Control of TITO Process using Internal Model Control Technique

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Control of TITO Process using Internal Model Control Technique

Shashi Minz

Gaurav Kumar

Department of Electrical Engineering BIT, Sindri

Jharkhand, Dhanbad 828123, INDIA

Atul Sharma

Department of Electrical Engineering BIT, Sindri

Jharkhand, Dhanbad 828123, INDIA

Department of Instrumentation & Control Engineering University of Calcutta

West Bengal, Kolkata 700009, INDIA

Abstract – This paper presents an Internal Model Control technique for TITO process with different decoupling schemes and comparing the response with each other. A system which has multiple inputs & multiple outputs is called as MIMO system. In industries most of the systems are of MIMO type e.g. Chemical reactors, heat exchangers, distillation column. In this paper Binary distillation column has been taken as a TITO process and its reference model is taken from model given by Wood and Berry. Conventional and Inverted Decoupling schemes are used to reduce the interactions and by varying the single tuning parameter () of Internal Model Control (IMC) technique the set-point is trying to achieve.

Index Terms TITO process, Binary Distillation Column, Conventional Decoupler, Inverted Decoupler, Internal Model Control, Disturbances.

  1. INTRODUCTION

    Distillation is a process in which a mixture consisting of two or more miscible components is separated out on the basis of their volatility or B.P. If a mixture of methanol and water undertakes distillation process, the higher volatile component (methanol) will vaporize rapidly than water under the same atmospheric pressure [1, 2]. There are 3 sections in distillation column Feed section, rectifying/enriching section, stripping/exhaust section. The internal column is used to enhance separation quality, it consists different types of trays like sieve tray, chimney tray, valve tray, bubble cap etc. and also has different packing like structural packing and random packing, they provide maximum surface area and maximize the heat & mass transfer between downward flowing liquid & vapor. The trays between the feed and top of vertical column is called rectifying section & vice-versa is stripping section. In rectifying section lighter components (more volatile) is removed and in stripping section the heavier component (less volatile) is removed. Firstly in preheater mixture is heated under pressure just below the B.P. the pressure in tower is kept lower than that of preheater, so when feed enters the tower it starts boiling, the vapors from boiling liquid which contains lighter component in feed, rises up in the tower the remaining liquid which contain primarily the heavier component in feed goes down the tower & collected to bottom some of the liquid is drawn off as a bottom product & some of it is given to re-boiler, which is connected to bottom of a tower [3, 4]. The re-boiler is mainly a heat exchanger

    which is considered to vaporize lighter components that remain in liquid from the bottom of the tower. Vapors from re-boiler, reenters the tower & rises up, these vapors & the heat they contain is called boil up, the hot boil up provides heat that needed for distillation, the vapor which rises up is gone to a condenser, the condenser cool & condense the vapors into liquid, a part of liquid is stored as overhead product & rest is pumped back into top of tower the reintroduced liquid is called external reflux , it is cooler than the liquid in the top of tower so vapor made of heavier fractions are condensed & liquid made of heavier fraction flows down the tower & called as internal reflux [4,5].

  2. DISTILLATION COLUMN

    The basic diagram is shown below. The temperature of distillation column decreases as materials moves higher in tower the steady decrease in temperature from top bottom to top is called as temperature gradient [6]. Here the controlled variable is mole fraction of top & bottom product manipulated variables are external reflux & boil up, and the disturbances are feed compositions & feed flow rate. The middle loop creates interactions between upper & lower loop, also in between manipulated variables & controlled output.

    Fig. 1 Distillation Column

    G(S)

    K

    (S 1)n

    K

    eDS

    Where n belongs to integer (2.1)

      1. Model is accurate and Disturbance effects the Process

        If the model is accurate which gives Gm (s) GP (s) and

        G(S)

        ( 1 S 1)( 2 S 1)

        e DS

        (2.2)

        there is disturbance,

        Hence, feedback signal is

        d ' (s) d(s)

        then output is

        Where 1 , 2 & are constants

        given by

        y(s) GP (s)Q(s)r(s) 1 Gm (s)Q(s)d(s)

        (3.2)

  3. INTERNAL MODEL CONTROL SYSTEM

    The block diagram of feedback control system is given in Fig. 2

      1. Model Ambiguity & zero Disturbance

    If there is zero disturbance

    d (s)

    but model ambiguity

    GP (s) Gm (s) occurs then the feedback signal is

    P

    P

    d ' (s) G (s) G (s)u(s)

    m

    (3.3)

    Hence process output

    y(s)

    GP (s)Q(s)

    r(s)

    1 Gm (s)Q(s)

    d (s)

    (3.4)

    1 Q(s)G (s) G (s) 1 Q(s)G (s) G (s)

    P m P m

    Fig. 2 Block diagram of IMC

    where Q(s) is the primary controller (IMC) transfer function,

    3.5. Design Procedure of IMC

    We discern that dynamic controller [9] gives faster reaction than the static controller so we use dynamic control by-law.

    GP (s) is the process transfer function, Gm (s) is the

    Hence

    process model transfer function, r(s) is set point, e(s) is

    Q(s) 1

    (3.5)

    error c(s) is manipulated variable,

    d (s) is

    GP (s)

    This is only applicable for stable process which has zero time

    disturbance, ym (s) is model output and y(s) is controlled variable (process output), d ' (s) is estimated disturbances.

    Internal Model Control Systems are categorized by a control system involving of the controller and of a simulation of the process, the internal model [7, 8]. IMC controller is an advance model based controller in which dynamics of model is also incorporated in its control law, it has one only one degree of freedom (). It tracks set point and also care about process- model mismatch & disturbance rejection, where as a

    delay. Now we have to attention on designing the IMC for time delay system. The controller strategy has been comprehensive to the following step.

    Firstly we identify the process model & convert it into invertible (decent stuff) and non-invertible (useless stuff which is demarcated by time delays and right hand plane Zeroes) by applying all pass factorization or simple formulation [10]. Inverse the invertible portion of the process model and to mark appropriate we multiply it with the filter transfer function [11].

    general PID control is a model free controller and it only tries

    to tracks given set point and doesnt care about the above.

    Q(s)

    1

    Gm (s)

    f (s)

    (3.6)

      1. IMC strategy

        Where

        f (s)

        1

        (s 1)n

        filter transfer function [11], n is a

        There are three IMC strategies and these are as follows:

      2. Model is accurate and has zero Disturbance

    constant (1, 2, 3 ..).We choose it accordingly to make the controller proper or semi proper.

    3.6 Calculation of tuning parameter of PI controlle

    If the model is accurate then we have Gm (s) Gp (s) and there is no effect of disturbance (d (s) 0) then according to

    KP, Ki

    in terms of IMC tuning parameter ().

    Fig. 2 feedback signal becomes zero. Hence correlation

    Let

    GIMC (S)

    is transfer function of IMC controller,

    between input and output is given by the expression which is

    given below

    Gm (s) is transfer function of reference model of process,

    y(s) GP (s)Q(s)r(s)

    (3.1)

    Q(S) is open loop gain of IMC controller, GP (s) is transfer

    This is corresponding to open loop control strategy proposal

    function of a process, F(S) is filter transfer function. Now we find the expression of PI controller tuning parameter in terms of tuning parameter of IMC controller [12, 13].

    G Q(s)

    (3.7)

    Decoupler transfer function matrix = D(s)

    IMC 1 G (s)Q(s)

    m

    Output matrix = X (s)

    Q(s) 1

    f (s)

    (3.8)

    Filter transfer functions = f1 (s), f 2 (s)

    Gm (s)

    k eSTD

    (3.9)

    Controller output matrix = C(s)

    The relation between input and output matrix is given by

    G(s)

    s 1

    X (s) G(s)D(s)C(s)

    (4.1)

    e STD is non invertible term so we will neglect it [14].

    G (s)

    G (s)

    1 Where G(s) 11 12

    (4.2)

    f (s)

    (s 1)n

    (3.10)

    G21 (s)

    G22 (s)

    f (s) is used to make the transfer function

    Q(s) at least

    D(s) 1

    R12 (s)

    (4.3)

    R (s) 1

    semi proper, n is the order of plant here we take n=1, and is chosen in between 0.5 to 0.67 times dominant time constant [15].

    21

    Here we take R11 (s) & R22 (s) 1

    Q(s) s 1 1

    (3.11)

    X (s) X D (s) & C(s) C1

    (4.4)

    K s 1

    s 1

    X

    X

    C

    C

    B

    B

    So,

    (s)

    2

    G ks 1 s 1

    (3.12)

    X D (s)

    G11 (s)

    G12 (s) 1

    R12 (s)C1

    IMC

    k s 1

    ks

    X (s) G

    (s)

    G (s)R

    (s)

    1 C

    (4.5)

    1 s 1 k s 1

    B 21

    22 21

    2

    1 1 1

    X (S) G11 (s) G12 (s)R21 (s)

    G12 (s) G11 (s)R12 (s) C

    P

    P

    1 k 1

    k ks k s s

    D

    1

    (4.6)

    i

    X B (s) G21 (s) G22 (s)R(s) G22 (s) G21 (s)R12 (s)C2

    By comparing this result with transfer function of PI controller we get

    k and , , So k 1 (3.13)

    Decoupling of the TITO process requires the design of a transfer matrix D(s) such that G(s)D(s) P(s) is a

    i

    i

    P k i i

    diagonal matrix [17]. Only then we can remove the interactions between upper and lower loop of TITO, for this

  4. DECOUPLER DESIGN

    we have to make following terms equal to zero,

    G12 (s) G11 (s)R12 (s) 0

    (4.7)

    4.1 Conventional Decoupling

    TITO is basically a coupled system, so we use decoupler to

    G21

    (s) G22 (s)R21

    (s) 0

    (4.8)

    reduce the process & control loop interactions. We have designed a decoupler, the best input output pair is found by

    Then we get,

    R

    (s) G12 (s)

    (4.9)

    relative gain array method which is introduced by Bristol in

    1966 [16]. After converting the TITO process into two SISO

    11

    11

    12 (s)

    G

    G

    by using decoupler we uses nyquist criterion for finding

    R (s) G21 (s)

    (4.10)

    stability. Consider a TITO process with Conventional Decoupler,

    21

    22

    (s)

    G

    G

    By putting the values of R12 (s) & R21 (s)

    in equation

    of D(s)

    decoupler.

    we can easily find the decoupling matrix for any

    Now calculation of IMC controller transfer function for TITO process.

    P(s) G(s)D(s) G(s) G(s)R(s)

    0

    (4.11)

    0

    Gm11 (s) 0

    G(s) G(s)R(s)

    Fig 3. Conventional Decoupler

    Or,

    P(s)

    0

    Gm22

    (s)

    (4.12)

    Now we can easily find transfer function of IMC controller Gc1 (s) & Gc2 (s)for upper and lower loop of TITO process by using formulae,

    5.1 With Load disturbance & with Conventional Decoupler I have taken a transfer matrix of a well-accepted reference model of a binary distillation column as a TITO process and by using different decoupling technique like conventional and

    Gc1

    (s)

    1 G

    Q1 (s)

    (s)Q (s)

    (4.13)

    inverted I have made simulation model, we also add two disturbances at a particular time of 150 sec in upper loop of

    G (s)

    m11 1

    Q2 (s)

    (4.14)

    TITO and at 75 sec in lower loop of TITO then I have recorded the responses by running the simulation model for

    c 2

    Where

    Q (s)

    1 G

    1

    m22

    1. Q2

      f (s)

      (s)

      (4.15)

      different values of IMC tuning parameter (). I have shown the simulation graphs and tables for both top and bottom composition of binary distillation column below.

      1

      Q (s)

      Gm11

      1

      (s)

      1

      f (s)

      (4.16)

      5.1.1 Top Composition

      Top Composition (Y1)

      Top Composition (Y1)

      1.2

      2 G (s) 2

      1

      m22

      0.8

      Lambda=4

      Lambda=2.1

      4.2 Inverted Decoupling

      0.6 Lambda=1.72

      0.4

      0.2

      0

      Inverted decoupling is also a method to remove interactions, it gives more accurate result than conventional decoupling technique.

      0 50 100

      0

      Control Response (U1)

      Control Response (U1)

      -0.5

      -1

      -1.5

      Time (S)

      150 200 250

      -2

      0 50 100

      Time (S)

      150 200 250

      Fig 5 – Top Composition Response & Controller Response with Conventional Decoupler

      Fig. 4 Inverted Decoupler

      Its decoupling transfer matrix is

      Fig 5 shows top composition (Y1) and manipulated variable (U1) when we apply with load disturbance & with conventional decoupler. The load disturbance or interaction of loop 2 on loop 1 is minimized by conventional decoupler D21 here we want to observe the minimization of interaction parts thats why we keep the same values of lambda, proportional control action and integral control actions [19].

      D(s) 1

      D12 (s)

      (4.17)

      Table 1: Performance Indices (Top Composition)

      21

      21

      D (s) 1

      lambda k P

      ki IAE ITAE TV

      4

      0.326

      0.019

      9.516

      413.4

      0.129

      2.1

      0.604

      0.030

      7.909

      561.8

      0.403

      1.72

      0.760

      0.045

      10.10

      795.7

      0.700

      4

      0.326

      0.019

      9.516

      413.4

      0.129

      2.1

      0.604

      0.030

      7.909

      561.8

      0.403

      1.72

      0.760

      0.045

      10.10

      795.7

      0.700

      The value of D12 (s) & D21 (s) are find by similar method as used for conventional decoupling, and we get the same results.

      D (s) G12 (s)

      (4.18)

      Table above shows IAE and ITAE values for loop 1 for

      G

      G

      11

      11

      12 (s)

      dfferent types of lambda values. For different values of lambda, we get different values of proportional and integral

      D (s) G21 (s)

      (4.19)

      control action values.

      G

      G

      21

      22

      (s)

      5.1.2 Bottom Composition

  5. SIMULATION RESULTS OF WOOD AND BERRY DISTLLATION COLUMN

    The transfer matrix of wood & berry distillation column is shown below its all transfer functions are FOPDT type.

    1.5

    Bottom Composition (Y2)

    Bottom Composition (Y2)

    Lambda=9

    Lambda=5.84

    Lambda=5.12

    0 50

    100

    Time (S)

    150

    200 25

    Lambda=9

    Lambda=5.84

    Lambda=5.12

    0 50

    100

    Time (S)

    150

    200 25

    1

    0.5

    0 0

    Control Response (U2)

    Control Response (U2)

    2

    12.8

    es

    18.9 e3s 1

    G(S ) 16.7s 1

    6.6

    21s 1

    19.4

    [18] 0

    -1

    e7 s e3s

    0 50 100

    Time(s)

    150 200 250

    10.9s 1 14.4s 1

    Fig 6- Bottom Composition Response & Controller Response with Conventional Decoupler

    Fig 6. Shows top composition (Y2) and manipulated variable (U2) when we apply both load disturbances and conventional decoupler. The load disturbance or interaction of loop 1 on loop 2 is minimized by conventional decoupler D12. Here we observed that the interaction can be minimized by using conventional decoupler.

    Here we want to observe the minimization of interaction parts thats why we keep the same values of lambda, proportional control action and integral control actions.

    Table 2: Performance Indices (Bottom Composition)

    lambda

    k P

    ki

    IAE

    ITAE

    TV

    9

    -0.082

    -0.006

    18.09

    466.7

    0.0257

    5.84

    -0.127

    -0.009

    13.04

    346.3

    0.058

    5.12

    -0.125

    -0.010

    11.09

    348.8

    0.071

    loops. Inverted decoupler provides better response than conventional decoupler.

    For Inverted decoupler we had chosen same values of lambda. So that we can compare both the responses.

    Table 3: Performance Indices (Top Composition)

    lambda

    k P

    ki

    IAE

    ITAE

    TV

    4

    0.326

    0.019

    5.986

    301.4

    0.108

    2.1

    0.604

    0.037

    4.389

    289.4

    0.237

    1.72

    0.760

    0.045

    4.353

    307.2

    0.339

    Table above shows IAE and ITAE values for loop 1 for different types of lambda values. For different values of lambda, we get different values of proportional and integral control action values.

    Table above shows IAE and ITAE values for loop 2 for different types of lambda values. For different values of lambda, we get different values of proportional and integral control action values [20]. We also find stability using Bode plot, thats why we also attached the values of gain margin and phase margin. The values of GM and PM shows the

    5.2.2 Bottom Composition

    Bottom Composition (Y2)

    Bottom Composition (Y2)

    1.5

    1

    0.5

    Lambda=9

    Lambda=5.84

    Lambda=5.12

    system stable.

    0

    0 50 100

    Time (S)

    150 200 250

    0.05

      • Lambda=4, Lambda=9 (Recommended by B. Wayne Bequette)

      • Lambda=2.1, Lambda=5.84 (Recommended by Dale E. Seborg & Thomas F. Edgard)

      • Lambda=1.72, Lambda=5.12 (Recommended by Morari & Zafirion)

      • 1st Load Disturbance is +ve step of (0 to 0.5) at 75 sec &

        -ve step of (0 to 0.5) at 77 sec

      • 2nd Load Disturbance is ve step of (0 to 0.5) at 150 sec & +ve step of (0 to 0.5) at 152 sec

      1. With Load disturbance & With Inverted Decoupler

        1. Top Composition

          0

          Control Response (U2)

          Control Response (U2)

          -0.05

          -0.1

          -0.15

          -0.2

          0 50 100 Time (S) 150 200 250

          Fig. 8 – Bottom Composition Response & Controller Response with Inverted Decoupler

          Fig.8 shows the top composition (Y2) and controller response (U2) for loop 2 when inverted decoupler use. Inverted decoupler used to minimize of interactions between two loops.

          Inverted decoupler provides better response than conventional decoupler.

          For Inverted decoupler we had chosen same values of

          Top Composition (Y1)

          Top Composition (Y1)

          1.5

          1

          0.5

          0

          Control Response (U1)

          Control Response (U1)

          0.8

          0.6

          0.4

          0.2

          0

          -0.2

          -0.4

          Lambda=4

          Lambda=2.1

          Lambda=1.72

          0 50

          100

          Time (S)

          150

          200

          0 50

          100

          Time (S)

          150

          200

          0 50

          100

          Time (S)

          150

          200

          0 50

          100

          Time (S)

          150

          200

          250

          250

          lambda. So that we can compare both the responses.

          Table 4: Performance Indices (Bottom Composition)

          lambda

          k P

          ki

          IAE

          ITAE

          TV

          9

          -0.082

          -0.006

          11.18

          241.8

          0.025

          5.84

          -0.127

          -0.009

          9.19

          224.9

          0.045

          5.12

          -0.145

          -0.010

          9.146

          234.1

          0.055

          Table above shows IAE and ITAE values for loop 2 for different types of lambda values. For different values of lambda, we get different values of proportional and integral

          Fig 7 – Top Composition Response & Controller Response with Inverted Decoupler

          Fig 7 shows the top composition (Y1) and controller response (U1) for loop 1 when inverted decoupler use. Inverted decoupler used to minimize of interactions between two

          control action values. We also found the stability using Bode plot, thats why we also attached the values of gain margin and phase margin. The values of GM and PM shows the system is stable [21].

          • Lambda=4,Lambda=9 (Recommended by B. Wayne Bequette)

          • Lambda=2.1, Lambda=5.84 (Recommended by Dale E. Seborg & Thomas F. Edgard)

          • Lambda=1.72, Lambda=5.12 (Recommended by Morari & Zafirion)

          • 1st Load Disturbance is +ve step of (0 to 0.5) at 75 sec &

            -ve step of (0 to 0.5) at 77 sec

          • 2nd Load Disturbance is ve step of (0 to 0.5) at 150 sec & +ve step of (0 to 0.5) at 152 sec

  6. CONCLUSION

PID controller has 3 degree of freedom ( kp , ki & kd ) for TITO system it becomes 6 (including both PID) so it is difficult to synchronise all the parameters at the same time,

also it is model free controller means it doesnt care about

process- model mismatch & disturbance rejection it only tries to track given set point – for its solution we use IMC controller which is an advance model based controller in which dynamics of model is also incorporated in its control law, it has one only one degree of freedom (), and it also solves the above problem [21]. Moreover using IMCs, rise time will be decreased, faster response and disturbance compensation and able to compensate the model uncertainty. We also observed from the graph if the values of lambda is high (that means if the values of proportional control action and integral control action low) then the overshoot of the response will be low but rise time will be high and oscillation will be less. But if the value of lambda is low (that means if the value of proportional control action and integral control action high) then the overshoot of the response will be high but rise time will be low and oscillation will be more. We also observed from the graph that inverted decoupler is better than the conventional decoupler. When we use inverted decoupler then it can eliminate the interaction between two loops completely. It is an advantage of inverted decoupler over conventional decoupler.

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