Multiloop Adaptive Controllers for a Nonlinear Interacting Coupled Tank Process

Multiloop Adaptive Controllers for a Nonlinear Interacting Coupled Tank Process

D. Gayathri *,

* PG Student,

Dept. of Electronics and Instrumentation Engineering, Annamalai University, Annamalai Nagar

1. Rathikarani**, L. Thillai Rani***, D. Sivakumar**

   ** Faculty,

   Dept. of Electronics and Instrumentation Engineering, Annamalai University, Annamalai Nagar Chidambaram, India

   Abstract This paper presents the design and implementation of Direct Adaptive Controller (DAC) to control the level of liquid in a nonlinear two tank interacting process. Mean Square error is chosen as minimization criteria for the design of the controller. The objective of this work includes performance comparison of Adaptive MIT and Adaptive PI controller with two tank interacting process. The Constant PI, Adaptive MIT and Adaptive PI controllers are implemented for four different operating regions. White box model of the process is used in this work. The design and simulation studies are carried out in MATLAB/SIMULINK.

   Keywords Nonlinear system, Interacting two tank system, MRAC, MIMO, Mathematical modeling, White box model

   1. INTRODUCTION

      In process industries one of the major problem is to control the liquid level in tanks. Vital industries such as Petro- chemical industries, Paper industries, Water treatment industries have tanks used for chemical treatment and/or mixing the process fluids. In two tank interacting process, the level of liquid in two tanks must be controlled to improve the quality of the product. The difficulty in level control of Multiple Input Multiple Output (MIMO) process is due to its complex dynamics and the interacting nature. Control of the nonlinear process is a difficult task by itself. Deepa et al.[1] have compared the performances of MRAC with fuzzy control for a Single Input Single Output (SISO) system. Anna Joseph et al.[2] and Rathikarani et al.[3] have used Model Reference Adaptive Controller (MRAC) to control nonlinear process . For a coupled tank process, a Model Predictive Controller is designed by Gireesh Kumar et al.[4].

      First principle based model of the MIMO process is used in this work. The most widely used PI controllers in the industrial applications have simple structures and good dynamic performances. These Constant PI (CPI) controller are popular in industrial applications, as they are easy to install and reasonably robust. It is necessary to develop advanced PI controllers for controlling nonlinear processes. Adaptive controllers parameters are adjusted automatically to compensate the variation in the process characteristics. These controllers performs better when compared to Constant Gain PI controllers for Nonlinear processes. Hence Adaptive MIT and Adaptive PI controller are designed and implemented in this work. The MRAC based Adaptive MIT and Adaptive PI controllers are designed to control the liquid level in interacting tanks. When production rate changes, the dynamics

      of the process along with the amount of interaction varies. The adaptive controller has to decrease the error vector between the reference model and plant to zero. The proposed method can adjust the controller parameters in response to changes in plant and disturbances by referring to the reference model that satisfies properties of the desired closed loop control system

   2. DESCRIPTION OF LEVEL PROCESS

      In this work, two tank interacting process available in the laboratory is considered. This process is a MIMO process with two controlled variables. Hence two controllers are designed and implemented to control the level in two tank interacting process. The level in tank1 and tank2 are the controlled variables.Pump1 and pump2 are used to feed inflow to the tank1 and tank2. The Hand Valves (HV) are adjusted so that the levels in both the tanks are brought to nominal condition initially. Disturbances are applied to the tanks by varying the position of the hand valves HV21 and HV26. When the flow to the tank 1 is varied, the inflow to tank 2 also varies. When the level and/or flow of tank2 varies, tank1 level change, due to interaction between the tanks (Fig. 1).

      . The volumetric inflow rate into the tank1 and tank2 are qin1 and qin2. The volumetric flow rate from the tank1 and tank2 are q01 and q02. Flow rate between tank1 and tank2 is q12. The height of he liquid level is p in tank1 and p in tank2.

      Fig.1 Piping and Instrumentation Diagram for two tank interacting level process

      The schematic diagram of the two tank interacting level process is shown in Fig. 2. The controlled variables in the

      process are level in tank1 (p) and level in tank2 (p). The 20

      Manipulated variables to the process are qin1 (l/hr) for tank1,

      qin2 (l/hr) for tank2. 15

      Level(cm)

      10

      qin1 qin2 p

      TANK2

      TANK1

      qin1 5 p

      q12

      p

      p

      RESERVOIR TANK

      q01 q02

      00 200 400 600 800 1000 1200 1400 1600 1800 2000

      Sampling instants

      Fig. 3 Open loop responses of p and p for +10LPH change in qin1

      20

      Level(cm)

      15

      10

      Fig. 2 Schematic diagram of Two tank interacting level process

      The mathematical model of the process is obtained from mass balance equations, and are given below,

      p

      5 p

      00 200 400 600 800 1000 1200 1400 1600 1800 2000

      Sampling instants

      Fig. 4. Open loop responses of p and p for +10LPH change in qin2

      A1 dp

      dt

      A2 dp

      dt

      qin1 q01 q12

      qin2 q02 q12

      (1a)

      (1b)

      In the same manner, step change is given in qin2 maintaining qin1 in nominal condition [Fig. 4]. Using these responses the model of the process in continuous time domain for the change in qin1 and qin2 are computed using

      where A1= A2=1130.4cm2 are the cross sectional area of the tank1 and tank2. p=p=25cm are the height of the tank1 and tank2. a1=5.3 cm2; a2=10.6 cm2 are the restriction areas in the outlet pipes of tank1 and tank2. g=9.81cm2/s acceleration due to gravity and cd= 0.8 the discharge co-efficient.

      1. Modelling of Level Process

         The four models relating the two controlled outputs p and p with two manipulated inputs qin1 and qin2 are essential to design the multi-loop controllers [5]. The model transfer functions with the flow rates as manipulated inputs and the levels as controlled outputs can be written as follows:

         process reaction curve method and are tabulated in Table I.

         TABLE I. IDENTIFIED MODELS

         Operating Regions	qin (LPH)	Level (cm)	Models
         1	-5	12.6 to 9.8	0.5486e5.5S 0.338e4.5s c c 11 57s 1 12 48s 1 0.449e10..5s 0.6084e3S c c 21 58.5s 1 22 27s 1
         2	+5	12.6	0.586e4S 0.502e8.5s c c 11 54s 1 1 2 73.5s 1 0.387e6.5s 0.5806e2S c c 21 64s 1 22 71s 1
         		to
         		16.02
         3	-10	12.6	0.5635e3.5S 0.3742e145s c c 11 34.5s 1 1 2 43.5s 1 0.4607e8s 0.6345e4.5S c c 21 52.5s 1 22 22.5s 1
         		to
         		7.45
         4	+10	12.6 to 20	0.737e3S 0.508e3s c c 11 69s 1 12 75s 1 0.4607e7.5s 0.587e2S c c 21 52.5s 1 22 75s 1

         p p

         G11(s) ; G21(s)

         qin1 qin1

         qin2

         h

         qin2

         h

         G12(s) 1 ; G22(s) 2

         q q

         in2 qin1

         in2 qin1

      2. Validation of the Models

      Interacting two tank process is modeled from the process reaction curve [Fig. 3 and Fig. 4]. The levels in the tanks are initially maintained at nominal operating condition (p=12.6cm, p=12.1cm). In 800th sampling instant 10LPH change is given in qin1 . which causes p to change from 12.6 to 20cm. Due to interaction between the tanks p has reached the steady state value of 16cm from its nominal value.

      Time domain validation of the models are shown in Fig. 5 and Fig. 6. To evaluate the degree of closeness of the model with actual process the validation is done. The actual response (white box) of the process is compared with model response (black box) for the same input.

      Fig. 5 Time domain validation for the Model (Tank1)

      Fig. 6 Time domain validation for the Model(Tank2)

   3. DESIGN OF CONTROLLERS

      The process considered in this work is a Two Input Two Output (TITO) process. The constant PI, Adaptive MIT and Adaptive PI are designed and implemented to control level in the tanks. Hence two control loops are designed and implemented.

      1. Constant PI (CPI) Controller

         The CPI controller can be used to improve the dynamic response as well as reduce or eliminate the steady state error. The strategy used for controlling the interacting process with controllers are shown in Fig.7. The reference set points are hsp1 and hsp2. Manipulated inflow rates to tank1 and tank2 are qin1 and qin2. The process outputs from tank1 and tank2 are p and p. The variables e1 and e2 are the modeling errors to the controllers. Gc1 and Gc2 are the controllers in loop1 and loop2.

         p

         p

         Table II

         PI Controller Parameters

         Operating Regions	PI Controller parameters
         	Gc1	Gc2
         1	K c1 =14.084; Ti1 4.05	K c2 =12.344; Ti2 =3.86
         2	K c1 =7.771; Ti1 =5.65	K c2 =7.728; Ti2 =3.90
         3	K c1 =12.450; Ti1 =2.54	K c2 =15.567; Ti2 =1.22
         4	K c1 =9.056; Ti1 =2.89	K c2 =7.997; Ti2 =1.08

      2. Adaptive Controller

         To control level in each tank, MRAC is used. The structure of the MRAC system with MIT rule used in this work is shown in Fig. 8. Each control loop consists of a reference model, adjustment mechanism and controller. The reference model describes the desired input/output character of the closed loop system. The controller drives the control signal so that the plants closed loop characteristics from the command signal; hspi to the plant output hi is equal to the dynamics of the reference model, hm. The suffix i in the variables represents the control loop, nos. 1 and 2.

         Matching the plant and the reference model characteristics guarantees the convergence of the modeling error to zero for any given command signal (hspi). The controller drives the difference between the process response and desired model output to zero asymptotically at a rate constrained by the adaptation gain [6,7].

         The designed controller has a conventional inner loop followed by a separate adaptive outer loop to adjust the controllers feedback gains ( 1i , 2i ) based on equating the coefficients of closed loop plant to coefficients of the desired model.

         The advantage is that the proposed technique can deal with the nonlinear nature of the process and also retain the designers intuition and insight through the relatively simple design scheme that is proposed. This controller design is based on grey box model which combines both black and white box models.

         The outer loop adjusts the controller parameter in such a way that the model error (ei), the difference between process output hi (i=1,2) and model output hm is small.

         ei hi hm

         (2)

         Fig. 7 Block Schematic representation of closed loop system

         The controller parameters for various operating ranges of the taken up process using the Ziegler Nichols tuning method are presented in Table II.

         e bm u

         1 s 2 s

         2 2 c

         n

         n

         n

         n

         e

         2

         bm s 2 2

         n

         s 2

         ( y)

         (7)

         n

         The controller parameters are obtained

         1

         n

         n

         1 s s 2 2

         s 2

         uc e

         1

         n

         n

         2 s s 2 2

         s 2

         ye

         Fig. 8 Block Schematic diagram of the system with adaptive control

      3. Adaptive MIT Controller

         The controller parameters () may be adjusted with the

      4. Adaptive PI (API) Controller

      The design of API controllers leads to large improvement in industries. API controllers are simple and easy to implement [8,9]. Hence an API based on MRAC is designed and implemented in this work. The API algorithm used in this work is given by equation (8).

      following loss function,

      u k (u y) ki (u

      • y)

      (8)

      J ( ) 1 e2

      (3)

      p c s c

      ni 2 i

      where kp and ki are the proportional and integral gains of the

      where n=1,2 represents controller parameter number. In order to minimize the loss function J, the parameters can be changed in the direction of negative gradient of J.

      controller[10,11]. Based on apriori knowledge the process considered for control is represented by equation (5). The closed-loop transfer function is given by

      The control law is

      u(t) 1uc (t) 2 y(t)

      (4)

      y

      uc

      bk p ki

      s2 (a bk p )s bki

      (9)

      The closed loop transfer function is given by equation

      For perfect model matching

      n

      s 2 s(a bK p ) bKi s 2 2 n s 2

      y b

      1

      c

      2

      u s2 a s a

      (5)

      The adapted PI Controller parameters based on MIT

      where uc is the command signal (input). The controllers algorithm are shown in equations (10) and (11).

      parameters are to be adapted such that the process output (equation 5) follows the model output (equation 6)

      k e

      p s

      s

      n

      s2 2

      s 2 n

      [uc

      • y]

        (10)

        ym

        uc

        bm s 2 2

        n

        s 2

        (6)

        k e

        i s

        1

        n

        s 2 2

        s 2 n

        [uc

      • y]

      (11)

      n

      The modeling error is as follows

      e y ym

      Substituting u(t) in equation (5)

   4. SIMULATION RESULTS

      The servo and regulatory responses of interacting tank

      y b1

      1

      2

      s 2 a s (b

      uc

      a2 )

      (white box model) are plotted in Fig. 9. The damping ratio ( ) of the reference model is 0.7.

      ym

      bm s 2 2

      s 2 uc

      The set point tracking for level p with Conventional PI , Adaptive MIT and Adaptive PI are presented in Fig. 9. The

      n

      n

      For perfect model following the controller parameters are chosen as(when e=0)

      bm

      1 b

      am a

      2 b

      The sensitivity derivatives are obtained by the partial derivatives of modeling error with respect to the controller parameters

      psp is the set point for tank1 a nominal operating condition. The set point variation of p from 12.6 to 17.6cm is applied at 950th sampling instant. Due to interaction the level of tank 2 increases. The p-API is the level of the tank1 using API controller. The p-CPI is the level of tank1 with CPI controller. The p-MIT is the level of tank1 when Adaptive MIT controller is used.

      The set point tracking for level p with Conventional PI , Adaptive MIT and Adaptive PI are shown in Fig.10. Due to set point variation in tank1, the level in tank 2 varies at 950th

      sampling instant. Due to controllers in loop2, the level variations are nullified and brought to nominal operating condition (12.1cm). A set point variation for level p from

      12.1 to 17.1cm is applied at 1800th sampling instant, due to

      1.4

      1.2

      Controller parameters

      1

      kc1-CPI

      interaction there is considerable rise in p.

      kc2-CPI

      0.8 kc1-API

      kc2-API

      21 0.6

      20

      0.4

      19

      18

      Level(cm)

      17 psp

      p-MIT

      0.2

      0

      800 1000 1200 1400 1600 1800 2000 2200 2400

      16 p-API

      p-CPI

      Sampling instants

      15

      X: 802

      Y: 12.58

      14

      13

      12

      800 1000 1200 1400 1600 1800 2000 2200 2400

      Sampling instants

      Fig. 9 Servo and Regulatory responses of the Interacting Coupled tank process for p( = 0.7)

      19

      18

      17

      1.4

      Controller parameters

      1.2

      1

      0.8

      0.6

      Fig. 12 Adaptation of Proportional Gains

      ki1-CPI ki2-CPI ki1-API ki2-API

      Level(cm)

      16

      15 psp

      p-MIT

      14 p-API

      p-CPI

      13

      12

      0.4

      800 1000 1200 1400 1600 1800 2000 2200 2400

      Sampling instants

      Fig. 13 Adaptation of Integral Gains

      The vanishing nature of adapted controller

      11

      800

      1000 1200 1400 1600 1800 2000 2200 2400

      Sampling instants

      parameters(1, Fig. 14.

      2 ) of MIT1 and MIT2 can be visualized from

      Fig. 10 Servo and Regulatory responses of the Interacting Coupled tank 60

      50

      process for p( =0.7)

      Fig. 11 shows the response of the controllers for tank1

      Controller Parameters

      and tank2. At 950th sampling instant, the inflow rate to the 40

      30

      tank 1 increases as well as inflow rate to tank2 decreases. In tank 2 , the 1800th sampling instant the inflow rate of tank 2

      increases due to this change the flow rate of the tank 20

      1decreases in order to bring back the level p to set point. 10

      Tp-MIT1 Tp-MIT1 Tp-MIT2 Tp-MIT2

      55

      50 qin1-API

      qin2-API

      45 qin1-CPI

      qin2-CPI

      40 qin1-MIT

      Flow(LPH)

      qin2-MIT

      35

      30

      25

      20

      15

      10

      800 1000 1200 1400 1600 1800 2000 2200 2400

      Sampling Instants

      Fig. 11 Response of controllers for tank1 and tank 2

      The adaptation of Proportional and Integral gains(Kc,Ki) for CPI and API can be visualized from Fig. 12 and Fig. 13.

      0

      800 1000 1200 1400 1600 1800 2000 2200 2400

      Sampling instants

      Fig. 14 Adaptation of the controller parameters(MIT)

      The reference models damping ratio( ) is changed from

      0.7 to 1 and 0.7 to 2. The set point tracking for level p with Conventional PI, Adaptive MIT and Adaptive PI for ( =1)

      are presented in Fig. 15. The set point tracking for level p with Conventional PI , Adaptive MIT and Adaptive PI for ( =1) are shown in Fig.16. Fig.17 shows the corresponding response of the controllers for tank1 and tank2. The adaptation of Proportional and Integral gains(Kc,Ki) for CPI and API can be visualized from Fig. 18 and Fig. 19. The vanishing nature of adapted controller parameters (1, 2 ) of

      MIT1 and MIT2 can be visualized from Fig. 20.

      20

      Level(cm)

      18

      16 psp

      p-MIT

      14 p-API

      p-CPI

      1.4

      Controller parameters

      1.2

      1

      0.8

      0.6

      0.4

      ki1-CPI

      ki2-CPI ki1-API ki2-API

      12

      800 1000 1200 1400 1600 1800 2000 2200 2400

      Sampling instants

      Fig. 15 Servo and Regulatory responses of the Interacting Coupled tank

      process for p ( = 1)

      800 1000 1200 1400 1600 1800 2000 2200 2400

      Sampling instants

      Fig. 19 Adaptation of Integral Gains

      60

      19

      50

      18

      Controller Parameters

      17 40

      Level(cm)

      16

      30

      15 psp

      p-MIT

      14 p-API

      p-CPI 20

      13

      Tp-MIT1 Tp-MIT1 Tp-MIT2 Tp-MIT2

      12 10

      11

      800 1000 1200 1400 1600 1800 2000 2200 2400

      Sampling instants

      Fig. 16 Servo and Regulatory responses of the Interacting Coupled tank process for p ( =1)

      55

      50 qin1-API

      qin2-API

      45 qin1-CPI

      qin2-CPI

      40 qin1-MIT

      Flow(LPH)

      qin2-MIT

      35

      30

      25

      20

      15

      0

      800 1000 1200 1400 1600 1800 2000 2200 2400

      Sampling instants

      Fig. 20 Adaptation of the controller parameters(MIT)

      Table IIIA and IIIB shows the Time Integral Criteria of the process for various controllers with various reference model parameters

      Table IIIA

      Performance comparison of Adaptive controllers

      Parameters		IAE			ISE
      		CPI	API	MIT	CPI	API	MIT
      0.7	Tank1	2025	2165	911.5	18550	19450	7673
      	Tank2	7829	2361	883.8	115700	2142000	6923
      1.0	Tank1	2024	2167	908.7	18540	19310	7520
      	Tank2	7817	2367	867.4	117400	2134000	7626
      2.0	Tank1	2022	2175	904	18070	18920	7091
      	Tank2	7795	2388	804	116900	2121000	7268

      10

      800 1000 1200 1400 1600 1800 2000 2200 2400

      Sampling Instants

      Fig. 17 Response of controllers for tank1 and tank

      1.4

      1.2

      Controller parameters

      1

      0.8

      0.6

      0.4

      0.2

      0

      kc1-CPI

      kc2-CPI kc1-API kc2-API

      Table IIIB

      Performance comparison of Adaptive controllers

      Parameters		ITAE
      		CPI	API	MIT
      0.7	Tank1	5063000	5413000	2279000
      	Tank2	19650000	5902000	2038000
      1.0	Tank1	5062000	5418000	2268000
      	Tank2	19540000	5814000	20136000
      2.0	Tank1	506000	5438000	2254000
      	Tank2	19480000	5468000	1934000

      800 1000 1200 1400 1600 1800 2000 2200 2400

      Sampling instants

      Fig. 18 Adaptation of Proportional Gains

   5. CONCLUSION

This paper has presented aGradient approach based MRAC to control level in the interacting tanks. Identification of the first principle based process is done in simulation. The ISE and IAE values of the process with Adaptive MIT controller has lesser values compared to the process with conventional and Adaptive PI controller. From the implementation of MRAC based MIT, it is inferred that by increasing damping factor the time integral absolute error are minimized. Peak overshoot and undershoot are minimum in conventional PI controller. Hence Adaptive MIT controller is suitable for control of the Interacting coupled tank process when compared to Conventional and Adaptive PI controllers.

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