Controller Design for MIMO Processes Based on Simple Decoupled

Article pubs.acs.org/IECR

Controller Design for MIMO Processes Based on Simple Decoupled Equivalent Transfer Functions and Simplified Decoupler C. Rajapandiyan and M. Chidambaram* Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India ABSTRACT: A method for the independent design of proportional-integral/proportional-integral derivative (PI/PID) controllers is proposed based on the equivalent transfer function (ETF) model of the individual loops and the simplified decoupler matrix. It is shown that the conventional effective open-loop transfer function (EOTF, derived from the dynamic relative gain array (DRGA)) is equivalent to the ETF (derived from the relative normalized gain array (RNGA) and relative average residence time array (RARTA)). This relation is used to approximate the decoupled process models as ETF models. The simplified decoupler is shown to decompose the multiloop systems into independent loops (multi-single loop systems) with the ETFs as the resulting decoupled process model. The concept of the ETF (perfect control approximation) is validated by introducing the decoupler. Based on the corresponding ETFs, the decentralized PI controllers are designed using the simplified internal model control (SIMC) method. Three simulation examples of multi-input multi-output (MIMO) process models are considered to demonstrate the simplicity and effectiveness of the proposed method. The performance of the proposed control system is compared with the ideal, normalized, inverted decoupling, and centralized control systems.

1. INTRODUCTION Most industrial processes are multi-input multi-output (MIMO) systems. Controller design for MIMO processes is difficult compared to that of the single-input single-output (SISO) processes, due to the interactions between the input/ output variables. Since the control loops interact with each other, the tuning of one loop cannot be carried out independently. The MIMO process can be controlled by decentralized (multiloop) controllers, or decoupled controllers, or centralized controllers. In multiloop control, the MIMO processes are treated as a collection of multi-single loops. The controller is designed and implemented on each loop by considering the loop interactions. Multiloop controllers have been widely used due to their reasonable performances, simplicity, and robustness. Many design methods are reported in the literature such as detuning method,1 sequential loop closing method,2 relay auto tuning method,3 and independent design method.4 Decentralized controllers work well when the interactions among the loops are modest. If the interactions are significant, centralized controllers are desirable. It is difficult to design a controller for each loop independently.5 The number of tuning parameters of proportional-integral derivative (PID) controllers for centralized control system is 3n2, where n is the number of inputs. The next approach5 is to design the decoupler with a decentralized control system. This method allows us to use SISO controller design methods and the number of tuning parameters (for PID controller), in this case is 3n. There are three types of basic decoupling techniques available,6 ideal, simplified, and inverted decoupling methods. The ideal decoupling scheme needs to calculate the inverse of the process transfer function matrix. It may result in too complicated decoupling elements, and the ideal decoupling is sensitive to modeling errors.7 The simplified decoupling technique has a simple decoupler form, but the controller cannot be designed directly from the decoupled process model © 2012 American Chemical Society

without introducing the model reduction technique. Weischedel and McAvoy8 have compared the ideal and simplified decoupling control techniques, and they reported that the simplified decoupling is more robust than the ideal decoupling. The inverted decoupling method has the advantages of both the ideal and simplified decoupling techniques. The implementation of the higher dimension system could result in to physical nonrealizability,7,9 and this technique is more sensitive to modeling errors.7,8,10 Gagnon et al.7 and Wade11 discussed the implementation issues in the inverted decoupling method. Tavakoli et al.,5 Kumar et al.,12 and Maghade and Patre13 have used the simplified decoupler plus a decentralized controller. First, the simplified decoupler matrix is introduced in to the process model to decouple the multiloop process into equivalent independent single loops. Second, the resultant decoupled process model is approximated as a first order plus time delay (FOPTD) or second order plus time delay (SOPTD) model using a suitable model reduction technique (Maclaurin series,4 open-loop transient response plot, graphical approach12 and frequency response fitting13). Third, the proportional-integral (PI)/PID controllers are designed independently based on the corresponding reduced decoupled process model. Recently, several researchers have introduced the concept of equivalent transfer functions/effective open-loop transfer functions (ETFs/EOTFs)/effective open-loop process (EOP) to take into account the loop interactions in the design of multiloop control systems.14,15,18−20 Vu and Lee4 have proposed an independent design method for the design of multiloop controllers. Based on the assumption of perfect controllers (other loop), the EOTF is first derived to Received: Revised: Accepted: Published: 12398

March 17, 2012 July 5, 2012 August 27, 2012 August 28, 2012 dx.doi.org/10.1021/ie301448c | Ind. Eng. Chem. Res. 2012, 51, 12398−12410

Industrial & Engineering Chemistry Research

Article

equivalent to the ETF (derived from RNGA and RARTA). This relation is used to approximate the decoupled process model (by the introduction of simplified decoupler matrix) as the ETF model and the SISO controller is designed based on the corresponding ETF. The main advantage of this proposed method compared to the conventional simplified decoupling approach5,12,13 is not requiring the model reduction steps (to reduce the decoupled process model). The formulation of EOTFs for the higher dimension systems is difficult. Further, the model reduction methods also pose problems for higher dimension systems. On other hand, the derivation of ETFs is easy for higher dimension systems. Cai et al.6 have calculated the decoupler matrix by choosing the decoupled process model parameters (g*) as

decompose multiloop control system in to a set of equivalent independent single loops. By using the model reduction technique (Maclaurin series), the EOTF model is reduced to a FOPDT/SOPTD model. Based on the reduced EOTF model, the decentralized controllers are designed independently. Cai et al.6 have proposed a normalized decoupling control for the TITO process. The ETF matrix is derived from the concepts of relative gain array (RGA), relative normalized gain array (RNGA), and relative average residence time array (RARTA). The relationship between the ETF and process transfer function matrices is derived. For deriving the decoupler matrix, the ETF matrix is used instead of the inverse of process transfer function. On the basis of the ETF matrix, the criteria for determining stable, proper, and casual ideal diagonal decouplers are established. In the performance of the decoupled control system, it is found that the interaction is a considerable one and the response is sluggish. Choosing of the decoupled process model involves some ad-hoc procedures. Huang et al.14 formulated EOPs without the prior knowledge of controller dynamics of other loops and that the controllers are independently designed based on the derived EOPs for the corresponding equivalent single loops. Xiong et al.15 have derived the effective transfer function in terms of effective relative gain array for the multivariable system. This effective transfer function model incorporates the interactions among the loops. The decentralized control system is designed by using the single loop design approaches based on the derived effective transfer function. The effective transfer function provides both gain and phase information required for the design of the decentralized controllers. He et al.16 have used the relative normalized gain array (RNGA) concept to determine the interaction measurements and loop pairing. This RNGA includes both the steady state and the transient information of the transfer function matrix. Naini et al.17 introduced effective relative energy array (EREA) concept. EREA is utilized to propose a new loop pairing method for the MIMO systems. Shen et al.18 obtained a full controller matrix by the independent design approach. Controllers for the selected loops are designed independently based on the gain and phase margin (GPM) method. The effective transfer function is obtained by utilizing the RNGA and the relative average residence time array (RARTA) concepts. Hu et al.19 have proposed a decentralized controller design method for the integrating MIMO systems. In this method, based on the ETF, the decentralized controller is designed by the simple internal model control (SIMC) method. Recently, Kumar et al.20 have proposed two methods of designing centralized control system for multi-input, multi-output (MIMO) processes. Centralized PI controllers are designed based on the direct synthesis approach. In the direct synthesis method, the inverse of the process transfer function matrix is approximated by the ETF matrix. In Kumar et al.'s paper,20 it is shown that the centralized control system works better for the systems having RGA less than 1. From the literature, it is observed that the ETF matrix is mainly used to find the inverse of the process model6,18,20 and the perfect control approximation is used to incorporate the loop interactions of multivariable systems in the form of ERGA/EOTF/ETF. In the present work, we combine the simplified decoupler approach5,12,13 with the ETF model approximations6,18,20 in order to get the advantages of both the methods. The conventional EOTF (derived from DRGA) is shown to be

T D(s) = G−1(s)G*(s) = Ĝ (s)G*(s)

whereas, in the present method, there is no need to assume the expression for g*; instead, one can take the ETFs as g*. In the proposed method, the method of designing decoupling control system consists of three steps. First, the multiloop process is decoupled in to independent loops (multi-single loop system) by using the simplified decoupler matrix. Next, the resulting decoupled process model is approximated using the concept of ETF. Based on the corresponding ETFs, the decentralized PI/ PID controllers are designed using the simplified internal model control (SIMC) method. Three simulation examples of MIMO industrial process models are considered to show the simplicity and effectiveness of the proposed method. The objective of the present work is to show the performance improvement and robustness of the combination of simplified decoupler matrix and ETF based controller is better than the existing methods (ideal and normalized decoupling, and centralized controllers).

2. EFFECTIVE OPEN-LOOP TRANSFER FUNCTION (EOTF) Consider the two-input two-output (TITO) systems with the decentralized control system as shown in Figure 1. G(s) and

Figure 1. Simplified decentralized control system of a TITO process.

Gc(s) are process transfer function matrix and decentralized controller matrix of TITO systems respectively and are represented as ⎡ g11 g12 ⎤ ⎥ G (s ) = ⎢ ⎣ g21 g22 ⎦ (1) 12399

dx.doi.org/10.1021/ie301448c | Ind. Eng. Chem. Res. 2012, 51, 12398−12410

Industrial & Engineering Chemistry Research ⎡g 0 ⎤ c11 ⎥ Gc(s) = ⎢ ⎢⎣ 0 gc22 ⎥⎦

Article eff g22 =

(2)

K pij e−θijs (τijs + 1)

i = 1, 2

j = 1, 2 (3)

This input-output relation can be expressed as Y (s ) = G (s )U (s ) ⎡ y (s ) ⎤ 1 ⎥ Y (s ) = ⎢ ⎢ y (s)⎥ ⎣ 2 ⎦

(4)

⎡ u1(s) ⎤ ⎥ U(s) = ⎢ ⎢⎣ u 2(s)⎥⎦

where Y(s) and U(s) are output and input vectors respectively. The input−output relationship for the TITO system can be written as21 (6)

y2 (s) = g21(s)u1 + g22(s)u 2(s)

(7)

u1

In the TITO system, when the second loop is closed, the input from ui to yi has two transmission paths. The combination of two transmission paths is considered as effective open-loop dynamics. If the second feedback controller is in the automatic mode, with yr2 = 0, then the overall closed-loop transfer function between y1 and u1 is given by21 g g g y1 = g11 − 12 21 c2 1 + gc2g22 u1

KNij =

kij

=

u1

= g11 −

σij

τij + θij

⎡ KN,11 KN = ⎢ ⎢⎣ KN,21

u2

= g22 −

(8)

g21g12(gc1g11) g11(1 + gc1g11)

(12)

⎡ k11 k12 ⎤ ⎢ ⎥ KN,12 ⎤ ⎢ τ11 + θ11 τ12 + θ12 ⎥ ⎥= KN,22 ⎥⎦ ⎢ k 21 k 22 ⎥⎥ ⎢ ⎢⎣ τ21 + θ21 τ22 + θ22 ⎥⎦

(13)

In these expressions, σij = τij + θij is the average residence time and it represents the response speed of the controlled variable yi to manipulated variable uj. The relative normalized gain array is expressed using the normalized gain matrix, and hence, the RNGA (denoted as ϕ) can be obtained as

Similarly, for the second loop y2

(11)

and normalized gain matrix is expressed as

g12g21(gc2g22) g22(1 + gc2g22)

g11

kij

This can be written as y1

g21g12

3. EQUIVALENT TRANSFER FUNCTION Because most processes are open-loop stable and exhibit nonoscillatory behavior for step inputs, the response is simplified either by analytical or empirical methods to a FOPDT model for designing the controllers. To describe the dynamic properties of a transfer function, the normalized gain6,16 (KNij) for a particular transfer function, gij(s), is defined as

(5)

y1(s) = g11(s)u1 + g12(s)u 2(s)

= g22 −

eff eff Here, g11 and g22 are the effective open-loop transfer 4 functions (EOTF). These EOTFs are complicated transfer function models, and it is difficult to directly use them for the controller design. For the purpose of controller design, resulting EOTFs are reduced to FOPTD models using Maclaurin series.4 This method poses complications in higher dimension systems, in the formulation of EOTFs and in the model reduction step. In the present work, by using RGA, RNGA, and RARTA concepts,6,16,18−20 the expression for ETF can be derived easily for higher dimension systems also.

The process transfer function models are expressed as FOPDT models gij(s) =

y2

ϕ = KN ⊗ KN−T

(9)

(14)

where

In MIMO systems, the open-loop dynamics between controlled variable (yi) and manipulated variable (ui) not only depend on the corresponding transfer function model (gii) but also depend on the other processes and controllers in all other loops. This implies that the tuning of one controller cannot be done independently and it depends on the other controllers. The complicated relations of eqs 8 and 9 can be simplified by assuming two assumptions:4,19 First, the perfect controller approximation for the other loop (the output attains steady state with no transient) was used to simplify the eqs 8 and 9), that is, gcigii = 1 i = 1, 2 (1 + gcigii)

⎡ϕ ϕ ⎤ 11 12 ⎥ ϕ=⎢ ⎢⎣ ϕ21 ϕ22 ⎥⎦

(15)

Relative average residence time6,16 (γij), which is defined as the ratio of loop yi−uj average residence times, when other loops are closed and when other loops are open: γij =

σiĵ σij

=

ϕij Λij

(16)

When the relative average residence times are calculated for all the elements in the transfer function matrix, the results are in array form, and it is called a relative average residence time array6,16 (RARTA):

Second, ETFs have the same structure of the corresponding open-loop model. By using the perfect controller approximation, eqs 8 and 9 can be approximated as follows: g g y g11eff = 1 = g11 − 12 21 g22 u1 (10)

⎡ γ11 γ12 ⎤ Γ=⎢ ⎥ ⎣ γ21 γ22 ⎦ 12400

(17)



Article

⎡ϕ ϕ ⎤ ⎡ Λ Λ ⎤ 11 12 ⎥ ⊙ ⎢ 11 12 ⎥ Γ=ϕ⊙Λ=⎢ ⎢⎣ ϕ21 ϕ22 ⎥⎦ ⎢⎣ Λ 21 Λ 22 ⎥⎦

(18)

where ⊙ is Hadamard division. Using the definition of the relative average residence time, it is possible to write6,16 σiĵ =γijσij =γijτij + γijθij

(19)

=τiĵ + θiĵ

(20)

By using RGA, RNGA, and RARTA, it is possible that a transfer function element of a MIMO process when other loops are closed6,16 can be approximated by a transfer function element having the same form as open-loop transfer function element: 1 ̂ giĵ (s) = kiĵ e−θijs τiĵ s + 1 giĵ (s) =

Figure 2. Simplified decoupled control system of a TITO process.

(21)

kij

1 e−γijθijs Λij γijτijs + 1

(22)

3.1. Relationship between ĝii(s) and The EOTF can be expressed in terms of elements of dynamic relative gain array (DRGA) as4 g giieff = ii Λii (23) geff ii .

∫0

∫0

/[ =

ωc, ij

ωc, ij

⎡g* 0 ⎤ 11 ⎥ G(s)D(s) = ⎢⎢ ⎥ * ⎣ 0 g22 ⎦

(28)

* 0⎤ ⎡ g11 g12 ⎤ ⎡ 1 d12 ⎤ ⎡ g11 ⎥ ⎥=⎢ ⎢ ⎥⎢ ⎢ g g ⎢ ⎥ ⎣ 21 22 ⎦ ⎣ d 21 1 ⎦ ⎣ 0 g * ⎥⎦ 22

(29)

d12(s) = −

(∂yi /∂ui)dω]all loops open

(∂yi /∂ui)dω]all other loops closed except for loop y − uj i

d 21(s) = −

gij(s) giĵ (s)

DRGA is expressed as Λ (s ) = G (s ) ⊗ ⎡ g (s ) 11 =⎢ ⎢ g (s ) ⎣ 21

Ĝ (s) ⎡1/g ̂ (s) 1/g ̂ (s)⎤ g12(s)⎤ 12 ⎥ ⎥ ⊗ ⎢ 11 ⎥ ⎥ ⎢ g21(s)⎦ 1/ g ( s ) 1/ g ( s ) ̂ ̂ ⎥ ⎢ 21 22

*= g11 (24)

* = g22

By comparing eqs 23 and 24, it is visible that = giĵ (s)

(25)

g11(s)

(30)

g21(s) g22(s)

(31)

y1 u1

y2 u2

= g11 −

= g22 −

g12g21 g22

(32)

g21g12 g11

(33)

By comparing eqs 10 and 11 to 32 and 33, the equations are found to be equivalent with each other. Perfect controller assumption is used for deriving eqs 10 and 11. The decoupler is introduced for deriving eqs 32 and 33. The controllers designed based on the EOTFs/ETFs give a better performance when introducing the decoupler. g*11 and g*22 can be approximated directly by ETF models. Both the formulations of EOTFs and ETFs (to decompose individual loops) are based on the perfect control approximations, and this assumption is fully validated only by introducing the decouplers among the loops. Wang et al.23 have suggested a method to obtain the realizable decoupler, that is,

From eq 25, it is shown that the conventional EOTF (from DRGA) is equivalent to the ETF (from RNGA).

4. DECOUPLING CONTROL DESIGN Consider the two-input two-output systems with the decoupled control as shown in Figure 2. The relationship between the input vector and process output vector is given by Y (s) = G(s)D(s)U (s)

g12(s)

In the presence of the decoupler, the TITO system behaves like two independent loops for which the controllers can be designed independently, and it is expressed as22

6

giieff

(27)

Decouplers can be designated as22

The dynamic relative gain array (DRGA) is defined as6 Λij = [

⎡ y (s ) ⎤ ⎡ g g ⎤ ⎡ 1 d ⎤ ⎡ u ( s ) ⎤ 12 1 ⎢ 1 ⎥ = ⎢ 11 12 ⎥ ⎢ ⎥ ⎥⎢ ⎢ y (s)⎥ ⎣ g21 g22 ⎦ ⎢⎣ d 21 1 ⎥⎦ ⎢⎣ u (s)⎥⎦ 2 ⎣ 2 ⎦

(26)

For the two-input two-output (TITO) process22 12401



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⎡ ⎤ g (s ) ⎢ − 12 e−v(θ12 − θ11)s ⎥ e−v(θ22 − θ21)s g11(s) ⎢ ⎥ ⎥ D(s) = ⎢ ⎢ g21(s) −v(θ − θ )s ⎥ −v(θ11− θ12)s 21 22 e e ⎢− ⎥ ⎣ g22(s) ⎦

⎡−2.2 1.3 ⎤ K=⎢ ⎣ −2.8 4.3⎥⎦ ⎡ 1.6254 − 0.6254 ⎤ Λ = K ⊗ K −T = ⎢ ⎥ ⎣−0.6254 1.6254 ⎦ ⎡ 1.5537 − 0.5537 ⎤ ϕ = KN ⊗ KN−T = ⎢ ⎣−0.5537 1.5537 ⎥⎦

where

v(θ ) =

1,

if θ ≥ 0

0,

if θ < 0

⎡ 8 7.3 ⎤ Tar = ⎢ ⎥ ⎣11.3 9.55⎦ ⎡−0.2750 0.1781⎤ KN = K ⊙ Tar = ⎢ ⎣−0.2478 0.4503⎥⎦

23

In the Wang et al. method, the extra time delay is to be incorporated to the decoupler matrix in the decoupler (in nonrealizable cases). In such cases, the incorporated extra time delay to the decoupler matrix will also change the corresponding decoupled process model. Hence, the extra time delay is to be added with the corresponding ETF model and it will be used for controller design purposes. In this present work, the controllers are designed based on the corresponding ETFs of each individual loops using the SIMC tuning method.24 This method is used here for its simplicity and robustness. The SIMC method PI controller forms as ⎛ 1⎞ gcii(s) = kc⎜1 + ⎟ τIs ⎠ ⎝

⎡ 0.9559 0.8853 ⎤ Γ=ϕ⊙Λ=⎢ ⎥ ⎣ 0.8853 0.9559 ⎦

By using the RGA and RNGA concepts, the ETF model parameters are deduced by ⎡−1.3535 − 2.0786 ⎤ K̂ = K ⊙ Λ = ⎢ ⎥ ⎣ 4.4769 2.6455 ⎦ ⎡ 6.6910 6.1971⎤ T̂ = Γ ⊗ T = ⎢ ⎥ ⎣ 8.4103 8.7940 ⎦

⎡ 0.9559 0.2656 ⎤ L̂ = Γ ⊗ L = ⎢ ⎥ ⎣1.5935 0.3345 ⎦

(34)

T̂ = [τiĵ ]

For the FOPTD SISO systems, the SIMC-PI24 settings are given by ⎛1 ⎞ τii kci(s) = ⎜ ⎟ ⎝ kii (τci + θii) ⎠

(35)

τIi = min(τii , 4(τci +θii))

(36)

L̂ = [θiĵ ]

The ETF model matrix is expressed as ⎡ −1.3535 e−0.9559s −2.0786 e−0.2656s ⎤ ⎢ ⎥ (6.1971s + 1) ⎥ ⎢ (6.6910s + 1) Ĝ (s) = ⎢ ⎥ −1.5935s 2.6455 e−0.3345s ⎥ ⎢ 4.4769 e ⎢⎣ (8.4103s + 1) (8.7940s + 1) ⎥⎦

In the SIMC design method, the tuning parameter τci is suggested to be taken as time delay θii for better performance. In the present work, the single-input single-output (SISO) SIMC tuning method is used for the demonstration purpose only; other methods such as the Ziegler−Nichols tuning rule can also be used.

(38)

The actual EOTF models are derived using eqs 10 and 11 and are expressed as g11eff =

y1 u1

=

(7.7878s + 0.8465) e−1.75s −2.2 e−s + (7s + 1) (66.5s 2 + 16.5s + 1) (39)

5. SIMULATION EXAMPLES

eff g22 =

We consider three simulation examples to show the simplicity and effectiveness of the proposed method. 5.1. Example 1. The transfer function matrix of VL column given by Luyben1 is considered and it is expressed as ⎡ −2.2 e−s 1.3 e−0.3s ⎤ ⎢ ⎥ (7s + 1) ⎥ ⎢ (7s + 1) G (s ) = ⎢ ⎥ −1.8s 4.3 e−0.35s ⎥ ⎢ −2.8 e ⎢⎣ (9.5s + 1) (9.2s + 1) ⎥⎦

K̂ = [kiĵ ]

y2 u2

=

4.3 e−0.35s 1.65 e−1.1s − (9.2s + 1) (9.5s + 1)

(40)

It is found that the open-loop time response of the ETF (derived from RNGA) model is found to adequately match the actual EOTF (derived from DRGA). Simplified decoupler matrix (by Wang et al.23 method) is expressed as ⎡ 1 0.5909 ⎤ ⎢ ⎥ − 1.45 s D(s) = ⎢ (5.9907s + 0.6512) e −0.7s ⎥ e ⎢⎣ ⎥⎦ (9.5s + 1)

(37)

The normalized gain matrix (KN), RGA (Λ), RNGA (ϕ), average residence time (Tar), and RARTA (Γ) are calculated as

(41)

The extra time delay is incorporated in to the decoupler matrix. 12402



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Table 1. Decoupling Control System Elements Using Different Techniques6 for VL Column decoupling schemes ideal decoupling

decoupled process model −s * = − 2.2 e g11 (7s + 1)

* = g22

inverted decoupling (Nx= e−0.7s)

normalized decoupling

proposed method

4.3 e−1.05s (9.2s + 1)

*= g11

− 2.2 e−s (7s + 1)

* = g22

4.3 e−1.05s (9.2s + 1)

*= g11

2.0785 e−0.9558s (6.6910s + 1)

* = g22

4.4769 e−1.5935s (8.7939s + 1)

*= g11

− 1.3535 e−0.9559s (6.6910s + 1)

* = g22

− 2.6455 e−1.5935s (8.7943s + 1)

decoupler matrix

SIMC controller param.

⎛ 1⎞ gc11 = − 1.5909⎜1 + ⎟ ⎝ 7s ⎠

(89.87s + 9.46) d11 = (25.116s 2 + 59.112s + 5.82) d 22 =

(89.87s + 9.46)e−0.7s (25.116s 2 + 59.112s + 5.82)

d12 =

(53.105s + 5.59) (25.116s 2 + 59.112s + 5.82)

d 21 =

(− 42.504s 2 + 52.052s + 6.16) e−0.7s (25.116s 2 + 59.112s + 5.82)

⎛ 1 ⎞⎟ gc22 = 1.0188⎜1 + ⎝ 8.4s ⎠

⎡ 1 0.5909 ⎤ ⎢ ⎥ D(s) = ⎢ (5.9907s + 0.6512) e−0.75s ⎥ 1 ⎢⎣ ⎥⎦ (9.5s + 1)

⎛ 1⎞ gc11 = − 1.5909⎜1 + ⎟ ⎝ 7s ⎠

⎡ (8.4103s + 1) ⎤ − 1.5357 ⎢ ⎥ (8.7939s + 1) ⎥ ⎢ D(s) = ⎢ ⎥ −0.6903s ⎢ − (6.1970s + 1) e 1.6923 e−1.2590s ⎥ ⎢⎣ ⎥⎦ (6.6910s + 1)

⎛ 1 ⎞ ⎟ gc11 =1.6841⎜1 + ⎝ 6.6910s ⎠ ⎛ 1 ⎞ ⎟ gc22 =0.6163⎜1 + ⎝ 8.7939s ⎠

⎡ 1 0.5909 ⎤ ⎢ ⎥ D(s) = ⎢ (5.9907s + 0.6512) e−1.45s −0.7s ⎥ e ⎢⎣ ⎥⎦ (9.5s + 1)

⎛ 1 ⎞ ⎟ gc11 = − 2.5858⎜1 + ⎝ 6.6910s ⎠

⎛ 1 ⎞⎟ gc22 =1.0188⎜1 + ⎝ 8.4s ⎠

⎛ 1 ⎞⎟ gc22 =1.6066⎜1 + ⎝ 8.276s ⎠

Figure 3. Closed-loop responses (sequential step changes in the set point) for the VL column.

* 0⎤ ⎡ g11 g12 ⎤ ⎡ 1 d12 e−0.7s ⎤ ⎡ g11 ⎥ ⎥=⎢ ⎢ ⎥⎢ ⎣ g21 g22 ⎦ ⎢⎣ d 21 e−0.7s ⎥⎦ ⎢⎣ 0 g * ⎥⎦ 22

*= g11

* = g22

2.6455 e−1.0345s (8.7940s + 1)

Using the tuning rules of eqs 35 and 36, the controller

The incorporated extra time delay in the decoupler matrix will make the decoupled process model as * = ĝ g11 11

−1.3535 e−0.9559s (6.6910s + 1)

parameters are estimated based on the corresponding ETF models. In the SIMC24 tuning formula, the closed-loop time

* = g ̂ e−0.7s g22 22

constant τci is taken as same as θii and the PI parameters are given by

The decoupled process models are expressed as 12403



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Table 2. Comparison of IAE and ISE Values for Different Decoupling Control Schemes for VL Column (Example 1) ideal performance indices yr1 IAE

ISE

y1

normalized

inverted

y2

y1

y2

y1

y2

2.1830

0.7186

2.4370

2.2640

2.1790

0.0111

1.9330

0.0156

0.0049

2.4870

0.4360

3.6120

0.0020

2.4320

0.0027

2.1150

disturbance yr1

0.9950 1.6910

1.438 0.0947

0.9573 1.6890

2.3840 0.7161

0.9950 1.6930

1.4380 1.54 × 10−5

0.9540 1.4480

1.4600 3.49 × 10−5

yr2

2.5 × 10−6

1.8840

0.0249

2.6930

5.71 × 10−7

1.7750

1.21 × 10−6

1.5370

disturbance

0.0621

0.2279

0.0591

0.4063

0.0621

0.2179

0.0531

0.1928

of input uncertainty plot, the proposed and normalized decoupled control systems show the same stability in low and high frequency regions. However, the ideal decoupling shows the less stability in low frequency region and better stability in high frequency region. Overall, the proposed method has better robust stability than that of the normalized and ideal decoupling control schemes. 5.2. Example 2. The transfer function matrix of industrialscale polymerization (ISP) reactor proposed by Chien et al.27 is given by

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For the VL distillation column system, the decoupler, controller matrix, and decoupled process models for the proposed method and for the three decoupling control techniques are listed in Table 1. Figure 3 shows the closed-loop responses of the VL distillation column subject to the sequential unit step changes at t = 0 and t = 50, respectively, and to show the disturbance rejection performance, a step disturbance d = 0.5 at t = 100 is introduced in both the loops. It can be seen that the interaction is reduced and the set point response is improved in the proposed decoupling control method compared to that of ideal, inverted, and normalized decoupling control techniques. The IAE and ISE values are listed in Table 2, and it is shown that the IAE and ISE values are low for the proposed method (designed based on ETFs). This indicates that response is fast and less oscillations in both response and interactions. For quantitative performance measurement, the sum of IAE and ISE values are listed in Table 3. The overall response in terms

⎡ 22.89 e−0.2s −11.64 e−0.4s ⎤ ⎢ ⎥ ⎢ 4.572s + 1 1.807s + 1 ⎥ (s ) = ⎢ ⎥ −0.2s 5.8 e−0.4s ⎥ ⎢ 4.689 e ⎣ 2.174s + 1 1.801s + 1 ⎦

performance indices a

ideal

normalized

inverted

proposed

5.3935 3.6697

8.7490 5.1230

4.6241 3.4681

4.0663 2.9850

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The normalized gain matrix (KN), RGA (Λ), RNGA (ϕ), average residence time (Tar), and RARTA (Γ) are calculated as ⎡ 22.89 − 11.64 ⎤ K=⎢ ⎥ ⎣ 4.689 5.8 ⎦ ⎡ 0.7087 0.2913 ⎤ Λ = K ⊗ K −T = ⎢ ⎥ ⎣ 0.2913 0.7087 ⎦ ⎡ 0.5482 0.4518 ⎤ ϕ = KN ⊗ KN−T = ⎢ ⎣ 0.4518 0.5482 ⎥⎦

Table 3. Performance Indices for VL Column (Example 1)

a

y2

yr2

⎡ ⎤ ⎛ 1 ⎞ ⎟ 0 ⎢−2.5858⎜1 + ⎥ ⎝ 6.6910s ⎠ ⎢ ⎥ Gc(s) = ⎢ ⎥ ⎛ ⎞ 1 ⎢ ⎟⎥ 0 1.6066⎜1 + ⎢⎣ ⎝ 8.276s ⎠ ⎥⎦

IAE ISEa

proposed

y1

⎡ 4.772 2.207 ⎤ Tar = ⎢ ⎣ 2.374 2.201⎥⎦

Sum of main and interaction responses.

⎡ 4.7967 − 5.2741⎤ KN = K ⊙ Tar = ⎢ ⎥ ⎣ 1.9751 2.6352 ⎦

of IAE and ISE values are better in the case of proposed method. The inverted decoupling method is also found to be sensitive to modeling errors,7,8 and it cannot be directly implement to the higher dimension systems.7,9 Robustness. Among the various methods available for the study of the robustness analysis in multivariable systems,25 the method based on the inverse of maximum singular value is easy to use and to compare the different control system stabilities.20,26 The stability bound of the VL column is shown in Figure 4. Figure 4 shows the frequency plot of inverse of maximum singular value, indicating the stability bounds of VL column. The region below the curve denotes the stability region and the region above the curve denotes the instability region. More area under the curve indicates high stability of the system. At low frequency region, the proposed, normalized, and ideal decoupling control systems have same stability. As the frequency increases, the proposed method shows more stability in the output uncertainty plot. In the case

⎡ 0.7736 1.5508 ⎤ Γ=ϕ⊙Λ=⎢ ⎥ ⎣1.5508 0.7736 ⎦

By using the RGA and RNGA concepts, the ETF model parameters are deduced by ⎡ 32.3003 − 39.9535⎤ K̂ = K ⊙ Λ = ⎢ ⎥ ⎣16.0947 8.1844 ⎦ ⎡ 3.5368 2.8022 ⎤ τ̂ = Γ ⊗ T = ⎢ ⎥ ⎣ 3.3713 1.3932 ⎦ ⎡ 0.1547 0.6203 ⎤ θ̂ = Γ ⊗ L = ⎢ ⎥ ⎣ 0.3102 0.3094 ⎦

ETF model is expressed as 12404



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Figure 4. Stability regions of output and input uncertainties for the VL column. (Left figure: for output uncertainty.)

Table 4. Decoupling Control System Elements Using Different Techniques for ISP Reactor decoupling schemes inverted decoupling29

proposed method

decoupled process model −0.4s

22.89 e (4.572s + 1) −0.4s * = 5.8 e g22 (1.801s + 1) *= g11

*= g11

32.3003 e−0.3547s (3.5368s + 1)

* = g22

8.1844 e−0.3094s (1.3932s + 1)

⎡ 32.3003 e−0.1547s −39.9535 e−0.6203s ⎤ ⎢ ⎥ (2.8022s + 1) ⎥ ⎢ 3.5368s + 1 Ĝ (s) = ⎢ ⎥ −0.3102s 8.1844 e−0.3094s ⎥ ⎢ 16.0947 e ⎢⎣ (3.3713s + 1) 1.3932s + 1 ⎥⎦

decoupler matrix

y1 u1

=

*= g11

y2 u2

=

⎛ 1 ⎞⎟ gc22 = 0.3881⎜1 + ⎝ 1.801s ⎠

32.3003 e−0.3547s 3.5368s + 1

* = g22

⎛ 1 ⎞⎟ gc11 = 0.1544⎜1 + ⎝ 2.8376s ⎠ ⎛ 1 ⎞ ⎟ gc22 = 0.2751⎜1 + ⎝ 1.3932s ⎠

8.1844 e−0.3094s 1.3932s + 1

The controller parameters for the proposed and inverted decoupling methods are estimated by using eqs 35 and 36. The controller and decoupler elements of the proposed and inverted decoupling methods are listed in Table 4. The centralized control parameters for the ISP reactor system are estimated by Xiong et al.28 method using the SIMC tuning method.

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(16.948s + 9.4103) e−0.2s 22.89 e−0.2s + (4.572s + 1) (3.9284s 2 + 3.981s + 1)

⎡ ⎛ ⎛ 1 ⎞⎟ 1 ⎞⎤ ⎟⎥ 0.3504⎜1 + ⎢ 0.4993⎜1 + ⎝ ⎝ 1.6s ⎠ 1.5416s ⎠ ⎥ ⎢ Gc(s) = ⎢ ⎛ ⎞ ⎛ ⎞⎥ ⎢−0.0586⎜1 + 1 ⎟ 0.3881⎜1 + 1 ⎟ ⎥ ⎝ ⎝ ⎣ 1.807s ⎠ 1.801s ⎠ ⎦

(45) eff g22 =

⎛ 1 ⎞⎟ gc11 =0.2496⎜1 + ⎝ 3.2s ⎠

⎡ (2.3249s + 0.5085)e−0.2s ⎤ ⎢ ⎥ e−0.2s (1.807s + 1) ⎢ ⎥ D(s) = ⎢ ⎥ ⎢ (− 1.4560s − 0.8084) ⎥ 1 ⎢⎣ ⎥⎦ (2.174s + 1)

The actual EOTF models are derived using eqs 10 and 11 and are expressed as g11eff =


⎡ (2.3249s + 0.5085) ⎤ 1 ⎢ ⎥ (1.807s + 1) ⎥ ⎢ D(s) = ⎢ ⎥ ⎢ (− 1.4560s − 0.8084) ⎥ 1 ⎢⎣ ⎥⎦ (2.174s + 1)

(10.9017s + 2.3844) e−0.4s 5.8 e−0.4s + (1.801s + 1) (3.9284s 2 + 3.981s + 1)

(47)

(46)

Figure 5 shows the closed-loop step responses of the ISP reactor subject to the sequential unit step changes at t = 0 and t = 25, respectively. It can be seen that the interaction is reduced and set point response is improved in the proposed decoupled control system compared to the centralized control system (Xiong et al.:28 controller tuned by the SIMC method). Compared to centralized control system, the proposed

It is found that the open-loop time response of the ETF (derived from RNGA) model adequately matches that of the actual EOTF (derived from DRGA). The decoupler matrix elements are estimated by Wang23 et al. method. The decoupled process models are expressed as (g11 * = ĝ11 e−0.2s), 12405



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Figure 5. Closed-loop response (sequential step changes in the set point) for the ISP reactor system.

⎡ −0.098 ⎢ −0.043 K=⎢ ⎢−0.012 ⎢ ⎣ −0.013 ⎡139 ⎢ 172 Tar = ⎢ ⎢184 ⎢ ⎣188

decoupling control system has lead-lag filter term with the diagonal PI control elements. This lead-lag term improves the performance better than the centralized control system. The IAE values are listed in Table 5, and it is shown that the IAE values are considerably reduced for the proposed decoupled control system compared to that of inverted decoupling and centralized control systems. Table 5. Comparison of IAE Values for the Different Control Schemes for ISP Reactor

−0.036 −0.014 − 0.017 ⎤ ⎥ −0.092 −0.011 −0.012 ⎥ −0.016 −0.102 −0.033 ⎥ ⎥ −0.015 −0.029 − 0.108 ⎦ 176 190 185 ⎤ ⎥ 146 189 191⎥ 185 134 172 ⎥ ⎥ 190 169 146 ⎦

IAE values method used proposed method

change in set point yr1

y1

y2

IAEa

0.9674

0.0024

1.8948

centralized controller inverted decoupler a

yr2

0.0141

0.9109

yr1

0.7270

0.4039

yr2

0.5917

0.9864

yr1

1.0813

0.0001

yr2

0.0014

0.8776

Λ = K ⊗ K −T ⎡ 1.2207 − 0.2051 − 0.0053 − 0.0103 ⎤ ⎢ ⎥ −0.1947 1.2198 − 0.0136 − 0.0116 ⎥ ⎢ = ⎢ −0.0106 − 0.0085 1.1095 − 0.0904 ⎥ ⎢ ⎥ ⎣−0.0154 − 0.0062 − 0.0907 1.1124 ⎦

2.7090 1.9604


KN = K ⊙ Tar

5.3. Example 3. The transfer function matrix of the temperature control of the four room process proposed by Shen et al.30 is given by ⎡ − 0.098 e−17s ⎢ ⎢ 122s + 1 ⎢ −25s ⎢ − 0.043 e ⎢ 147s + 1 G(s) = ⎢ ⎢ − 0.012 e−31s ⎢ 153s + 1 ⎢ ⎢ − 0.013 e−32s ⎢ ⎣ 156s + 1

= 1.0 e−03 ⎡−0.7050 ⎢ −0.2500 ×⎢ ⎢−0.0652 ⎢ ⎣ −0.0691

− 0.036 e−27s − 0.014 e−32s − 0.017 e−30s ⎤ ⎥ 149s + 1 158s + 1 155s + 1 ⎥ ⎥ − 0.092 e−16s − 0.011 e−33s − 0.012 e−34s ⎥ 130s + 1 156s + 1 157s + 1 ⎥ ⎥ −34s −16s − 0.016 e − 0.102 e − 0.033 e−26s ⎥ 151s + 1 118s + 1 146s + 1 ⎥ ⎥ − 0.015 e−31s − 0.029 e−25s − 0.108 e−18s ⎥ ⎥ 159s + 1 144s + 1 128s + 1 ⎦

− 0.2045 − 0.0737 − 0.0919 ⎤ ⎥ − 0.6301 − 0.0582 − 0.0628 ⎥ − 0.0865 − 0.7612 − 0.1919 ⎥ ⎥ − 0.0789 − 0.1716 − 0.7397 ⎦

ϕ = KN ⊗ KN −T ⎡ 1.1389 − 0.1287 − 0.0036 − 0.0067 ⎤ ⎢ ⎥ −0.1238 1.1389 − 0.0078 − 0.0073 ⎥ =⎢ ⎢−0.0057 − 0.0055 1.0710 − 0.0597 ⎥ ⎢ ⎥ ⎣−0.0094 − 0.0046 − 0.0596 1.0737 ⎦

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The normalized gain matrix (KN), average residence time (Tar), RGA (Λ), RNGA (φ), and RARTA (Γ) are calculated as 12406



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Table 6. Decoupling Control System Elements Using Normalized30 and Proposed Decoupling Method for Example 3 decoupling schemes proposed method

normalized30

⎡ 0.9330 ⎢ 0.6361 Γ=ϕ⊙Λ=⎢ ⎢ 0.5404 ⎢ ⎣ 0.6083

decoupled process model


−15.8615s

*= g11

− 0.0803 e (113.8299s + 1)

⎛ ⎞ 1 ⎟ gc11 = − 44.6855⎜1 + ⎝ 113.8299s ⎠

* = g22

− 0.0754 e−14.9383s (121.3735s + 1)

⎛ ⎞ 1 ⎟ gc22 = − 53.8792⎜1 + ⎝ 119.5064s ⎠

* = g33

− 0.0919 e−15.4444s (113.9022s + 1)

⎛ ⎞ 1 ⎟ gc33 = − 40.1251⎜1 + ⎝ 113.9022s ⎠

* = g44

− 0.0971 e−17.3739s (123.5480s + 1)

⎛ ⎞ 1 ⎟ gc44 = − 36.6175⎜1 + ⎝ 118.6090s ⎠

*= g11

e−21.8281s (113.8299s + 1)

⎛ ⎞ 1 ⎟ gc11 = 2.6086⎜1 + ⎝ 113.8299s ⎠

* = g22

e−21.3160sd (121.3735s + 1)

⎛ ⎞ 1 ⎟ gc22 = 2.8470⎜1 + ⎝ 121.3735s ⎠

* = g33

e−22.2075s (113.9022s + 1)

⎛ ⎞ 1 ⎟ gc33 = 2.5645⎜1 + ⎝ 113.9022s ⎠

* = g44

e−23.125s (118.609s + 1)

⎛ ⎞ 1 ⎟ gc44 = 2.6713⎜1 + ⎝ 123.5480s ⎠

⎡113.8299 ⎢ 93.5129 τ̂ = Γ ⊗ T = ⎢ ⎢ 82.6888 ⎢ ⎣ 94.8909

0.6274 0.6821 0.6461 ⎤ ⎥ 0.9336 0.5731 0.6269 ⎥ 0.6532 0.9653 0.6604 ⎥ ⎥ 0.7460 0.6575 0.9652 ⎦

93.4854 107.7763 100.1438 ⎤ ⎥ 121.3735 89.4106 98.4299 ⎥ 98.6274 113.9022 96.4196 ⎥ ⎥ 118.6090 94.6824 123.5480 ⎦

By using the RGA and RNGA concepts, the ETF model parameters are deduced by ⎡15.8615 ⎢ 15.9036 ̂ θ=Γ⊗L=⎢ ⎢16.7539 ⎢ ⎣19.4648

⎡−0.0803 0.1755 2.6653 1.6468 ⎤ ⎢ ⎥ 0.2209 − 0.0754 0.8097 1.0363 ⎥ ⎢ ̂ K=K⊙Λ= ⎢ 1.1315 1.8867 − 0.0919 0.3648 ⎥ ⎢ ⎥ ⎣ 0.8420 2.4132 0.3197 − 0.0971⎦

16.9403 21.8281 19.3827 ⎤ ⎥ 14.9383 18.9138 21.3160 ⎥ 22.2075 15.4444 17.1760 ⎥ ⎥ 23.1250 16.4379 17.3739 ⎦

⎡ − 0.0803 e−15.8615s 0.1755 e−16.9403s 2.6653 e−21.8281s 1.6468 e−19.3827s ⎤ ⎢ ⎥ 93.4854s + 1 107.7763s + 1 100.1438s + 1 ⎥ ⎢ 113.8299s + 1 ⎢ ⎥ −15.9036s − 0.0754 e−14.9383s 0.8097 e−18.9138s 1.0363 e−21.3160s ⎥ ⎢ 0.2209 e ⎢ 93.5129s + 1 121.3735s + 1 89.4106s + 1 98.4299s + 1 ⎥ Ĝ(s) = ⎢ ⎥ ⎢ 1.1315 e−16.7539s 1.8867 e−22.2075s − 0.0919 e−15.4444s 0.3648 e−17.1706s ⎥ ⎢ ⎥ 98.6274s + 1 113.9022s + 1 96.4196s + 1 ⎥ ⎢ 82.6888s + 1 ⎢ 0.8420 e−19.4648s 2.4132 e−23.1250s 0.3197 e−16.4379s −0.0971 e−17.3739s ⎥ ⎢ ⎥ ⎣ 94.8909s + 1 118.6090s + 1 94.6824s + 1 123.5480s + 1 ⎦

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The decoupler is expressed as ⎡ (− 44.8163 − 0.3673)e−10s (− 17.4338 − 0.1429)e−15s (− 21.1633 − 0.1735)e−13s ⎤ ⎢ ⎥ 1 149s + 1 158s + 1 155s + 1 ⎢ ⎥ ⎢ −9s −17s −18s ⎥ ( 60.7609 0.4674)e ( 15.5485 0.1196)e ( 16.952 0.1304)e − − − − − − ⎢ ⎥ 1 ⎢ ⎥ 147s + 1 156s + 1 157s + 1 D(s) = ⎢ ⎥ −15s −18s −10s ( 13.8824 0.1176)e ( 18.5142 0.1569)e ( 38.173 0.3235)e − − − s − − s − ⎢ ⎥ 1 ⎢ ⎥ 153s + 1 151s + 1 146s + 1 ⎢ ⎥ ⎢ (− 15.4112s − 0.1204)e−14s (− 17.7792s − 0.1389)e−13s (− 34.3704s − 0.2685)e−7s ⎥ 1 ⎢ ⎥ ⎣ ⎦ 156s + 1 159s + 1 144s + 1

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Figure 6. Closed-loop response plot for example 3 (solid, proposed method; dash, Shen et al.30).

Table 7. Comparison of IAE Values for Both Control Schemes of Example 3 normalized30 performance indices yr1 IAE

IAEa a

proposed method

y1

y2

y3

y4

y1

y2

y3

y4

48.3300

6.4670

1.9900

2.4620

34.3100

2.1870

5.3270

5.3070

yr2

6.7320

47.2300

4.8830

3.6730

3.0770

32.0900

4.8850

5.1690

yr3

2.4360

2.2010

48.6100

3.4640

3.8350

3.6666

33.5900

2.5010

yr4

2.6990

2.3990

3.9540

50.6700

3.8380

4.0400

2.5020

36.9400

238.2

183.2640


The unit set-point changed in r1 at t = 0, r2 at t = 1000, r3 at t = 2000, and r4 at t = 3000. Compared to Shen et al.29 method, the overall performances are improved by the proposed method. IAE values are listed in Table 7. The three simullation examples

Decoupled process models and controller settings for the proposed and normalized30 decoupling methods are listed in Table 6. Figure 6 shows the closed-loop responses for the proposed method and for the normalized decoupling method.30 12408



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θij = element of relative normalized gain σij = average residence time σ̂ij = average residence time when other loops are closed Λij = relative gain array element

show that the nominal performance of the proposed simplified decoupler with ETF based controller method is better than the ideal, inverted, and normalized decoupling and centralized control systems. The performance by the robustness analysis shows that the proposed method gives a better robust performance compared to that of the other decoupling control techniques.

Subscripts

■

6. CONCLUSIONS A decoupler with a decentralized control system is designed based on ETF models of MIMO systems. It is shown that, for the examples considered here, the decoupler with decentralized control system (designed based on ETFs) reduced the interaction and gives better responses when compared to that of the centralized control system, ideal, inverted, and normalized decoupling control methods. The proposed decoupler with the decentralized control system is easy to design when compared to the method of EOTF (derived using Maclaurin series) and Cai et al.6 It is shown that the perfect control assumption4 is valid when introducing the decoupler in to the control system. The independent controllers are designed using the SIMC method based on the calculated ETFs. The simulation examples show the better performance of the proposed method compared with the centralized control system, ideal, inverted, and normalized decoupling methods.

■

i,j = loop representation

REFERENCES

(1) Luyben, W. L. Simple Method for Tuning SISO Controllers in Multivariable Systems. Ind. Eng. Chem. Proc. Des. Dev. 1986, 25, 654− 669. (2) Shen, S. H.; Yu, C. C. Use of Relay Feedback Test for Automatic Tuning of Multivariable Systems. AIChE J. 1994, 40, 627−646. (3) Loh, P. A.; Hang, C. C.; Quek, K. C.; Vasnani, U. V. Auto-tuning of Multi-loop Proportional-Integral Controllers Using Relay Feedback. Ind. Eng. Chem. Res. 1993, 32, 1102−1107. (4) Vu, T. N. L.; Lee, M. Independent Design of Multi-loop PI/PID Controllers for Interacting Multivariable Processes. J. Process Control 2010, 922−933. (5) Tavakoli, S.; Griffin, I.; Fleming, P. J. Tuning of Decentralized PI (PID) Controllers for TITO Processes. Control Eng. Pract. 2006, 14 (9), 1069−1080. (6) Cai, W. J.; Ni, W.; He, M. J.; Ni, C. Y. Normalized Decoupling A New Approach for MIMO Process Control System Design. Ind. Eng. Chem. Res. 2008, 47 (19), 7347−7356. (7) Gagnon, E.; Pomerleau, A.; Desbiens, A. Simplified, Ideal, or Inverted Decoupling? ISA Trans. 1998, 37 (4), 265−276. (8) Weischedel, K.; McAvoy, T. J. Feasibility of Decoupling in Conventionally Controlled Distillation Columns. Ind. Eng. Chem. Fundam. 1980, 19 (4), 379−384. (9) Yunhui, L. U. O.; Hongbo, L. I. U.; Lei. J. I. A. Improved Inverted Decoupling Control Using Dead-time Compensator for MIMO Processes. Proceedings of the 29th Chinese Control Conference, China, 2010. (10) McAvoy, T. J. Interaction Analysis: Principles and Applications; Instrument Society of America: Research Triangle Park, NC, 1983. (11) Wade, H. L. Inverted Decoupling: A Neglected Technique. ISA Trans. 1997, 36 (1), 3−10. (12) Kumar, N.; Pandit, M.; Chidambaram, M. Multivariable Control of Four-Tank System, . Indian Chem. Eng., Sect. A 2004, 46 (4), 216− 221. (13) Maghade, D. K.; Patre, B. M. Decentralized PI/PID Controllers based on Gain and Phase Margin Specifications for TITO Processes. ISA Trans. 2012, 51 (4), 550−558. (14) Huang, H. P.; Jeng, J. C.; Chiang, C. H.; Pan, W. A Direct Method for Multi-Loop PI/PID Controller Design. J. Process Control 2003, 13 (8), 769−786. (15) Xiong, Q.; Cai, W. J. Effective Transfer Function Method for Decentralized Control System Design of Multi-input Multi-output Processes. J. Process Control 2006, 16 (8), 773−784. (16) He, M. J.; Cai, W. J.; Ni, W.; Xie, L. H. RNGA Based Control System Configuration for Multivariable Processes. J. Process Control 2009, 19 (6), 1036−1042. (17) Naini, N. M.; Fatehi, A.; Sedigh, A. K. Input−Output Pairing Using Effective Relative Energy Array. Ind. Eng. Chem. Res. 2009, 48 (15), 7137−7144. (18) Shen, Y.; Cai, W. J.; Li, S. Multivariable Process Control: Decentralized, Decoupling, or Sparse? Ind. Eng. Chem. Res. 2010, 49 (2), 761−771. (19) Hu, W.; Cai, W. J.; Xiao, G. Decentralized Control System Design for MIMO Processes with Integrators/Differentiators. Ind. Eng. Chem. Res. 2010, 49 (24), 12521−12528. (20) Kumar, V. V.; Rao, V. S. R.; Chidambaram, M. Centralized PI Controllers for Interacting Multivariable Processes by Synthesis Method. ISA Trans. 2012, 51 (3), 400−409. (21) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control, 2nd ed.; John Wiley & Sons Asia Pte. Ltd.: Singapore, 2009.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

NOMENCLATURE DRGA = dynamic relative gain array IAE = integral of absolute error IMC = internal model control ISE = integral of the squared error ISP = industrial scale polymerization SIMC = simplified IMC G, Gc = process and controller transfer function matrices gij, gc,i,j, ĝij = process, controller, and equivalent transfer function models geff ij , g* ii = effective open-loop transfer function and decoupled process model dij = decoupler elements ̂ = process and effective steady-state gain kpij, kpij τij, τ̂ij = process and effective time constant θij, θiĵ = process and effective time delay

Y = closed-loop process response vector yi = closed-loop process response tc = closed time constant of the model Kci = Controller gain τIi, τDi = integral and derivative time ϕ = relative normalized gain array (RNGA) Λ = relative gain array (RGA) Γ = relative average residence time array (RARTA) KN = normalized gain matrix KNij = normalized gain s = laplace domain t = time uj = manipulated variable γij = relative average residence time 12409



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Controller Design for MIMO Processes Based on Simple Decoupled

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