 G 3 l 6l 7 33 l 4 5 kn1^g n1 1 ðsÞ kn2^g n2 1 ðsÞ 3 3 3 knn^g nn 1 ðsÞ where κij is a selector ( 1 if the element is selected kij ¼ 0 if the element is not selected

^ T ðsÞ ~I ðsÞ ¼ GðsÞ G 2 ~g 11 ðsÞ ~g 12 ðsÞ 6 6 ~g ðsÞ ~g ðsÞ 6 21 22 ¼6 6l l 6 4 ~g n1 ðsÞ ~g n2 ðsÞ

ð14Þ

for i, j = 1, 2, ..., n, where ^kij, ^τij, and θ^ij are the closed-loop gain, time constant, and dead time, respectively. ETF Matrix. The ETF matrix is a matrix made up of ETFs, which is in the form of 3 2 1=^g 1n ðsÞ 1=^g 11 ðsÞ 1=^g 12 ðsÞ ::: 7 6 1=^g 2n ðsÞ 7 6 1=^g 21 ðsÞ 1=^g 22 ðsÞ ::: ^ ðsÞ ¼ 6 7 ð15Þ G 3 7 6l l 33 l 5 4 1=^g n1 ðsÞ 1=^g n2 ðsÞ ::: 1=^g nn ðsÞ

^ T ðsÞ ¼ G1 ðsÞ G

To further determine the value of κij, the following matrix is calculated

n

ik ½ðθ e ∑ ^ k ¼ 1 kjk

k

ik

^ þ ^τ Þs þ τik Þ  ðθ jk jk

,

i, j ¼ 1, 2, ... , n

ð25Þ

To weight the effect of each ETF element in decoupling control, an index parameter is introduced, which is defined as 8  9 0.05

Because GR(s) is a specified diagonal matrix, the structure of partial decoupler is determined as 3 2 k11 gi, 11 ðsÞ k21 gi, 12 ðsÞ 3 3 3 kn1 gi, 1n ðsÞ 7 6 k g ðsÞ k g ðsÞ 6 12 i, 21 22 i, 22 3 3 3 kn2 gi, 2n ðsÞ 7 7 ð20Þ 6 GI ðsÞ ¼ 6 l 3 7 l 33 l 5 4 k1n gi, n1 ðsÞ k2n gi, n2 ðsÞ 3 3 3 knn gi, nn ðsÞ

4. PARAMETERIZATION OF ETFS UNDER CONSTRAINT

where κij is a selector ( 1 if the element is selected kij ¼ 0 if the element is not selected

^ T ðsÞ GR ðsÞ GI ðsÞ ¼ G _ Substituting eq 28 into eq 27, G R(s) is obtained as _ ^ T ðsÞ GR ðsÞ G R ðsÞ ¼ GðsÞ G

ð21Þ

4.1. Effects of ETF Approximation. According _ to Figure 1, the actual decoupled transfer function matrix, G R(s), is calculated as _ ð27Þ G R ðsÞ ¼ GðsÞ GI ðsÞ

Substituting eq 16 into eq 3, we have

7382

ð28Þ

ð29Þ

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To further _ analyze the constraints imposed on the ETF parameters, G R(s) is expanded as 2 3 g r, 11 ðsÞ g r, 12 ðsÞ 3 3 3 g r, 1n ðsÞ 6 7 6 g ðsÞ g ðsÞ 7 _ 6 r, 21 r, 22 3 3 3 g r, 2n ðsÞ 7 7 G R ðsÞ ¼ 6 ð30Þ 3 6l 7 l 33 l 4 5 g r, n1 ðsÞ g r, n2 ðsÞ 3 3 3 g r, nn ðsÞ

it is expected that ^ ¼ θβ2  θ ^ ¼ ^ θβ1  θ j1 j2 3 3 3 ¼ θβn  θjn where 8  9   k   θij  θ ^ ðj 6¼ iÞ θii  θ ð33Þ ii

^  ¼ θβi  θβj ^ θ θ ii ij

and ^ ^τar, ij ¼ γij τar, ij } ^τij þ θ ð36Þ ij _ From eq 31, the off-diagonal elements of G R(s) are expressed as " # n k ^ ^ Þs il τil s þ 1 ðθil  θ jl e g r, ij ðsÞ ¼ gr, ii ðsÞ ð37Þ ^ l ¼ 1 kjl τ jl s þ 1

∑

for i 6¼ j = 1, 2, ..., n. Because it has been shown that8 n

il ¼ 0, ∑ ^ k l ¼ 1 jl

k

i 6¼ j ¼ 1, 2, ... , n

ð38Þ

ð41Þ

Therefore, eqs 34 and 39 will unite into ^ ¼ θβi  θβj ^ θ θ ii ij

ð42Þ

where κij = 1(j 6¼ i). In the second case, θβi  θβj* < θii  θij*. In this case, eq 34 is in conflict with eq 39. To guarantee the stability of the overall system, eq 41 is modified as ^  ¼ θii  θij ^ θ θ ii ij

g iij(s)

where λij* < 0. 4.2. Parameterization of ETFs. From eqs 5 and 11, the closed-loop steady-state gain and average resident time are easily obtained as ^kij ¼ kij ð35Þ λij

ð40Þ

where β denotes the loop on which the impact of the jth loop is more aggressive than that on the other loops, if eq 39 does not hold. To further determine the closed-loop dead time and time constant, two different cases are discussed, respectively: In the first case, θβi  θβj* g θii  θij*. This indicates that eq 34 holds when

ij

then will respond first to any external excitation in this loop. Even_ though controller is designed based on GR, it actually acts on G R. In such cases, the loop will exhibit an inverse response, and the dynamic performance will deteriorate. Thus, the controller has to be detuned for the performance and stability of the overall systems. However, the detuning of the controllers for multi-inputmulti-output (MIMO) processes is very complex and will make the overall system more sluggish. In some cases, performance is even worse than decentralized controls. Therefore, a constraint on the ETF parameters is obtained as ^ e θij  θ ^  ðj 6¼ iÞ ð34Þ θii  θ ii ij

ð39Þ

Then, eqs 34 and 43 are unified as 8 ^ ¼ θβi  θβj , ^ θ 0

ð51Þ

Once GR(s) is selected, GI(s) is determined by eq 48. The controller GC(s) can be determined from the elements of GR(s) such that gC,ii(s) gR,ii(s), for i = 1, 2, ..., n, has good dynamic responses. Normally, the gain and phase margin (GPM) method15 is recommended for balanced performance in set-point tracking and disturbance rejection. However, if the loop transfer function has a high D/τ ratio, the gain and phase margin (GPM) method will result in a very aggressive PI/PID controller, which will deteriorate the system stability. In such a case, internal model control (IMC)Maclaurin is recommended for processes with high D/τ ratios (D/τ g 8),16,17 where the PID controller takes the form ! τD, ij s 1 þ gC, ij ðsÞ ¼ kC, ij 1 þ ð52Þ τI, ij s RτD, ij s þ 1

The design procedure of partial decoupling controller is summarized as follows: Step 1: Do loop paring using RGA-based rules. Step 2: Calculate ηij (i 6¼ j) by eq 26 and select the structure of the partial decoupler according to selection criteria in section 3. Step 3: Determine the parameters of the ETF on the basis of ^ T is obtained. eq 37, 45, or eq 46 and eq 47; then, the matrix G Step 4: Choose the desired forward transfer function matrix GR(s) and calculate the decoupler GI(s) by eq 19. Step 5: Design the decentralized controller GC(s) for GR(s). Remark 3: To deal with process uncertainties, an online adjustment could be made monotonically for τR,ii, which was constrained by the expression 0:2θR , ii e τR , ii e 5:0θR , ii

ð53Þ

6. CASE STUDIES In this section, several examples are employed to illustrate the design procedures and the effectiveness of the proposed design technique. Example 1 provides the results of different ways to determine the ETF parameters with the same decoupler structure. The effect of different decoupler structures is compared in Example 2. The last example illustrates the power of the proposed method compared to another existing method. Example 1. Consider a 2  2 process given by Luyben18 2 3 2:2es 1:3e0:3s 6 7s þ 1 7s þ 1 7 7 GðsÞ ¼ 6 4 2:8e1:8s 4:3e0:35s 5 9:5s þ 1 9:2s þ 1 The RGA (Λ) and RARTA (Γ) can be calculated as " # " # 1:6254 0:6254 0:9559 0:8853 Λ¼ , Γ¼ 0:6254 1:6254 0:8853 0:9559 ^ ar are calculated ^ and T According to eqs 35 and 36, the matrices K as " # " # 7:6469 6:4627 1:3535 2:0786 ^ ar ¼ ^ ¼ , T K 10:0038 9:1285 4:4769 2:6455 From eq 26, the index parameters ηij (i 6¼ j) are obtained as η12 ¼ 0:9605,

η21 ¼ 1:0584

^ is determined Following the structure selection procedures, G as " # ^g 11 1 ðsÞ ^g 21 1 ðsÞ T ^ G ðsÞ ¼ ^g 12 1 ðsÞ ^g 22 1 ðsÞ T

From eqs 34 and 39, the relationships between the ETF parameters are ^ > 0:7, ^ θ θ 11 12 ^ ¼ 1:45, ^ θ θ 11 12

^ θ ^ >  1:45 θ 22 21 ^ θ ^ ¼  0:7 θ 22 21

Thus, these relationships can be united as ^ θ ^ ¼ 1:45, θ 11 12 7384

^ θ ^ ¼  0:7 θ 22 21

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By taking θ11 = 2.45 and θ22 = 0.3, the ETF matrix is calculated as 3 2 5:1969s þ 1 2:45s 9:0038s þ 1 s e e 7 6 1:3535 4:4769 7 ^ T ðsÞ ¼ 6 G 5 4 5:4627s þ 1 8:8285s þ 1 s 0:3s e e 2:0786 2:6455 According to eqs 50 and 51, the decoupled forward transfer function is selected as 2 3 1:3535 2:45s e 0 6 5:1969s þ 1 7 7 GR ðsÞ ¼ 6 4 5 2:6455 s e 0 8:8285s þ 1 which gives the stable, causal, and proper decoupler ^ T ðsÞ GR ðsÞ GI ðsÞ ¼ G 2 3 2:6455 9:0038s þ 1 0:7s e 1 6 7 4:4769 8:8285s þ 1 7 ¼6 4 1:3535 5:4627s þ 1 1:45s 5 e 1 2:0786 5:1969s þ 1 Using the gain and phase margin method, we have 2 3 0:09474 0 0:4923 þ 6 7 s 7 GC ðsÞ ¼ 6 4 0:1188 5 0 1:048 þ s

Figure 2. Closed-loop response for Example 1.

The results for the closed-loop step response are given in Figure 2, together with the results from decentralized control and Cai et al.5 It shows that the ETF parameters chosen under the constraints are more reasonable than those obtained by proportional distribution. It indeed makes a great improvement in the performance compared with the decoupler obtained following the same design procedure. In addition, a step signal with a magnitude of 0.2 was added to the inputs of the process at t = 200 s, and the results are included in Figure 2. Example 2. Consider a 3  3 process given by Wang et al.6 2 3 0:26978 27:5s 1:978 0:07724 56s 53:5s e e e 6 7 118:5s þ 1 96s þ 1 6 97:5s þ 1 7 6 0:4881 7 5:26 0:19996 117s 26:5s 35s 7 6 e e e GðsÞ ¼6 7 56s þ 1 58:5s þ 1 51s þ 1 6 7 4 0:6 16:5s 5 5:5 0:5 e e15:5s e17s 40:5s þ 1 19:5s þ 1 18s þ 1 2

0:5s

e 6 6 6 0:4565 34:2498  60s þ 1 ^ T ðsÞ GR ðsÞ ¼ 6 e1:5s GI ðsÞ ¼ G 6 13:7238 76:0594s þ 1 6 4 0:4565 2:4190s þ 1 0:2915 76:0594s þ 1

From eq 26, the index parameters ηij (i 6¼ j) are obtained as η12 ¼ 0:4008, η21 ¼ 0:4982, η31 ¼ 0:0031,

η13 ¼ 1:7150, η23 ¼ 0:0872, η32 ¼ 0:2831

In light of the structure selection criterion, the ETF matrix is reduced to 2 3 ^g 11 1 ðsÞ ^g 21 1 ðsÞ 0 6 ^g 1 ðsÞ ^g 1 ðsÞ ^g 1 ðsÞ 7 7 ^ T ðsÞ ¼ 6 G 22 32 4 12 5 1 1 1 ^g 13 ðsÞ ^g 23 ðsÞ ^g 33 ðsÞ The decoupler GI and controller GC are then designed, respectively, as 3 9:3954 24:6022s þ 1 0:5s e 0 7 1:2044 38:9654s þ 1 7 7 0:7141 13:3991s þ 1 1:5s 8:5s 7 e e 7 18:5798 4:5112s þ 1 7 5 9:3954 302:5445s þ 1 1 5:7334 38:9654s þ 1

and 2

6 0:5784 þ 6 GC ðsÞ ¼ 6 60 6 4 0

0:007604 s

3 0

0

0:0005482 0:02136 þ s

0

0

0:06615 þ 7385

7 7 7 7 7 0:01466 5 s

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Figure 3. Closed-loop responses for Example 2.

Another two different structures for the decoupler, GI1(s) and GI2(s), are also given for comparison; they are expressed as 2 3 ^g 11 1 ðsÞ ^g 21 1 ðsÞ 0 6 7 ^g 22 1 ðsÞ ^g 32 1 ðsÞ 7 GI1 ðsÞ ¼ 6 40 5 ^g 13 1 ðsÞ ^g 23 1 ðsÞ ^g 33 1 ðsÞ and 2

3 ^g 11 1 ðsÞ ^g 21 1 ðsÞ ^g 31 1 ðsÞ 6 1 7 1 1 7 GI2 ðsÞ ¼ 6 4 ^g 12 ðsÞ ^g 22 ðsÞ ^g 32 ðsÞ 5 1 1 1 ^g 13 ðsÞ ^g 23 ðsÞ ^g 33 ðsÞ The results for closed-loop step response of the decoupled process under the same design criterion are shown in Figure 3. Indeed, the simulation results in Figure 3 show that the proposed decoupler structure did provide the best overall system performance. However, upon addition of gi,23 or elimination of gi,12, the performance becomes worse because of the violation of rule 2.

Figure 4. Closed-loop responses for Example 3.

Example 3. Consider a 3  3 process given by Ogunnaike and Ray19 2

0:66 2:6s 6 6:7s þ 1 e 6 6 1:11 e6:5s GðsÞ ¼ 6 6 3:25s þ 1 6 4 34:68 9:2s e 8:15s þ 1

0:61 3:5s e 8:64s þ 1 2:36 3s e 5s þ 1 46:2 e9:4s 10:9s þ 1

3 0:0049 s e 7 9:06s þ 1 7 0:01 1:2s 7 7 e 7 7:09s þ 1 7 0:87ð11:61s þ 1Þ s 5 e ð3:89s þ 1Þð18:8s þ 1Þ

The partial decoupler and decentralized controller are given, respectively, as 2

^ T ðsÞ GR ðsÞ GI ðsÞ ¼ G e0:2s

6 6 6 0:2296s þ 1 ¼6 6 0:3889 0:7483s þ 1 6 4 5:7917s þ 1 8:4s e 19:2164 0:7483s þ 1 7386

0:7527

0:5108s þ 1 0:2s e 1:1328s þ 1

1 23:0964

5:5720s þ 1 8:4s e 1:1328s þ 1

3 0

7 7 7 07 7 7 5 1

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and GC ðsÞ 2 0:1593 6 0:1192 þ s 6 ¼6 60 6 4 0

3 0

0

0:04858 0:05503 þ s

0

0

2:34 þ

7 7 7 7 7 0:3692 5 s

The results for the closed-loop step response are given in Figure 4. It shows that partial decoupling control indeed makes a great improvement in the performance compared with the decoupling controller given by Xiong et al.14 and Lee et al.20 To test the capability of partial decoupling control against the input disturbance, a step signal with a magnitude of 0.2 was added to the input of the processes at t = 1400 s; the simulation results are included in Figure 4.

5. CONCLUSIONS By analyzing the system characteristics after decoupling, two constraints on ETF parameters are derived to guarantee the main-loop performance and achieve a satisfactory decoupling effect. By measuring the effect of each ETF element in decoupling control, a novel partial decoupling control scheme for highdimensional multivariable processes is presented in this work. The structure selection criterion provides a compromise between system performance and structure complexity while guaranteeing close-loop stability. In addition, the ETF parameters are then chosen under these constraints. This method is very easy to understand and implement. The simulation results for several industrial processes show that the partial decoupling control scheme provides an overall system performance that is better than that obtained with decentralized control and, in many cases, comparable to that achieved with full decoupling control. However, the decoupling effects can sometimes be limited by the approximation of the equivalent transfer function by FOPDT. Our further research will focus on the improvement of the overall performance by optimizing the model structure of ETFs. This topic is currently under investigation and will be reported later. ’ AUTHOR INFORMATION Corresponding Author

*Tel./Fax: þ86-21-3420 4011. E-mail: [email protected].

’ ACKNOWLEDGMENT This work was supported by the National Nature Science Foundation of China (Grants 60825302, 60934007, 61074061), by the High Technology Research and Development Program of China (Grant 2009AA04Z162), in part by the Program of Shanghai Subject Chief Scientist and the “Shu Guang” project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation, and by the Key Project of Shanghai Science and Technology Commission (10JC1403400). The authors thank the anonymous reviewers and the Associate Editor for their constructive and valuable comments that have improved the article.

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