Design of Proportional Integral Controllers with Decouplers for

Article pubs.acs.org/IECR

Design of Proportional Integral Controllers with Decouplers for Unstable Two Input Two Output Systems Sukanya Hazarika and M. Chidambaram* Department of Chemical Engineering, Indian Institute Of Technology Madras, Chennai 600 036, India ABSTRACT: A method is proposed based on the equivalent transfer function (ETF) model to design multivariable proportional integral (PI) controllers for unstable multivariable systems with delay. The simplified decoupler decomposes the unstable multiloop systems into independent loops with ETFs as the resultant decoupled process model having unstable poles. PI controllers are designed for the diagonal elements of ETFs by the synthesis method meant for unstable first order plus time delay (FOPTD) systems. Since the overshoot is higher for the unstable systems, a double-loop control structure (inner-loop diagonal proportional (P) controllers with the decoupler and outer loop only with diagonal PI controllers) is proposed to get a reduced overshoot. Two examples of two input two output (TITO) unstable systems are considered to demonstrate the simplicity and effectiveness of the proposed method.

1. INTRODUCTION Unstable single input single output (SISO) systems with time delays are more difficult to control than that of the stable systems. The performance specifications such as overshoot and settling time for such systems are larger for unstable systems than those of the stable systems.1 Several methods for the design of PI/PID controllers for SISO unstable systems are available.1 A review of the control of unstable systems is recently given by Rao and Chidambaram.2 To reduce the overshoot, Jacob and Chidambaram3 have proposed a two stage design of controllers for unstable SISO systems. First, the unstable system is stabilized by a simple proportional controller. For the stabilized system, a PI controller is designed. It is shown that an improved performance is obtained. The method of designing the multivariable PI controllers is complicated due to the interactions among the loops. A review on the design of decentralized PI controllers for a stable MIMO system is given by Luyben and Luyben4 and Maciejowski.5 Tanttu and Lieslehto6 and Katebi7 have given simple methods of tuning centralized PI controllers for stable systems. These methods include Davison’s method8 (requires only the steady state gain matrix, SSGM) and the Tanttu and Lieslehto method6 (requires the transfer function matrix). More rigorous methods of designing multivariable PI controllers are reviewed by Wang et al.9 and Wang and Nie.10 Only a few methods are reported for the design of controllers for unstable multivariable processes. Georgiou et al.11 have suggested an optimization method. However the system considered does not have a significant time delay. Agamenoni et al.12 have proposed a method of designing controllers based on an optimization method. In the above methods, the systems considered have unstable components only for one of the input variables. Govindhakannan and Chidambaram13 have applied the method of Tanttu and Lieslehto to unstable MIMO processes.The interactions are found to be significant. Decentralized PI controllers are also designed by the detuning method proposed by Luyben and Luyben.4 The decentralized PI controllers do not stabilize the system if the unstable pole is present in each of the transfer functions of the system. Only centralized © 2014 American Chemical Society

PI controllers stabilize such systems. Govindhakannan and Chidambaram14 have applied the two stage P−PI controllers for the unstable systems based on the Tanttu and Lieslehto method. The method gives significant interactions. There are three types of decoupling methods reported for stable systems:15 ideal, simplified, and inverted decoupling. The ideal decoupling method needs to calculate the inverse of the process transfer function matrix. It results in complicated decoupling elements, and the ideal decoupling is sensitive to modeling errors. The simplified decoupler has a simple decoupler form, but the controller cannot be designed directly from the decoupled process model without any model reduction method. The inverted decoupling method is also sensitive to modeling errors. Jevtovic and Matausek16 have proposed an optimization method under the constraints on robustness and sensitivity to measurement noise based on the ideal decoupler method.17 Recently several works18−20 appeared for stable systems, that introduced the concepts of equivalent transfer functions/effective open-loop transfer functions (ETFs/EOTFs) to take into account the loop interactions in the design of multiloop control systems. Rajapandiyan and Chidambaram21 recently proposed a method of designing controllers for MIMO stable systems by combining the simplified decoupler approach with the ETF model approximation.This method is applicable when the decoupler is a stable one. Both the formulation EOTFs and ETFs (decomposed into individual loops) are based on perfect control approximations and the assumption is fully validated only by the decouplers. This method gives less interactions and better performances when compared to the ideal and inverted decoupling methods. In the present work, the applicability of this method to unstable TITO systems is evaluated. Received: Revised: Accepted: Published: 6467

November 9, 2013 January 16, 2014 February 21, 2014 February 21, 2014 dx.doi.org/10.1021/ie403791q | Ind. Eng. Chem. Res. 2014, 53, 6467−6476

Industrial & Engineering Chemistry Research

Article

⎡ϕ ϕ ⎤ 11 12 ⎥ ϕ = K N ⊗ K N −T = = ⎢ ⎢⎣ ϕ21 ϕ22 ⎥⎦

2. DESIGN BASED ON ETF MODEL In the two input two output (TITO) system, 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 by y1 u1

= g p,11 −

where ⊗ denotes Hadamard multiplication. RARTA, which is defined as the ratio of loop yi − uj average residence times, when other loops are closed and when other loops are open, is given by

(g p,12g p,21gc,2) (1 + gc,2g p,22)

⎡ γ11 γ12 ⎤ Γ=ϕ⊙Λ=⎢ ⎥ ⎣ γ21 γ22 ⎦

(1)

This can be further written as y1 u1

= g p,11 −

̂

e−θijs g p,̂ ij (s) = k p,̂ ij τiĵ s − 1

(2)

Similarly, for the second loop y2 u2

= g p,22 −

(3)

Equations 2 and 3, being complicated, can be simplified by two assumptions:22 the first assumption is the perfect controller approximation for the other loop (the output attains steady state with no transient) which is used to simplify both equations; i.e., gc, ig p, ii =1 i = 1, 2 (1 + gc, ig p, ii) (4)

3. PAIRING CRITERIA The problem of loop interaction can be minimized by a proper choice of input output pairings. The degree of interaction is quantified using the RGA analysis which helps in choosing the manipulated−controlled variable pairings that best suits the control problem. For stable plants, the pairing is selected corresponding to the positive values of NI (Niederlinski index) and RGA, where

The second is assuming ETFs have the same structure of the corresponding open-loop model. By using the perfect controller approximation, eqs 2 and 3 can be approximated as g p,12g p,21 y eff g p,11 = 1 = g p,11 − u1 g p,22 (5) eff g p,22 =

y2 u2

= g p,22 −

NI =

(11)

(12) 23

The pairing criteria for unstable systems will differ when the number of open-loop unstable poles of Gp(s) is different from G̅p(s) = diag[gp,ii(s)] . Hence pairing is carried out by the following ways: If for an (n × n) plant (1) with one unstable pole which appears in all elements of Gp(s), pairing should be such that NI is positive if n is odd and negative if n is even; or (2) with P unstable poles which appears in all elements of Gp(s), pairing should be such that NI is positive if (n − 1)P is even and negative if (n − 1)P is odd.

(6)

eff Here, geff p,11 and gp,22 are the effective open-loop transfer 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 the Maclaurin series.22 This method leads to complications in higher dimension systems, in the formulation of EOTFs, and in the model reduction step. By using relative gain array (RGA), relative normalized gain array (RNGA), and relative average residence time array (RARTA) concepts, the expression for ETF can be derived easily for higher dimension systems also.21 The controllers are designed on the diagonal elements of the ETF, and its closed-loop responses have to be matched with the eff closed-loop response of geff p,11 and gp,22. The result of getting the same closed-loop response would signify that both ETF and EOTF are the same. The basic steps required to obtain ETF for a TITO system are as follows: The normalized gain KN,ij is defined as

⎡ KN,11 KN,12 ⎤ ⎥ K N = K ⊙ Tar = ⎢ ⎢⎣ KN,21 KN,22 ⎥⎦

det[K ] ∏ K ii

⎡ λ11 λ12 ⎤ ⎥ RGA(Λ) = K ⊗ K −T = ⎢ ⎢⎣ λ 21 λ 22 ⎥⎦

g p,12g p,21 g p,11

(10)

̂ = kp,ij/Λij, τ̂ij = γijτij, and θ̂ij = γijθij. where kp,ij It should be noted that the method is applicable when EOTF = ETF.

g p,21g p,12(gc,1g p,11) g p,11(1 + gc,1g p,11)

(9)

Hence for an unstable system, the ETF can be expressed as

g p,12g p,21(gc,2g p,22) g p,22(1 + gc,2g p,22)

(8)

4. DESIGN OF CONTROLLERS Figure 1 shows the block diagram of the process, simplified decoupler, and the controllers for a TITO process. For the preceding TITO system, we have Y (s) = Gp(s) D(s) U (s)

(13)

* 0 ⎤ ⎡ g p,11 g p,12 ⎤⎡1 d12 ⎤ ⎡ g p,11 ⎥ ⎢ ⎥⎢ ⎥= Gp(s) D(s) = ⎢ ⎥ ⎢⎣ g p,21 g p,22 ⎥⎦⎢⎣ d 21 1 ⎥⎦ ⎢⎢ 0 * g p,11⎥⎦ ⎣ (14)

(7)

where the simplified decoupler is designed as

where ⊙ denotes Hadamard division, K is the steady state gain, and Tar = τij + θij is defined as the average residence time which signifies the response speed of the controlled variable yi to manipulated variable uj. The RNGA can be obtained as

d12(s) = − 6468

g p,12(s) g p,11(s)

;

d 21(s) = −

g p,21(s) g p,22(s)

(15)

dx.doi.org/10.1021/ie403791q | Ind. Eng. Chem. Res. 2014, 53, 6467−6476


Article

Figure 1. Simplified decoupled control system of a TITO process. (d11 = 1;d22 = 1).

⎡ 3.6 4 ⎤ ⎡ 0.444 0.1500 ⎤ Tar = ⎢ ⎥; KN = ⎢ ⎥; ⎣ 4.5 3.2 ⎦ ⎣ 0.1556 0.5313 ⎦ ⎡ 0.9383 0.6005⎤ ⎡1.1097 −0.1097 ⎤ ϕ=⎢ ⎥ ⎥; Γ = ⎢ ⎣−0.1097 1.1097 ⎦ ⎣ 0.6005 0.9383⎦

For these systems, having time delay may lead to unwanted situations (nonrealizable cases) by eq 15. Hence, an extra time delay (θ) is to be incorporated into the decoupler matrix which is further added to the corresponding ETF.21 For any uncertainty in time delay, time constant, or process gain, the robustness of the closed-loop system is to be evaluated by simulation. In the presence of the decoupler, the TITO system behaves like two independent loops for which the controllers can be designed independently. In the present work, diagonal PI controllers are designed by the synthesis method1 based on the unstable ETFs: ⎛ 1 ⎞ ⎟⎟ gc, ii(s) = kc, ii⎜⎜1 + τI, ii(s) ⎠ ⎝ kc, iik p, ii = τI, ii τii

0.8668εii−0.8288

= 0.1523e7.9425εii

(21)

By using the preceding concepts, the ETF matrix is obtained as ⎡ 1.3529e−0.9383s −3.2857e−0.9008s ⎤ ⎢ ⎥ ⎢ ( −2.4396s + 1) (1.5013s + 1) ⎥ Ĝp(s) = ⎢ ⎥ −0.9008s 1.4375e−0.9383s ⎥ ⎢ −3.8333e ⎢⎣ (1.8016s + 1) ( −2.0643s + 1) ⎥⎦

(16)

for 0.1 ≤ εii ≤ 0.7 for 0 ≤ εii ≤ 0.7

According to eq 15, the decoupler is given by

(17)

⎡ (7.8s − 3)e−0.5s ⎤ ⎢1 ⎥ (20s + 8) ⎥ ⎢ D(s) = ⎢ ⎥ −0.5s ⎢ (15.4s − 7)e ⎥ 1 ⎢⎣ (51s + 17) ⎥⎦

(18)

where εij = θii/τii.

5. SIMULATION EXAMPLES To demonstrate the ETF method, two simulation examples of unstable systems are considered. 5.1. Example 1. Consider an example given by Flesch et al.,24 which has diagonal elements of unstable and unequal poles: ⎡ 1.6e−s 0.6e−1.5s ⎤ ⎢ ⎥ ⎢ ( −2.6s + 1) (2.5s + 1) ⎥ Gp(s) = ⎢ ⎥ −1.5s 1.7e−s ⎥ ⎢ 0.7e ⎢⎣ (3s + 1) ( −2.2s + 1) ⎥⎦

(23)

The PI controllers for the diagonal elements of ETF are calculated by eqs 16−18 as ⎡ ⎤ ⎛ 1 ⎞ ⎟ 0 ⎢−1.3797⎜1 + ⎥ ⎝ 8.4s ⎠ ⎢ ⎥ Gc(s) = ⎢ ⎛ ⎞ 1 ⎟ ⎥⎥ ⎢0 −0.98⎜1 + ⎝ ⎣ 11.62s ⎠ ⎦ (24)

The actual EOTF models for Gp(s)D(s) can be derived as

(19)

According to section 3, the pairing criteria for this system will be the same as that for a stable one since the number of openloop unstable poles is the same for both Gp(s) and G̅p(s) = diag[gp,ii(s)]. To determine the pairing for this system, K and RGA are calculated as ⎡1.6 0.6 ⎤ K=⎢ ⎥; ⎣ 0.7 1.7 ⎦

(22)

eff g p,11 =

y1 u1

=

⎡ 15.4s − 7 ⎤ −2s 1.6e−s 0.6 + ⎥e ⎢ ( −2.6s + 1) (2.5s + 1) ⎣ 51s + 17 ⎦ (25)

eff g p,22 =

⎡1.1826 − 0.1826 ⎤ RGA(Λ) = ⎢ ⎥ ⎣−0.1826 1.1826 ⎦

y2 u2

=

1.7e−s 0.7 ⎡ 7.8s − 3 ⎤ −2s + ⎥e ⎢ ( −2.2s + 1) (3s + 1) ⎣ 20s + 8 ⎦ (26)

As stated earlier, the PI controllers are designed based on ETF. The corresponding closed-loop responses are obtained by simulation . The same controller settings are applied on EOTF, and the closed-loop responses are found to be same. Hence the reduced model (ETF) depicts the EOTF. Figure 2 shows the performance of the controlled system for a step change in set point y1 and y2 separately. The overshoot is found to be large

(20)

Since NI = 0.6969 is positive as calculated by eq 11, the pairing can be kept as it is. The dynamic elements such as normalized gain matrix (KN), RNGA (ϕ), average residence time (Tar), and RARTA (Γ) to obtain ETF matrix are calculated by using eqs 7,8, 9, and 10: 6469



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Figure 2. Performance of the PI controllers with decouplers for a step input in y1 and y2 separately.

Figure 3. Responses of the manipulated variable for a step input in y1 and y2 separately. ′

which is the typical performance of any unstable system with delay. However, no interactions among the loops are observed. Figure 3 shows the manipulated variable versus time behavior corresponding to Figure 2. In order to reduce the overshoot, a double-loop control structure is applied. As shown in the block diagram of Figure 4, first the decouplers are to be used. The

Gp,′ ii =

k p,′ ii e−θiis τii′s + 1

(28)

Here k′p,ii is the steady state value reached, θ′ii is the initial value noted, and τ′ii = t − θ′ii where t is the time taken to (0.63k′p,ii) from Figure 5. The fitted parameters are kp,11 ′ = 2.628, τ11 ′ = 7, kp,22 ′ = 3.225, τ22 ′ = 7, and θ11 ′ = θ22 ′ = 1 . For the stabilized FOPTD system, diagonal PI controllers by the synthesis method are designed and placed in the outer loop. kc, ii =

⎞ 1 ⎛ τii′ 0.5 ⎜⎜ + θii⎟⎟ ≈ ; k p,′ ii ⎝ τc, ii ⎠ k p,′ ii

τI, ii = τii′ (29)

Hence the obtained diagonal PI controller matrix for the outer loop is given by

Figure 4. Block diagram for the two stage control scheme: Gp, multivariable system; GD, decouplers; Gc,I, inner-loop stabilizing decentralized proportional controllers; Gc,O, outer-loop decentralized PI controllers; Y, output variables; Yr, set point values; V, load variables.

⎡ ⎤ ⎛ 1⎞ ⎢ 0.1902⎜1 + ⎟ 0 ⎥ ⎝ ⎠ 7s ⎢ ⎥ Gc(s) = ⎢ ⎛ 1 ⎞⎥ ⎢0 0.155⎜1 + ⎟ ⎥ ⎝ ⎣ 7s ⎠ ⎦

resulting system is to be then stabilized by diagonal proportional controllers which are designed on the diagonal elements of ETF by using this formula25 1 k p, iikc, ii = 0.5 εii (27)

(30)

Figure 6 shows that the responses of the double-loop controller are much superior to that of a single-loop PI controller. The interactions are found to be negligible (not shown in Figure 6). Figures 7 and 8 show the responses of single- and double-loop controllers when the load variables v1 and v2 enter the system along with the inputs u1 and u2. Stable responses with high oscillations and large interactions are observed for the singleloop systems, whereas the double-loop method reduces the overshoots and eliminates the interactions.The robustness of the double-loop system is evaluated by perturbing each process gain by +10% of the normal value in the process.The same controller

The proportional controller tuning parameters are kc,11 = −1.19 and kc,22 = −1.007. Further for the stabilized inner-loop system, diagonal PI controllers are designed based on the identified simple FOPTD model. Figure 5 shows the step responses of the single loop alone. Based on the above step response simple FOPTD models are fitted: 6470



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Figure 5. Responses of single-loop P controllers with decouplers for a step input in y1 and y2 separately.

Figure 6. Comparison of servo responses of double-loop and single-loop multivariable PI controllers with the decoupler. (No interactions for doubleloop system.

Figure 7. Responses and interactions of the single-loop PI controllers with decoupler for regulatory problems: (left) v1 = 1, v2 = 0; (right) v1 = 0, v2 = 1.

Figure 8. Comparison of responses of double-loop and single-loop multivariable PI controllers with the decoupler for regulatory problems (no interactions for the double-loop system): (left) v1 = 1, v2 = 0; (right) v1 = 0, v2 = 1). 6471



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Figure 9. Robustness of double-loop controllers for perturbation in each kp (10%) in the process: (top panels) servo problem; (bottom panels) regulatory problem.

Table 1. Comparison of IAE and ISE Values for Perturbations in the Model Parameters for the Double-Loop Control System (Example 1) servo (y1)

servo (y2)

regulatory (y1)

regulatory (y2)

perturbation

IAE

ISE

IAE

ISE

IAE

ISE

IAE

ISE

kp 1.1kp 0.9kp θ 1.1θ 0.9θ τ 1.1τ 0.9τ

13.98 16 11.86 13.98 13.98 13.99 13.98 14.02 13.98

6.378 7.071 5.62 6.38 6.43 6.37 6.37 6.5 6.2

16.77 14.11 11.92 14.11 14.35 14.13 14.11 14.2 14.2

7.664 6.7 5.69 6.7 6.8 6.69 6.7 6.8 6.71

36.4 36.2 36.7 36.47 36.46 36.4 36.45 36.49 36.47

29.7 27.9 32.5 29.74 31.97 28.74 31.16 29.12 29.74

53.08 52.6 54.28 53.08 53.6 53.1 53.08 53.14 53.29

63.2 58.6 71.49 63.27 73.62 60.37 63.27 61.04 71.43

The NI for this 2 × 2 system is 0.5833, and hence the columns are

setting and decouplers are used. Similar studies are also carried out for −10% perturbations. The robust performance is shown in Figure 9. Similar studies are carried out for perturbations in time delay (θ) and time constant (τ) separately, and the results are given in Table 1 in terms of IAE/ISE values. 5.2. Example 2. Consider the example given by Govindhakannan and Chidambaram,13,14 containing all elements unstable with equal poles: ⎡ −1.6667e−s ⎤ −1e−s ⎢ ⎥ ⎢ ( −1.6667s + 1) ( −1.6667s + 1) ⎥ Gp(s) = ⎢ −s −1.6667e−s ⎥⎥ ⎢ −0.8333e ⎢⎣ ( −1.6667s + 1) ( −1.6667s + 1) ⎥⎦

interchanged to get the correct pairing. Therefore the newly paired system is given by ⎡ −1e−s −1.6667e−s ⎤ ⎢ ⎥ ⎢ ( −1.6667s + 1) ( −1.6667s + 1) ⎥ Gp(s) = ⎢ −s −0.8333e−s ⎥⎥ ⎢ −1.6667e ⎢⎣ ( −1.6667s + 1) ( −1.6667s + 1) ⎥⎦

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

(31)

⎡ 2.6667 2.6667 ⎤ ⎡− 0.4283 1.4283 ⎤ Λ=⎢ ⎥; ⎥; Tar = ⎢ ⎣1.4283 − 0.4283⎦ ⎣ 2.6667 2.6667 ⎦ ⎡− 0.375 − 0.625 ⎤ ⎡− 0.4283 1.4283 ⎤ KN = ⎢ ⎥; ϕ = ⎢ ⎥; ⎣1.4283 − 0.4283⎦ ⎣− 0.625 − 0.3124 ⎦ ⎡1 1⎤ Γ=⎢ ⎣1 1⎦⎥ (34)

Since the number of open-loop unstable poles of Gp(s) is different from G̅p(s) = diag[gp,ii(s)], the pairing criteria for this system will differ from that of a stable system. Hence, the pairing is carried out according to point 1 of section 3: ⎤ ⎡−1.6667 − 1 K=⎢ ⎥; ⎣−0.8333 − 1.6667 ⎦

(33)

⎡1.4283 − 0.4283⎤ Λ=⎢ ⎣−0.4283 1.4283 ⎥⎦

The ETF model matrix is given by

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Figure 10. Performance of the PI controllers with decouplers for a step input in y1 and y2 separately (example 2).

Figure 11. Responses of a single loop with the set point filter (example 2).

Figure 12. Responses of single-loop P controllers with decouplers for a step input in y1 and y2 separately.

⎡ 2.3348e−s −1.16711e−s ⎤ ⎢ ⎥ ⎢ ( −1.6667s + 1) ( −1.6667s + 1) ⎥ ̂ Gp(s) = ⎢ −s 1.9456e−s ⎥⎥ ⎢ −1.16711e ⎢⎣ ( −1.6667s + 1) ( −1.6667s + 1) ⎥⎦

The obtained diagonal PI controller matrix by the synthesis method is given by ⎡ ⎤ ⎛ ⎞ 1 ⎟ 0 ⎢−0.5681⎜1 + ⎥ ⎝ ⎠ 24.3798s ⎢ ⎥ Gc(s) = ⎢ ⎥ ⎛ ⎞ 1 ⎢0 ⎟⎥ −0.68⎜1 + ⎢⎣ ⎝ 24.3798s ⎠ ⎥⎦

(35)

The simplified decoupler matrix is given by ⎡1 −1.6667 ⎤ D(s) = ⎢ ⎥ ⎣−2.001 1 ⎦

Figure 10 shows the responses of the single-loop PI controller with the decoupler. The overshoots are large but with negligible interactions. To reduce the overshoot, we use a set point filter with τI,ii as the time constant of the filter. The designed filter is given as follows: 1/(1 + τI,iis) where τI,ii is the integral time constant of the diagonal PI controller. The performances of the single-loop system with the set point filters are shown in Figure 11. The overshoot is reduced. However, the set point filter can reduce overshoot only for a servo problem. In order to reduce overshoot for a regulatory problem, the double-loop control

(36)

The combined model Gp(s)D(s) is the same as that obtained by the ETF method. 2.3348e−s ; ( −1.6667s + 1) 1.9456e−s * = = g p,22 ( −1.6667s + 1)

eff * = g p,11 = g p,11 eff g p,22

(38)

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Figure 13. Comparison of servo responses of double-loop and single-loop multivariable PI controllers with the decoupler (no interactions for the double-loop system.).

Figure 14. Responses and interactions of the single-loop PI controllers with decoupler for regulatory problems: (left) v1 = 1, v2 = 0; (right) v1 = 0, v2 = 1.

Figure 15. Comparison of double-loop and single-loop multivariable PI controllers with the decoupler for regulatory problems (no interactions for the double-loop system): (left) v1 = 1, v2 = 0; (right) v1 = 0, v2 = 1).

Table 2. Comparison of IAE and ISE Values for Perturbations in the Model Parameters for the Double-Loop Control System (Example 2) servo (y1)

servo (y2)

regulatory (y1)

regulatory (y2)

perturbation

IAE

ISE

IAE

ISE

IAE

ISE

IAE

ISE

kp 1.1kp 0.9kp θ 1.1θ 0.9θ τ 1.1τ 0.9τ

8 10.49 4.94 8 8 8 8 8 8

3.78 4.65 3.12 3.78 3.89 3.80 3.78 3.934 3.75

7.99 10.5 4.93 7.99 7.99 8 7.99 8 7.99

3.78 4.66 3.12 3.78 3.89 3.8 3.78 3.93 3.756

27.48 27.46 27.48 27.48 27.62 27.48 27.48 27.48 27.48

41.41 35.08 57.35 41.41 50.3 37.95 41.41 38.98 49.19

22.95 22.93 22.95 22.95 23.04 22.95 22.95 22.95 22.94

28.8 24.4 39.93 28.8 34.98 26.42 28.82 27.14 34.12

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Figure 16. Robustness of double-loop controllers for perturbation in each kp (10%) in the process: (top panels) servo problem; (bottom panels) regulatory problem.

■

structure is required. The design of a double-loop controller is carried out as stated in Example 1. Also given are the responses of single-loop P controllers with decouplers in Figure 12. The proportional controller tuning parameters are kc,11 = −0.55 and kc,22 = −0.66. The fitted parameters of the simplified FOPTD model for the inner loop are obtained as follows: k′p,11 = 4.435, τ11 ′ = 4, kp,22 ′ = 4.444, τ22 ′ = 4, and θ11 ′ = θ22 ′ = 1. The diagonal PI controller matrix by the synthesis method is calculated as ⎡ ⎤ ⎛ 1⎞ ⎢ 0.112⎜1 + ⎟ 0 ⎥ ⎝ ⎠ 4s ⎢ ⎥ Gc(s) = ⎢ ⎥ ⎛ 1⎞ ⎢0 0.112⎜1 + ⎟ ⎥ ⎝ ⎣ 4s ⎠ ⎦

AUTHOR INFORMATION

Corresponding Author

*E-mail:[email protected]. Notes

The authors declare no competing financial interest.

■

(39)

Figures 13 and 14 show the responses of the single- and the double-loop systems for step changes in the load variables v1 and v2 entering along with u1 and u2. The double-loop system gives an improved performance. The robustness of the double-loop control system is studied by perturbing each gain, time delay, and time constant in the process as explained earlier in the previous simulation example. Figure 15 shows the response when the uncertainty is given in the process gain. Table 2 shows the IAE/ ISE values for the above robustness studies (Figure 16) for the servo and regulatory problems.

6. CONCLUSION Based on the equivalent transfer function (ETF) model, multivariable PI controllers are designed for unstable multivariable systems with time delay. The method uses simplified decouplers which decompose the unstable multiloop systems into independent loops with ETFs as the resulting decoupled process model having unstable poles. This step is followed by the design of PI controllers for the diagonal elements by the synthesis method. Since the overshoot is found to be larger, a double-loop control structure is proposed to reduce the overshoot. To demonstrate the simplicity and effectiveness of the proposed method, two examples of two input two output TITO unstable systems, one with mild interactions and another with large interactions, are given. 6475

NOMENCLATURE dij = decoupler elements EOTF = effective open-loop transfer function ETF = equivalent transfer function FOPTD = first order plus time delay Gp, Gc = process and controller transfer function matrixes gp,ij, gc,ij, ĝij = process, controller, and equivalent transfer function models geff * = effective open-loop transfer function and decoupled p,ij, gp,ii process model IAE = integral of absolute error ISE = integral of the squared error K = process steady state gain KN = normalized gain matrix KNij = normalized gain kc,ii = diagonal proportional gain ̂ = process and effective steady state gain kp,ij, kp,ij NI = Niederlinski index P = proportional PI = proportional integral RARTA = relative average residence time array RGA = relative gain array RNGA = relative normalized gain array s = Laplace domain SSGM = steady state gain matrix t = time TITO = two input−two output uj = manipulated variable vk = load variable Y = closed-loop process response vector yi = closed-loop process response Γ = relative average residence time array (RARTA) θij, θ̂ij = process and effective time delay dx.doi.org/10.1021/ie403791q | Ind. Eng. Chem. Res. 2014, 53, 6467−6476


Article

Λ = relative gain array (RGA) τij, τ̂ij = process and effective time constant τI,ii = diagonal integral time constant τIi = integral time ϕ = relative normalized gain array (RNGA) Γij = relative average residence time ϕij = element of relative normalized gain Λij = relative gain array element

(23) Hovd, M.; Skogestad, S. Pairing criteria and decentralized control of unstable plant. Ind. Eng. Chem. Res. 1994, 33, 2134. (24) Flesch, R. C. C.; Torrico, B. C.; Normen-Rico, J. E.; Cavalcante, M. U. Unified aapproach for minimal output dead time compensation in MIMO processes. J. Process Control 2011, 21, 1081. (25) DePaor, A. M.; OMalley, M. Controllers of Ziegler-Nichols type for unstable process with time delay. Int. J. Control 1989, 49, 1273.

Subscripts

i, j= loop representation

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REFERENCES

(1) Padmasree, R.; Chidambaram, M. Control of Unstable Systems, 1st ed.; Narosa Publishing House: New Delhi, India, 2006. (2) Rao, A. S.; Chidambaram, M. PI/PID Controllers Design for Integrating and Unstable Systems. In PID control in the third Milleneium; Vilanova, R., Visioli, A., Eds.; Springer Verlag: London, 2012; p 75 . (3) Jacob, E. F.; Chidambaram, M. Control of unstable delay plus first order systems. Comput. Chem. Eng. 1996, 20, 579. (4) Luyben, W. L.; Luyben, M. L. Essentials of Process Control; McGrawHill: Singapore, 1997. (5) Maciejowski, J. M. Multivariable Feedback design; Addison-Wesley: New York, 1989. (6) Tanttu, J. T.; Lieslehto, J. A. Comparative study of some multivariable PI controller tuning methods. In Intelligent tuning and adaptive control methods; Devanathan, R., Ed.; Pergamon: Oxford, U.K., 1991; p 357. (7) Katebi, R. Robust multivariable tuning methods. In PID Control in the Third Millennium, Vilanova, R., Visioli, A., Eds.; Springer Verlag: London, 2012; p 255. (8) Davison, E. J. Multivariable tuning regulators: The feed-forward and robust control of general servo-mechanism problem. IEEE Trans Autom. Control 1976, AC-21, 35. (9) Wang, Q.-G.; Ye, Z.; Cai, W.-J.; Hang, C.-C. PID Control for Multivariable Processes; Springer-Verlag: Berlin, Heidelberg, 2008. (10) Wang, Q.-G.; Nie, Z.-Y. PID Control for MIMO Processes. In PID Control in the Third Milleneium; Vilanova, R., Visioli, A., Eds.; Springer-Verlag: London, 2012; p 177. (11) Georgiou, A.; Georgakis, C.; Luyben, W. L. Control of a multivariable open loop unstable process. Ind. Eng. Chem. Res. 1989, 28, 1481. (12) Agamenoni, O. E.; Deagas, A. C.; Romagnoli, J. A. A multivariable delay compensation scheme. Chem. Eng. Sci. 1992, 47, 1173. (13) Govindhakannan, J.; Chidambaram, M. Multivariable PI control of unstable systems. Process Control Qual. 1997, 10, 319. (14) Govindhakannan, J.; Chidambaram, M. Two stage multivariable PID controllers for unstable plus time delay systems. Indian Chem. Eng. 2000, 42, 34. (15) Gagnon, E.; Pommerleau, A.; Desbiens, A. Simplified, Ideal or Inverted Decoupling? ISA Trans. 1998, 37, 265. (16) Jevtovic, B. T.; Matausek, M. R. PID controller design of TITO system based on ideal decoupler. J. Process Control 2010, 20, 869. (17) Nordfeldt, P.; Hagglund, T. Decoupler and PID controller design of TITO systems. J. Process Control 2006, 16, 923. (18) Huang, H. P.; Jeng, J. C.; Chiang, C. H.; Pan, W. A. Direct method for multi loop P/PID controller design. J. Process Control 2003, 13, 769. (19) Xiong, Q.; Cai, W. J. Effective transfer function matrix method for decentralized control system design of multi-input multi-output processes. J. Process Control 2006, 16, 773. (20) Shen, Y.; Cai, W. J.; Li, S. Multivariable Process Control: Decentralized, Decoupling or Sparse? Ind. Eng. Chem. Res. 2010, 49, 761. (21) Rajapandiyan, C.; Chidambaram, M. Controller design for MIMO processes based on simple decoupled equivalent transfer functions and simplified decoupler. Ind. Eng. Chem. Res. 2012, 51, 12398. (22) Vu, T. N. L.; Lee, M. Independent design of multi loop PI/PID controllers for interacting multivariable processes. J. Process Control 2010, 922. 6476


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