DYNAMIC MODELLING, SIMULATION AND CONTROL OF mimo

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Jun 4, 2012 - Consequently, the design of decentralized controller for MIMO processes can be converted to the design of single loop controllers. The method ...
International Journal of Computer Trends and Technology (IJCTT) – volume 3 Issue 3 Number 4 – Jun 2012

DYNAMIC MODELLING, SIMULATION AND CONTROL OF mimo SYSTEMS M.Bharathi*,C.Selvakumar** *HOD/ Department of electronics and instrumentation, Bharath University, Chennai-73 **Prof & Head, St.Joseph’s College Of Engineering, Chennai-119

ABSTRACT Designing control systems for complete plants is the ultimate goal of a control designer. The problem is quite large and complex. It involves a large number of theoretical and practical considerations such as quality of controlled response; stability; the safety of the operating plant; the reliability of the control system; the range of control and ease of startup, shutdown, or changeover; the ease of operation; and the cost of the control system. The difficulties are aggravated by the fact that most of the industrial and chemical processes are largely nonlinear, imprecisely known, multivariable systems with many interactions. The measurements and manipulations are limited to a relatively small number of variables, while the control objectives may not be clearly stated or even known at the beginning of the control system design. Thus, the presence of process inputoutput time delay of different magnitude in multi-input-multi-output systems have drawn attention to research as the processes are difficult to control. Increase in complexity and interactions between inputs and outputs yield degraded process behavior. KEYWORDS multivariable systems, interaction, control system design, nonlinear, ISSN: 2231-2803

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OBJECTIVE In recent years all the methodologies adapted to solve for the parameters of individual controllers in which the loop interactions are taken into account have not guaranteed a solution. In addition, the extension for higher dimensional systems seems difficult because of the complicated and non-linear computation. It has been found that the independent design of decentralized controllers based on model based method is simple and effective only for low dimensional processes. For high dimensional processes this design has to be more conservative due to the inevitable modeling errors encountered in formulation. To overcome all these drawbacks and to include interactions in the control design, a novel method based on the equivalent transfer function method (ETF) is proposed. By considering four combination modes of gain and phase changes for a particular loop when all other loops are closed, this equivalent transfer function can effectively approximate the dynamic interactions among loops. Consequently, the design of decentralized controller for MIMO processes can be converted to the design of single loop controllers. The method is simple, straightforward, easy to understand and implement. Several multivariable industrial processes with different interaction characteristics are employed to demonstrate the effectiveness and simplicity of the design method compared to the existing methods.

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CONTROLLER DESIGN METHODOLOGIES: There are three major controller designs that are available. They are mainly a) Centralized controller b) Decentralized controller and c) Decoupler Of all the three configurations discussed above, the centralized controller is not used very widely because of the complexity and time constraints in computation. In addition to it the design is less transparent and can be damaging the entire plant during failures, thus not being highly reliable. The figure below shows the block diagram of a decentralized controller and with its representation.

 GC11 GC12 G GC 22 Gc ( s)   C 21     GCN1 GCN 2

 GC1N   GC 2 N       GCNN 

Figure 1: Centralized Control

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The decoupler though profitable and realistic is also very complex and degrades the load rejection. It has to be applied carefully and is often recommended only for the servo operations. The Figure 2 shows the block diagram for the decoupler.

Figure 2: Decoupling Controller Design

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The Decentralized controllers are widely used because of their simplicity in hardware, design and tuning simplicity, flexibility in operation and maintenance. The block diagram of the decentralized controller is shown in figure 3. The decentralized controller is represented as

0  0  GC1  0 G  0  C2  Gc ( s)         0  0 G C1  

Figure 3: Decentralized controller design All the three controller configurations are being shown for a 2x2 system with interactions. The decentralized controllers consist of multi loop SISO controllers with one control variable paired with one manipulated variable. The major idea in this design approach is that the SISO controllers should be tuned simultaneously with the interactions in the process taken into account. ISSN: 2231-2803

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PROCEDURE FOR DECENTRALISED CONTROLLER DESIGN Most industrial control systems use the multi loop SISO diagonal control structure. It is the most simple and understandable structure. Operators and plant engineers can use it and modify it when necessary. It does not require an expert in applied mathematics to design and maintain it. In addition, the performance of these diagonal controller structures is usually quite adequate for process control applications. In fact, there has been little quantitative unbiased data showing that the performances of the more sophisticated controller structures are really any better! The slight improvement is seldom worth the price of the additional complexity and engineering cost of implementation and maintenance. A number of critical questions must be answered in developing a control system for a plant. What should be controlled? What should be manipulated? How should the controlled and manipulated variables be paired in a multivariable plant? How do we tune the controllers? The procedure discussed in this chapter provides a practical approach to answering these questions. It was developed to provide a workable, stable, simple SISO system with only a modest amount of engineering effort. The resulting diagonal controller can then serve as a realistic benchmark, against which the more complex multivariable controller structures can be compared. The limitations of the procedure should be pointed out. It does not apply to open loop-unstable systems. It also does not work well when the time constants of the transfer functions are quite different, i.e., some parts much faster than others. The fast and slow sections should be designed separately in such a case. The procedure has been tested primarily on realistic distillation column models. This choice was deliberate because most industrial processes have similar gain, dead time, and lag transfer functions. Undoubtedly, some pathological transfer ISSN: 2231-2803

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functions can be found that the procedure cannot handle. But we are interested in a practical engineering tool, not elegant, rigorous, all-inclusive mathematical theorems.

The steps in the procedure are summarized below. Each step is discussed in more detail in later sections of this chapter. 1. Select controlled variables. Use primarily engineering judgment based on process understanding. 1. 2. Select manipulated variables. Find the set of manipulated variables that gives the largest minimum singular value of the steady-state gain matrix. 3. Eliminate unworkable variable pairings. The pairing can be done with RGA ERGA or using NI indices. 4. Find the best pairing from the remaining sets. a. Tune all combinations using a efficient tuning methodology. b. Select the pairing that gives the lowest-magnitude closed loop regulator transfer function.

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EFFECTIVE TRANSFER FUNCTION

COMPUTATION OF ETF Consider an open loop stable multivariable system within inputs and n outputs as shown in Fig. 1, where 𝑟𝑖 , 𝑖 =1, 2, . . . ,n, are the reference inputs; 𝑢𝑖 , 𝑖 = 1,2,. . . ,n, are the manipulated variables; 𝑦𝑖 , 𝑖 = 1,2,. . . ,n, are the system. Outputs, G(s) and 𝐺𝑐 𝑠 are process transfer function matrix And decentralized controller matrix with compatible dimensions, expressed by

Figure 4: Closed-loop multivariable control system. 𝑔11 (𝑠) 𝑔 (𝑠) 𝐺 𝑠 = 21 … 𝑔𝑛1 (𝑠) ISSN: 2231-2803

𝑔12 (𝑠) … ⋯ 𝑔𝑛2 (𝑠)

… … … …

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𝑔1𝑛 (𝑠) 𝑔2𝑛 (𝑠) … 𝑔𝑛𝑛 (𝑠) Page 43

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And 𝑔𝑐1 (𝑠) 0 𝐺𝑐 (𝑠) = … 0

0 𝑔𝑐2 (𝑠) ⋯ 0

… 0 … 0 … … … 𝑔𝑐𝑛 (𝑠)

respectively.

Let

𝑔𝑖𝑗 𝑗𝜔 = 𝑘𝑖𝑗 𝑔𝑖𝑗0 𝑗𝜔 ,

Where 𝑘𝑖𝑗 and 𝑔𝑖𝑗0 𝑗𝜔 ,are steady state gain and normalized transfer function of 𝑔𝑖𝑗 𝑗𝜔 ,i.e., 𝑔𝑖𝑗0 0 = 1 , respectively. The interaction among individual loop is described by ERGA, the main result of ERGA is summarized as follows. Define 𝑒𝑖𝑗 of a particular transfer function as 𝑒𝑖𝑗 = 𝑘𝑖𝑗

𝜔 𝑐,𝑖𝑗 0

𝑔𝑖𝑗0 (𝑗𝜔) 𝑑𝜔,

where 𝜔𝑐,𝑖𝑗 for i,j = 1,2,. . . ,n are the critical frequency of the transfer function 𝑔𝑖𝑗 𝑗𝜔 and ∎ is the absolute value of ∎. In order to calculate 𝑒𝑖𝑗 , the critical frequency can be defined in two ways: 𝜔𝑐,𝑖𝑗 = 𝜔𝐵,𝑖𝑗 , where 𝜔𝐵,𝑖𝑗 for i,j = 1,2,. . . ,n is the bandwidth of the transfer function 𝑔𝑖𝑗0 (𝑗𝜔) and determined by the frequency where the magnitude plot of frequency response reduced to 0.707 time, i.e., 𝑔𝑖𝑗 (𝑗𝜔𝐵,𝑖𝑗 ) = 0.707 𝑔𝑖𝑗 (0) .

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𝜔𝑐,𝑖𝑗 = 𝜔𝑢,𝑖𝑗 , where 𝜔𝑢,𝑖𝑗 for i,j = 1,2,. . . ,n is the ultimate of the transfer function 𝑔𝑖𝑗0 (𝑗𝜔) and determined by the frequency where the phase plot of frequency response across -𝜋, i.e., 𝑎𝑟𝑔 𝑔𝑖𝑗 (𝑗𝜔𝑢,𝑖𝑗 ) = −𝜋.

For transfer function matrices with some elements without phase crossover frequencies, such as first order or second order without time delay, it is necessary to use corresponding bandwidths as critical frequencies to calculate 𝑒𝑖𝑗 .However, it is worth to point out that the phase crossover frequency information, i.e., ultimate frequency (𝜔𝑢,𝑖𝑗 ) is recommended if applicable for calculation of 𝑒𝑖𝑗 , since it is closely linked to system dynamic performance and control system design. Without loss of generality, we will use 𝜔𝑢,𝑖𝑗 as the bases for the following development. For the frequency response of 𝑔𝑖𝑗 𝑗𝜔 as shown in Fig. 5, 𝑒𝑖𝑗 is the area covered by 𝑔𝑖𝑗 𝑗𝜔 up to 𝜔𝑢,𝑖𝑗 .

Since 𝑔𝑖𝑗0 (𝑗𝜔) represents the magnitude of the transfer

function at various frequencies, 𝑒𝑖𝑗 is considered to be the energy transmission ratio from the manipulated variable 𝑢𝑗 to the controlled variable 𝑦𝑖 . Express the energy transmission ratio array as 𝑒11 𝑒21 𝐸= … 𝑒𝑛1

𝑒12 … ⋯ 𝑒𝑛2

… 𝑒1𝑛 … 𝑒2𝑛 … … … 𝑒𝑛𝑛

To simplify the calculations, we approximate the integration of 𝑒𝑖𝑗 by a rectangle area, i.e., ISSN: 2231-2803

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𝑒𝑖𝑗 ≈ 𝑘𝑖𝑗 𝜔𝑢,𝑖𝑗 i,j = 1,2,.....,n.

Then, the effective energy transmission ratio array is given as: 𝐸 = G(0) ⊗ Ω, Where the operator ⊗ is the Hadamard product, and 𝑘11 𝑘 𝐺(0) = 21 … 𝑘𝑛1

𝑘12 … ⋯ 𝑘𝑛2

… … … …

𝑘1𝑛 𝑘2𝑛 … 𝑘𝑛𝑛

… … … …

𝜔𝑢,1𝑛 𝜔𝑢,2𝑛 … 𝜔𝑢,𝑛𝑛

And 𝜔𝑢,11 𝜔𝑢,21 𝛀= … 𝜔𝑢,𝑛1

𝜔𝑢,12 … ⋯ 𝜔𝑢,𝑛2

Are the steady state gain and the critical frequency array, respectively. Since 𝑒𝑖𝑗 is an indication of energy transmission ratio for loop 𝑦𝑖 − 𝑢𝑗 , the bigger the 𝑒𝑖𝑗 value is, the more dominant of the loop will be.

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Figure 5: Frequency response and effective energy of 𝑔𝑖𝑗 𝑗𝜔 Similar to the definition of relative gain , the effective relative gain,𝜙𝑖𝑗 between output variable 𝑦𝑖 and input variable 𝑢𝑗 , is define as the ratio of two effective energy transmission ratio: 𝜙𝑖𝑗 =

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𝑒𝑖𝑗 𝑒𝑖𝑗

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where 𝑒𝑖𝑗 is the effective energy transmission ratio between output variable 𝑦𝑖 and input variable 𝑢𝑗 , when all other loops are closed. When the effective relative gains are calculated for all the input/output combinations of a multivariable process, it results in an array, ERGA, which can be Calculated by 𝜙 = 𝐸 ⊗ 𝐸 −𝑇

𝜙11 𝜙 = 21 … 𝜙𝑛1

𝜙12 … ⋯ 𝜙𝑛2

… … … …

𝜙1𝑛 𝜙2𝑛 … 𝜙𝑛𝑛

The introduction of energy transmission ratio is to mathematically represent the effectiveness of a control loop which is affected by two key factors, i.e., the steady state gain of the transfer function reflecting the effect of the manipulated variable 𝑢𝑗 , to the controlled variable 𝑦𝑖 and the response speed reflecting the sensitivity of the controlled variable 𝑦𝑖 to the manipulated variable 𝑢𝑗 , and, consequently, the ability to reject the interactions from other loops. Since ERGA is a relative measure, using the multiplication of the two parameters to approximate the energy transmission ratio in 𝜙𝑖𝑗 can simplify the calculation while captures the key elements in a multivariable control system. The introduction of energy transmission ratio is to mathematically represent the effectiveness of a control loop which is affected by two key factors, i.e., the steady state gain of the transfer function reflecting the effect of the manipulated variable 𝑢𝑗 , to the controlled variable 𝑦𝑖 and the response speed reflecting the sensitivity of the controlled variable 𝑦𝑖

to the manipulated variable 𝑢𝑗 , and,

consequently, the ability to reject the interactions from other loops. Since ERGA is a relative measure, using the multiplication of the two parameters to approximate the energy transmission ratio in 𝜙𝑖𝑗 can simplify the calculation while captures the ISSN: 2231-2803

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key

elements

in

a

multivariable

control

system.

The ERGA is used to determine the best variable paring. In the following sections, we will employ this interaction measure to develop effective transfer functions (ETFs) under decentralized control structure

EFFECTIVE TRANSFER FUNCTION Suppose that the best loop configuration has been determined and the best pair is diagonally placed in the transfer function matrix as shown in Fig. 7. Similar to the open loop gain, we let the effective energy transmission ratio, 𝑒𝑖𝑗 when all other loops are closed be𝑒𝑖𝑗 = 𝑔𝑖𝑗 0 𝜔𝑢,𝑖𝑗 i,j= 1; 2; . . . ; n, where𝑔𝑖𝑗 0 and 𝜔𝑢,𝑖𝑗 are the steady state gain and ultimate frequency between output variable 𝑦𝑖 and input variable 𝑢𝑗 , when all other loops are closed, respectively. Then, from Eq.(1) 𝑔𝑖𝑗 0 𝜔𝑢,𝑖𝑗 =

𝑔 𝑖𝑗 (0)𝜔 𝑢 ,𝑖𝑗 , 𝜙 𝑖𝑗

(2)

By the definition of RGA, we have 𝑔𝑖𝑗 0 =

𝑔 𝑖𝑗 (0) 𝜆 𝑖𝑗

,

(3)

Substitute Eq.(3) into (2) and rearrange to result 𝜙 𝑖𝑗 𝜆 𝑖𝑗

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=

𝜔 𝑢 ,𝑖𝑗 , 𝜔 𝑢 ,𝑖𝑗

≡ 𝛾𝑖𝑗

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(4)

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where 𝛾𝑖𝑗 represents the critical frequency change of loop 𝑖 − 𝑗 when other loops are closed, defined as relative critical frequency. When the relative frequencies are calculated for all the input/output combinations of a multivariable process, it results in an array, i.e., relative frequency array (RFA). Since control loop transfer functions when other loops closed will have similar frequency properties with when other loops open if it is well paired , we can let the ETFs have the same structures as the corresponding open loop transfer functions but with different parameters. 𝑔𝑖𝑗 𝑠 = 𝑔𝑖𝑗 0 𝑔𝑖𝑖𝑟 (𝑠)𝑒 −𝑑 𝑖𝑖 𝑠

(5)

Where 𝑔𝑖𝑖𝑟 (𝑠) is defined by, 𝑔𝑖𝑖𝑟 𝑠 = 𝑔𝑖𝑖0 (𝑠) 𝑒 −𝑑 𝑖𝑖 𝑠 And 𝑑𝑖𝑖 is the time delay of the ETF. As the change in ultimate frequency of a control loop is generally affected by changes in both time constant and time delay when other loops are closed, and they are exchangeable by linear approximation, it is reasonable to change only time delay to reflect the phase changes. In Eq. (5), 𝑔𝑖𝑖0 (0) can be determined by using Eq. (3), while by the definition of the ultimate frequency,

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−d𝑖𝑖 𝜔𝑢,𝑖𝑖 + ∠𝑔𝑖𝑖𝑟 𝑗𝜔𝑢,𝑖𝑖 = −𝑑𝑖𝑖, 𝜔𝑢,𝑖𝑖, + ∠𝑔𝑖𝑖𝑟 (𝜔𝑢,𝑖𝑖, ) = −𝜋. 𝑑𝑖𝑖, can be easily determined by

𝑑𝑖𝑖, =

𝜋+∠𝑔𝑖𝑖𝑟 𝑗 𝜔 𝑢 ,𝑖𝑖 𝜔 𝑢 ,𝑗𝑖

= 𝛾𝑖𝑗 ×

𝜋+∠𝑔𝑖𝑖𝑟 𝑗 𝜔 𝑢 ,𝑖𝑖

(6)

𝜔 𝑢 ,𝑖𝑗 ,

Notice that 𝑔𝑖𝑖𝑟 (𝑠) is usually low order transfer functions, their contribution to the phase change at low frequency range are small and can be equivalently represented by the additional time delay term. In many decentralized control system designs, such as gain and phase margin method, an individual loop is tuned around the critical frequency region of each control loop. Accurate estimation of overall variation is required around the critical frequency, not who contribute to the change. By letting 𝑔𝑖𝑖𝑟 𝑗𝜔𝑢,𝑖𝑖

≈ 𝑔𝑖𝑖𝑟 𝑗𝜔𝑢,𝑖𝑖 , we can make further simplification to

Eq. (6) as −𝑑𝑖𝑖 𝜔𝑢,𝑖𝑖 ≈ −𝑑𝑖𝑖, 𝜔𝑢,𝑖𝑖,

which results by considering Eq. (4)

𝑑𝑖𝑖 ≈

𝜔 𝑢 ,𝑖𝑗 , 𝜔 𝑢 ,𝑖𝑗

𝑑𝑖𝑖, = 𝛾𝑖𝑖 𝑑𝑖𝑖,

(7)

This is the practical formula which will be used to derive the ETFs. Even though Eq. (7) is less accurate than Eq. (6), several simulation results have showed that the control system performances are comparable by the two approximations, ISSN: 2231-2803

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but Eq. (7) is much more straightforward ,easier explainable and understandable than Eq. (6). Since it is necessary that the controlled system possesses integrity property; that is, the overall control system remained to be stable regardless put in and/or taken out of other control loops, 𝑔𝑖𝑖 0 and 𝑑𝑖𝑖 in ETF must take different values for different combination of 𝜆𝑖𝑖 and 𝛾𝑖𝑖 . For the four different combinations of 𝜆𝑖𝑗 and𝛾𝑖𝑖 , 𝑔𝑖𝑖 𝑠

may take different modes shown in Figs. 6-9, and are

discussed below: Case 1: 𝜆𝑖𝑖 ≤ 1, 𝛾𝑖𝑖 ≤ 1 In this case,

1 𝜆 𝑖𝑖

− 1 ≥ 0 and 𝛾𝑖𝑖 − 1 ≤ 0. According to Eqs. (3) and (7), we

have 𝑔𝑖𝑖 0 ≥ 𝑔𝑖𝑖 (0) and 𝑑𝑖𝑖 ≤ 𝑑𝑖𝑖, . 

𝑔𝑖𝑖 0 ≥ 𝑔𝑖𝑖 (0) , this means that the magnitude of the frequency response when the other loops closed is not less than that of when the other loops open. Since the retaliatory effect from the other loops magnifies the main effect of 𝑢𝑗 , on 𝑦𝑖 , we need to reduce the controller gain to assure system stability. In this case, the gain is by Eq. (3) 𝑔𝑖𝑖 0 =

𝑔 𝑖𝑖 (0) 𝜆 𝑖𝑖

 𝑑𝑖𝑖 ≤ 𝑑𝑖𝑖, .this means that the time delay when the other loops closed is not bigger than that of when other loops open. The reduced time delay will increase the phase margin. However, by considering the control system integrity, the time delay needs to be kept as before, i.e., 𝑑𝑖𝑖 = 𝑑𝑖𝑖,

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Figure 6: Interaction mode with case 1 Case 2: 𝜆𝑖𝑖 ≤ 1, 𝛾𝑖𝑖 > 1 In this case

1 𝜆 𝑖𝑖

− 1 ≥ 0 and 𝛾𝑖𝑖 − 1 > 0. According to Eqs. (3) and (7), we

have 𝑔𝑖𝑖 0 ≥ 𝑔𝑖𝑖 (0) and 𝑑𝑖𝑖 > 𝑑𝑖𝑖, .  𝑔𝑖𝑖 0 ≥ 𝑔𝑖𝑖 (0), same as in Case 1, 𝑔𝑖𝑖 0 =

𝑔𝑖𝑖 (0) 𝜆𝑖𝑖

 𝑑𝑖𝑖 > 𝑑𝑖𝑖, ., this means that the time delay when the other loops closed is bigger than that of when the other loops open. The enlarged time delay will reduce the phase margin. In this case, the time delay is determined by Eq. (7) 𝑑𝑖𝑖 = 𝛾𝑖𝑖 𝑑𝑖𝑖

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Figure 7: Interaction mode with case 2 Case 3: 𝜆𝑖𝑖 > 1, 𝛾𝑖𝑖 ≤ 1 In this case,

1 𝜆 𝑖𝑖

− 1 < 0 and 𝛾𝑖𝑖 − 1 ≤ 0. According to Eqs. (3) and (7), we

have 𝑔𝑖𝑖 0 < 𝑔𝑖𝑖 (0) and 𝑑𝑖𝑖 ≤ 𝑑𝑖𝑖, . 

𝑔𝑖𝑖 0 ≥ 𝑔𝑖𝑖 (0) this means that the magnitude of the frequency response when the other loops closed is smaller than that of when the other loops open. Even if the retaliatory effect from other loops acts in opposition to the main effect of 𝑢𝑖, on 𝑦𝑖, we cannot enlarge the controller gain for better performance due to the system integrity consideration. Hence, the gain should be unchanged, i.e., 𝑔𝑖𝑖 0 = 𝑔𝑖𝑖 0 . ISSN: 2231-2803

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 𝑑𝑖𝑖 > 𝑑𝑖𝑖, ., same as in case 1, 𝑑𝑖𝑖 = 𝑑𝑖𝑖, .,

Figure 8: Interaction mode with case 3 Case 4: 𝜆𝑖𝑖 > 1, 𝛾𝑖𝑖 > 1 In this case,

1 𝜆 𝑖𝑖

− 1 < 0 and 𝛾𝑖𝑖 − 1 > 0. According toEqs. (3) and (7), we

have 𝑔𝑖𝑖 0 < 𝑔𝑖𝑖 (0) and 𝑑𝑖𝑖 > 𝑑𝑖𝑖, . 

𝑔𝑖𝑖 0 < 𝑔𝑖𝑖 (0) same as in Case 3, 𝑔𝑖𝑖 0 = 𝑔𝑖𝑖 (0) , same as in Case 2,

 𝑑𝑖𝑖 > 𝑑𝑖𝑖 same as in Case 2, ISSN: 2231-2803

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𝑑𝑖𝑖 = 𝛾𝑖𝑖 𝑑𝑖𝑖

Figure 9: Interaction mode with case 4

A unique problem for decentralized control of MIMO processes is the zero crossing: stable or unstable zeros might be introduced into a particular control loop when other loops are closed. If an unstable zero is introduced, it will result phase shift to the left in the frequency domain. In order to guarantee the entire system stability, the controllers are normally conservatively designed by conventional detuning approaches. By introducing the relative critical frequency,𝛾𝑖𝑖 , to indicate phase changes after the other loops closed, the effects of unstable zeros can be accurately estimated in each control loop. Consequently, the resultant control systems will be much less conservative.

Mathematically, the equivalent transfer function should incorporate the controllers of all other loops. ISSN: 2231-2803

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To solve such a complex problem, recursive solution is required by first assigning initial controllers, then finding the equivalent loop transfer functions and designing controllers again. This process is continuous until a stable solution is obtained. To simplify the problem, both detuning and independent methods proposed so far assume that all other closed loops are under perfect control when designing the controller for a particular loop and consider only the gain change. In the proposed method, the changes are considered for both gain and frequency. Especially, Eq. (3) focuses on the gain impact while Eq. (7) contributes to time delay portion, i.e., frequency impact. As will be shown later, it is far more accurate than those existing methods

3.3 DECENTRALISED CONTROL SYSTEM DESIGN Without loss of generality, we assume that each main loop, i.e., diagonal element in the transfer function matrix is represented by a second order plus dead time (SOPDT) model, which can be used to describe most of the industrial processes:

𝑔𝑖𝑖 𝑠 =

𝑏0,𝑖𝑖 𝑒 −𝑑 𝑖𝑖 𝑠 𝑎2,𝑖𝑖 𝑠 2 + 𝑎1,𝑖𝑖 𝑠 + 1

Similarly, ETF is represented as,

𝑔𝑖𝑖 𝑠 =

𝑔𝑖𝑖 0 𝑒 −𝑑 𝑖𝑖 𝑠 2 𝑎2,𝑖𝑖 𝑠 + 𝑎1,𝑖𝑖 𝑠 + 1

The decentralized controllers can then be independently designed by single loop approaches based on the corresponding ETFs. Here we employ the gain and phase margins approach. This is primary because the frequency response method ISSN: 2231-2803

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provides good performance in the face of uncertainty in both plant model and disturbances. The PID controller of each loop is supposed of the following standard form:

𝑔𝑐,𝑗 𝑠 = 𝑘𝑝,𝑖 +

𝑘𝑖,𝑖 + 𝑘𝑝,𝑖 𝑠 𝑠

The controller can be rewritten as

𝑔𝑐,𝑗

Where A=

𝑘 𝑑 ,𝑖 𝑘

, B=

𝑘𝑖,𝑖 + 𝑘𝑝,𝑖 𝑠 + 𝑘𝑑,𝑖 𝑠 2 𝐴𝑠 2 + 𝐵𝑠 + 𝐶 𝑠 = = , 𝑠 𝑠

𝑘 𝑝 ,𝑖 𝑘

, c=

𝑘 𝑖,,𝑖

By selecting A=𝑎2 , B=𝑎1 , C=1, the open loop

𝑘

transfer function becomes 𝑔𝑐,𝑗 𝑠 𝑔𝑖𝑖 𝑠 = 𝑘

𝑔𝑖𝑖 0 −𝑑 𝑠 𝑒 𝑖𝑖 𝑠

Denoting the gain and phase margin specifications as 𝐴𝑚 ,𝑖 and 𝜓𝑚 ,𝑖 and their crossover frequencies as and 𝜔𝑝,𝑖 , respectively, we have 𝑎𝑟𝑔 𝑔𝑐,𝑖 𝑗𝜔𝑔,𝑖, 𝑔𝑖𝑖 𝑗𝜔𝑔,𝑖,

= −𝜋,

𝐴𝑚 ,𝑖 𝑔𝑐,𝑖 𝑗𝜔𝑔,𝑖, 𝑔𝑖𝑖 𝑗𝜔𝑔,𝑖,

= 1, 𝑔𝑐,𝑖 𝑗𝜔𝑝,𝑖𝑢,𝑖𝑗 , 𝑔𝑖𝑖 𝑗𝜔𝑝,𝑖,

= 1,

𝜓𝑚 ,𝑖 = 𝜋 + 𝑎𝑟𝑔 𝑔𝑐,𝑖 𝑗𝜔𝑢,𝑖𝑗 , 𝑔𝑖𝑖 𝑗𝜔𝑢,𝑖𝑗 , . ISSN: 2231-2803

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By substitution and simplification to above equations, we obtain 𝜔𝑔,𝑖, 𝑑𝑖𝑖 =

𝜋

𝐴𝑚 ,𝑖 =

2

𝜔 𝑔,𝑖, 𝑔 𝑖𝑖 0 𝑘

,

𝜋

𝑔𝑖𝑖 0 𝑘 = 𝜔𝑝,𝑖,

𝜓𝑚 ,𝑖 = − 𝜔𝑔,𝑖, 𝑑𝑖𝑖 , 2

Which results 𝜓𝑚 ,𝑖 =

𝜋 2

1−

1

,

𝐴 𝑚 ,𝑖

𝑘=

𝜋 2𝐴𝑚 ,𝑖 𝑑 𝑖𝑖 𝑔 𝑖𝑖 0

By this formulation, the gain and phase margins are interrelated to each other, some possible gain and phase margin selections are given in Table 1. The PID parameters are given by 𝑘𝑝,𝑖 𝜋 𝑘𝑖,𝑖 = 2𝐴 𝑚 ,𝑖 𝑑 𝑖𝑖 𝑔 𝑖𝑖 𝑘𝑑,𝑖

0

𝑎1,𝑖𝑖 0 𝑎2,𝑖𝑖

(8)

Applying Eq. (8) for each case discussed in Section 3, we can easily obtain both ETFs and the PID parameters which are summarized in Table 2.

Table 1: Typical gain and phase margin values: 𝝍𝒎,𝒊 𝑨𝒎,𝒊

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𝝅/𝟒 2

𝝅/𝟑 3

𝟑𝝅/𝟖 4

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𝟐𝝅/𝟓 5

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Table 2: Decentralized PID controller design Mode

𝒈𝒊𝒊 𝒔

𝒌𝒑,𝒊𝒊

𝒌𝒊,𝒊𝒊

𝒌𝒅,𝒊𝒊

Case

𝑔𝑖𝑖 0 / 𝜆𝑖𝑖 𝑒 −𝑑 𝑖𝑖 𝑠 𝑎2,𝑖𝑖 𝑠 2 + 𝑎1,𝑖𝑖 𝑠 + 1

𝜋𝑎1,𝑖𝑖 𝜆𝑖𝑖 2𝐴𝑚 ,𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0

𝜋𝜆𝑖𝑖 2𝐴𝑚 ,𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0

𝜋𝜆𝑖𝑖 𝑎2,𝑖𝑖 2𝐴𝑚 ,𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0

𝜋𝜆𝑖𝑖 2𝐴𝑚 ,𝑖 𝛾𝑖𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0

𝜋𝜆𝑖𝑖 𝑎2,𝑖𝑖 2𝐴𝑚 ,𝑖 𝛾𝑖𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0

𝜋 2𝐴𝑚 ,𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0

𝜋𝑎2,𝑖𝑖 2𝐴𝑚 ,𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0

𝜋 2𝐴𝑚 ,𝑖 𝛾𝑖𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0

𝜋𝑎2,𝑖𝑖 2𝐴𝑚 ,𝑖 𝛾𝑖𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0

1 Case

𝜋𝑎1,𝑖𝑖 𝜆𝑖𝑖 𝑔𝑖𝑖 0 / 𝜆𝑖𝑖 𝑒 −𝛾 𝑖𝑖 𝑑 𝑖𝑖 𝑠 2𝐴𝑚 ,𝑖 𝛾𝑖𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0 𝑎2,𝑖𝑖 𝑠 2 + 𝑎1,𝑖𝑖 𝑠 + 1

2 Case

𝑔𝑖𝑖 0 𝑒 −𝑑 𝑖𝑖 𝑠 2 𝑎2,𝑖𝑖 𝑠 + 𝑎1,𝑖𝑖 𝑠 + 1

𝜋𝑎1,𝑖𝑖 2𝐴𝑚 ,𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0

3 Case

𝜋𝑎1,𝑖𝑖 𝑔𝑖𝑖 0 −𝛾 𝑖𝑖 𝑑 𝑖𝑖 𝑠 𝑒 2𝐴𝑚 ,𝑖 𝛾𝑖𝑖 𝑑𝑖𝑖 𝑔𝑖𝑖 0 𝑎2,𝑖𝑖 𝑠 2 + 𝑎1,𝑖𝑖 𝑠 + 1

4

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EXTENSION OF ETF-A PROPOSAL 4.1 PRELIMINARIES For most of the processes the steady state behavior can be defined with the help of the frequency response of the process. If the input u (t) is a sine wave 𝑢 𝑠 (t) of amplitude Ū and frequency ω then 𝑢 𝑠 (t)= Ū sin (ωt)

If the system is initially at rest (all derivatives equal to zero) and we start to force it with a sine wave 𝑢 𝑠 (t), the output y (t) will go through some transient period and then settle down to a steady sinusoidal oscillation. In the Laplace domain, the output is by definition

Y(s) =G(s) U(s) For the sine wave input u (t) = Ū sin (ωt). Laplace transforming,

U(s)= Ū

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𝜔 𝑠2+𝜔2

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Therefore, the output with this sine wave input is

Y(s) =G(s) U(s) = G(s) Ū

𝜔 𝑠2+𝜔2

When s=0, Y(s) = (G (0)*Ū)/ ω The above equation () indicates that the output Y(s) is dependent on the value of the input frequency ω. As the value of the frequency increases the output ceases to decrease and vice-versa. This means that whenever the system has to be forced in such a way that the output has to follow the input, we need to multiply the value of the steady state gain with the frequency. For a multi-variable system, when the independent design of the decentralized controller is designed with the ETF, the interactions present in the system plays a major role. The change in these interaction frequencies affects the dynamics of the system to a larger extent. This is because the output of the system is also dependent on the interaction frequency. During the design of the controllers these interaction frequencies may be incorporated in order to achieve a better overall response. When the frequency of the interaction present in the system varies then the energy transmission ratio will also vary as the energy transmission ratio for i,j is 𝜔

𝑒𝑖𝑗 = 𝑘𝑖𝑗

0

|𝑔𝑖𝑗 𝑗𝜔 |𝑑𝜔

This change in the value of the energy transmission ratio will affect the stability of the system as ERGA also is dependent on this energy transmission ratio. It is important to note that the ERGA plays a major role in the ETF computation. This can be overcome by taking into consideration the interaction frequencies. ISSN: 2231-2803

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MODIFIED CONTROLLER DESIGN During the decentralized controller design, the interactions of the systems are not taken into considerations. As mentioned earlier, considering these interactions would result in a better overall response for the system. These interactions can be incorporated directly in the controller design of the system. The dynamic change in the system due to the interactions can be estimated well using the RFA computed for the system. According to Y.G.Wang and W.J.Cai, the controller design for integrating and unstable processes with gain and phase margins is proposed as, 𝑎1,𝑖𝑖 𝑘𝑝,𝑖 𝜋 0 𝑘𝑖,𝑖 = 2𝐴 𝑑 𝑔 0 𝑚 ,𝑖 𝑖𝑖 𝑖𝑖 𝑎2,𝑖𝑖 𝑘𝑑,𝑖 Consider the equations below 𝜙𝑖𝑗 𝜔𝑢,𝑖𝑗 , = ≡ 𝛾𝑖𝑗 𝜆𝑖𝑗 𝜔𝑢,𝑖𝑗 −𝑑𝑖𝑖 𝜔𝑢,𝑖𝑖 ≈ −𝑑𝑖𝑖, 𝜔𝑢,𝑖𝑖, This clearly indicates that the delay of the effective transfer function is dependent only on the frequency of the diagonal elements but not on the interaction frequencies. Now by taking into account the interaction frequencies also into consideration equation (7) becomes,

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−𝑑𝑖𝑖 𝜔𝑢,𝑖𝑖 ≈ −𝑑𝑖𝑖, 𝜔𝑢,𝑖𝑖, /𝛾𝑖𝑗

𝑑𝑖𝑖 ≈

𝜔 𝑢 ,𝑖𝑗 , 𝜔 𝑢 ,𝑖𝑗

𝑑𝑖𝑖, = 𝛾𝑖𝑖 𝑑𝑖𝑖 /𝛾𝑖𝑗

(9)

Now from equations (8) and (9) we have, 𝑎1,𝑖𝑖 𝑘𝑝,𝑖 𝜋𝛾𝑖𝑗 0 𝑘𝑖,𝑖 = 2𝐴 𝑑 𝑔 0 𝑚 ,𝑖 𝑖𝑖 𝑖𝑖 𝑎2,𝑖𝑖 𝑘𝑑,𝑖 The controller design is performed after all the four cases as in table (2) has been evaluated.

4.3 CASE STUDIES In this section of the chapter the proposed design methods is being applied to a variety of industrial processes. To demonstrate the simplicity and effectiveness of the proposed method certain bench mark models have also been considered. The discussions are performed for four 2x2 systems and a 3x3 system. Example 1: Consider a process given by 5𝑒 −3𝑠 2.5𝑒 −5𝑠 1 15𝑠 + 1 𝐺 𝑠 = 4𝑠 +−6𝑠 −4𝑒 𝑒 −4𝑠 20𝑠 + 1 5𝑠 + 1

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This process seems easy to control. However both the variable pairing and control design are very difficult with the conventional processes. The computations of RGA (Relative Gain Array) Λ, ERGA (Effective Relative Gain Array) Φ, CFA (Critical Frequency Array) Ω and RFA (Relative Frequency Array) Γ are shown below. G (0) =

5 −4

2.5 1

Λ= G (0).* [G(0)]−𝑇

Λ=

𝟎. 𝟑𝟑𝟑𝟑 𝟎. 𝟔𝟔𝟔𝟕 𝟎. 𝟔𝟔𝟔𝟕 𝟎. 𝟑𝟑𝟑𝟑

The Critical Frequency Array is computed by determining the cross over frequency for the transfer functions describing the process. The cross over frequency for 𝐺11 is found to be 0.6308 from the following figure [10]. The same procedure is performed for each element in G(s) to determine the CFA. Bode Diagram(Example System) Gm = 2.84 dB (at 1.72 rad/sec) , Pm = 31.4 deg (at 1.22 rad/sec)

Magnitude (dB)

20 10 0 -10

Phase (deg)

-20 0 -180 -360 -540 -720 -2 10

10

-1

0

10

1

10

Frequency (rad/sec)

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Figure 10

Thus the CFA is given as

Ω=

𝟎. 𝟔𝟑𝟎𝟖 𝟎. 𝟐𝟖𝟓𝟔

𝟎. 𝟐𝟕𝟑𝟐 𝟎. 𝟒𝟗𝟎𝟗

The ERGA is computed as shown below E=G (0).* Ω

Thus E=

3.154 −1.1424

0.683 0.4909

Φ=E.*[E]−T

Φ=

𝟎. 𝟔𝟔𝟒𝟗 𝟎. 𝟑𝟑𝟓𝟏

𝟎. 𝟑𝟑𝟓𝟏 𝟎. 𝟔𝟔𝟒𝟗

The Relative Frequency Array can be obtained by performing the Hadamard division of ERGA by RGA. The RFA was found to be,

Γ= ISSN: 2231-2803

𝟏. 𝟗𝟗𝟒𝟖 𝟎. 𝟓𝟎𝟐𝟔

𝟎. 𝟓𝟎𝟐𝟔 𝟏. 𝟗𝟗𝟒𝟖

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It can be found that the best pairing according to RGA (0.6667) is off-diagonal. But the above pairing leads to a very oscillatory response. However following ERGA method, it falls under case 2. The equivalent process for two loops are calculated using the ETF method as, 15 𝑒 −5.9844 𝑠

and

4𝑠+1

3𝑒 −7.9792 𝑠 5𝑠+1

For loop1 and loop2 respectively. The resultant PI controllers using the ETF approach and the proposed method are tabulated below. The closed loop response of the proposed method is shown with the ETF response.

Table 3: Controllers for Example 1 Controller

Proposed

Xiong and Cai

Kpii

Kpii

τIii

τIii

Loop 1

0.0117

4

0.0233

4

Loop 2

0.0550

5

0.1094

5

Figure 11: Closed loop response for example 1 1.4

1.2

1

0.8

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0.6

0.4

Q.Xiang,W-J Cai

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Time(secs)

0.2

0.1

0

y2 -0.1

-0.2

Q Xiang,W-J Cai Proposed

-0.3

Time(secs)

It can be seen that the proposed method results in better performance with a lesser value of peak overshoot and minimized oscillations. Example 2: Consider an industrial-scale polymerization reactor given by

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22.89𝑒 −0.2𝑠 +1 𝐺 𝑠 = 4.572𝑠 −0.2𝑠 4. 689𝑒 2.174𝑠 + 1

−11.64𝑒 −0.4𝑠 1.807𝑠 + 1 5.80𝑒 −0.4𝑠 1.801𝑠 + 1

The time scales are in hours, so it is a quite slow process. In addition, it is easy to verify that it is not diagonally dominant. The RGA, ERGA, CFA and RFA are computed and shown below. G (0) =

22.89 4.689

−11.64 5.80

Λ= G (0).* [G(0)]−𝑇

Λ=

𝟎. 𝟕𝟎𝟖𝟕 𝟎. 𝟐𝟗𝟏𝟑 𝟎. 𝟐𝟗𝟏𝟑 𝟎. 𝟕𝟎𝟖𝟕

The Critical Frequency Array is computed by determining the cross over frequency for the transfer functions describing the process. The cross over frequency for 𝐺11 is found to be 8.0554 from the following figure [12]. The same procedure is performed for each element in G(s) to determine the CFA.

Bode Diagram (Polymerization Reactor) Gm = 4.06 dB (at 7.99 rad/sec) , Pm = 35.2 deg (at 5 rad/sec) 30 Magnitude (dB)

20 10 0 -10 -20 -30 0 Phase (deg)

-180 -360 -540 -720 -900 -1080 -1260 -2 10

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-1

10

0

10 Frequency (rad/sec)

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10

1

2

10

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Figure 12 Thus the CFA is given as Ω=

𝟖. 𝟎𝟓𝟓𝟒 𝟕. 𝟖𝟓𝟒𝟎

𝟒. 𝟏𝟖𝟖𝟖 𝟒. 𝟑𝟎𝟑𝟔

The ERGA is computed as shown below E=G (0).* Ω

Thus E=

184.3873 36.8273

−48.7575 24.9606

Φ=E.*[E]−T

Φ=

𝟎. 𝟕𝟏𝟗𝟑 𝟎. 𝟐𝟖𝟎𝟕

𝟎. 𝟐𝟖𝟎𝟕 𝟎. 𝟕𝟏𝟗𝟑

The Relative Frequency Array can be obtained by performing the Hadamard division of ERGA by RGA. The RFA was found to be,

Γ=

𝟏. 𝟎𝟏𝟓𝟏 𝟎. 𝟗𝟔𝟑𝟑

𝟎. 𝟗𝟔𝟑𝟑 𝟏. 𝟎𝟏𝟓𝟏

Both ERGA and RGA indicate diagonal pairing. This process falls under the case 2 and according to ETF method the equivalent process for the two loops are calculated as 32.3003 𝑒 −0.2030 𝑠 4.572𝑠+1 ISSN: 2231-2803

and

8.1844 𝑒 −0.4060 𝑠 1.801𝑠+1

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The PI controllers are determined with the proposed method and the response is compared with ETF method, RGA based tuning approach by Chien et al, BLT tuning approach by Luyben and the relay based auto-tuning approach proposed by Loh et al. The controller values computed are tabulated below.

Table 4: Controllers for Example 2 Controller

Proposed Kpii

Xiong and Cai τIii

Kpii

τIii

Luyben Kpii

τIii

Chien et al.

Loh et al.

Kpii

Kpii

τIii

τIii

Loop 1

0.2109 4.5720 0.2190 4.5720 0.210 2.26 0.263 1.42 0.620 0.60

Loop 2

0.1640 1.8010 0.1703 1.8010 0.175 4.25 0.163 1.77 0.247 1.78

Figure 13 : Closed loop response for example 2

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Q Xiang W-J CAi Luyben Chien et al. Loh et al. Proposed

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Q Xiang W-J CAi Luyben Chien et al. Loh et al. Proposed

It can be seen that the proposed method can also be applied for those processes that are not diagonally dominant. To make it evident that this method is applicable for all processes with complex interactions, let us extend the approach to certain bench mark models such as the WB and the VL column. Example 3: Consider the WB process given by 12.8𝑒 −𝑠 −18.9𝑒 −3𝑠 +1 21𝑠 + 1 𝐺 𝑠 = 16.7𝑠−7𝑠 6.6𝑒 −19.4𝑒 −3𝑠 10.9𝑠 + 1 14.4𝑠 + 1 ISSN: 2231-2803

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There are many tuning methods for the control of the benchmark WB distillation column. The ETF parameters for the column are computed as below. The RGA, ERGA, CFA and RFA are computed and shown below.

G (0) =

12.8 6.6

−18.9 −19.4

Λ= G (0).* [G(0)]−𝑇

Λ=

𝟐. 𝟎𝟎𝟏 −𝟏. 𝟎𝟎𝟏

−𝟏. 𝟎𝟎𝟏 𝟐. 𝟎𝟎𝟏

The Critical Frequency Array is computed by determining the cross over frequency for the transfer functions describing the process. The cross over frequency for 𝐺11 is found to be 1.6080 from the following figure [14]. The same procedure is performed for each element in G(s) to determine the CFA. Bode Diagram( Wood and Berry column) Gm = 6.44 dB (at 1.61 rad/sec) , Pm = 50.7 deg (at 0.764 rad/sec) 30 Magnitude (dB)

20 10 0 -10 -20 -30 0 Phase (deg)

-90 -180 -270 -360 -450 -540 -630 -3 10

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-2

10

-1

10 Frequency (rad/sec)

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10

0

1

10

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Figure 14 Thus the CFA is given as Ω=

𝟏. 𝟔𝟎𝟖𝟎 𝟏. 𝟏𝟑𝟑𝟓

𝟎. 𝟓𝟓𝟐𝟑 𝟎. 𝟓𝟒𝟕

The ERGA is computed as shown below E=G (0).* Ω

Thus 20.5824 7.4811

E=

−10.4385 −10.6118

Φ=E.*[E]−T

Φ=

𝟏. 𝟓𝟓𝟔𝟓 −𝟎. 𝟓𝟓𝟔𝟓

−𝟎. 𝟓𝟓𝟔𝟓 𝟏. 𝟓𝟓𝟔𝟓

The Relative Frequency Array can be obtained by performing the Hadamard division of ERGA by RGA. The RFA was found to be, Γ=

𝟎. 𝟕𝟕𝟒𝟔 𝟎. 𝟓𝟓𝟏𝟑

𝟎. 𝟓𝟓𝟏𝟑 𝟎. 𝟕𝟕𝟒𝟔

Both ERGA and RGA indicate diagonal pairing. This process falls under the case 3 and according to ETF method the equivalent process for the two loops are calculated as

12.8 𝑒 −𝑠 16.7𝑠+1 ISSN: 2231-2803

and

−19.4 𝑒 −3𝑠 14.4𝑠+1

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The PI controllers computed using the ETF method as well as the proposed method are tabulated as follows.

Table 5 Controllers for Example 3 Controller

Proposed

Xiong

and BLT

ZN

Emprical

Cai Kpii

tuning

Kpii

τIii

Sequential

τIii

Kpii

τIii

Kpii

τIii

Kpii

τIii

Kpii

τIii

8.3

0.96

3.3

0.2

4.4

0.87

3.3

-0.19

9.2

-0.04 2.7

Loop 1

0.5381

16.7 0.976 16.7

0.38

Loop 2

-0.181

14.4 0.32

-0.08 80

14.4

-0.09 10.4

Figure 15 Closed loop responses of WB column 2

WB(Wood and Berry) (loop1)

1.8

1.6

1.4

y1

1.2

1

0.8

0.6

Proposed Emperical BLT Z-N Sequential

0.4

0.2

0

0

10

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20

30

Time(secs)

40

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50

60

70

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1.2

WB(Wood and Berry) (loop2)

1

Proposed BLT Z-N Emperical Sequential

0.8

0.6

y2

0.4

0.2

0

-0.2

-0.4

0

10

20

30

Time(secs)

40

50

60

70

The responses obtained have been compared with the Empirical tuning method, ZN tuning method BLT as well as with the sequential order tuning. The peak overshoot is fairly reduced in the proposed method with minimized oscillations. The overall response is found to be better even after the interaction is taken into consideration.

Example 4: Consider the VL column given by −2.2𝑒 −𝑠 1 𝐺 𝑠 = 7𝑠 +−1.8𝑠 −2.8𝑒 9.5𝑠 + 1

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1.3𝑒 −0.3𝑠 7𝑠 + 1 4.3𝑒 −0.35𝑠 9.2𝑠 + 1

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The RGA, ERGA, CFA and RFA are computed and shown below.

G (0) =

−2.2 1.3 −2.8 4.3

Λ= G (0).* [G(0)]−𝑇 𝟏. 𝟔𝟐𝟓𝟒 −𝟎. 𝟔𝟐𝟓𝟒

Λ=

−𝟎. 𝟔𝟐𝟓𝟒 𝟏. 𝟔𝟐𝟓𝟒

The Critical Frequency Array is computed by determining the cross over frequency for the transfer functions describing the process. The cross over frequency for 𝐺11 is found to be 4.73 from the following figure [15]. The same procedure is performed for each element in G(s) to determine the CFA. Bode Diagram( Vinate and Luyben) Gm = -6.85 dB (at 0 rad/sec) , Pm = -79 deg (at 0.28 rad/sec)

Magnitude (dB)

10 0 -10 -20 -30 -40 180 Phase (deg)

90 0 -90 -180 -270 -360 -450 -3 10

-2

10

-1

10 Frequency (rad/sec)

10

0

1

10

Figure 16 Thus the CFA is given as

Ω=

ISSN: 2231-2803

𝟒. 𝟕𝟑 𝟐. 𝟔𝟑

𝟓. 𝟑𝟐𝟓𝟑 𝟒. 𝟓𝟓𝟔𝟏

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The ERGA is computed as shown below E=G (0).* Ω Thus E=

−10.406 7.364

6.9229 19.5112

Φ=E.*[E]−T

Φ=

𝟏. 𝟑𝟑𝟑𝟓 −𝟎. 𝟑𝟑𝟑𝟓

−𝟎. 𝟑𝟑𝟑𝟓 𝟏. 𝟑𝟑𝟑𝟓

The Relative Frequency Array can be obtained by performing the Hadamard division of ERGA by RGA. The RFA was found to be,

Γ=

𝟎. 𝟖𝟐𝟎𝟒 𝟎. 𝟓𝟑𝟑𝟑

𝟎. 𝟓𝟑𝟑𝟑 𝟎. 𝟖𝟐𝟎𝟒

Both ERGA and RGA indicate diagonal pairing. This process falls under the case 3 and according to ETF method the equivalent process for the two loops are calculated as −2.2 𝑒 −𝑠 7𝑠+1

and

4.3 𝑒 −0.35𝑠 9.2𝑠+1

The PI controllers computed using the ETF method as well as the proposed method are tabulated below. The responses obtained have been compared with the ISSN: 2231-2803

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Empirical tuning method, Z-N tuning method BLT as well as with the sequential order tuning.

Table 6 Controllers for Example 4 Controller

Proposed

Xiong

and BLT

ZN

Emprical

Cai Kpii

Kpii

τIii

Sequential tuning

τIii

Kpii

τIii

Kpii

τIii

Kpii

τIii

Kpii

τIii

Loop 1

-0.50

7

-0.94

7

-1.07

7.1

-2.4

3.2

-2.4

3.2

-1.4

3

Loop 2

0.52

9.2

0.98

9.2

1.97

2.6

4.5

1.2

4.4

1.2

3.4

1.33

Figure 17 Closed loop response of VL column 1.4

Vinate and Luyben(loop1) 1.2

1

y1

0.8

0.6

Emperical Z-N BLT Proposed Sequential

0.4

0.2

0

0

10

20

30

40

50

60

70

Time(secs)

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0.5

Vinate and Luyben(loop2)

Emperical Sequential Proposed Z-N BLT

0.4

0.3

0.2

y2

0.1

0

-0.1

-0.2

-0.3

0

10

20

30

Time(secs)

40

50

60

70

The peak overshoot is fairly reduced in the proposed method with minimized oscillations and a better settling time. Example 5: Consider a 3x3 process given by 119𝑒 −5𝑠 21.7𝑠 + 1 37.0𝑒 −5𝑠 𝐺 𝑠 = 500𝑠 + 1 93.0𝑒 −5𝑠 500𝑠 + 1

153𝑒 −5𝑠 337𝑠 + 1 76.7𝑒 −5𝑠 28𝑠 + 1 −66.7𝑒 −5𝑠 166𝑠 + 1

−2.1𝑒 −5𝑠 10𝑠 + 1 −5.0𝑒 −5𝑠 10𝑠 + 1 −103.3𝑒 −5𝑠 23𝑠 + 1

This process was used by Loh and Vasnani to verify their design for high dimensional systems. It is also not a diagonally dominant process.

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The RGA, ERGA, CFA and RFA are computed and shown below. 119 𝐺 0 = 37 93

153 76.7 −66.7

−2.1 −5 −103.3

Λ= G (0).* [G(0)]−𝑇 𝟐. 𝟏𝟖𝟗𝟔 𝚲 = −𝟏. 𝟑𝟏𝟒𝟕 𝟎. 𝟏𝟐𝟓𝟐

−𝟏. 𝟏𝟒𝟒𝟔 𝟐. 𝟎𝟔𝟕𝟕 𝟎. 𝟎𝟕𝟔𝟗

−𝟎. 𝟎𝟒𝟒𝟗 𝟎. 𝟐𝟒𝟕𝟎 𝟎. 𝟕𝟗𝟕𝟗

The Critical Frequency Array is computed by determining the cross over frequency for the transfer functions describing the process. The cross over frequency for 𝐺 is found to be 0.0000 from the following figure []. The same procedure is performed for each element in G(s) to determine the CFA. Bode Diagram (3x3 System) Gm = -0.229 dB (at 5.34 rad/sec) , Pm = -40.5 deg (at 5.48 rad/sec) 50 Magnitude (dB)

40 30 20 10

Phase (deg)

0 -10 0 -360 -720 -1080 -1440 -1800 -2160 -2520 -2880 -3 10

-2

10

-1

10 Frequency (rad/sec)

10

0

1

10

Figure 18 ISSN: 2231-2803

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Thus the CFA is given as 𝟎. 𝟑𝟐𝟗𝟎 𝟎. 𝟑𝟐𝟐𝟐 𝟎. 𝟑𝟔𝟏𝟏 Ω = 𝟎. 𝟑𝟏𝟒𝟐 𝟎. 𝟑𝟑𝟎𝟕 𝟎. 𝟑𝟔𝟗𝟔 𝟎. 𝟑𝟏𝟒𝟐 𝟎. 𝟑𝟏𝟒𝟐 𝟎. 𝟑𝟒𝟗𝟏 The ERGA is computed as shown below E=G (0).* Ω Thus 𝟑𝟗. 𝟏𝟓𝟏 𝐄 = 𝟏𝟏. 𝟔𝟐𝟓𝟒 𝟐𝟗. 𝟐𝟐𝟎𝟔

𝟒𝟗. 𝟐𝟗𝟔𝟔 −𝟎. 𝟕𝟓𝟖𝟑 𝟐𝟓. 𝟑𝟔𝟒𝟕 −𝟏. 𝟖𝟒𝟖 −𝟐𝟎. 𝟗𝟓𝟕𝟏 −𝟑𝟔. 𝟎𝟔𝟐𝟎 Φ=E.*[E]−T

𝟐. 𝟎𝟎𝟗𝟓 Φ= −𝟏. 𝟏𝟐𝟐𝟔 𝟎. 𝟏𝟏𝟑𝟏

−𝟎. 𝟗𝟔𝟗𝟑 𝟏. 𝟖𝟗𝟕𝟔 𝟎. 𝟎𝟕𝟏𝟕

−𝟎. 𝟎𝟒𝟎𝟐 𝟎. 𝟐𝟐𝟓𝟎 𝟎. 𝟖𝟏𝟓𝟑

The Relative Frequency Array can be obtained by performing the Hadamard division of ERGA by RGA. The RFA was found to be, 𝟎. 𝟗𝟏𝟕𝟖 𝟎. 𝟖𝟒𝟔𝟖 𝟎. 𝟖𝟗𝟒𝟖 Γ= 𝟎. 𝟖𝟓𝟑𝟗 𝟎. 𝟗𝟏𝟕𝟕 𝟎. 𝟗𝟏𝟎𝟖 𝟎. 𝟗𝟎𝟑𝟒 𝟎. 𝟗𝟑𝟐𝟐 𝟏. 𝟎𝟐𝟏𝟕

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The beat pairing according to both RGA and ERGA is the diagonal elements. The process involves the first two diagonal elements falling under the case 3 and the 33 element under case 2.

Thus the effective transfer functions are given as

119𝑒 −5𝑠

76.7𝑒 −5𝑠

21.7𝑠+1

28𝑠+1

−129.4574 𝑒 −5.1084 𝑠

and

23𝑠+1

The resultant PI controllers by proposed method are listed in Table [7] together with ETF method and describing function matrix approach proposed by Loh and Vasnani. Table 7 Controllers for example 5 Controller

Proposed

Xiong and Cai

Kpii

τIii

Kpii

Loh and Vasnani τIii

Kpii

τIii

Loop 1

0.0163

21.7

0.0191

21.7

0.0181

14.49

Loop 2

0.0356

28

0.0382

28

0.0335

14.49

Loop 3

0.0165

23

-0.0182

23

-0.0260

14.49

Figure 19 Closed loop response of Example 5

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Time(secs)

Time(secs)

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Time(secs)

As can be seen, the proposed method results in better performance when compared to the other methods even after taking the interaction into account.

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CONCLUSION CONCLUSION Effective transfer function approach is a novel method for decentralized control system design of multivariable interactive processes. An extension of the effective transfer function approach by taking into consideration all the interactions was proposed and implemented successfully with improved responses. The simplicity and effectiveness of the method is based on the incorporation of the interaction frequency directly in the controller design. This approach ensures that all the necessary information of the gain and interaction frequency changes are provided. The decentralized controllers are obtained by simply using the single loop design approaches. Simulation results for the four 2x2 processes and a 3x3 process show that the proposed method provides a better overall performance compared to the other design approaches even after taking into account the interactions. The advantage of this method is more significant when applied to higher dimensional processes with complicated interaction modes. Since this is an extension of the ETF approach, it can also be easily integrated into an auto-tuning control structure. This method can also be successfully tested for the other MIMO processes. Also employment of BLT tuning after obtaining the effective transfer function can also be performed for better results.

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REFERENCES

1. W.L.Luyben, M.L.Luyben, Essentials of process control, McGraw-Hill Chemical Engineering Series, 1997. 2. Qiang Xiong, Wen-Jian Cai, Effective transfer function method for decentralized control system of MIMO processes, JPC 16(2006). 3. Deshpande, P.B. (1989), Multivariable Process Control, ISA, North Carolina. 4. W.L.Luyben, Simple Method for Tuning SISO Controllers in Multivariable Systems, I&EC Process Design and Development ,Vol. 25, No.3,1986 5. George Stephanopoulos, Chemical Process Control – An Introduction to theory and practice, Prentice Hall Of India,2008.

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