Mathematical Modeling and Fuzzy Adaptive PID Control of Erection Mechanism

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DOI:_{10.12928/TELKOMNIKA.v15i1.3568} ₂₅₄

Mathematical Modeling and Fuzzy Adaptive PID Control

of Erection Mechanism

Feng Jiangtao*, Gao Qinhe, Guan Wenliang

Xi’an High-tech Research Institute, Tongxin Road No. 2, Hongqing Town, Baqiao District, Xi’an, Shaanxi, China

*Corresponding author, e-mail: fengjt291082217@126.com

Abstract

This paper describes an application of fuzzy adaptive PID controller to erection mechanism. Mathematical model of erection mechanism is derived. Erection mechanism is driven by electro-hydraulic actuator system which is difficult to control due to its nonlinearity and complexity. Therefore fuzzy adaptive PID controller is applied to control the system. Simulation was performed in Simulink software and experiment was accomplished on laboratory equipment. Simulation and experiment results of erection angle controlled by fuzzy logic, PID and fuzzy adaptive PID controllers were respectively obtained. The results show that fuzzy adaptive PID controller can effectively achieve the best performance for erection mechanism in comparison with fuzzy logic and PID controllers.

Keywords: erection mechanism, electro-hydraulic system, fuzzy logic control, PID control

1. Introduction

Erection mechanism of large machineries and armaments is a complex mechanical, electronic and hydraulic integration system. It can be mainly divided into electro-hydraulic part and mechanical part. Electro-hydraulic actuator has become one of the most important actuators in the past decade. The main reasons why hydraulics is preferred to mechanical drives in industrial applications include their ability to produce large forces at high speeds, high power density and good dynamic performance [1]. Its application can be found in many engineering fields such as automotive industries, machine tools, aircrafts and manufacturing equipment [2-4].

The dynamic behaviour of hydraulic system is highly non-linear due to friction, fluid compressibility and associated stiffness, which cause difficulties in control of such systems [5]. Position tracking performance can be assured when its robustness and tracking accuracy are guaranteed. There are a number of problems such as nonlinearities, uncertainties and time varying parameters which can reduce the robustness and tracking accuracy. Therefore, the development of a suitable controller which could reflect robustness and tracking accuracy is very significant [6]. The problem of position tracking control has been addressed in many papers and numerous publications have proposed fuzzy logic control [7], model based control [8] and adaptive control [9-10] to improve the position tracking performance and have proven to be successful in their specific area. Jelali and Kroll summarized recent developments in nonlinear identification, control and application to hydraulic servo systems and arrived at the conclusion that advanced control methods were necessary for high-performance motion control [11]. Advanced control methods can compensate for the nonlinearities of electro-hydraulic systems. Conventional proportional-integral-derivative (PID) controller is commonly used in industry due to its simplicity. PID control algorithm has good stability and reliability. It has satisfactory control effect for linear system. However, the parameters of PID control cannot change online, so it is sensitive to variations of parameters. Fuzzy logic control has achieved desired goals in control of electro-hydraulic system. It is able of adaptation to parameter changes and to deal with nonlinear dynamic behaviour. Fuzzy control algorithm transforms experts and operators’ experience into control rules and don’t need precise system information. It has the advantages of adapting to uncertainty and complexity. It is usually associated with traditional control method and has practical value and great development potential. Fuzzy PID controller which combines

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the traditional PID control and fuzzy control has been applied to solve many engineering problems [12].

This paper is organized as follows. Mathematical model of the electro-hydraulic system is derived. Kinetic analysis of erection mechanism is accomplished. The relationships of erection force, erection angle and the length of hydraulic cylinder are obtained. Fuzzy adaptive PID controller is designed in combination of fuzzy logic and PID control method. Models of erection system and fuzzy adaptive controller are established in Simulink. Experimental studies are completed on laboratory equipment. The aim of this research is to investigate the effect of fuzzy adaptive PID controller on erection mechanism.

2. Mathematical Modeling of Erection System 2.1. Model of the Electro-Hydraulic System

Erection mechanism is mainly composed of hydraulic system and mechanical system. Mathematical models of each system are established separately. Hydraulic system includes hydraulic pump, relief valve, hydraulic cylinder and electro-hydraulic proportional valve, as is shown in Figure 1.

Figure 1. Hydraulic Principle of Erection System

The pump model can be built by the following equation.

p p p v

Q D w (1) Where Dp is pump displacement. wp is rotational speed of the motor. v is volumetric efficiency.

The model of the electro-hydraulic proportional directional valve is expressed by the following formulas.

1 1

2( _s )

d x

p p q C A





 (2)

2 2

d y

p q C A



 (3) Where q1 and q2 are fluid flows from and to cylinder. ps is the hydraulic supply pressure. Axand Ay are the spool valve areas. Cd is the discharge coefficient. ρ is the fluid density.

The hydraulic cylinder model is built using the cavity node method. The dynamic equation of the hydraulic cylinder is described by the second Newton’s Law.

01 1 1

1 1 ( ec ic) 1 ic 2

V A y dp dy

q A C C p C p

dt  dt



 ˄ ˅    (4)

5 M

1

4

3

2

(3)

02 2 2

2 2 ( ec ic) 2 ic 1

V A y dp dy

q A C C p C p

dt  dt



 ˄ ˅    (5)

1 1 2 2 2

d y

p A p A m F

   (6) Where A1 is cross-sectional area of the piston and A2 is the annular area of piston rod chamber. y is the piston displacement. V01 and V02 are the initial volumes of the two cylinder chambers. A1y and A2y represent the flow rates as a function of volume change due to the piston motion.

βe is the fluid bulk modulus. p1 and p2 are the pressures in the forward and return cylinder chambers.Cic and Cec are the internal and external leakage coefficients. m is mass of the load. F is the external force.

2.2. Kinetic Analysis of Erection Mechanism

Mechanical part of erection mechanism is shown in Figure 2. The load revolves around the point P2 driven by erection cylinder.

Figure 2. Kinetic Model of Erection Mechanism

We can obtain the following equation through Euler dynamic equations in coordinate oxy.

2 4 2 5 ( ) _p( )

Jt F t P P  G P P (7) Where J is the moment of inertia of the load to point P2. (t) is the erection angle. Fp(t) is the

thrust force of erection cylinder. G is gravity of the load.

Geometry relationships are described by the following equations.

2 3

2 4 2 2 3 2

3 sin( ( ) )

= sin P P t

P P OP P OP OP

 

  (8)

2 5= 2 Gcos( ( ) )

P P P P



t 



(9) The relationship of erection angle and cylinder length can be acquired by using cosine

theorem in ΔOP2P3.

2 2 2

2 3 2 3

2 3 2

cos( ( ) )

P P OP OP

P P OP

    

 (10)

Fp(t)

θ(t)

α β

(4)

3. Fuzzy Adaptive PID Controller Design

We adopt fuzzy adaptive PID control method in erection process to enhance control performance. PID control is connected with fuzzy control and then the parameters of PID control method can change based on circumstance [13]. Fuzzy adaptive PID control method can make

full use of operators’ successful nonlinear experience and excellent effect of PID control. It can

improve precision and achieve better effect compared with PID control and fuzzy control. The differential equation of PID control is as follows [14].









( ) _P ( ) ( ) ( )

T TD

u kT K e kT e kT e kT T

T T

 



  (11) Where k is sampling number. T is sampling period. u(kT) is output signal. KP is propotional

coefficient. TI is integral time constant. TD is differential time constant.

The structure of fuzzy adaptive PID controller is shown in Figure 3. Fuzzy logic controller is used to change the parameters of PID controller. Its input variables are error e(t) and error change rate c(t) and output variables are △KP, △KI, △KD, which are respectively added

to initial values KP0, KI0 and KD0.

Figure 3. The Structure of Fuzzy Adaptive PID Controller

Figure 4. Membership Function Cure of Input

All of the variables are implied as linguistic values and defined with seven linguistic values which are: NB-negative big, NM-negative medium, NS-negative small, ZO-zero, PS -positive small, PM-positive medium, PB-positive big [15]. The domain of error e(t) is [-xe, xe] and

the domain of error change rate c(t) is [-xc, xc]. The domain of fuzzy subset is {6, 5, 4, 3, 2,

-1, 0, -1, 2, 3, 4, 5, 6}.

Triangular shape function is chosen as input membership function which is shown in Figure 4 [16]. The relationships between e(t), c(t) and KP, KI, KD are summarized through much

operating experience [17].

1. When e(t) is relatively large, KP should be adopted relatively large value to speed up

the system response and reduce time constant with damping coefficient. KDshould be adopted

Electro-hydraulic proportional valve Fuzzy Inference Module P K  I K  D K    F uz zy PID controller de dt D efuz zi fi ca tion Fuzzy rules 0 0 0

P P P

I I I

D D D

K K K

      ec K e K P G I G D G Sensor Hydraulic cylinder E EC ec(t) e(t) e(t) u(t) c(t)

ZO PS PM PB

1.0 NS NM NB

0 2 4 6

-2 -4

(5)

relatively small value to avoid out of range control at the beginning. To avoid large overshoot, remove the integral action.

2. When e(t) is medium, KP should be adopted relatively small value to minimize

response overshoot. The value of KD is important and should be adopted medium value. The

value of KI should be appropriately increased.

3. When e(t) is relatively small, KP and KI should be adopted relatively large value to

have good steady state and avoid oscillation at equilibrium point. The value of KD should be

appropriate.

Based on the above relationships and the impact of c(t), fuzzy control rules are obtained in Table 1 through theoretical analysis.

Table 1. Fuzzy Control Rules △KP /△KI /△KD

NB NM NS ZO PS PM PB

NB PB/NB/NS PB/NB/PS PM/NM/PB PM/NM/PB PS/NS/PB ZO/ZO/PM ZO/ZO/NS NM PB/NB/NS PB/NB/PS PM/NM/PB PS/NS/PM PS/NS/PM ZO/ZO/PS NS/ZO/ZO NS PM/NB/ZO PM/NM/PS PM/NS/PM PS/NS/PM ZO/ZO/PS NS/PS/PS NS/PS/ZO ZO PM/NM/ZO PM/NM/PS PS/NS/PS ZO/ZO/PS NS/PS/PS NM/PM/PS NM/PM/ZO PS PS/NM/ZO PS/NS/ZO ZO/ZO/ZO NS/PS/ZO NS/PS/ZO NM/PM/ZO NM/PB/ZO PM PS/ZO/NB ZO/ZO/PS NS/PS/NS NM/PS/NS NM/PM/NS NM/PB/NS NB/PB/NB PB ZO/ZO/NB ZO/ZO/NM NM/PS/NM NM/PM/NM NM/PM/NS NB/PB/NS NB/PB/NB

Take the first rule as an example, the above rule can be interpreted as “if e is NB and c is NB, then △KP /△KI /△KD are PB/NB/NS”. Mamdani algorithm is introduced as fuzzy implication

relation [18].

( , ) ( )( , )

= min( ( ), ( )) ( ) ( )

R a u A U a u

A a U u A a U u

 

  (12) Triangular shape function is chosen as output membership function which is shown in Figure 5 [19].

Figure 5. Membership Function Cure of Output

Centroid is chosen as defuzzification method [20], which calculates the area center of the fuzzy set membership function curve surrounded by the horizontal coordinate. The horizontal coordinate value of the center is chosen as the representative value of the fuzzy set. The membership function of F set is A(u) on the domain U and the horizontal coordinate u0 corresponding to the area center is calculated by the following formula [21].

( ) ( ) U

A u udu u

A u du





(13)

ZO PS PM PB

1.0 NS NM NB

0 2 4 6

-2 -4

(6)

4. Simulation and Results

4.1. Modeling Erection Mechanism in Simulink

Figure 6 illustrates the top level Simulink diagram of the entire system. The pump flow and the desired angle signal are simulation inputs. The pressure and cylinder position are

simulation outputs. The model is organized as several subsystems blocks such as the ‘source’

model, the ‘pump’ model, the ‘valve/cylinder’ model and the ‘fuzzy adaptive PID controller’

model. The output of the controller is derived to control motion of the valve spool. Position of the spool influences the nominal flow into cylinder chamber. The nominal flow change will change the volume and pressure inside both cylinder chambers. Finally, the piston position is affected, therefore the mechanical mechanism moves driven by cylinder piston. Hence, the performance of the system is dependent on the output of the controller.

Figure 6. Simulink Block of the Complete System

The ‘source’ model is reference angle trajectory. We adopt composite sine function as

reference angle trajectory. 0 is initial value of erection angle and 1 is final value. Erection time is T, τ=t/T. (t) is determined by the following expressions:

1 0

( )t ( ) ( )s





 





(14)

sin(4 ) 1

4 4 8

2 4 1 7

( ) 9 cos( ) / 4

4 3 6 8 8

4 sin(4 2 ) 7

4 4 8

k k s



 

 

 

   

 

 

 

  

  _  _{ }

 _ _



 _ _

_ _    _  

 



 _ _ _

   

 _ _

 



(15)

Where k4

 

2 4.

Figure 7 shows the valve and hydraulic cylinder sub-block obtained from Equations (1) to (6). The control valve is modeled as turbulent flow through a variable area orifice. A system of differential-algebraic equations models the cylinder pressurization with the pressure p1 and p2, which appear as derivatives in Equations (4) to (6) and are used as the states.

Simulink block of fuzzy adaptive controller is shown in Figure 8. We establish the model based on the controller structure in Figure 3.

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Figure 7. Simulink Block of Hydraulic Cylinder and Valve

Figure 8. Simulink Block of Fuzzy Adaptive Controller

4.2. Simulation Results

Table 2. Parameter Values for Simulation

Symbol Explanation Value

Cd Discharge coefficient 0.61



Bulk modulus of hydraulic fluid. 6.85e8 Pa

A1 Cross-sectional area of the piston 1.23e-3 m2

A2 Annular area of piston rod chamber 5.91e-4 m

ρ Fluid density 800 kg/m3

Cic Internal leakage coefficient 3e-11

Cec External leakage coefficient 2e-11

m Equivalent mass 500 kg

Ps Hydraulic supply pressure 2.5e6 Pa

V01 Original total fluid volumes of the forward cylinder chamber 2.4e-3 m3

V02 Original total fluid volumes of the return cylinder chamber 0.0089 m3

OP The length of OP₂ 3.08 m

2 3

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Simulation was performed in Simulink software to investigate the performance of fuzzy adaptive PID controller for erection system. The simulation results are demonstrated in this section. The parameters used for numerical simulation are shown in Table 2.

The goal of the controllers is to get the system tracking the desired angle reference. Figure 9 shows the simulation results of erection angle controlled by the fuzzy logic, PID and fuzzy adaptive PID controllers respectively. Figure 9(a) presents desired and actual erection angle curves. Figure 9(b) illustrates the tracking errors of desired and actual erection angles. It is clearly shown that the angle tracking error of fuzzy adaptive PID controller is the smallest. The PID controller exhibits a phase lag and the tracking error of the fuzzy logic controller fluctuates wildly. Fuzzy adaptive PID controller clearly performs better than fuzzy logic and PID controllers. It has the advantage of both PID and fuzzy logic controllers and is able to cope with parameter changes. We can obtain that the erection system tracks the reference angle trajectory very well with the fuzzy adaptive controller.

(a) Desired and Actual Erection Angle of Simulation

(b) Tracking Error of Desired and Actual Erection Angle of Simulation Figure 9. Simulation Results of Erection Angle

5. Experimental Verification

Figure 10. Mechanical Constitution of the Platform

0 10 20 30 40 50 60 70 80 90

0 20 40 60 80 100

Time(s)

ngl

(de

Fuzzy adaptive PID PID

Fuzzy logic Desired angle

0 10 20 30 40 50 60 70 80 90

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

Time(s)

ngl

(de

Fuzzy adaptive PID PID

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Mechanical constitution of laboratory equipment is shown in Figure 10. It is mainly composed of erection arm and erection cylinder. Programs of data acquisition and fuzzy adaptive PID controller are completed in LabVIEW software.

Figure 11 shows experimental results of erection angle controlled by fuzzy logic, PID and fuzzy adaptive PID controllers respectively. Figure 11(a) presents desired and actual erection angle curves. Figure 11(b) illustrates the tracking error of desired and actual erection angle. From the results we can obtain that the tracking error of PID controller is the largest. The tracking error of fuzzy logic controller fluctuates wildly. The tracking error of fuzzy adaptive PID controller is small and steady. It can be concluded from the experimental results that the performance of fuzzy adaptive PID controller is the best compared with fuzzy logic and PID controllers.

(a) Desired and Actual Erection Angle of Experiment

(b) Tracking Error of Desired and Actual Erection Angle of Experiment Figure 11. Experimental Results of Erection Angle

6. Conclusion

In this paper, mathematical model of the electro-hydraulic system is derived. Kinetic analysis of erection mechanism is presented. Fuzzy adaptive PID controller is applied to erection mechanism in combination of fuzzy logic and PID control method. The model of erection mechanism is established in Simulink. Simulation and experimental results of erection angle controlled by fuzzy logic, PID and fuzzy adaptive PID controllers are respectively obtained. It is clearly shown that the angle tracking error of fuzzy adaptive PID controller is the smallest. Hence, it can be concluded that fuzzy adaptive PID controller is suitable for erection mechanism.

Acknowledgement

This research was financially supported by the National Natural Science Foundation of China (Grant No. 51475462).

0 10 20 30 40 50 60 70 80 90

0 20 40 60 80 100

Time(s)

ngl

(de

Fuzzy adaptive PID PID

Fuzzy logic Desired angle

0 10 20 30 40 50 60 70 80 90

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

Time(s)

ngl

(de

Fuzzy adaptive PID PID

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[7] Jones E, Dobson A, Roskilly AP. Design of a Reduced-rule Self-organizing Fuzzy Logic Controller for Water Hydraulic Applications. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering. 2000; 214(5): 371-382.

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[12] Truong DQ, Ahn KK. Force Control for Hydraulic Load Simulator Using Self-tuning Grey Predictor– fuzzy PID. Mechatronics. 2009; 19(2): 233-246.

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[15] Chiang MH, Chen CC, Kuo C F J. The High Response and High Efficiency Velocity Control of a Hydraulic Injection Molding Machine Using a Variable Rotational Speed Electro-hydraulic Pump-controlled System. The International Journal of Advanced Manufacturing Technology. 2009; 43(9-10): 841-851.

[16] Zheng J, Zhao S, Wei S. Application of Self-tuning Fuzzy PID Controller for a SRM Direct Drive Volume Control Hydraulic Press. Control Engineering Practice. 2009; 17(12): 1398-1404.

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relatively small value to avoid out of range control at the beginning. To avoid large overshoot, remove the integral action.

2. When e(t) is medium, KP should be adopted relatively small value to minimize response overshoot. The value of KD is important and should be adopted medium value. The value of KI should be appropriately increased.

3. When e(t) is relatively small, KP and KI should be adopted relatively large value to have good steady state and avoid oscillation at equilibrium point. The value of KD should be appropriate.

Based on the above relationships and the impact of c(t), fuzzy control rules are obtained in Table 1 through theoretical analysis.

Table 1. Fuzzy Control Rules △KP /△KI /△KD

NB NM NS ZO PS PM PB

Take the first rule as an example, the above rule can be interpreted as “if e is NB and c is NB, then △KP /△KI /△KD are PB/NB/NS”. Mamdani algorithm is introduced as fuzzy implication relation [18].

( , ) ( )( , )

= min( ( ), ( )) ( ) ( ) R a u A U a u

A a U u A a U u

 

  (12)

Triangular shape function is chosen as output membership function which is shown in Figure 5 [19].

Figure 5. Membership Function Cure of Output

corresponding to the area center is calculated by the following formula [21].

( ) ( ) UA u udu

A u du 



(13)

ZO PS PM PB 1.0

NS NM NB

0 2 4 6

-2 -4

(2)

4. Simulation and Results

4.1. Modeling Erection Mechanism in Simulink

Figure 6 illustrates the top level Simulink diagram of the entire system. The pump flow and the desired angle signal are simulation inputs. The pressure and cylinder position are simulation outputs. The model is organized as several subsystems blocks such as the ‘source’ model, the ‘pump’ model, the ‘valve/cylinder’ model and the ‘fuzzy adaptive PID controller’ model. The output of the controller is derived to control motion of the valve spool. Position of the spool influences the nominal flow into cylinder chamber. The nominal flow change will change the volume and pressure inside both cylinder chambers. Finally, the piston position is affected, therefore the mechanical mechanism moves driven by cylinder piston. Hence, the performance of the system is dependent on the output of the controller.

Figure 6. Simulink Block of the Complete System

The ‘source’ model is reference angle trajectory. We adopt composite sine function as reference angle trajectory. 0 is initial value of erection angle and 1 is final value. Erection time

is T, τ=t/T. (t) is determined by the following expressions:

1 0

( )t ( ) ( )s





 





(14)

sin(4 ) 1

4 4 8

2 4 1 7

( ) 9 cos( ) / 4

4 3 6 8 8

4 sin(4 2 ) 7

4 4 8

k k s



 

 

 

   

 

 

 

  

  _  _{ }

 _ _



 _ _

_ _    _  

 



 _ _ _

   

 _ _

 



(15)

Where k4

 

2 4.

which appear as derivatives in Equations (4) to (6) and are used as the states.

Simulink block of fuzzy adaptive controller is shown in Figure 8. We establish the model based on the controller structure in Figure 3.

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Figure 7. Simulink Block of Hydraulic Cylinder and Valve

Figure 8. Simulink Block of Fuzzy Adaptive Controller

4.2. Simulation Results

Table 2. Parameter Values for Simulation

Symbol Explanation Value

Cd Discharge coefficient 0.61



Bulk modulus of hydraulic fluid. 6.85e8 Pa

A1 Cross-sectional area of the piston 1.23e-3 m2

A2 Annular area of piston rod chamber 5.91e-4 m

ρ Fluid density 800 kg/m3

Cic Internal leakage coefficient 3e-11

Cec External leakage coefficient 2e-11

m Equivalent mass 500 kg

Ps Hydraulic supply pressure 2.5e6 Pa

V01 Original total fluid volumes of the forward cylinder chamber 2.4e-3 m3 V02 Original total fluid volumes of the return cylinder chamber 0.0089 m3

OP The length of OP₂ 3.08 m

(4)

(a) Desired and Actual Erection Angle of Simulation

(b) Tracking Error of Desired and Actual Erection Angle of Simulation Figure 9. Simulation Results of Erection Angle

5. Experimental Verification

Figure 10. Mechanical Constitution of the Platform

0 10 20 30 40 50 60 70 80 90

0 20 40 60 80 100

Time(s)

ngl

(de

Fuzzy adaptive PID PID

Fuzzy logic Desired angle

0 10 20 30 40 50 60 70 80 90 -0.1

-0.05 0 0.05 0.1 0.15 0.2 0.25

Time(s)

ngl

(de

Fuzzy adaptive PID PID

(5)

(a) Desired and Actual Erection Angle of Experiment

(b) Tracking Error of Desired and Actual Erection Angle of Experiment Figure 11. Experimental Results of Erection Angle

6. Conclusion

Acknowledgement

This research was financially supported by the National Natural Science Foundation of China (Grant No. 51475462).

0 10 20 30 40 50 60 70 80 90 0

20 40 60 80 100