Implementation of an adaptive fuzzy compensator for coupled tank liquid level control system

Measurement 91 (2016) 12–18

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Implementation of an adaptive fuzzy compensator for coupled tank liquid level control system Abdullah Basßçi a,⇑, Adnan Derdiyok b a b

Department of Electrical Electronics Engineering, Ataturk University, Turkey Department of Mechatronics Engineering, Sakarya University, Turkey

a r t i c l e

i n f o

Article history: Received 1 August 2015 Received in revised form 7 March 2016 Accepted 10 May 2016 Available online 10 May 2016 Keywords: Adaptive control Fuzzy control Nonlinear control Process control

a b s t r a c t In this paper, an adaptive fuzzy control (AFC) system is proposed to realize level position control of two coupled water tanks, often encountered in practical process control. The fuzzy control system includes an adaptive model identifier and controller. The gains of AFC are obtained by using the fuzzy identifier model which is defined by real system outputs and control inputs. The parameters of fuzzy identifier model are adjusted online by using recursive least square algorithm. Because the controller has a recursive form it treats model uncertainties and external disturbances in an implicit way. Thus there is no need to specify uncertainty and disturbances for this controller design in advance. A well-tuned conventional proportional integral (PI) controller is also applied to the two coupled tank system for comparison with the AFC system. Experimentation of the coupled tank system is realized in two different configurations, namely configuration #1 and configuration #2 respectively. In configuration #1, the water level in the top tank is controlled by a pump. In configuration #2, the water level in the bottom tank is controlled by the water flow coming out of the top tank. Experimental results prove that the AFC shows better trajectory tracking performance than PI controller in that the plant transient responses to the desired output changes have shorter settling time and smaller magnitude overshot/undershoot. Robustness of the AFC with respect to water level variation and capability to eliminate external disturbances are also achieved. Experimental results show that AFC is a strong and a practical choice for liquid level control. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Liquid level control is a typical representation of process control and is widely used in the area of water purification, chemical and biochemical processing, automatic liquid dispensing, food and beverage processing, and pharmaceutical industries. The control quality directly affects the quality of products and safety of equipment. However, the coupled tank liquid level control system is a large lag, nonlinear and complex characteristics, in which the control accuracy is directly affected by system status, system parameters, and the control algorithm. Therefore, it is quite difficult to perform a high precision servo control by using linear control methods. To perform high precision liquid level control and good tracking precision in the presence of the system nonlinearities and parameter uncertainties, it is needed to use nonlinear control method to solve these problems effectively and achieve precise control. As a solution, sliding mode techniques have been introduced to compensate the uncertainties in dynamics and/or kinematics [1–6]. The sliding ⇑ Corresponding author. E-mail address: [email protected] (A. Basßçi). http://dx.doi.org/10.1016/j.measurement.2016.05.026 0263-2241/Ó 2016 Elsevier Ltd. All rights reserved.

mode control is robust with respect to uncertainties in the system and external disturbances. However, this control methodology has some disadvantages associated with a large control chattering. Also neural network [7–11] and genetic algorithm [1,2,12,13] based controllers are proposed as an effective tool for nonlinear controller design. Both controllers offer exciting advantages such as adaptive learning, fault tolerance, generalization and disadvantages such as complex learning algorithm and computational requirement. Many existing experiments have demonstrated that an adaptive fuzzy controller can be applied to the system whose dynamic model is not well defined or not available at all and has proven to be a strong tool for controlling nonlinear systems [14– 18]. In addition to handling nonlinear problems, adaptive fuzzy control can also enhance the robustness of the system. In this paper, the proposed AFC and conventional PI controllers are used for level control of experimental setup of liquid level system, respectively. The coupled tank system is used in two different configurations, namely configuration #1 and configuration #2. Our study is focused on the level control of top tank in configuration 1 and level control of bottom tank in configuration 2. The experimental results obtained prove that the AFC is robust to liquid level

A. Basßçi, A. Derdiyok / Measurement 91 (2016) 12–18

13

changes as well as to disturbances as compared to PI controller and can also follow command trajectories very well.

Moreover using the mass balance principle for tank 1, we obtain the following first-order differential equation in L1

2. Modeling and control of the coupled tank system

At1

2.1. Single tank model (Configuration #1)

where At1 is tank 1 inside cross-section area. Substituting Eqs. (1) and (2) into Eq. (6) and it can be rearranged in the following form for the tank 1 system,

Single tank system which consisting of the top tank is shown in Fig. 1. It is reminded that in configuration #1, the pump feeds into tank 1 and that tank 2 is not considered at all. Therefore, the input to the process is the voltage to the pump and its output is the water level in tank 1. The mathematics model of the single tank system determined by relating the volumetric inflow rate f i1 into tank to the outflow rate f o1 leaving through the hole at the tank bottom. The volumetric inflow rate and the outflow rate to tank 1 can be expressed as [19],

f i1 ¼ K p V p

ð1Þ

f o1 ¼ Ao1 v o1

ð2Þ

where Ao1 is the outlet cross sectional area, v o1 is the tank 1 outflow velocity, K p is the pump volumetric flow constant and V p is the actual pump input voltage. The outflow velocity by using Bernoulli’s equation

v o1 ¼

@ L1 @t



¼ f i1  f o1

ð6Þ

pffiffiffipffiffiffiffiffiffiffi K p V p  Ao1 2 gL1 @ L1 ¼ At1 @t

ð7Þ

2.2. Coupled tank model (Configuration # 2) A schematic of the coupled tank plant is depicted in Fig. 2. In configuration #2 the pump feeds into tank 1, which in turn feeds into tank 2. As far as tank 1 is concerned, the same equation as the ones previously developed in Section 2.1 is applied. However, the water level equation of motion in tank 2 still needs to be derived. In the coupled tank, the system states are the level L1 in tank 1 and the level L2 in tank 2. The outflow rate from tank 2 can be expressed as;

f o2 ¼ Ao2 v o2

ð8Þ

Tank 2 outflow velocity by using Bernoulli’s equation

pffiffiffipffiffiffiffiffiffiffi 2 gL1

ð3Þ

where g is the gravitational constant on earth. As a remark, the cross-section area of tank 1 outlet hole can be calculated by,

Ao1 ¼



1 pD2o1 4

ð4Þ

in the Eq. (4) Do1 is the tank 1 outlet diameter. Using Eq. (3) the outflow rate from tank 1 given in Eq. (3) becomes,

pffiffiffipffiffiffiffiffiffiffi F o1 ¼ Ao1 2 gL1

ð5Þ

Fig. 1. Tank configuration #1.

v o2 ¼

pffiffiffipffiffiffiffiffiffiffi 2 gL2

ð9Þ

As a remark, the cross-section area of tank 2 outlet hole can be calculated by,

Ao2 ¼

1 pD2o2 4

ð10Þ

Using Eqs. (9) and (10) the outflow rate from tank 2 given in Eq. (8) becomes

pffiffiffipffiffiffiffiffiffiffi F o2 ¼ Ao2 2 gL2

ð11Þ

Fig. 2. Tank configuration #2.

A. Basßçi, A. Derdiyok / Measurement 91 (2016) 12–18

14

Using Eq. (5) the inflow rate to tank 2 is as follow

pffiffiffipffiffiffiffiffiffiffi f i2 ¼ Ao1 2 gL1

Parameter Adapter

ð12Þ

Moreover using the mass balance principle for tank 2, we obtain the following first-order differential equation in L2



At2

@ L2 @t

Lˆ1 ( k + 1), Lˆ 2 ( k + 1)

Fuzzy Identifier



¼ f i2  f o2

ð13Þ

Substituting Eqs. (12) and (11) into Eq. (13) and it can be rearranged in the following form for the tank 2 system,

pffiffiffipffiffiffiffiffiffiffi pffiffiffipffiffiffiffiffiffiffi Ao2 2 gL2 þ Ao1 2 gL1 @ L2 ¼ @t At2

ð14Þ

2.3. Control of coupled tank system A generalize control structure of coupled tank system is depicted in Fig. 3. As seen from the figure, there are two controllers that first one is used to calculate reference level of tank 1 and second one is used to determine pump voltage command to ensure the tank 1 water level to feed tank 2. In this study, for coupled tank, we use a linearized model determined by fuzzy rule base. Because the model parameters are calculated recursively in real time, the validity of the model highly depends on the accuracy of the measurement. The error in the measurement reflects on the model directly. In the coupled tank system, measurement noise is mostly created by the sensor’s environment consisting of turbulent flow and circulating air bubbles. In order to attenuate high-frequency noise content of level measurements a digitally designed low-pass filter of cut-off frequency 1.5 Hz and 0.33 Hz are added to the output signal of the tank 1 and tank 2 level pressure sensors respectively. In this way, it is achieved a model expressing the system behavior more accurately and depending on this model, the controller’s parameter are calculated. The properly defined model rather improves the performance of the controller.

Fuzzy Control

L2 r (k )

L1r (k ),V p (k )

approach proposed, the closed-loop system will be immune to disturbance, parameter uncertainties and load changes. The schematic diagram of fuzzy identifier based fuzzy control is shown in Fig. 4 for configuration 2. Fuzzy control produces two outputs that are V p and L1r commands. It is important to note that a Takagi–Sugeno fuzzy system may have any linear mapping as its output function. This fuzzy system can be considered as a nonlinear interpolator between R linear systems. One mapping is to have a linear dynamic system as the output function so that the ith rule has the form (one rule of identifier model would be) [20];

~ i Then ^L1i ðk þ 1Þ ¼ ai L1 ðkÞ þ bi V p ðkÞ IF L1 ðkÞ is A ~i Then ^L2i ðk þ 1Þ ¼ ci L2 ðkÞ þ di Lr1 ðkÞ IF L2 ðkÞ is B which is formed as discrete-time linear systems. Here, V p ðkÞ and ~i Lr1 ðkÞ are the plant inputs and L1 ðkÞ and L2 ðkÞ are plant outputs. A i ~ and B are the linguistic value; ai ; bi ; ci ; di , (i = 1, 2, . . . , R) are the parameters of the consequents. ^L1i ðk þ 1Þ and ^L2i ðk þ 1Þ are the identifier model outputs considering only rule i. Suppose that li denotes premise certainty for the rule i. Using center-average defuzzification, we get the identifier model outputs;

PR ^ i¼1 L1i ðk þ 1ÞlL1 i PR i¼1

In this section, an adaptive compensator using Takagi–Sugeno fuzzy systems is proposed for the liquid level control system. A Takagi–Sugeno model of the plant is found by the system identification for using in the controller. From this model, a compensator is constructed which could provide a global asymptotically stable equilibrium for the closed-loop system. An on-line method is used to adjust the parameters of a Takagi–Sugeno identifier model to match the behavior of the liquid level system. Then, using the certainty equivalence principles, the parameters of the identifier model are employed in a standard parallel-distributed compensator. In this way, the identifier model does not need to have any initial knowledge about the liquid level system. As identifier becomes more accurate, the controller parameters of the compensator will be adjusted, and the controller will succeed. With this

L2 r

Controller I

L2 m

L1r

Controller II

Vp

Tank I

^L2 ðk þ 1Þ ¼

l L1 i

PR ^ i¼1 L2i ðk þ 1ÞlL2 i PR i¼1

l L2 i

ð15Þ

ð16Þ

Let’s define followings,

lL1 i i¼1 lL1 i

kL1 i ¼ PR

ð17Þ

Tank II

L1m

Fig. 3. Schematic diagram of the level control system.

L1 (k ), L2 (k )

Fig. 4. Block diagram of fuzzy identifier based fuzzy control of the liquid level system.

^L1 ðk þ 1Þ ¼

3. Fuzzy identifier model based fuzzy controller

Two Tank System

Fig. 5. The coupled tank experiment.

A. Basßçi, A. Derdiyok / Measurement 91 (2016) 12–18

l L2 i i¼1 lL2 i

kL2 i ¼ PR

ð18Þ

^L1 ðk þ 1Þ ¼ rT kL L1 1

ð19Þ

^L2 ðk þ 1Þ ¼ rT kL L2 2

ð20Þ

where ^L1i ðk þ 1Þ and ^L2i ðk þ 1Þ are the identifier model outputs. rL1 ; rL2 ; kL1 and kL2 are defined as

rL1 ¼ ½ai . . . aR bi . . . bR 

T

PR

PR

T

ð24Þ

...

V p ðkÞkL1 R 

kL2 ¼ ½L2 ðkÞkL2 i

...

L2 ðkÞkL2 R

Lr1 ðkÞkL2 i

...

Lr1 ðkÞkL2 R 

An on-line method, Recursive Least Square (RLS) is used to adjust the ai ; bi ; ci and di parameters since they enter linearly. For the controller, we use Takagi–Sugeno rules in the form

Tank 1 Level (cm)

10 5

15 10 5 0

80

Tank 1 level error (cm)

Tank 1 level error (cm)

20

40

60

80

20

15 10 5 0 -5 0

20

40

60

15 10 5 0 -5 -10

80

0

20

-5 0

20

Time (sec) 5

40 Time (sec)

60

80

5

Pump Voltage (V)

Pump Voltage (V)

0

Time (sec)

20

0

-5

Reference Measured

20

Time (sec)

-10

ð26Þ

lL2i ð/ðkÞÞ ¼ lL2 ðuðkÞÞ 25

15

l

lL1i ðv ðkÞÞ ¼ lL1 ðv ðkÞÞ

Reference Measured

60

lr1i

The Gaussian input membership functions on the universes of discourse L1 ðkÞ and L2 ðkÞ are used. Notice that since there is only one input, the membership function certainty is the premise membership function certainty for a rule;

V p ðkÞkL1 i

40

i¼1 Lr1i ðk þ 1Þ PR i¼1 r1i

ð23Þ

L1 ðkÞkL1 R

20

ð25Þ

T

...

0

þ 1Þlpi

lpi

ð22Þ

kL1 ¼ ½L1 ðkÞkL1 i

20

i¼1 V pi ðk PR i¼1

V p ðk þ 1Þ ¼

Lr1 ðk þ 1Þ ¼

rL2 ¼ ½ci . . . cR di . . . dR 

Tank 1 Level (cm)

where Lr1 ðkÞ and Lr2 ðkÞ are the reference values of the L1 ðkÞ and L2 ðkÞ. V pi ðkÞ and Lr1i ðkÞ are the controller outputs considering rule i, they are linear functions of their arguments that depend on past plant outputs and reference inputs. Our controllers that are tuned are given by

ð21Þ T

0

~ i Then V pi ðkÞ ¼ k1i L1r ðkÞ  k2i L1 ðkÞ IF L1 ðkÞ is A ~ i Then L1ri ðkÞ ¼ l1i L2r ðkÞ  l2i L2 ðkÞ IF L2 ðkÞ is B

then,

25

15

0

20

40

60

80

Time (sec) Fig. 6. AFC experimental results for step + square reference in configuration #1.

0

40

60

80

Time (sec) Fig. 7. PI experimental results for step + square reference in configuration #1.

A. Basßçi, A. Derdiyok / Measurement 91 (2016) 12–18

16

Using a certainty equivalence approach for the controller system, we view each rule of the controller as if it controls only one rule of the plant and the identifier is assumed to be accurate. In particular, we assume that ^L1 ðkÞ ¼ L1 ðkÞ and ^L1i ðkÞ ¼ L1i ðkÞ, where L1i ðkÞ represents ith component of the plant model, so that the identifier is also perfectly estimating the dynamics represented by each rule in the plant. If the plant is operating near its ith rule and there is little or no affect from its other rules, then L1 ðkÞ ¼ L1i ðkÞ. Thus we obtain,

^L1 ðk þ 1Þ ¼ L1i ðk þ 1Þ ¼ ai L1i ðkÞ þ bi ½k1i Lr1 ðkÞ  k2i L1i ðkÞ

ð27Þ

We pick k1i and k2i for each i = 1, 2, . . . , R so that the pole of the closed-loop system is at 0.1 and the steady state error between Lr1 ðkÞ and L1 ðkÞ is zero. In particular, if Lz1i ðkÞ and Lzr1 ðkÞ are the z-transforms of L1 ðkÞ and Lr1 ðkÞ respectively, then,

Lz1i ðzÞ bi k1i ¼ Lzr1 ðzÞ z þ bi k2i  ai

ð28Þ

k2i ¼

k2i bi  ai ¼ p

 1 < p < 1;

ð29Þ

ai  p bi

ð30Þ

where subscript i = 1, 2, . . . , R at each time step using the estimates of ai and bi from the identifier. Notice that we must ensure bi > 0  > 0.By adding a that could be done by specifying a priori some b rule to the adaptation scheme that if at some time the RLS updates  we let it be equal to b.  In this way, any bi so that it becomes less b,  the lowest value bi will take b. Next, to have a zero steady state error, we want L1i ðk þ 1Þ ¼ L1i ðkÞ ¼ L1r ðkÞ for a large k and i = 1, 2, . . . , R. From Eq. (27) we want

1:0 ¼ ai þ bi k1i  bi k2i 1:0  ai þ bi k2i bi

Tank 1 level (cm)

Reference Measured

10

5

0 0

20

30

40

50

Reference Measured

10

5

0 10

ð32Þ

Eqs. (29) and (32) specify the controller designer for the indirect adaptive scheme, and the identifier will provide the values of ai 15

15

ð31Þ

thus our controller designer will choose,

k1i ¼

choose,

Tank 1 level (cm)

to get the placement of the pole, our controller designer in our indirect adaptive scheme will pick,

60

0

10

20

12

10

10

Tank 1 level error (cm)

Tank 1 level error (cm)

Time (sec) 12

8 6 4 2 0

0

10

20

30

40

50

60

40

50

60

40

50

60

6 4 2 0

-4

60

0

10

20

30

Time (sec)

5

5

4

4

Pump voltage (V)

Pump voltage (V)

50

8

Time (sec)

3 2 1 0 -1

40

-2

-2 -4

30

Time (sec)

3 2 1 0

0

10

20

30

40

50

60

Time (sec) Fig. 8. AFC experimental results for step + sinusoidal reference in configuration #1.

-1

0

10

20

30

Time (sec) Fig. 9. PI experimental results for step + sinusoidal reference in configuration #1.

A. Basßçi, A. Derdiyok / Measurement 91 (2016) 12–18

and bi at each time step so that the k1i and k2i can be updated at  will each time step. Notice that the modifications to Eq. (29) with b also ensure that Eq. (32) will not be divide by zero. The above approach that is used to calculate the gains of the V p ðkÞ can be used to obtain the gains of the controller L1r ðkÞ and they can be defined as follows

l2i ¼

ci  p ; di

l1i ¼

1:0  ci þ di l2i di

ð33Þ

4. Experimental results The experimental of coupled tank plant is shown in Fig. 5. The couple tank specialty module is a ‘‘Two-Tank” module consisting of a pump with a water basin and two tanks of uniform cross sections. Such an apparatus forms an autonomous closed and recirculating system. The two tanks, mounted on the front plate, are configured such that flow from the first (top) tank can flow into second (bottom) tank. Flow from the second tank flows into the main water reservoir. In each one of the two tanks, liquid is withdrawn from the bottom through an outflow orifice. The outlet pressure is atmospheric. In order to introduce a disturbance flow, the first tank is also equipped with a drain tap so that, when opened, flow can be released directly into the water basin. The water level in each tank is measured using a pressure-sensitive sensor located at the bottom of the each tanks. Experimental results are presented in this section to show the performance of the AFC and PI controllers for both single and coupled tank system and results are shown in Figs. 6–11. In the first experiment, the performance of the controllers is tested for a step + square level reference for single tank conditions. The square wave reference is important because the control system is tested for a sudden increasing and decreasing liquid level period. As shown in Figs. 6 and 7, two disturbances, in which occurred with opening drain tap so that water flow released directly in to the water basin, are applied between 20 and 40 s to show the proposed

25

Reference Measured

12

controllers response against the undesired circumstances. It can be seen that, the AFC gives a fast response with having no overshoot and better tracking performance than the PI controller, and the control signal adjusts itself so that the output converges to its desired value. As seen from Figs. 6 and 7 the AFC controller increases the pump voltage to overcome the disturbances quickly and follows the reference level with small error when it is compared to the responses of PI. The errors are defined as the errors between the reference and measured values of the tank 1 water level. As shown in Figs. 6 and 7, if we disregard the error occurred at step changes in the reference and given disturbances, the maximum percentage errors of square wave reference for AFC is 2.92% and for PI 14.3%, respectively. It also can be observed that, when the square wave reference changes, the AFC gives fast response with having no overshoot and better tracking performance than the PI controller, and the water level tracking error in AFC is profoundly reduced in comparison with PI. In the second experiment, the step + sinusoidal level reference, as shown in Figs. 8 and 9, is chosen for water level control to determine the controller responses for reference changing continuously. Comparing graphs of both controllers, the AFC gives fast response with having no overshoot and better tracking performance than the PI controller and adapt itself quickly to changing conditions and follows the reference trajectory with small error. On the other hand the pump voltage produced by AFC has more chattering than PI. As shown in Figs. 8 and 9, if we disregard the error occurred at the starting point of sine wave, the maximum percentage errors of sine wave reference for AFC is 1.23% and for PI 7.86%, respectively. Finally, the step + sinusoidal reference level is chosen for tank 2 to show performance of the proposed controllers in coupled tank conditions. As seen from Fig. 3, the first controller calculates the reference level of tank 1 and the second one calculates the pump voltage command to guarantee that the actual liquid level in tank 2 tracks the given reference. Figs. 10 and 11 show the step + sinusoidal level position tracking output and the level tracking error, and pump voltage responses. It can be seen that the AFC gives faster response with having no overshoot and better tracking performance than the PI controller, and the figure shows how the control

Tank 1 Level (cm)

Tank 2 level (cm)

14

10 8 6 4 2 0

0

20

40

60

80

100

120

20 15 10 5 0

140

Reference Measured

0

20

40

10

25

8

20

6 4 2

140

100

120

140

0

-2 0

Time (sec)

120

5

-5 80

100

10

-10

60

80

15

0 40

60

Time (sec)

Pump Voltage (V)

Tank 2 level error (cm)

Time (sec)

20

17

100

120

140

0

20

40

60

80

Time (sec)

Fig. 10. AFC experimental results for step + sinusoidal reference in configuration #2.

A. Basßçi, A. Derdiyok / Measurement 91 (2016) 12–18

18

12

25

Reference Measured

Tank 1 level (cm)

Tank 2 level (cm)

14

10 8 6 4 2 0 0

20

40

60

80

100

120

Reference Measured

20 15 10 5 0 0

140

20

40

8

Pump Voltage (V)

Tank 2 level error (cm)

10

6 4 2 0 -2 0

20

40

60

80

60

80

100

120

140

100

120

140

Time (sec)

Time (sec)

100

120

140

Time (sec)

25 20 15 10 5 0 -5 -10 -15 -20 -25

0

20

40

60

80

Time (sec)

Fig. 11. PI experimental results for step + sinusoidal reference in configuration #2.

signals adjust themselves so that the output converges to its desired value. In Figs. 10 and 11, it is observed that the maximum steady state error of tank 2 using the AFC is 6.2%. Meanwhile, the tracking error for the conventional PI control is 9.7%, which is obviously much larger. However, the controller 1 which used to calculate the reference value of tank 1 causes a fluctuation because of amplifying tank 2 level error. Also note that the reference signal of tank 1 is generated after tank 2 level measurement, therefore a delay is occurred between the two levels and the error of the tank 2 level is lower than tank 1. The control system is focused on arranging tank 2 levels. As a result, the experimental results show that the proposed control scheme possesses a remarkably better tracking precision performance than PID control and the responses of the liquid level control system are very good and satisfactory. 5. Conclusions In this paper, we have presented an adaptive fuzzy controller and the proposed controller has been validated in two tank system and compared from the robustness point of view. The experimental results strongly show that the proposed approach provides better position tracking performance than PI with high tracking precision as well as better robustness against disturbance and changes of references. This means that the proposed identification model and control structure adapt themselves to changed conditions and parameter uncertainties quickly, and working very well. References [1] K.C. Ng, Y. Li, D.J. Murray-Smith, K.C. Sharman, Genetic algorithms applied to fuzzy sliding mode controller design, in: Presented at First International Conference on Genetic Algorithms in Engineering Systems: Innovations and Applications, 1995. [2] B. Moshiri, M. Jalili-Kharaajoo, F. Besharati, Application of fuzzy sliding mode based on genetic algorithms to control of robotic manipulators, Emerg. Technol. Factory Automa. 2 (2003) 169–172 (IEEE).

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