Water Level Control by Fuzzy Logic and NeuralNetworks

Daniel WuDept. of Systems Design Eng.

University of WaterlooWaterloo, ON N2L 3G1

Fakhreddine KarrayDept. of Electrical & Computer Eng.

University of WaterlooWaterloo, ON N2L 3G1

Insop SongDept. of Systems Design Eng.

University of WaterlooWaterloo, ON N2L 3G1

Abstract— The objective of this paper is to investigate and finda solution by designing the intelligent controllers for controllingwater level system, such as fuzzy logic and neural network. Thecontrollers also can be specifically run under the circumstanceof system disturbances. To achieve these objectives, a prototypeof water level control system has been built and implementationsof both fuzzy logic and neural network control algorithms areperformed. In fuzzy logic control, Sugeno model is used tocontrol the system. In neural network control, the approach ofModel Reference Adaptive Neural Network Control based on thebackpropagation algorithm is applied on training the system.Both control algorithms are developed to embed into a stand-alone DSP-based micro-controller and their performances arecompared.

I. I NTRODUCTION

Traditionally, accurate mathematical model-based strategieshave been applied to deal with control problems. However,water level control system, for example, is very complexsystem, because of the nonlinearities and uncertainties ofasystem. Conventional control approaches are not convenientto solve the complexities. Fuzzy logic and neural networkscontrol have emerged over the years and become one of themost active and fruitful areas of the research in the intelligentcontrol applications. There are two major different types ofthe control rules in fuzzy control: the Mamdani type and theSugeno type. The Mamdani control rules are significantly morelinguistically intuitive while Sugeno rules appear to havemoreinterpolation power even for a relative small number of controlrules. In neural network control, the most commonly usedones are supervised control, direct inverse control and neuraladaptive control.

There are many papers addressed the fuzzy or neuralnetworks control in the water or liquid level control system.Niimura et al.utilized the fuzzy logic for water level control insmall hydro-generating units[1]. Rouboset al. chose Sugenofuzzy model as the model structure for a linear model basedpredictive control of the liquid level [2]. Ghwanmehet al.showed a self-learning fuzzy logic control applied to a non-linear process and demonstrated the robust and repeatableperformance [3]. Namanet al. presented an adaptive model-reference fuzzy controller for controlling the water levelin awater tank [4]. Xiaoet al. provided a backpropagation neuralnetwork algorithm used to adjust the parameters of the PIDcontroller and control the liquid level of molten steel smelling

3

3 2

DSP Controller

Sensor

Pump

Valve2

Valve1

Servor

Motor

Water Level

Water Tank

Valve3

Fig. 1. Configuration of the Water Level Control System. The system consistsof the servo motor, valves, a pump, a infra-red sensor and a DSP controller.

non-crystalloid [5].In this paper, we first elaborate the configuration of the

water level control system. Then, we introduce Sugeno fuzzycontrol [7], [8] and model reference adaptive neural networkcontrol (MRANNC) [6], [9], [10] strategies based on back-propagation algorithm. Finally, some experimental and real-time results using DSP controller are presented.

II. SYSTEM CONFIGURATION

Figure 1 shows the system configuration. The system con-sists of the servo motor, valves, a pump, a infra-red sensor anda DSP controller. Water is constantly pumped into a verticaltube by a pump. Water can outflow from the horizontal tube.The general control objective is to reach and stabilize a certainwater level in the vertical tube. As shown in the figure, asensor measures the water level and sends height signal tothe DSP micro-controller. The controller controls the servomotor, which adjusts the valve 1 to maintain a desired waterlevel in the vertical tube. Valve 2 constantly opens and valve 3randomly opens to provide disturbances to the control system.

III. F UZZY LOGIC CONTROL

In Fuzzy control, two inputs for the system are chosen.They are an error (e) and an error derivative (e). The erroris calculated by taking the difference between the reference

1 If e is HN OPEN valve WIDE2 If e is N ande is NOT P OPEN valve A BIT3 If e is N ande is P LEAVE valve ALONE4 If e is S ande is N OPEN valve A BIT5 If e is S ande is S LEAVE valve ALONE6 If e is S ande is P CLOSE valve A BIT7 If e is P ande is N LEAVE valve ALONE8 If e is P ande is NOT N CLOSE valve A BIT9 If e is HP CLOSE valve A LOT

TABLE I

RULE BASE

Output Functions1 OPEN valve WIDE 900 · e + 0 · e − 20

2 OPEN valve A BIT −50 · e − 200 · e − 20

3 LEAVE valve ALONE −5 · e + 0 · e + 0

4 CLOSE valve A BIT −50 · e − 2000 · e + 20

5 CLOSE valve A LOT −100 · e + 0 · e + 20

TABLE II

OUTPUT FUNCTIONS.

signal and the current water height. The error derivative iscalculated by subtracting a previous error from the currenterror. The output of the system is a voltage that is sent to aservo motor to control the valve 1.

The Sugeno model is used in the system. The Sugeno modelis computationally efficient, and works well with optimizationand adaptive techniques, so it is popular for control problems,in particular for dynamic nonlinear systems. The Sugeno fuzzymodel takes the form:

IF e is A AND e is B, THEN SERVO isf(e, e)

The output functionf(e, e) takes the form:

A2e + A1e + A0

The parameters of the output functionA2, A1, andA0 canbe modified through the micro-controller interface.

The rule base used here is shown in Table I. Five mem-bership functions for the error are shown in Figure 2(a),and three for the error derivative are shown in Figure2(b). The rules mapping is shown in Figure 2(c). Labelsin Table I and Figure 2 are as follows: HN=High Nega-tive;N=Negative;S=Small;P=Positive; HP=High Positive;

For the Sugeno model, output functions must be defined.This will decide how the system provides the proper controlthrough the manipulation of the valve 1. Table II shows theoutput functions and nominal tuning values that the controlleruses on start up. Each of these parameters can be modifiedusing the user interface of the micro-controller.

IV. N EURAL NETWORK CONTROL

In this paper, the Model Reference Adaptive Neural Net-work Control approach based on backpropagation algorithmis applied to implement the water level control system. Figure

0 -2 2 5 -5 15 -15

S HP P HN N

0 0.5 -0.5

S P N

4 -4

HN N S P HP

N

S

P

1 2 3

4

5

1

1

2

2 3

3

4

4

5

5

(a) Error Membership Functions

(b) Error-Dot Membership Functions

(c) Mapping

error

error_dot

Fig. 2. Membership Functions and Mapping

Nonlinear

Plant

NN

controller

NN

identifier

TDL TDL

TDL TDL

TDL p y u

i e

c e

) ( t r

i e

c e +

+

-

-

i y

Fig. 3. Neural Network Control in Water Level Control Systems, TDLdenotes time delays of the input or output signals.

3 shows the control structure. TDL in Figure 3 denotes timedelays of the input or output signals.

A. Neural Network for Identifier

The neural network for identifier is designed as a three-layer neural network. It has a input layer, a hidden layer,and an output layer.N i

i ,N ij and N i

k are their output valuesrespectively. The neuron numbers in the hidden layers can bechosen depending on the practical training result.

The neural network identifier models are trained to learn theforward dynamics of the plant. Six inputs and one output areselected as the identifer model for the system. These six inputsare the control signals:u(t − i), i = 0...2, and the previousoutput signals:y(t − j), j = 1...3. These signals provide the

current and previous two control signals, and also incorporatein the historical trend from the last three time steps of theoutput of the system. A set of corresponding input-outputtraining patterns is selected from the open-loop continuoussignal response of the system. The control input signal isdirectly added to servo motor of the system and system outputsignal is reflected with the sensor, which is actual water level.

In training a neural network to learn a forward dynamicmodel of a plant, the backpropagation error signal betweenthe output and the hidden layer is expressed as

δk = Tk − N ik (1)

whereTk is the target pattern andN ik is the actual output of

the identifier, and between the hidden and input layers, it isexpressed as

δj = f′

(netj)∑

k

δk · wkj (2)

Here, f′

(netj) is the derivative of the activation functionf(netj) where

f(netj) =1

1 + exp(−netj)(3)

The weights between the input and hidden layers are updatedas

∆wji(t + 1) = η · δj · Nii + α · ∆wji(t) (4)

and the weights between the hidden and output layer areupdated as

∆wkj(t + 1) = η · δk · N ij + α · ∆wkj(t) (5)

whereN ii and N i

j are the outputs of the input and hiddenlayers, respectively,η is the learning rate, andα is themomentum coefficients. The constantsη, α are all chosenempirically.

B. Neural Network for Controller

The neural network controller is created directly based onthe neural network identifier. Its design is fully incorporatedthe learning strategy into the trained identifier. The weightsof the neural network identifier are constantly verified againstthe actual plant output. This ensures that the weights allowthe neural network identifier to properly predict actual plantoutput. The neural network identifer is used as means tobackpropagate the performance error to get the equivalent errorat the output of the neural network controller.

The accuracy of the plant model is critical in ensuring thatthe controller is accurate as well. The error between the plantoutput and the identifier output is also checked for the accuracylevel of the identifier. This error is used to backpropagate andadjust the weights of the identifier to provide the most accuraterepresentation of the plant. The neural network for controlleris also designed as a three-layer neural network. It has a inputlayer, a hidden layer, and an output layer.N c

i , N cj andN c

k aretheir output values respectively. The neuron numbers in thehidden layers can be chosen also depending on the practicaltraining result.

c

i N c

j N c

k N i

i N

i

j N

i

k N

c

ji w c

kj w i

ji w i

kj w

NN Contorller NN identifier

Fig. 4. The Connection Between the Controller and the Identifier

To illustrate the derivation of the error signals for the neuralnetwork controller, Figure 4 provides the connections betweenthe controller and identifier networks.

The adaptation of the weights of the neural network con-troller between the hidden and output layers can be derived asfollows:

∆wckj = η ·

∂Ei

∂wckj

(6)

where

Ei =1

2· (yp − r)2 (7)

whereyp andr are the actual and desired plant output. Thesuperscriptc denotes the variables belonging to the neuralnetwork controller and superscripti denotes the variablesbelonging to the neural network identifier.

Using chain rule, equation (6) can be expanded as follows:

∆wckj = η ·

∂Ei

∂netck·

∂netck∂wc

kj

so∆wc

kj = η · δck · N c

j (8)

where

δck =

∂Ei

∂netck

andN c

j =∂netck∂wc

kj

δck is the error signal between the hidden and output layers

of the neural network controller. Linear functions are usedatthe input and output neurons of the neural networks betweenthe controller and identifier. Therefore,δc

k can be representedas follows:

δck =

∂Ei

∂N ck

=∂Ei

∂netii=

∂Ei

∂N ii

(9)

wherenetii andN ii are the input and output of the identifier

input layer neurons. Further using chain rule

δck =

∂Ei

∂netij·

∂netij

∂N ii

(10)

Hence,∂Ei

∂netij= δi

j

NN In ni nj nk η α

Identifier 6 10 1 0.00001 0.8Controller 10 20 1 0.00005 0.65

TABLE III

PARAMETERS SETUP

and∂netij

∂N ii

= wiji

so,

δck = δi

j · wiji (11)

The error signal between the input and hidden layer of theneural network controller can be derived as follows:

δcj =

∂Ei

∂netcj=

∂Ei

∂N cj

·

∂N cj

∂netcj(12)

By using chain rule

∂Ei

∂N cj

=∑ ∂Ei

∂netck·

∂netck∂N c

j

and∂N c

j

∂netcj= f

′

(netcj)

where

f′

(netcj) = N cj · (1 − N c

j )

so,

δcj = N c

j · (1 − N cj ) · (

∑δck · wc

kj) (13)

The weights of the neural network identifier can be furtherimproved online if necessary. This can be reached by back-propagation of the following error equation through the neuralnetwork identifier at every sample.yp and yi are the outputsof the actual plant and neural network identifier, respectively.

e =1

2· (yp − yi)

2 (14)

V. EXPERIMENTAL RESULTS

The neural network control algorithm were first trained andtested with C language with a normal PC. Then, the algorithmwere ported to DSP-based controller and tested.

A. Training and Testing for Neural Network Control

1) Parameters Setup:In order to have quicker calculationwithout sacrificing performance, some parameters were setupas shown in Table III. In Table III,ni,nj and nk denotethe number of nodes in the input layer, hidden layer andoutput layer, respectively,η and α denote learning rate andmomentum term, respectively.

Plant Disturbance Training Data

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 201 401 601 801 1001 1201 1401 1601 1801 2001 2201 2401 2601

Sample

Volt

age

(V)

Y

U

Fig. 5. Plant Disturbance Training Data, U is an input signaland Y is aoutput of the sensor

Plant Test Data

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 401 801 1201 1601 2001 2401 2801 3201 3601 4001 4401 4801 5201 5601

Sample

Volt

age

(V)

Y

U

Fig. 6. Plant Test Data

2) Identifier Training and Testing:The trained identifierwill be used as a representation of the system plant toeffectively create a controller for the system. Then, the off-line training of the controller will be feasible to reach.

1. Training DataAn open loop system was used to obtain a system response

curve for the plant response to a random input signal. Figure5 shows the input signal used for training of the NN iden-tifier. This was done by sending a voltage signal through apotentiometer into the servo motor, and it controls the valve1. By creating a random input control signal and measuringthe sensor output while a random and a constant disturbanceexists. A set of data was created and indicated the behaviorof the plant response to a characteristic input signal. It wasdesigned to fall within the operating range of the sensor. Asshown in the figure, the input signalu varies between valuesof 2V and4V .

2. Testing DataOnce the training portion of the Identifier is completed, it

needs to be verified with the training results. A selection ofsimilar data was created for testing the trained network. Figure6 shows a set of testing data used for the Identifier.

3. Identifier TrainingIdentifier training resulted to a level of error of 7.9842

after 150,000 epochs. At the cut-off of 150,000 epochs, theidentifier was still converging on an error, but extremelyslowly. Then, this level of an error was used to move onto

Identifier Training Error

1

10

100

1000

100 1000 10000 100000 1000000

Epoch

Err

or

(V)

Error

Fig. 7. Identifier Training Error

Identifier Test Data

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 501 1001 1501 2001 2501 3001 3501 4001 4501 5001 5501

Epoch

Volt

age

(V)

U

Y

Y-PRED

Fig. 8. Identifier Testing Results

controller training. Figure 7 shows the trend of decreasingerror in the identifier training.

4. Identifier TestingThe error seen as a result of testing is shown in Figure 8.

Since the level of error seen is on the same scale of that for thetraining, then moving to controller training became the nextstep. It is believed that a more rigorous training could resultin better error and better performance for controller trainingand final implementation.

3) Controller Training and Testing:In order to effectivelycreate a controller for the system, the existing system wasanalyzed in an off-line setup. The goal in the project isto develop a controller through simulation and use morecomputer resources in a highly effective manner. It proved farmore effective to design the training and testing in an off-lineformat.

1. Training DataFigure 9 shows the reference signal used for training of the

neural network controller. It was designed to maintain a levelwithin the operating range of the sensors. A random signalwas created that would provide the controller with typicalvalues that a user could choose during normal operation. Thesevalues were held within the ideal operating range for whichthe identifier was trained.

2. Testing DataFigure 10 shows a set of testing data used for the neural

network controller. As can be seen, this pattern is somewhatsimilar to that seen in the training. As was the case inthe identifier, once the training portion of the controller is

Controller Training Data

0

5

10

15

20

25

30

35

1 101 201 301 401 501 601 701 801 901 1001 1101 1201 1301

Test Sample

Wa

ter

Lev

el (

cm)

Desired

Fig. 9. Controller Training Data

Controller Test Data

0

5

10

15

20

25

30

35

40

1 26 51 76 101 126 151 176 201 226 251 276 301 326 351 376

Sample

Wate

r L

evel

(cm

)

Test Data

Fig. 10. Controller Testing Data

completed, it needs to verify that training. A set of datawas selected for testing the controller training. Since thetraining data used was designed to be a component of typicaloperational values, a similar set of data was created for testingthe trained network.

3. Controller TrainingIn Figure 11, convergence towards the goal of small error

is clearly seen in the controller training algorithm. However,the error does not reach the desired cut-off. Despite this, theweights trained as a result of this training provide a very goodresponse when the testing data is used to evaluate the controllerweights.

4. Controller TestingAs shown in Figure 12, the weights for both the identifier

and controller had been trained to a level that provided anacceptable level of prediction, and showed that the controllerwas working as expected.

Controller Training Error

1

10

100

1000

0 1000 2000 3000 4000 5000 6000 7000 8000

Epoch

Err

or

(V)

Error

Fig. 11. Controller Training Error

Controller Test Data

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 26 51 76 101 126 151 176 201 226 251 276 301 326 351 376

Sample

Volt

age

(V)

R

Y

Fig. 12. Controller Testing

Fuzzy Control Result

0

5

10

15

20

25

30

1

11

21

31

41

51

61

71

81

91

101

111

121

131

141

151

161

171

181

191

201

211

221

231

Time (s)

Wate

r L

evel

Sensor

Setpoint

Fig. 13. Online Fuzzy Control Result

B. Experiments on micro-controller

The fuzzy and neural network control algorithms wereported to a stand-alone VRP MTSC32 micro-controller. Themain board of micro-controller is based on Texas Instruments’TMS320C32 DSP with a CAN bus communication port thatfacilitates the communication with a PC for monitoring onlineparameters. The control algorithms were developed with Clanguage, complied withTI Compiler and run from the micro-controller.

To provide an objective comparison between the fuzzy andneural network control algorithms, two tests were carried outin the VRP MTSC32 DSP micro-controller. Figure 13 showsfuzzy control result and Figure 14 shows the neural controlresult.

VI. CONCLUSION

This paper proposed two control schemes for the water levelcontrol system. From the results of the experimental studies,the following summary can be obtained.

For the fuzzy control, in order to ensure the best perfor-mance, a number of factors and values need to be onlinedetermined heuristically or by trial and error, for example,the membership functions. For neural network control, thelearning parameters and prior well-training is essential forthe success of the control. Once trained, the neural networkdoes not require for tuning. Table IV shows a summary of

Neural Network Control Results

0

5

10

15

20

25

30

1

9

17

25

33

41

49

57

65

73

81

89

97

10

5

11

3

12

1

12

9

13

7

14

5

15

3

16

1

16

9

17

7

18

5

19

3

20

1

20

9

21

7

Time (s)

Wa

ter

Le

ve

l (c

m)

Setpoint

Sensor1

Fig. 14. Online Neural Network Control Result

comparisons between the two control schemes based on theexperimental results.

FLC NNCPlant Math Model Unnecessary UnnecessaryComputation Light HeavyTracking Performance Good BetterDisturbance Rejection Good Better

TABLE IV

COMPARISON BETWEEN THE FUZZY CONTROL AND NEURAL NETWORK

CONTROL BASED ON THE EXPERIMENTAL RESULTS

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[5] Xiao,Y., Hu,H., Jiang,H., Zhou,J.and Yang,Q., “A Adaptive Control BasedNeural Network for Liquid Level of Molten Steel Smelting NoncrystlloidFlimsy Alloy Line”, Proc. of4th World Congress on Intelligent Controland Automation, China, 2002.

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