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Centralized PI/PID Controller Design for Multivariable Processes Yuling 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 nonsquare processes. First, the relationship between equivalent transfer function (ETF) and the pseudo-inverse of multivariable transfer matrix is derived. Second, the relative normalized gain array (RNGA)-based ETF parametrization method is extended to the nonsquare processes. Finally, a centralized proportional integral/proportional integral derivative (PI/PID) multivariable controller is obtained from the Maclaurin expansion. The effectiveness of the proposed approach is verified by analysis of several multivariable industrial processes; better overall performance is demonstrated compared with other centralized control methods.

1. INTRODUCTION It is known that not all the processes are square and processes with unequal number of inputs and outputs are frequently encountered in the chemical industry. The traditional approach for a nonsquare multiple input−multiple output (MIMO) process is to square up or square down the original system into square process, which is realized by the addition or removal of inputs or outputs. It proves to be costly and difficult to get satisfactory performance.1 For multivariable square systems, Davison has formulated a centralized PI controller design method based on the steady state gain matrix.2,3 Sarma and Chidambaram extended it for multivariable nonsquare systems.4 For MIMO nonsquare systems with multiple time delay, the Smith delay compensator is used to enhance the control performance, and the controller is composed of a static decoupler and a decentralized controller.5,6 Many equivalent transfer function (ETF) related methods can be found in the literature. An ETF-based centralized controller design is simple and easy to implement in engineering processes,7,8 but it is not concerned with MIMO nonsquare systems, in which the difficulty lies in the ETF model parametrization. Jin et al. developed the application of ETF into nonsquare systems.9 To improve the availability of the ETF models, model reduction is involved, and then the neighborhood search-assisted particle swarm optimization (NPSO) algorithm is used to design the internal model control-propotional integral derivative (IMC-PID) controller. However, it is a partial decoupling method technically. For square processes (n = m), the parameters of the ETF model can be easily determined by a relative normalized gain array (RNGA)-based method. When n < m, the inverse operation of matrix, which is involved in this method, does not exist any more. In this paper, the RNGA-based ETF parametrization method is extended to nonsquare processes. In addition to this, the ETF for process transfer function matrix with zero terms is also discussed. The proposed method is based on the concept of ideal decoupling. The traditional decoupling techniques include a decoupler and a controller, or an integrated controller with the aforementioned two functions.10−15 Satisfactory control results for square systems are available from both of them. © 2014 American Chemical Society

As the decoupler or controller is designed based on a model inverse, it may result in a very complicated structure. In contrast, the centralized controller is composed of PI/PID controllers, and each of them is designed independently. The resulting controller is simple and easy for application. Above all, the presented approach is suitable for both square and nonsquare processes. Since nonsquare systems with more outputs than inputs are generally not desirable, the nonsquare systems with more inputs than outputs still often arise in the chemical process. Therefore, this article mainly focuses on the centralized controller design for multivariable processes with control inputs not 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 ⎡ g (s ) ⎢ 11 ⎢ g (s ) G(s) = ⎢ 21 ⎢ ... ⎢ g (s ) ⎣ n1

g12(s) ... g1n(s) ⎤ ⎥ g22(s) ... g2n(s)⎥ ⎥ ... ... ... ⎥ gn2(s) ... gnn(s)⎥⎦

(1)

and GC(s)is a m × n multivariable centralized PI/PID controller, represented by ⎡ g (s ) ⎢ c,11 ⎢ g (s ) GC(s) = ⎢ c,21 ⎢ ... ⎢ ⎣ gc, n1(s) Received: Revised: Accepted: Published: 10439

gc,12(s) ... gc,1n(s)⎤ ⎥ gc,22(s) ... gc,2n(s)⎥ ⎥ ... ... ... ⎥ ⎥ gc, n2(s) ... gc, nn(s)⎦

(2)

January 27, 2014 May 23, 2014 May 27, 2014 May 27, 2014 dx.doi.org/10.1021/ie501541s | Ind. Eng. Chem. Res. 2014, 53, 10439−10447

Industrial & Engineering Chemistry Research

Article

Figure 1. Block diagram of multivariable feedback structure.

2.1. Equivalent Transfer Function (ETF) Matrix. Let Ĝ (s) be the equivalent transfer function (ETF) matrix of G(s), i.e. ⎡1/g ̂ (s) ⎢ 11 ⎢ ̂ (s ) ̂ s) = ⎢1/g21 G( ⎢ ... ⎢1/g ̂ (s) ⎣ 2n

1/g12 ̂ (s) ... 1/g1̂ m (s) ⎤ ⎥ 1/g22 ̂ (s) ... 1/g2̂ m (s)⎥ ⎥ ... ... ... ⎥ 1/g2̂ n (s) ... 1/gnm ̂ (s)⎥⎦

Table 1. Average Residence Times of FOPDT and SOPDT Models

FOPDT

k τs + 1

kiĵ τiĵ s + 1

k

(3)

k (τ1s + 1)(τ2s + 1)

SOPDT

kωn2 s 2 + 2ξωns + ωn2

e−θs (τ1 ≠ τ2)

θ+τ1+τ2

e−θs (ξ < 1)

θ + 2ω

ξ n

(8)

where K = [gij(0)]n×m and K+ is the generalized inverse matrix of K. Similar to GRGA, the generalized relative normalized gain array (GRNGA) can be defined as ΛN = KN ⊙ K̂ N

(5)

and calculated by

(9)

17

ΛN = KN ⊗ K+NT

(10)

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

ΛN = Λ ⊙ Γ

(6)

(11)

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

where K̂ = [ĝij(0)]n×m, T̂ AR = [τ̂ar.ij]n×m and τ̂ar.ij is the average resident time of ĝij(s). As is known, the generalized relative gain array (GRGA) for multivariable process is defined as

Λ = K ⊙ K̂

θ+2τ

Λ = K ⊗ K +T

(4)

where K = [gij(0)]n×m, TAR = [τar.ij]n×m, and τar.ij is the average resident time of gij(s). The calculating formulas for τar.ij are given in Table 1. Correspondingly, the normalized steady-state gain matrix of Ĝ (s) is defined as ̂ K̂ N = K̂ ⊙ TAR

e−θs

θ+τ

and calculated by16

̂

e−θijs

where kiĵ , τ̂ij and θ̂ij are the steady-state gain, time constant, and dead time, respectively. 2.2. Parameterization for ETF. Define the normalized steady-state gain matrix of G(s), calculated as KN = K ⊙ TAR

e−θs

(τs + 1)2

where ĝij(s) is the equivalent transfer function of gij(s) when all the other loops are closed under control. For simplicity, the form of first order plus dead time (FOPDT) model is adopted for ĝij(s), that is, giĵ (s) =

τar

g(s)

̂ ⊙ TAR Γ = [γij]n × m = TAR

(12)

Then, the RARTA can be obtained from eq 11 as Γ = ΛN ⊙ Λ

(7) 10440

(13)

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


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From eq 21, the gain from uj(s) to yi(s) can be taken as 1/g+ji (s). By definition, the RDGA can be obtained as

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

K̂ = K ⊙ Λ

⎡ g (s) g (s) ... g (s)⎤ 12 1m ⎢ 11 ⎥ ⎢ g (s) g (s) ... g (s)⎥ 22 2m Λ(s) =⎢ 21 ⎥ ... ... ... ⎥ ⎢ ... ⎢ g (s) g (s) ... g (s)⎥ ⎣ n1 ⎦ n2 nm ⎡1/g +(s) 1/g + (s) ... 1/g + (s)⎤ 21 m1 ⎢ 11 ⎥ ⎢1/g + (s) 1/g + (s) ... 1/g + (s)⎥ 22 m2 ⎥ ⊙ ⎢ 12 ... ... ... ⎥ ⎢ ... ⎢ + ⎥ + + ⎣1/g1n(s) 1/g2n(s) ... 1/gmn(s)⎦

(14)

and

̂ = TAR ⊗ Γ TAR

(15)

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

̂ = T̂ + Θ̂ TAR

(16)

where T̂ = [τ̂ij]n×m and Θ̂ = [θ̂ij]n×m. Generally, the dead time of ETF is determined by Θ̂ = Θ ⊗ Γ

(17)

=G(s) ⊗ G+T(s)

Then, the time constant of ETF is calculated as

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

̂ − Θ̂ T̂ = TAR

(18)

Ĝ (s) = G+T(s)

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

giĵ (s) = kiĵ

(19)

Λ(s) = G(s) ⊗ Ĝ (s) ⎡ g (s ) g (s ) 12 ⎢ 11 ⎢ g (s ) g (s ) 22 = ⎢ 21 ... ⎢ ... ⎢ g (s ) g (s ) ⎣ n1 n2

... g1m(s) ⎤ ⎥ ... g2m(s)⎥ ⎥ ... ... ⎥ ... gnm(s)⎥⎦

1/g12 ̂ (s) ... 1/g1̂ m (s) ⎤ ⎥ 1/g22 ̂ (s) ... 1/g2̂ m (s)⎥ ⎥ ... ... ... ⎥ 1/g2̂ n (s) ... 1/gnm ̂ (s)⎥⎦

⎡ g (s) g (s) c,12 ⎢ c,11 ⎢ g (s) g (s) c,22 ⎢ c,21 ⎢ ⋮ ⋮ ⎢ ⎢ g (s) g (s) ⎣ c, m1 c, m2

(20)

In ideal control, the following relationship holds for control variable and output variable as U(s) = G+(s) Y(s)

(21)

where T

U(s) = [u1(s), u 2(s), ···, um(s)]

(22)

Y(s) = [y1(s), y2 (s), ···, yn (s)]T

(23)

⎡ 1 1 · ⎢ ̂ (s) s ⎢ g11 ⎢ 1 1 ⎢ · ̂ (s) s = ⎢ g12 ⎢ ⋮ ⎢ ⎢ 1 1 ⎢ · ⎣⎢ g1̂ m (s) s

+

and G (s) is the generalized inverse matrix of G(s), which is calculated as ⎡ g + (s ) g + (s ) 12 ⎢ 11 ⎢ g + (s ) g + (s ) 22 G+(s) = ⎢ 21 ⎢ ⋮ ⋮ ⎢ + + ⎢⎣ g (s) g (s) m1 m2

··· g1+n(s) ⎤ ⎥ ··· g2+n(s) ⎥ ⎥ ⋱ ⋮ ⎥ ⎥ + ··· gmn (s)⎥⎦

= G̅ ′(s)(G(s)G̅ ′(s))−1

(26)

3. MULTIVARIABLE PI/PID CONTROLLER DESIGN Theorem 1. The ideal control objective for multivariable process G(s) is equivalent to that for multiple single processes ĝij(s), that is, I 1 GC(s) G(s) = ⇔ gc, ij(s)gjî (s) = (27) s s where GC(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 I GC(s) G(s) = (28) s Multiplying eq 26 by eq 28 gives I T GC(s) = Ĝ (s) (29) s Then, eq 29 can be further expanded as

2.3. Relationship between G(s) and Ĝ (s). As for Nonsquare Multivariable Processe, The relative dynamic gain array (RDGA) can be defined as18

⎡1/g ̂ (s) ⎢ 11 ⎢1/g ̂ (s) 21 ⊗⎢ ⎢ ... ⎢1/g ̂ (s) ⎣ 2n

(25)

··· gc,1n(s) ⎤ ⎥ ··· gc,2n(s) ⎥ ⎥ ⋱ ⋮ ⎥ ⎥ ··· gc, mn(s)⎥⎦ 1 1 1 1⎤ · ··· · ⎥ g21 gn̂ 1(s) s ⎥ ̂ (s ) s 1 1 1 1⎥ · ··· · ⎥ g22 gn̂ 2 (s) s ⎥ ̂ (s) s ⎥ ⋮ ⋱ ⋮ ⎥ 1 1 1 1⎥ · ··· · ⎥ g2̂ m (s) s gnm ̂ (s) s ⎥⎦

(30)

According to one-to-one correspondence rule, the following relationship exists

gc, ij(s)gjî (s) = (24)

1 s

(31)

for i = 1,2,···,m and j = 1,2,···,n. 10441



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Figure 2. Step response of Example 1.

Lemma 1. The multivariable process control problem with the goal of ⎡ k e−l1s ⎤ ⎢ α ,1 ⎥ ⎢ ⎥ s ⎢ ⎥ kα ,2 e−l2s ⎢ ⎥ ⎥ GC(s)G(s) = ⎢ s ⎢ ⎥ ⋱ ⎢ ⎥ ⎢ −lns ⎥ kα , n e ⎥ ⎢ ⎢⎣ ⎥⎦ s

Substituting eq 35 into eq 36, the controller can be further represented as ⎛ g ̂ ′ 2(0) kα , j/gjî (0) ⎡ g ̂ji′(0) ji ⎢ 2⎜ gc, ij(s) = 1−s +s 2 2 ⎢ ⎜ s gjî (0) ⎝ g ̂ji (0) ⎣ −

(32)

gc, ij(s) = kP, ij +

(33)

1 F (s ) s

g jî ′(0) gjî (0)

(35)

1 [Fji(0) + sF′ji(0) + s 2F″ji(0) + ···] s

(39)

= −τar,̂ ji + l j (40)

Therefore, the PI controller is designed as

Applying the Maclaurin series expansion to the above equation, the controller can be expressed as gc, ij(s) =

(38)

When the ETF elements take the model form of FOPDT, it follows that

(34)

where Fji(s) = kα , j e−ljs /gjî (s) = kα , j/gjî (s)

s

⎧ k = k g ̂ ′(0)/g ̂ 2(0) α , j ji ⎪ P, ij ji ⎨ ⎪ kI, ij = kα , j/gjî (0) ⎩

where lj = min {θ̂ji, I = 1,2···,n} ; kα,j, j = 1,2···,n are the regulation parameters and 0 < kα,j ≤ 1. Equation 33 can be rewritten as

gc, ij(s) =

kI, ij

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

kα , j e−ljs s

(37)

(1) The standard PI controller form is

can be solved by designing the single loop controllers so as to gc, ij(s) gjî (s) =

⎤ g ̂ji″(0) ⎞ ⎟ + ···⎥ ⎥ gjî (0) ⎟⎠ ⎦

⎧ k = k (τ ̂ − l )/k ̂ α , j ar, ji j ji ⎪ P, ij ⎨ ⎪ kI, ij = kα , j/kjî ⎩

(36) 10442

(41)



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Figure 3. Manipulated variable responses of Example 1. 2 2 ∞ where ISEyi−ri = ∫ ∞ 0 [1 − yi] and ISEyi−rj = ∫ 0 [0 − yi] (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 1 Step 2: calculate GRGA, GRNGA, and RARTA by eq 8, eq 10. and eq 13 Step 3: determine the ETF parameters K̂ , T̂ AR and T̂ by eq 14, eq 15, and 18 Step 4: design the PI/PID controller according to eqs 41 or eqs 45 Remark 2: kα,j is tuned online to get a good compromise between the features of fast track and small overshoot.

(2) The standard PID controller form is gc, ij(s) = kP, ij +

kI, ij s

+ kD, ijs

(42)

where kP,ij, kI,ij and kD,ij are the controller parameters. Comparing eq 37 with eq 42, the controller parameters are derived as ⎧ k = k g ̂ ′(0)/g ̂ 2(0) α , j ji ji ⎪ P, ij ⎪ ⎨ kI, ij = kα , j/gjî (0) ⎪ ⎪ k = k (2g ̂ ′ 2(0) − g ̂ (0)g ̂ ″(0))/g ̂ 2(0) α ,j ji ji ji ji ⎩ D, ij

(43)

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

For ETF in the form of FOPDT, it is obtained that gji″̂ (0) gjî (0)

⎡ 4.05 e−81s 1.77 e−84s 5.88 e−81s ⎤ ⎢ ⎥ 60s + 1 50s + 1 ⎥ ⎢ 50s + 1 G(s) = ⎢ ⎥ −54s 5.72 e−42s 6.9 e−45s ⎥ ⎢ 5.39 e ⎣ 50s + 1 60s + 1 40s + 1 ⎦

= (τar,̂ ji − l j)2 + τji2̂ (44)

Substituting eq 40 and eq 44 into eq 43, the PID controller is designed as ⎧ k = k τ /k ̂ α , j ar, ji ji ⎪ P, ij ⎪ ⎨ kI, ij = kα , j/kjî ⎪ ⎪ k = k ((τ ̂ − l )2 − τ 2̂ )/k ̂ α ,j ar, ji j ji ji ⎩ D, ij

Table 2. ISE Values of Centralized Controller for Example 1 ISE values method

(45)

Jin

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

ISE =

Davison

m

∑ ∑ ISE y − r i

i=1 j=1

proposed

j

(46) 10443

step in

y1

y2

sum of ISE

r1 r2 r1 r2 r1 r2

132.68 15.10 115.79 0.36 111.89 0.84

73.47 69.84 6.41 81.40 1.61 73.75

206.15 84.94 122.20 81.76 113.50 74.59



Article

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

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

Table 4. ISE Values of Centralized Controller for Crude Distillation Column Example

ISE values method Jin Davison proposed

ISE values

step in

y1

y2

sum of ISE

method

step in

y1

y2

y3

y4

sum of ISE

r1 r2 r1 r2 r1 r2

146.50 24.12 210.68 1.82 142.85 2.65

108.04 102.37 32.88 90.72 3.21 82.90

254.54 126.49 243.56 92.54 146.06 85.55

Davison

r1 r2 r3 r4 r1 r2 r3 r4 r1 r2 r3 r4

2.27 1.00 0.09 0.07 11.87 0.21 0.12 0.26 5.47 0.15 0.04 0.02

0.23 6.18 0.10 0.09 1.48 17.20 0.31 0.97 0.12 4.94 0.01 0.07

0.19 3.34 3.80 0.20 1.48 0.35 17.45 1.18 0.1 0.16 5.83 0.15

0.20 0.95 0.34 2.10 0.02 0.03 0.12 14.87 0.15 0.16 0.34 5.26

2.89 11.47 4.33 2.46 14.85 17.79 18.00 17.28 5.84 5.41 6.22 5.5

Tanttu

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

proposed

⎡ 0.3203 − 0.5946 1.2744 ⎤ Λ=⎢ ⎥, ⎣−0.0170 1.5733 − 0.5563⎦ ⎡ 0.6662 − 0.6248 0.9585 ⎤ ΛN = ⎢ ⎥, ⎣−0.3174 1.6153 − 0.2978 ⎦ ⎡ 2.0803 1.0507 0.7522 ⎤ Γ=⎢ ⎥ ⎣18.6639 1.0267 0.5354 ⎦

Then, the ETF parameter matrices are determined by eq 14 and eq 15, respectively.

Chart 1

10444



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Figure 5. Step response of Example 2.

⎡12.64 − 2.9766 4.6140 ⎤ K̂ = ⎢ ⎥, ⎣−316.9045 3.6356 − 12.4033⎦

The simulation results of the other two centralized control methods3,9 are compared in Figure 2 and Figure 3, and ISE values are given in Table 2. It is shown that the proposed approach gets the best control performance and less interactions between loops. The manipulated variable curves are relatively smooth, which is valuable for the control technique when it is applied in practice, because the abrupt change 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, six time constants, and six time delays by a factor of 1.2, separately. Meanwhile, all the controllers are kept the same as before. The comparison results are shown in Figure 4, and the ISE values are given in Table 3. It shows that under such model mismatches, the deterioration is reasonable compared with the size of the perturbation.

⎡ 272.5140 151.2953 98.5330 ⎤ ̂ =⎢ TAR ⎥ ⎣1941.0417 104.7211 45.5088 ⎦

Set kα,1 = 0.008, kα,2 = 0.012 and solve the eqs 41; the multivariable controller is designed as ⎡ 0.0006326 0.00003787 ⎤ − 0.06712 − ⎥ ⎢ 0.0658 + s s ⎥ ⎢ ⎥ ⎢ 0.002688 0.003301 0.2033 + GC = ⎢− 0.2907 − ⎥ s s ⎥ ⎢ ⎢ 0.001734 0.0009675 ⎥ − 0.02072 − ⎥ ⎢ 0.1291 + ⎦ ⎣ s s 10445



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Example 2. Consider the crude distillation process (4 × 5)21 ⎡ 3.8(16s + 1) ⎢ ⎢ 140s 2 + 14s + 1 ⎢ ⎢ 3.9(4.5s + 1) ⎢ 96s 2 + 14s + 1 G(s) = ⎢ ⎢ 3.8(0.8s + 1) ⎢ 2 ⎢ 23s + 13s + 1 ⎢ − 1.62(5.3s + 1) e−s ⎢ 2 ⎣ 13s + 13s + 1

̂ TAR

2.9 e−6s 10s + 1 6.3 20s + 1

⎡ 0.1170 ⎢ 2.9156 T̂ = ⎢ ⎢ 3.0461 ⎢ ⎣ 0.1685

6.1(12s + 1) e−s 337s 2 + 34s + 1 − 1.53(3.1s + 1) 5.1s 2 + 7.1s + 1

0

0

0

0

3.4e−2s 6.9s + 1

0

derivative element. Sinceg25(0) = g35(0) = 0, so we consider them as disturbance terms and then the process transfer

■

function is simplified into ⎡ 3.8 e−3.2s 2.9 e−6s ⎢ 10s + 1 ⎢ 0.17s + 1 ⎢ −1.07s 6.3 ⎢ 3.9 e ⎢ 12.68s + 1 20s + 1 G(s) = ⎢ −0.826s 6.1 e−1.37s ⎢ 3.8 e ⎢ 11.51s + 1 24.91s + 1 ⎢ 0.166 − s ⎢ −1.62 e −1.53 e−0.0978s ⎢ ⎣ 6.66s + 1 4.58s + 1 0

0

0 − 2s

3.4 e 6.9s + 1

0

−1.3 e−0.0575 −0.6 e−s 0.41s + 1 2s + 1

3.4874 ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ 3.7839 ⎦

1.9708 ⎤ ⎥ 13.2277 0 0 0 ⎥ ⎥ 4.5011 6.8517 0 0 ⎥ 0.2460 5.6090 1.9915 2.2072 ⎦ 2.1097

0

0

AUTHOR INFORMATION

*Tel./Fax.: +86-21-2602 7776. E-mail: shenyuling2011@gmail. com. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We are very grateful to the editors and anonymous reviewers for their valuable comments and suggestions to help improve our paper. This work is supported by the Key Program of National Natural Science Foundation of China (No. 61333007) and the Major Program of National Natural Science Foundation of China (No. 61290321).

⎤ −0.73e−19.8s ⎥ ⎥ 25.73s + 1 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ −10.5s ⎥ 0.32 e ⎥ 14.70s + 1 ⎥ ⎦

■

REFERENCES

(1) Loh, E. J.; Chiu, M. S. Robust decentralized control of non-square systems. Chem. Eng. Commun. 1997, 158, 157−180. (2) Davison, E. Some properties of minimum phase systems and “squared-down” systems. IEEE Trans. Autom. Control 1984, 28, 221− 222. (3) Davison, E. J. Multivariable tuning regulators: The feed forward and robust control of a general servomechanism problem. IEEE Trans. Autom. Control 1976, 21, 35−47. (4) Sarma, K. L. N.; Chidambaram, M. Centralized PI/PID controllers for nonsquare systems with RHP zeros. J. Indian Inst. Sci. 2005, 85, 201−204. (5) Seshagiri Rao, A.; Chidambaram, M. Smith delay compensator for multivariable non-square systems with multiple time delays. Comput. Chem. Eng. 2006, 30, 1243−1255. (6) Chen, J.; He, Z.-F.; Qi, X. A new control method for MIMO first order time delay non-square systems. J. Process Control 2011, 21, 538− 546.

matrices are determined as − 1.7078 − 6.5227 ⎤ ⎥ 2.0383 12.8340 ⎥ − 69.2379 3.4019 − 1.5883 15.6073 ⎥ ⎥ − 26.7346 − 2363.3397 − 0.6072 6.0264 ⎦ − 3.6466 3.5297

0 0 0 2.9873

Corresponding Author

Following the above design procedure, the ETF parameter ⎡ 2.2574 ⎢ − 4.9692 K̂ = ⎢ ⎢ 42.8616 ⎢ ⎣16.5500

0 0 8.8377 6.3957

5. CONCLUSION In this work, a novel multivariable centralized controller design method is proposed, which is effective for both square and nonsquare processes. The RNGA-based ETF parametrization method is extended to all multivariable processes. The multivariable PI/PID controller is determined by Maclaurin expansion. Each controller is designed independently for corresponding ETF. The major advantage of the proposed approach is that it can achieve satisfactory performance with simple control structure, which is demonstrated by two simulation examples.

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

0

3.3755 13.2277 4.7487 0.2512

Set kα,1 = kα,2 = kα,3 = kα,4 = 0.1 and solve the eqs 45; the multivariable 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 with the other two controllers.3,22 Thus, the proposed method is still effective for high-dimension processes. Especially, simple design is one of its biggest advantages.

− 0.73(− 16s + 1) e−4s⎤ ⎥ 150s 2 + 20s + 1 ⎥ ⎥ 16s e−2s ⎥ (5s + 1)(14s + 1) ⎥ ⎥ ⎥ 22s e−2s ⎥ (5s + 1)(10s + 1) ⎥ 0.32(− 9.1s + 1)e−s ⎥ ⎥ ⎦ 12s 2 + 15s + 1

− 1.3(7.6s + 1) − 0.6 e−s 4.7s 2 + 7.1s + 1 2s + 1

⎡ 2.3206 ⎢ 3.1616 =⎢ ⎢ 3.2647 ⎢ ⎣ 0.1727

− 322.3574 − 3.5289

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