PSO TUNING OF A CDM BASED PID CONTROLLER FOR BALL AND … · 2018. 9. 29. · The CDM-PID control system consist of main controller C(s) and feed forward controller C f (s) as shown

PSO TUNING OF A CDM BASED PIDCONTROLLER FOR BALL AND

BEAM SYSTEM

Dr. B.Meenakshipriya1, Mr. C.Naveen2,Ms. K.Kalpana3

Mechatronics Engineering,Kongu Engineering College,

Perundurai - 638060, Tamilnadu, India

August 6, 2018

Abstract

This paper proposes a design strategy of a PID controllerfor a ball and beam system by combining the CoefficientDiagram Method (CDM) and Particle Swarm Optimization(PSO). Developing a PID controller for double integratingunstable system is uncommon. The PSO-CDM-PID con-troller parameters are predicted based on the dynamics ofball and beam system developed by Euler Lagrangian Ap-proach. The effectiveness of PSO-CDM-PID controller istested with ball and beam system. The performance ofPSO-CDM-PID controller is assessed through the experi-mental runs carried out at different operating points of ball’sposition through servo and regulatory analysis. The resultsare compared with classical Ziegler Nichols PID controller(ZN-PID) in terms of Error indices and Quality indices.The results evident that the proposed PSO tuned CDM-PID controller upholds the ball at the desired position onthe beam with good stability and reduced percentage of er-ror with better transient characteristics than ZN-PID con-troller.

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International Journal of Pure and Applied MathematicsVolume 120 No. 6 2018, 10905-10927ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/

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Key Words: Ball and Beam system, PID, CoefficientDiagram Method, Particle Swarm Optimization, Performanceindices

1 INTRODUCTION

PID controller plays a significant role in most of the industrialplants. [25]. Tuning rules and a variety of tuning structures imple-mented for different processes[1]. It’s used for different applicationlike seat suspension of a quarter car [25] and the movement of Stew-art platform [2]. Tuning rules that were proposed are related to un-stable, stable and integrating process, whereas only few literatureis available for double integrating process[14] [23]. Hence, it is nec-essary to design a simple and robust control strategy to provide su-perior performance with double integrating unstable process.CDMtechniqueis developed by Prof. ShunjiManabe in the year 1991. Itinvolves simultaneous design of characteristic and controller polyno-mials. Applications of CDM are found in several literaturessuch assteel mill control [4], control of robotic manipulators [3], space craftcontrol [8], control of DC motor [13], bio-reactor process [6],Con-trol Surface Missile [21], Control of Bio-reactor Process [24]and pHcontrol [4][5].Hence, based on the above features, CDM based PIDcontrol strategy is proposed for double integrating unstable ball andbeam system. Unfortunately, proper tuning of CDM parameters toobtain optimum PID gains has become quietly difficult due to non-linearities, higher order and delay[10]. A Nonlinear PID Controllerwas used to improve the Control Performance of the PneumaticArtificial Muscle Manipulator and the experimental results showedthat the effectiveness and superior performance [11][12] [13]. Thefuzzy, fuzzy MA and PID controller was used to control the po-sition of the muscle-like actuated arm and the conventional PIDcontroller performs better than the other controllers [8] [26]. To re-duce such complexity, many random search methods are available.Particle Swarm Optimization (PSO) is a population based opti-mization technique andsuccessful proven optimum tool in modelidentification. Hence, in the present work, PSO tuned CDM-PIDcontroller[18] [19][20] [22] is designed to optimally controlthe ball atthe desired position on the beam.Literatures reveal limited intelli-gent controllers for ball and beam system, such as neural control [2],

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sliding mode[15], and ACO-tuning of Fuzzy controller [16]. Hence,so far no real time work on the proposed control technique.

2 SYSTEM DYNAMICS

2.1 SYSTEM DESCRIPTION

It consists of a base, support block,lever arm, motorized gear,beamand ball, asshown in Fig.1. The beam tilts about its axis by certainangle through DC motor excitation. Built-in rotary incrementalencoder provides feedback information on the current position ofthe motor shaft. The linear potentiometer sensor senses the actualposition of the ball. These two positions are feedback to the controlsystem to formulate a closed loop control.

Figure 1. Lab-scale Ball and Beam System

Graphical User Interface (GUI) is illustrated in Fig.1. Itcommu-nicate via IPM 100 servo drive unit. The drive reads the feedbackand applies appropriate voltage to the DC motor windings to pro-vide sufficient torque. The system is an open loop unstable,the ballposition increases without limit for a fixed beam angle. To keepthe ball at the desired position it is necessary to develop a suitablecontroller.

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Figure 2. GUI for experimental work in MATLAB/SIMULINK

2.2 MATHEMATICAL MODELING

The parameters to develop the dynamics of the Ball and Beamsystem is shown in Fig.3

Figure 3. Dynamics of ball and beam

Parameters of ball and beam system are Ball mass (M) =0.11Kg,Ball radius (R) =0.015m, Lever arm- offset (D) =0.04m, Accelera-tion due to gravity (G) =9.8 m/s2, Beam length (L) =40cm, Ball’s

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moment of inertia (Jb) = 2mR2/5 Kgm2, Ball’s position coordinate(R), Coordinate of beam angle(α), Gear angle of servo gear (θ).Thesimplified Lagrangian equation of motion for the ball [7] is given by

(m+jbR2r +

jbRα−mrr2 +mgsinα) = (1)

By linearizing Eq. (1) about the beam angle, sinα = α for smallα,gives us the following linear approximation of the system

(m+jbR2

)r = mgα (2)

By equating the arc distance, the equation which relates theangle of the gear and the beam angle can be approximated as linearand it is specified as

α = (d

L) (3)

By substituting Eq. (3) in Eq. (2) gives

(m+jbR2

)r = mg(d

L) (4)

By taking Laplace transform of Eq.(4), the equation in terms ofinput and output is obtained as

(m+jbR2

)s2r(s) = mg(d

L) (5)

On rearranging, the transfer function of the ball position (r(s)) tothe gear angle (θ(s)) is represented as

r(s)

θ(s)=

mgd

L(m+ jbR2 )

(6)

By substituting the values of ball and beam system parametersin Eq. (6), the transfer function is obtained as

r(s)

θ(s)= (7)

From Eq. (7), it is clear that the system is a double integratingunstable process.

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3 DESIGN OF CONTROLLERS

3.1 ZN-PID CONTROLLER

Ziegler and Nicholas closed loop method also known as UltimateCycle Method originated in 1942[6]. It can be easily implementedin MATLAB-SIMULINK platform to find the tuning parameterssuch as Proportional Gain (Kc), Integral Time constant (Ti) andDerivative Timeconstant (Td).

• By setting Ti = ∞ and Td = 0, Kc value is increased from 0to a particular value of Ku at which the output exhibits sus-tained oscillation, from which period of sustained oscillationPu is found.

• ZN-PID controller settings are computed from Eq. (8) andtabulated in Table 1.

Kc = 0.6Ku;Ti = 0.5PuandTd = 0.125 (8)

From Table 1, the sustained oscillation is at Ku = 8 to 50.Controller parameters of Ku = 30 provides good response. Hencethe optimized ZN-PID controller parameters are Kc = 18, Ti = 2.2sec and Td = 0.55 sec.

Table I Optimized tuning parameters for ZN-PID controller basedon ultimate gain value

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3.2 CIDM-PID CONTROLLER

3.2.1 DESIGN OF CDM CONTROLLER

The output of the CDM control system is given by [18, 22],

y =N(s)F (s)

P (s)r +

A(s)N(s)

P (s)(9)

Where, characteristic polynomial P(s)

P = A(s)D(s) +B(s)N(s) =n∑

i=0

aisi, ai > 0 (10)

Here, the controller polynomials A(s) and B(s) are given as,

A(s) =P∑

i=0

1isiB(s) =

q∑

i=0

k (11)

When polynomial F(s) is chosen as,

F (s) = P (s)/N(s) (12)

The design parameters of CDM are the stability indices (γi) andequivalent time constant (τ) [9]. In addition, they determine therobustness of the system to parameter variations. The equivalenttime constant, which is closely related to the bandwidth and it de-termines the rapidity of the time response. The design parametersare defined as follows,

τ = ts/2.5 (13)

Where, ts is the user specified settling time.

γi = [2.5, 2, 2, ..], wherei = 1, ..., n− 1, γ0 = γ∞ (14)

If necessary, the designer can modify the values of stability indices.By using Eqs. (13) and (14), a target characteristic polynomial(Ptarget(s)) is derived as

Ptarget(s) = a0

[{n∑

i=2

(i−1∏

j=1

1

rji−j)(τs)i

}+ τs

](15)

The closed loop characteristic polynomial P(s) are attained by sub-stituting the controller polynomials in Eq. (11) into Eq. (10). Thispolynomial is compared with Eq. (15) to obtain the coefficients ofCDM controller polynomials (li, ki and ai).

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3.2.2 DESIGN OF CDM CONTROLLER PARAMETERSFOR DOUBLE INTEGRATING PROCESS

The general mathematical model of double integrating ball andbeam system is given below [17].

Gp(s) =Y (s)

U(s)(16)

The transfer function is defined by two independent polynomialsare

N(s) = KandD(s) = (17)

Where,A(s) = I1s+ I0andB(s) = k2s

2 + k1s+K0 (18)

Thus, let thethe structure of the controller be chosen with I0 = 0as follows,

Gc(s) =B(s)

A(s)=k2s

2 + k1s+ k0I1s+ I0

(19)

By substituting Eqs. (17) and (18) in Eq. (10)

P (s) = (I1s)s2 + (k2s

2 + k1s+ k0) (20)

By substituting and i= 1, 2in Eq. (15)

Ptarget(s) =τ 3

γ2γ21S3 +

τ 2

γ1S2 + τs (21)

By equating the coefficients of like power terms of Eqs. (20) and(21), CDM controller parameters (k2, k1, k0 and l1) are computedas follows

k2 = (τ 2

γ1K); k1 = (

τ

K)and, l1 = (

τ 3

γ21k) (22)

The pre-filter element (F(s)) are chosen by Eq. (12) and calculatedas

F (s) = P (s)/N(s)|s=0 = Kk0/K (23)

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3.2.3 Design of CDM-PID controller parameters fordou-ble integrating process

The CDM-PID control system consist of main controller C(s) andfeed forward controller Cf (s) as shown in Fig.4. C(s) is given by,

C(s) = Kc(1 +1

Tis) (24)

Figure 4. CDM-PID control system

Equivalent CDM block diagram is obtained by block diagramreduction methodto compute the CDM-PID parameter as shown inFig.5

Figure 5. Equivalent CDM block diagram

By comparing the Figs.4 and 5, the following relations are ob-tained

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3.3 Design of PSO tuned CDM-PIDcontrollerfor Ball and Beam system

To find the time constant (τ) and stability indices (γ1 and γ2). Theobtained optimal values are used to find the PID parameters usingEq. (28). Fig. 6 shows proposed controller.

Figure 6. PSO tuned CDM-PID controller with ball and beamsystem

In PSO technique,the particles are reorganized by global best Gi(t)

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and local best Pi(t) in the population. The particles thereby searchfor optimal solution by updating its position Xi(t) and velocityVi(t) using Eqs. (30) and (31),

Inertia weight (W) helps to maintain the velocity and positionwithin the bounded search space. The parameters Wmax favorthe global search and Wminsupports higher ability for local nearsearch. ThePseudo code of PSO based tuning of CDM-PID Param-eters is as follows.Step 1:Initialize: number of particles (n); cognitive and social learn-ing rates (C1, C2), dimension for search space (d); inertia weight(W)Step 2:Current position= rand(n,d), velocity= rand (n,d)Step 3:Compute and evaluate desired fitness function.Step 4:Evaluate the fitness of particle and compare it with its Pi(t).Then,replace the local best value as given below for i = 1 : nIf current fitness (i) ¡ local best fitness (i);then,local best fitness = current fitness;local best position = current position (i);endStep 5: Update current position and velocity of theparticle.forit=1:iter;W=(Wmax-it)∗((Wmax-Wmin)/iter);Vi(t+1) =W∗Vi(t)+ C1∗rand∗(Pi(t) Xi(t))+C2∗rand∗(Gi(t)Xi(t));Xi(t+1)=Xi(t) +Vi(t + 1);endStep 6:Repeat step 2-5 for defined number of iterations.Step 7:Perform closed-loop test with the optimized values of CDMparameters and compute the performance indices.

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3.3.1 OBJECTIVE FUNCTION

It defines the convergence speed, efficiency and optimization accu-racy of the PSO performance. It either maximizes or minimizes thedomain or preference constrains respectively.

Figure 7. Closed Loop structure of Ball and Beam system

The output of the closed loop unity feedback system is

Since ITSE requires analytical formula, its complex and time con-suming. On the other hand, IAE minimization can result longsettling time with relatively small overshoot. Hence, minimizationof ISE is considered to be an objective function.

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3.3.2 Optimization of CDM parameters using PSO tech-nique

The CDM parameters are optimized through simulation runs usingMATLAB programming. The C1 and C2 are varied from 0.5 to 2with an interval of 0.5. Figs.8-11 show the convergence graphs ofτ, γ1 and γ2 for different values of C1 and C2 against number ofiterations.

Figre 8. Convergence graph of , 1 and 2 for C1=C2=0.5

Figure 9. Convergence graph of , 1 and 2 for C1=C2=1

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Figure 10. Convergence graph of , 1 and 2 for C1=C2=1.5

Figure 11. Convergence graph of , 1 and 2 for C1=C2=2

From the above figures, CDM parameters attained an earlyconvergent at 1500th iteration for the values of C1 = C2 = 1.5.CDM parameters are thereby optimized as 1 = 4, 2 = 4and τ =3 for further analysis are based on the Hybrid PSO based CDM-PID controller parameters. The parameters chosen for PSO basedCDM PID controller are Members of each individual- τ , γ1 and γ2,Swarm size (n) -10, Number of iterations (iter)-1500, Dimensionalsearch space (d)- 3, Learning rate-Cognitive (C1)- 1.5, Learningrate-Social (C2) -1.5, Inertia weight-Maximum (Wmax)- 0.9, Inertiaweight-Minimum (Wmin) -0.4.By considering the model parametersand choosing the stability indices values 1 = 4, 2 = 4, and τ = 3,the proposed Hybrid PSO based CDM-PID settings are determinedusing Eq. (28) as Kp =10.1578, Ti =2.999 sec and Td = 0.7499 sec.

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4 EXPERIMENTAL RESULTS AND

DISCUSSION

4.1 PERFORMANCE ANALYSIS

The performance measures are calculated with the sampling rateof 0.02 sec to ensure the results and it’s tabulated in Table 2. Theexperimental runs are carried out as shown in Figs. 12 14.It gives areasonable transient responsewith minimum error indices and goodquality indicescompared to ZN-PID controller with less overshoot,fastest rise time and smallest settling time due to its two degreefreedom structure.

Figure 12. Closed loop response at 10cm

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Table II Performance measures at different balls position

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4.2 ROBUSTNESS TEST

The controller is enforced to appropriate set point with minimumovershoot,early settling time, stability with most consistent androbust response for different operating points compared to that ofZN-PID controller.

Figure 15. Set point tracking analysis (ball’s position- 20, 10 and30 cm)

4.3 LOAD REJECTION TEST

The load rejection test is carried out at an operating point of 20cm.A step disturbance is introduced at 29th sec by decreasing the op-erating point to 11cm as shown in the Fig.16. From the result,ZN-PID controller verge to instability whereas PSO-CDM-PID con-troller damp the disturbance in shorter time with fewer peaks andmaintains the stabilityand also not affected by nonlinear behaviorof the system.

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Figure 16. Closed loop response at the operating point of 20cmballs position with disturbance

5 CONCLUSION

The performance of proposed controller was tested and comparedwith the bench mark controller (ZN-PID). Based on the results, it isclear that PSO-CDM-PID controller accomplishes reliable and sta-ble performance on unstable ball and beam system with minimumerror and maintains robustness. Moreover it is evident that PSO-CDM-PID controller has improved quality measure in rejecting thedisturbance compared to that of ZN-PID controller. Thus it can beconcluded, if no overshoot collectively with early settling time andminimum error is required out of unstable double integrating balland beam system.

References

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[2] AhmetSumnu, Ibrahim HalilGuzelbey and Mehmet Veysel-Cakr,Simulation and PID control of a Stewart platform withlinear motor, Journal of Mechanical Science and Technology31 (1), 2017;pp. 345 356.

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[3] A. Ucar and S.E. Hamamci, A Controller Based on CoefficientDiagram Method for the Robotic Manipulators, In: 7th IEEEInternational Conference on Electronics Circuits and Systems,Lebanon,2000;Vol. 2, pp.777-780.

[4] B. Meenakshipriya, K. Saravanan, K. Krishnamurthy and P.K.Bhaba, Design and Implementation of CDM-PI Control Strat-egy in pH Neutralization System, Asian Journal of ScientificResearch 2012; Vol. 5:pp78-82.

[5] B. Meenakshipriya, K. Saravanan, S. Somasundaram and P.Kanthabhabha, CDM Based PIP Control Strategy in pH Neu-tralization System, Instrumentation Science and Technology2011; Vol. 39, Issue 3, pp.273-287.

[6] B. Wayne Bequette, Process control: Modeling, Design andSimulation, Prentice-Hall of India Pvt. Ltd.:New Delhi, 2003.

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[13] Kyoung Kwan Ahn and TU Diep Cong Thanh, Nonlinear PIDControl to Improve the Control Performance of the PneumaticArtificial Muscle Manipulator Using Neural Network, Journalof Mechanical Science and Technology, 2005 Vol. 19, No. 1, pp.106115.

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[22] S.Manabe, The Coefficient Diagram Method, In: 14th IFACsymposium on Automatic control in Aerospace, 1998, pp.24-28.

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1.Dr. B. Meenakshipriya received her B.E. degree in Electricaland ElectronicsEngineering from Thangavelu Engineering Col-lege, University of Madras, Chennai in 1999 and M.E. degreein Mechatronics engineering from Kongu Engineering college,Tamilnadu, India in 2006 and Ph.D from Anna University,Chennai in 2012. Her research area includes Process control,Intelligent Techniques, Robotics and Control.

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2. Mr. C. Naveen received his B.E. degree in Electronicsand Communication Engineering from Sengunthar engineeringcollege, affiliated to Anna University, Chennai in 2007 and re-ceived the M.E. degree in Mechatronics engineering from konguengineering, Tamilnadu, India in 2011, where he is currentlypursuing the part time Ph.D. degree with the Anna university,Chennai. He has been working as an assistant professor forthe Mechatronics department, Kongu engineering college since2012. His current research interests include control system, pro-grammable logic controller, and machine vision systems.

3. Ms. K. Kalpana received her Masters degree in Mecha-tronics Engineering from Anna University, Chennai. She grad-uated her Bachelors degree in Electronics and CommunicationEngineering from Anna University, Chennai in the year 2012.At present she is working in Infosys, Mysore as a System Engi-neer. She has participated and presented papers in 2 Interna-tional and 2 National Conferences. She has also participated in3 Workshops/Seminars sponsored by various funding agencies.Her areas of research interest include Control System

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Documents

PSO TUNING OF A CDM BASED PID CONTROLLER FOR BALL AND … · 2018. 9. 29. · The CDM-PID control system consist of main controller C(s) and feed forward controller C f (s) as shown