Centralized PI/PID Controller Design for Multivariable Processes

Centralized PI/PID Controller Design for Multivariable ProcessesYuling Shen,*,† Youxian Sun,† and Wei Xu‡

†State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, China‡Shanghai Electric Group Co., Ltd. Central Academe, Shanghai 20070, China

ABSTRACT: A novel centralized controller design method is proposed for multivariable systems, whether square or nonsquareprocesses. First, the relationship between equivalent transfer function (ETF) and the pseudo-inverse of multivariable transfermatrix is derived. Second, the relative normalized gain array (RNGA)-based ETF parametrization method is extended to thenonsquare processes. Finally, a centralized proportional integral/proportional integral derivative (PI/PID) multivariablecontroller is obtained from the Maclaurin expansion. The effectiveness of the proposed approach is verified by analysis ofseveral multivariable industrial processes; better overall performance is demonstrated compared with other centralized controlmethods.

1. INTRODUCTION

It is known that not all the processes are square and processeswith unequal number of inputs and outputs are frequentlyencountered in the chemical industry. The traditional approachfor a nonsquare multiple input−multiple output (MIMO)process is to square up or square down the original system intosquare process, which is realized by the addition or removal ofinputs or outputs. It proves to be costly and difficult to getsatisfactory performance.1

For multivariable square systems, Davison has formulated acentralized PI controller design method based on the steadystate gain matrix.2,3 Sarma and Chidambaram extended it formultivariable nonsquare systems.4 For MIMO nonsquare systemswith multiple time delay, the Smith delay compensator is used toenhance the control performance, and the controller is composedof a static decoupler and a decentralized controller.5,6

Many equivalent transfer function (ETF) related methodscan be found in the literature. An ETF-based centralizedcontroller design is simple and easy to implement in engineeringprocesses,7,8 but it is not concerned with MIMO nonsquaresystems, in which the difficulty lies in the ETF modelparametrization. Jin et al. developed the application of ETF intononsquare systems.9 To improve the availability of the ETFmodels, model reduction is involved, and then the neighborhoodsearch-assisted particle swarm optimization (NPSO) algorithm isused to design the internal model control-propotional integralderivative (IMC-PID) controller. However, it is a partial decouplingmethod technically. For square processes (n = m), the parametersof the ETF model can be easily determined by a relativenormalized gain array (RNGA)-based method. When n < m, theinverse operation of matrix, which is involved in this method, doesnot exist any more. In this paper, the RNGA-based ETFparametrization method is extended to nonsquare processes. Inaddition to this, the ETF for process transfer function matrix withzero terms is also discussed.The proposed method is based on the concept of ideal

decoupling. The traditional decoupling techniques include adecoupler and a controller, or an integrated controller withthe aforementioned two functions.10−15 Satisfactory controlresults for square systems are available from both of them.

As the decoupler or controller is designed based on a modelinverse, it may result in a very complicated structure. In contrast,the centralized controller is composed of PI/PID controllers, andeach of them is designed independently. The resulting controlleris simple and easy for application. Above all, the presentedapproach is suitable for both square and nonsquare processes.Since nonsquare systems with more outputs than inputs are

generally not desirable, the nonsquare systems with moreinputs than outputs still often arise in the chemical process.Therefore, this article mainly focuses on the centralizedcontroller design for multivariable processes with control inputsnot less than outputs.

2. PRELIMINARIES

A general multivariable control system is depicted as in Figure 1,where G(s) is an n × m (n ≤ m) process transfer function,described by

=

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥s

g s g s g s

g s g s g s

g s g s g s

G( )

( ) ( ) ... ( )

( ) ( ) ... ( )... ... ... ...

( ) ( ) ... ( )

n

n

n n nn

11 12 1

21 22 2

1 2 (1)

and GC(s)is a m × n multivariable centralized PI/PID controller,represented by

=

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥s

g s g s g s

g s g s g s

g s g s g s

G ( )

( ) ( ) ... ( )

( ) ( ) ... ( )... ... ... ...

( ) ( ) ... ( )

n

n

n n nn

C

c,11 c,12 c,1

c,21 c,22 c,2

c, 1 c, 2 c, (2)

Received: January 27, 2014Revised: May 23, 2014Accepted: May 27, 2014Published: May 27, 2014

Article

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© 2014 American Chemical Society 10439 dx.doi.org/10.1021/ie501541s | Ind. Eng. Chem. Res. 2014, 53, 10439−10447

pubs.acs.org/IECR

2.1. Equivalent Transfer Function (ETF) Matrix. LetG(s) be the equivalent transfer function (ETF) matrix ofG(s), i.e.

=

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥s

g s g s g s

g s g s g s

g s g s g s

G( )

1/ ( ) 1/ ( ) ... 1/ ( )

1/ ( ) 1/ ( ) ... 1/ ( )... ... ... ...

1/ ( ) 1/ ( ) ... 1/ ( )

m

m

n n nm

11 12 1

21 22 2

2 2 (3)

where gij(s) is the equivalent transfer function of gij(s) when allthe other loops are closed under control. For simplicity, theform of first order plus dead time (FOPDT) model is adoptedfor gij(s), that is,

τ =

+θ− g s

k

s( )

1eij

ij

ij

sij

(4)

where kij, τij and θij are the steady-state gain, time constant, anddead time, respectively.2.2. Parameterization for ETF. Define the normalized

steady-state gain matrix of G(s), calculated as

= ⊙K K TN AR (5)

where K = [gij(0)]n×m, TAR = [τar.ij]n×m, and τar.ij is the averageresident time of gij(s). The calculating formulas for τar.ij aregiven in Table 1. Correspondingly, the normalized steady-stategain matrix of G(s) is defined as

= ⊙ K K TN AR (6)

where K = [gij(0)]n×m, TAR = [τar.ij]n×m and τar.ij is the averageresident time of gij(s).As is known, the generalized relative gain array (GRGA) for

multivariable process is defined as

Λ = ⊙ K K (7)

and calculated by16

Λ = ⊗ +K K T (8)

where K = [gij(0)]n×m and K+ is the generalized inverse matrixof K. Similar to GRGA, the generalized relative normalized gainarray (GRNGA) can be defined as

Λ = ⊙ K KN N N (9)

and calculated by17

Λ = ⊗ +K KN N NT

(10)

Substituting eq 5 and 6 into eq 9, it is finally obtained as

Λ = Λ ⊙ ΓN (11)

Where the relative average resident time array (RARTA), Γ, isexpressed as

γΓ = = ⊙× T T[ ]ij n m AR AR (12)

Then, the RARTA can be obtained from eq 11 as

Γ = Λ ⊙ ΛN (13)

Figure 1. Block diagram of multivariable feedback structure.

Table 1. Average Residence Times of FOPDT and SOPDTModels

g(s) τar

FOPDTτ

θ+

−eks

s1

θ+τ

SOPDT

τθ

+−ek

ss

( 1)2θ+2τ

τ τ≠τ τ

θ+ +

−e ( )ks s

s( 1)( 1) 1 21 2

θ+τ1+τ2

ξ <ωξω ω

θ+ +

−e ( 1)k

s ss

2n

n n

2

2 2 θ + ξω

2n

Industrial & Engineering Chemistry Research Article

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From eq 8 and eq 12, the ETF parameter matrix can bedetermined as

= ⊙ ΛK K (14)

and

= Γ⊗T TAR AR (15)

According to Table 1, the average resident time matrix of gij(s)is calculated as

= + ΘT TAR (16)

where T = [τij]n×m and Θ = [θij]n×m. Generally, the dead time ofETF is determined by

Θ = Θ Γ⊗ (17)

Then, the time constant of ETF is calculated as

= − ΘT TAR (18)

Remark 1: When gij(s) = 0, there is a situation, in which zerois divided by zero. In this case, the ETF of gij(s) is modeled as

= g s k( )ij ij (19)

2.3. Relationship between G(s) and G(s). As forNonsquare Multivariable Processe, The relative dynamic gainarray (RDGA) can be defined as18

Λ =

=

⊗

⊗

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

s s s

g s g s g s

g s g s g s

g s g s g s

g s g s g s

g s g s g s

g s g s g s

G G( ) ( ) ( )

( ) ( ) ... ( )

( ) ( ) ... ( )... ... ... ...

( ) ( ) ... ( )

1/ ( ) 1/ ( ) ... 1/ ( )

1/ ( ) 1/ ( ) ... 1/ ( )... ... ... ...

1/ ( ) 1/ ( ) ... 1/ ( )

m

m

n n nm

m

m

n n nm

11 12 1

21 22 2

1 2

11 12 1

21 22 2

2 2 (20)

In ideal control, the following relationship holds for controlvariable and output variable as

= +s s sU G Y( ) ( ) ( ) (21)

where

= ···s u s u s u sU( ) [ ( ), ( ), , ( )]m1 2T

(22)

= ···s y s y s y sY( ) [ ( ), ( ), , ( )]n1 2T

(23)

and G+(s) is the generalized inverse matrix of G(s), which iscalculated as

=

···

···

⋮ ⋮ ⋱ ⋮

···

= ′ ′

+

+ + +

+ + +

+ + +

−

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

s

g s g s g s

g s g s g s

g s g s g s

s s s

G

G G G

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )( ( ) ( ))

n

n

m m mn

11 12 1

21 22 2

1 2

1(24)

From eq 21, the gain from uj(s) to yi(s) can be taken as 1/gji+(s).

By definition, the RDGA can be obtained as

Λ =

⊙

= ⊗

+ + +

+ + +

+ + +

+

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

s

g s g s g s

g s g s g s

g s g s g s

g s g s g s

g s g s g s

g s g s g s

s sG G

( )

( ) ( ) ... ( )

( ) ( ) ... ( )... ... ... ...

( ) ( ) ... ( )

1/ ( ) 1/ ( ) ... 1/ ( )

1/ ( ) 1/ ( ) ... 1/ ( )... ... ... ...

1/ ( ) 1/ ( ) ... 1/ ( )

( ) ( )

m

m

n n nm

m

m

n n mn

11 12 1

21 22 2

1 2

11 21 1

12 22 2

1 2

T(25)

In contrast with eq 20, the following relationship is derived as

= +s sG G( ) ( )T(26)

3. MULTIVARIABLE PI/PID CONTROLLER DESIGNTheorem 1. The ideal control objective for multivariableprocess G(s) is equivalent to that for multiple single processesgij(s), that is,

= ⇔ =s sIs

g s g ss

G G( ) ( ) ( ) ( )1

ij jiC c, (27)

whereGC(s) = [gc,ij(s)]m×n is the multivariable controller for G(s).Proof: In ideal control,19 it is hoped that the multivariable

process is decoupled into

=s ss

G GI

( ) ( )C (28)

Multiplying eq 26 by eq 28 gives

= s ss

G GI

( ) ( )CT

(29)

Then, eq 29 can be further expanded as

···

···

⋮ ⋮ ⋱ ⋮···

=

·

· ···

·

·

· ···

·

⋮ ⋮ ⋱ ⋮

·

· ···

·

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

g s g s g s

g s g s g s

g s g s g s

g s s g s s g s s

g s s g s s g s s

g s s g s s g s s

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1( )

1 1( )

1 1( )

1

1( )

1 1( )

1 1( )

1

1( )

1 1( )

1 1( )

1

n

n

m m mn

n

n

m m nm

c,11 c,12 c,1

c,21 c,22 c,2

c, 1 c, 2 c,

11 21 1

12 22 2

1 2 (30)

According to one-to-one correspondence rule, the followingrelationship exists

=g s g ss

( ) ( )1

ij jic, (31)

for i = 1,2,···,m and j = 1,2,···,n.



Lemma 1. The multivariable process control problem withthe goal of

=⋱

α

α

α

−

−

−

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

s s

k

s

k

s

k

s

G G( ) ( )

e

e

e

l s

l s

nl s

C

,1

,2

,n

1

2

(32)

can be solved by designing the single loop controllers so as to

= α−

g s g sk

s( ) ( )

eij ji

jl s

c,,

j

(33)

where lj = min {θji, I = 1,2···,n} ; kα,j, j = 1,2···,n are theregulation parameters and 0 < kα,j ≤ 1. Equation 33 can berewritten as

=g ss

F s( )1

( )ijc, (34)

where

= = α α−F s k g s k g s( ) e / ( ) / ( )ji j

l sji j ji, ,

j(35)

Applying the Maclaurin series expansion to the above equation,the controller can be expressed as

= + ′ + ″ + ···g ss

F sF s F( )1

[ (0) (0) (0) ]ij ji ji jic,2

(36)

Substituting eq 35 into eq 36, the controller can be furtherrepresented as

=

−′

+

′

−″

+ ···

α⎡

⎣⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦⎥⎥

g sk g

ss

g

gs

g

g

g

g

( )/ (0)

1(0)

(0)2

(0)

(0)

(0)

(0)

ij

j ji ji

ji

ji

ji

ji

ji

c,

, 22

2

(37)

(1) The standard PI controller form is

= +g s kk

s( )ij ij

ijc, P,

I,

(38)

where kP,ij and kI,ij are the controller parameters. Comparingeq 37 with eq 38, the controller parameters are derived as

= ′

=

α

α

⎧⎨⎪⎩⎪

k k g g

k k g

(0)/ (0)

/ (0)

ij j ji ji

ij j ji

P, ,2

I, , (39)

When the ETF elements take the model form of FOPDT, itfollows that

τ′

= − +

g

gl

(0)

(0)ji

jiji jar,

(40)

Therefore, the PI controller is designed as

τ= −

= α

α

⎧⎨⎪⎩⎪

k k l k

k k k

( )/

/

ij j ji j ji

ij j ji

P, , ar,

I, , (41)

Figure 2. Step response of Example 1.




(2) The standard PID controller form is

= + +g s kk

sk s( )ij ij

ijijc, P,

I,D, (42)

where kP,ij, kI,ij and kD,ij are the controller parameters.Comparing eq 37 with eq 42, the controller parameters arederived as

= ′

=

= ′ − ″

α

α

α

⎧

⎨⎪⎪

⎩⎪⎪

k k g g

k k g

k k g g g g

(0)/ (0)

/ (0)

(2 (0) (0) (0))/ (0)

ij j ji ji

ij j ji

ij j ji ji ji ji

P, ,2

I, ,

D, ,2 2

(43)

For ETF in the form of FOPDT, it is obtained that

τ τ″

= − +

g

gl

(0)

(0)( )ji

jiji j jiar,

2 2

(44)

Substituting eq 40 and eq 44 into eq 43, the PID controller isdesigned as

τ

τ τ

=

=

= − −

α

α

α

⎧

⎨⎪⎪

⎩⎪⎪

k k k

k k k

k k l k

/

/

(( ) )/

ij j ji ji

ij j ji

ij j ji j ji ji

P, , ar,

I, ,

D, , ar,2 2

(45)

To evaluate the control system performance, the ISEperformance index is introduced, which is calculated as follows

∑ ∑== =

−ISE ISEi

n

j

m

y r1 1

i j(46)

where ISEyi−ri = ∫ 0∞[1 − yi]

2 and ISEyi−rj = ∫ 0∞[0 − yi]

2 (j ≠ i).The detailed steps to design the multivariable PI/PID

controllers are given as follows:Step 1: obtain the average resident time matrix TAR from Table 1Step 2: calculate GRGA, GRNGA, and RARTA by eq 8,

eq 10. and eq 13Step 3: determine the ETF parameters K, TAR and T by

eq 14, eq 15, and 18Step 4: design the PI/PID controller according to eqs 41 or

eqs 45Remark 2: kα,j is tuned online to get a good compromise

between the features of fast track and small overshoot.

4. CASE STUDY

Example 1. Consider the shell control problem (2 × 3)20

=+ + +

+ + +

− − −

− − −

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥s

s s s

s s s

G( )

4.05 e50 1

1.77 e60 1

5.88 e50 1

5.39 e50 1

5.72 e60 1

6.9 e40 1

s s s

s s s

81 84 81

54 42 45

Figure 3. Manipulated variable responses of Example 1.

Table 2. ISE Values of Centralized Controller for Example 1

ISE values

method step in y1 y2 sum of ISE

Jin r1 132.68 73.47 206.15r2 15.10 69.84 84.94

Davison r1 115.79 6.41 122.20r2 0.36 81.40 81.76

proposed r1 111.89 1.61 113.50r2 0.84 73.75 74.59




The RGA, RNGA, and RARTA are calculated for Example 2 as

Λ =−

− −

Λ =−

− −

Γ =

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

0.3203 0.5946 1.27440.0170 1.5733 0.5563

,

0.6662 0.6248 0.95850.3174 1.6153 0.2978

,

2.0803 1.0507 0.752218.6639 1.0267 0.5354

N

Then, the ETF parameter matrices are determined by eq 14

and eq 15, respectively.

Figure 4. Step response of Example 1: Perturbed case.

Table 3. ISE Values of Centralized Controller for Example 1with Perturbation

ISE values

method step in y1 y2 sum of ISE

Jin r1 146.50 108.04 254.54r2 24.12 102.37 126.49

Davison r1 210.68 32.88 243.56r2 1.82 90.72 92.54

proposed r1 142.85 3.21 146.06r2 2.65 82.90 85.55

Chart 1

Table 4. ISE Values of Centralized Controller for CrudeDistillation Column Example

ISE values

method step in y1 y2 y3 y4 sum of ISE

Davison r1 2.27 0.23 0.19 0.20 2.89r2 1.00 6.18 3.34 0.95 11.47r3 0.09 0.10 3.80 0.34 4.33r4 0.07 0.09 0.20 2.10 2.46

Tanttu r1 11.87 1.48 1.48 0.02 14.85r2 0.21 17.20 0.35 0.03 17.79r3 0.12 0.31 17.45 0.12 18.00r4 0.26 0.97 1.18 14.87 17.28

proposed r1 5.47 0.12 0.1 0.15 5.84r2 0.15 4.94 0.16 0.16 5.41r3 0.04 0.01 5.83 0.34 6.22r4 0.02 0.07 0.15 5.26 5.5




http://pubs.acs.org/action/showImage?doi=10.1021/ie501541s&iName=master.img-004.png&w=500&h=110

=−

− −

=

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

K

T

12.64 2.9766 4.6140316.9045 3.6356 12.4033

,

272.5140 151.2953 98.53301941.0417 104.7211 45.5088AR

Set kα,1 = 0.008, kα,2 = 0.012 and solve the eqs 41; themultivariable controller is designed as

=

+ − −

− − +

+ − −

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

s s

s s

s s

G

0.06580.0006326

0.067120.00003787

0.29070.002688

0.20330.003301

0.12910.001734

0.020720.0009675

C

The simulation results of the other two centralized controlmethods3,9 are compared in Figure 2 and Figure 3, and ISEvalues are given in Table 2. It is shown that the proposedapproach gets the best control performance and lessinteractions between loops. The manipulated variable curvesare relatively smooth, which is valuable for the controltechnique when it is applied in practice, because the abruptchange of control signal is undesirable for the actuator.To test the robustness of the proposed method, we mismatch

the process model by increasing all six steady-state gains, sixtime constants, and six time delays by a factor of 1.2, separately.Meanwhile, all the controllers are kept the same as before. Thecomparison results are shown in Figure 4, and the ISE valuesare given in Table 3. It shows that under such modelmismatches, the deterioration is reasonable compared with thesize of the perturbation.

Figure 5. Step response of Example 2.




Example 2. Consider the crude distillation process (4 × 5)21

=

++ + +

++ + +

++ +

++ +

− ++ +

− ++ +

− − ++ +

+ +

+ + +

− ++ +

−+

− ++ +

−

−

−

−

−

− −

− −

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

s

ss s s

ss s s

ss s

ss s

ss s

ss s

ss s

ss s

ss

s s

ss s s

ss s

G( )

3.8(16 1)140 14 1

2.9 e10 1

3.9(4.5 1)96 14 1

6.320 1

3.8(0.8 1)23 13 1

6.1(12 1) e337 34 1

1.62(5.3 1) e13 13 1

1.53(3.1 1)5.1 7.1 1

0 00.73( 16 1) e150 20 1

0 016 e

(5 1)(14 1)

3.4e6.9 1

022 e

(5 1)(10 1)

1.3(7.6 1)4.7 7.1 1

0.6 e2 1

0.32( 9.1 1)e12 15 1

s

s

s

s

s

s s

s s

2

6

2

2 2

2 2

4

2

2

2 2

2 2

It is observed that g25(s) and g35(s) are the processes with

derivative element. Sinceg25(0) = g35(0) = 0, so we consider

them as disturbance terms and then the process transfer

function is simplified into

=

+ +

+ +

+ +

−+

−+

−+

+

−+

−+ +

− −

−

− −

− −

−

−

− − −

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

s

s s

s s

s s

s s

s

s

s s s

G( )

3.8 e0.17 1

2.9 e10 1

3.9 e12.68 1

6.320 1

3.8 e11.51 1

6.1 e24.91 1

1.62 e6.66 1

1.53 e4.58 1

0 00.73e

25.73 10 0 0

3.4 e6.9 1

0 0

1.3 e0.41 1

0.6 e2 1

0.32 e14.70 1

s s

s

s s

s s

s

s

s s

3.2 6

1.07

0.826 1.37

0.166 0.0978

19.8

2

0.0575 10.5

Following the above design procedure, the ETF parameter

matrices are determined as

=

− − − −− −

− −− − −

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥K

2.2574 3.6466 322.3574 1.7078 6.52274.9692 3.5297 3.5289 2.0383 12.8340

42.8616 69.2379 3.4019 1.5883 15.607316.5500 26.7346 2363.3397 0.6072 6.0264

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥T

2.3206 3.3755 0 0 3.48743.1616 13.2277 0 0 03.2647 4.7487 8.8377 0 00.1727 0.2512 6.3957 2.9873 3.7839

AR

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥T

0.1170 2.1097 0 0 1.97082.9156 13.2277 0 0 03.0461 4.5011 6.8517 0 00.1685 0.2460 5.6090 1.9915 2.2072

Set kα,1 = kα,2 = kα,3 = kα,4 = 0.1 and solve the eqs 45; themultivariable controller is designed as seen in Chart 1.It can be seen from Figure 5 and Table 4 that the proposed

method offers satisfactory control performance compared withthe other two controllers.3,22 Thus, the proposed method is stilleffective for high-dimension processes. Especially, simple designis one of its biggest advantages.

5. CONCLUSIONIn this work, a novel multivariable centralized controller designmethod is proposed, which is effective for both square andnonsquare processes. The RNGA-based ETF parametrizationmethod is extended to all multivariable processes. The multivariablePI/PID controller is determined by Maclaurin expansion. Eachcontroller is designed independently for corresponding ETF. Themajor advantage of the proposed approach is that it can achievesatisfactory performance with simple control structure, which isdemonstrated by two simulation examples.

■ AUTHOR INFORMATIONCorresponding Author*Tel./Fax.: +86-21-2602 7776. E-mail: [email protected] authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe are very grateful to the editors and anonymous reviewersfor their valuable comments and suggestions to help improveour paper. This work is supported by the Key Program ofNational Natural Science Foundation of China (No. 61333007)and the Major Program of National Natural Science Foundationof China (No. 61290321).

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mailto:[email protected]

mailto:[email protected]

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Centralized PI/PID Controller Design for Multivariable Processes